Topology I: general survey

Encyclopaedia of Mathematical Sciences Volume 12

Editor-in-Chief: R. V. Gamkrelidze

Springer Berlin Heidelbe rg New York Baree/ona Budapest Hong Kong London Mila n Paris Santa Clara Singapo re Tokyo

S. P. Novikov (Ed.)

Topology I General Survey

With 78 Figures

Springer

Consulting Editors of the Series: A. A. Agrachev , A. A. Gonchar, E.F. Mishchenko, N. M . Ostianu , V. P. Sakharova, A. B. Zhishchenko

Tille of lhe Russian origin al ed ition (only Part 1 i publi shed in thi s volume. PartIT will be published in volume 24 of the EMS seri e ): Itogi nauki i tekhniki , Sovremennye problemy matematiki, Fundamental'nye naprav leniya, Vol. 12, Topologiya- I (Part 1) Publisher VINITI , Moscow 1986

Cataloging.in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme TopolollY 1 S . P . Novikov (ed .l . - (Ausg . in Schriftenre ihe Encydopacdia in mathematica l sciences) . - Derlin ; Heidelberg ; New York; Barcelona ; Budapest; H ong Kong; London; Milan; Paris; Santa Clara; Singapore ; Tokyo ; Springer NE : No\'iko\'. S.,gej P. (Hrsg .l (Ausg. in Schriftenreihe Encyclopaedia in mathematical sciences) 1. General survey. - 1996 (Encyclop•• dia of m.lhemBlical sciences: Vol. 12) ISBN 3-540- 17007-3 (Berlin . .. ) ISBN 0-387-17007-3 (New York) NE: GT

Mathematics Subj ect CI as ifi cation ( 199 1): 19Lxx , 55-XX, 55Pxx , 55Qxx; 55Rxx , 55Sxx, 57-XX, 57N12, 8 1Txx

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Topology Topology Sergei Novikov Sergei P. N ovikov Translated ftom from the the Russian Russian Translated by Boris Botvinnik and Robert by Boris Botvinnik and Robert Burns Burns

Contents Contents Introduction . . .. .. . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. .. . . .. . .. Introduction ....................................................

44

Introduction the English Translation . .. .. . . .. . . . . . . .. . .. ........................... Introduction to to the English Translation

55

Chapter 1. Simplest Topological Topological Properties . . .. .. . . .. .. . . .. . .. 1. The The Simplest Properties .................... Chapter

5

Chapter 2. Topological Spaces. Fibrations. Homotopies Chapter 2. Topological Spaces. Fibrations. Homotopies $1. §l. $2. §2. $3. §3. 54. §4.

.............

15

15

............... Observations topology. Terminology Observations from from general general topology. Terminology ............... Homotopies. Homotopy type ................................. Homotopies. Homotopy type ................................. ............................. Covering homotopies. Fibrations Covering homotopies. Fibrations ............................. .... Homotopy groups and fibrations. Exact sequences. Examples

Homotopy groups and fibrations. Exact sequences. Examples

5

....

Chapter 3. Simplicial Complexes and CW-complexes. Homology and Chapter 3. Simplicial Complexes and CW-complexes. Homology and .. . .. Cohomology. Their Relation to Homotopy Theory. Obstructions

Cohomology. Their Relation to Homotopy Theory. Obstructions

$1. Simplicial complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simplicial complexes ........................................ . . . .. . .. The homology and cohomology groups. Poincare duality The homology and cohomology groups. Poincaré duaHty ........ Relative homology. The exact sequence of a pair. Axioms for Relative homology. The exact sequence of a pair. Axioms for homology theory. CW-complexes .. .. . . .. .. . . . . . .. . . .. . homology theory. CW-complexes ............................. $4. Simplicial complexes and other homology theories. Singular §4. Simplicial complexes and other homology theories. Singular homology. Coverings and sheaves. The exact sequence of sheaves homology. Coverings . . .and . . . .sheaves. . . . . . . . The . . . . exact . . . . . sequence . . . . . . . . .of. .sheaves . . .. . .. . . and cohomology and cohomology ........................................... $5. Homology theory of non-simply-connected spaces. Complexes §5. of Homology of non-simply--connected spaces. Complexes modules. theory Reidemeister torsion. Simple homotopy type . . . . . . . .

§l. $2. §2. 83. §3.

of modules. Reidemeister torsion. Simple homotopy type ........

15

15 18 18 19 19 23 23 40

40 40

40 47 47 57

57 64

64 70

70

Contents Contents

2

Simplicial and and cell cell bundles bundles with with aa structure structure group. group. Obstructions. Obstructions. §6. 93. Simplicial fiber bundles bundles and and the the universal universal Universal objects: universal fiber property of of Eilenberg-MacLane Eilenberg-MacLane complexes. complexes. Cohomology property The Steenrod algebra. algebra. The Adams Adams spectral spectral sequence sequence 79 operations. The 79 The classical classical apparatus of of homotopy homotopy theory. The Leray Leray §7. §7. The sequence.The The homology homology theory theory of fiber fiber bundles. bundles. The spectral sequence. Cartan-Serre method. The Postnikov Postnikov tower. The Adams spectral spectral Cartan-Serre sequence .................................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 103 sequence Definition and and properties properties of of K-theory. K-theory. The The Atiyah-Hirzebruch Atiyah-Hirzebruch §8. V3. Definition sequence.Adams operations. operations. Analogues Analogues of the the Thom Thorn spectral sequence. and the Riemann-Roch Riemann-Roth theorem. Elliptic Elliptic operators operators isomorphism and and K K-theory. Transformation groups. groups. Four-dimensional Four-dimensional and -theory. Transformation manifolds ................................................. . .. . .. . .. . .. . . .. .. . . .. .. . .. . .. . .. . .. . . .. .. . . .. . 113 manifolds 113 Bordism and cobordism theory theory as as generalized generalized homology homology 39. Bordism §9. and cobordism and cohomology. cohomology. Cohomology Cohomology operations operations in in cobordism. cobordism. The The and Adams-Novikov spectral spectral sequence. sequence.FormaI Formal groups. groups. Actions Actions of of Adams-Novikov cyclic groups and the circle on on manifolds manifolds ...................... .. .. . . . . .. . .. . . . . . 125 cyclic groups and the circle 125 Chapter 4. Manifolds .................................... . . . .. . .. . .. . .. . .. . .. . .. . .. . .. .. 142 Chapter 4. Smooth Smooth Manifolds 142 $1. Basic Basic concepts. Smooth fiber fiber bundles. bundles. Connexions. Connexions. Characteristic Characteristic §1. concepts. Smooth

classes .................................................... . .. . .. . .. . . .. . .. . .. .. . . .. .. . . .. .. . . .. . .. . .. . .. . . .. classes $2. The homology of smooth smooth manifolds. manifolds. Complex Complex §2. The homology theory theory of manifolds. The classical global calculus of variations. H-spaces. manifolds. The classical global calcul us of variations. H-spaces. Multi-valued functions and functionals . . . . . . . . . . . . . .. . . . .. Multi-valued functions and functionals ........................ 53. Smooth manifolds and homotopy theory. Framed manifolds. §3. Smooth manifolds and homotopy theory. Framed manifolds. Bordisms. The Hirzebruch of Bordisms. Thorn Thom spaces. spaces. The Hirzebruch formulae. formulae. Estimates Estimates of the orders of homotopy groups of spheres. Milnor’s example. the orders of homotopy groups of spheres. Milnor's example. The integral properties of cobordisms . . . . . . . . . . . . . . . . . . . . . . . The integral properties of cobordisms ......................... $4. Classification problems in the theory of smooth manifolds. The §4. Classification problems in the theory of smooth manifolds. The theory of immersions. Manifolds with the homotopy type of theory of immersions. Manifolds with the homotopy type of a sphere. Relationships between smooth and PL-manifolds. a sphere. Relationships between smooth and PL-manifolds. Integral Pontryagin classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integral Pontryagin classes .................................. $5. The role of the fundamental group in topology. Manifolds of §5. The role of the fundamental group in topology. Manifolds of low dimension (n = 2,3). Knots. The boundary of an open low dimension (n = 2,3). Knots. The boundary of an open manifold. The topological invariance of the rational Pontryagin manifold. The topologieal invariance of the rational Pontryagin classes.The classification theory of non-simply-connected classes. The classification theory of non-simply-connected manifolds of dimension > 5. Higher signatures. Hermitian manifolds of dimension?: 5. Higher signatures. Hermitian K-theory. Geometric topology: the construction of non-smooth K -theory. Geometrie topology: the construction of non-smooth homeomorphisms. Milnor’s example. The annulus conjecture. homeomorphisms. Milnor's example. The annulus conjecture. Topological and PL-structures .. . .. . . .. .. . . . .. . . .. .. .. .. .

244

Concluding Remarks

273

Topologieal and PL-structures

Concluding Remarks

142 142 165 165

203

203

227

227

.............................. 244

. . . . . . . . . . . . . . . . . . . . :. . . . . . . . . . . . .

........................................... 273

Contents Contents

3

Appendix. Appendix. Recent Recent Developments Developments in in the the Topology Topology of of 3-manifolds 3-manifolds 274 and . .. . . .. .. . . .. .. . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . . . and Knots Knots ..................................................... 274 . .. . . .. . .. . . $1. §l. Introduction: Introduction: Recent Recent developments developments in in Topology Topology ................. 52. §2. Knots: Knots: the the classical classical and and modern modern approaches approaches to to the the Alexander Alexander polynomial. .. . .. . .. . .. .. . .. . . . . polynomial. Jones-type Jones-type polynomials polynomials ........................... 53. . . .. . . . .. . .. . . .. . .. . .. . .. . . . . . . §3. Vassiliev Vassiliev Invariants Invariants ......................................... $4. §4. New New topological topological invariants invariants for for 3-manifolds. 3-manifolds. Topological Topological Quantum . . .. .. . . . . . . .. . . .. .. . . . . . . Quantum Field Field Theories Theories .................................... Bibliography Bibliography

274 274 275 275 289 289 291 291

............................................... .................................................. 299 299

Index Index ...................................................... ......................................................... 311 311

4

Introduction Introduction

Introduction Introd uction In the present some ide ideaa of of the the skeleton skeleton of of In the present essay, essay, we we attempt attempt to to convey convey sorne topology, and of various topological concepts. It must be said at once that, topology, and of various topological concepts. It must be said at once that, apart from necessary minimum, subject-matter of does apart from the the necessary minimum, the the subject-matter of this this survey survey does not include that subdiscipline known as “general topology” the theory of not include that subdiscipline known "general topology" the theory of general spaces and maps considered in the context of set theory and general general spaces and maps considered in the context of set theory and general category this subject subject will by others.) category theory. theory. (Doubtless (Doubtless this will be surveyed surveyed in in detail detail by others.) With it may may be claimed claimed that the "topology" “topology” dealt dealt with with in in the the With this this qualification, qualification, it that the present that mathematical mathematical subject 19th century century was was present survey survey is that subject which which in in the the late late 19th called and at various called Analysis A nalysis Situs, Situs, and at various various later later periods periods separated separated out out into into various subdisciplines: “Combinatorial “Differential subdisciplines: "Combinatorial topology”, topology", “Algebraic "Algebraic topology”, topology", "DifferentiaI (or smooth) “Homotopy theory", theory”, “Geometric (or smooth) topology”, topology", "Homotopy "Geometric topology”. topology" . With over aa long long period of topology topology to With the the growth, growth, over period of of time, time, in in applications applications of to other the following following further subdisciplines crystallized other areas areas of of mathematics, mathematies, the further subdisciplines crystallized out: variations, global the topology out: the the global global calculus calculus of of variations, global geometry, geometry, the topology of of Lie Lie groups spaces, the the topology topology of complex manifolds algegroups and and homogeneous homogeneous spaces, of complex manifolds and and algebraic qualitative (topological) (topological) theory of dynamical dynamical systems systems and and braie varieties, varieties, the the qualitative theory of foliations, the elliptic and and hyperbolic equations. foliations, the topology topology of of elliptic hyperbolic partial partial differential differential equations. Finally, 1970s and 80s aa whole complex of topological Finally, in in the the 1970s and 80s, whole complex of applications applications of of topological methods to problems problems of of modern modern physics; in fact fact in several instances methods was was made made to physics; in in several instances itit would the essence essence of the real real physical physical would have have been been impossible impossible to to understand understand the of the phenomena without the of concepts concepts from phenomena in in question question without the aid aid of from topology. topology. Since is not include treatments treatments of topics in in our our Since itit is not possible possible to to include of all aIl of of these these topies survey, we to content ourselves here with the general survey, we shall shall have have to content ourselves here with the following following general remark: found impressive impressive applications very wide wide range range of of to aa very remark: Topology Topology has has found applications to problems qualitative and stability properties properties of both mathematical problems concerning concerning qualitative and stability of both mathematieal and and the the algebraic apparatus that along with and physical physieal objects, objects, and algebraie apparatus that has has evolved evolved along with itit has led to the reorientation of the whole of modern algebra. has led to the reorientation of the whole of modern algebra. The years have shown that that the theory of of The achievements achievements of of recent recent years have shown the modern modern theory Lie groups and their representations, along with algebraic geometry, which Lie groups and their representations, along with algebraie geometry, which subjects have level of of development on the of an an subjects have attained attained their their present present level development on the basis basis of ensemble of deep algebraic ideas originating in topology, play a quite different ensemble of deep algebraic ideas originating in topology, play a quite different role applied for for the the most most part formurole in in applications: applications: they they are are applied part to to the the exact exact formulaic investigation of systems possessing a deep internal algebraic symmetry. laic investigation of systems possessing a deep internaI algebraic symmetry. In fact this earlier in exact sosoIn fact this had had already already been been apparent apparent earlier in connexion connexion with with the the exact lution of problems of classical mechanics and mathematical physics; however lution of problems of classical mechanics and mathematical physics; however itit became clear only only in in modern investigations of systems that that became unequivocally unequivocally clear modern investigations of systems are, in a certain well-defined sense, integrable. It suffices to recall for instance are, in a certain well-defined sense, integrable. It suffices to recall for instance the of inverse (algebro-geometric) finite-gap inthe method method of inverse scattering scattering and and the the (algebro-geometrie) finite-gap integration of non-linear field systems, the celebrated solutions of tegration of non-linear field systems, the celebrated solutions of of models models of statistical physics and quantum field theory, self-dual gauge fields, and string statistieal physics and quantum field theory, self-dual gauge fields, and string theory. aspect of is, however, of note, note, theory. (One (One particular particular aspect of this this situation situation is, however, worthy worthy of namely the need for a serious “effectivization” of modern algebraic geometry, namely the need for a serious "effectivization" of modern algebraic geometry,

Chapter Chapter 1. The The Simplest Simplest Topological Topological Properties Properties

5

which 19th which would would return return the the subject subject in in spirit spirit to to the the algebraic algebraic geometry geometry of of the the 19th century, when it was regarded as a part of formulaic analysis. cent ury, when it was regarded part of formulaic analysis. This This survey survey constitutes constitutes the the introduction introduction to to a series series of of essays essays on on topology, topology, in which the development of its various subdisciplines will be expounded in which the development of its various subdisciplines will be expounded in in greater greater detail. detail.

Introduction Introduction to to the the English English Translation Translation This This survey survey was was written written over over the the period period 1983-84, 1983-84, and and published published (in (in Russian) Russian) in 1986. The English translation was begun in 1993. In view of the appearance in 1986. The English translation was begun in 1993. In view of the appearance in in topology topology over over the the past past decade decade of of several several important important new new ideas, ideas, Il have have added added an appendix summarizing some of these ideas, and several footnotes, an appendix summarizing sorne of these ideas, and several footnotes, in in order or der to to bring bring the the survey survey more more up-to-date. up-to-date. Il am book: to M. am grateful grateful to to several several people people for for valuable valuable contributions contributions to to the the book: to M. Stanko, who performed a huge editorial task in connexion with the Russian Stanko, who performed a huge editorial task in connexion with the Russian edition; the edition; to to B. B. Botvinnik Botvinnik for for his his painstaking painstaking work work as as scientific scientific editor editor of of the English edition, in particular as regards its modernization; to R. Burns for English edition, in particular as regards its modernization; to R. Burns for making a very good English translation at high speed; and to C. Shochet for making a very good English translation at high speed; and to C. Shochet for advice and modernization text at the advice and and help help with with the the translation translation and modernization of of the the text at the University of Maryland. I am grateful also to other colleagues for their help University of Maryland. l am grateful also to other colleagues for their help with the text. text. with modernizing modernizing the Sergei Sergei P. P. Novikov, Novikov, November, 1995

November, 1995

Chapter Chapter 11 The Simplest Topological Properties The Simplest Topological Properties Topology is the study of topological properties or topological invariants of Topology is the study of topological properties or topological invariants of various kinds of mathematical objects, starting with rather general geometrivarious kinds of mathematical objects, starting with rather general geometrical figures. From the topological point of view the name “geometrical figures” cal figures. From the topological point of view the name "geometrical figures" signifies: general polyhedra (polytopes) of various dimensions (complezes); or signifies: general polyhedra (polytopes) ofvarious dimensions (complexes); or continuous or smooth “surfaces” of any dimension situated in some Euclidean continuous or smooth "surfaces" of any dimension situated or in sorne Euclidean space or regarded as existing independently (manifolds); sometimes subspace or regarded as existing independently (manifolds); or sometimes sets of a more general nature of a Euclidean space or manifold, or even ofsuban sets of a more generalspace nature of a Euclidean space itorismanifold, or even of ana infinite-dimensional of functions. Although not possible to give infinite-dimension al space of of “topological functions. Although is not possibleinvariant”) to give a precise general definition property” it (“topological precise general definition of "topological pro pert y" ("topological invariant") of a geometrical figure (or more general geometrical structure), we may de-

of a geometrical figure (or more general geometrical structure), we may de-

66

Chapter 1. 1. The The Simplest Simplest Topological Topological Properties Properties Chapter

scribe such such a property property intuitively intuitively as as one one which which is, is, generally generally speaking, speaking, "stable" “stable” scribe in sorne some well-defined well-defined sense, sense, i.e. remains remains unaltered unaltered under under small small changes changes or dedein formations (homotopies) (homotopies) of of the the geometrical geometrical object, object, no no matter matter how how this this is is given given formations to us. us. For For instance instance for for a general general polytope polytope (complex) (complex) the the manner manner in which which the the to polytope is given given may may be, and and often often is, changed changed by by means means of of an an operation operation polytope of subdivision, subdivision, whereby each each face of of whatever whatever dimension dimension is subdivided subdivided into into of whereby smaller parts, and and so converted converted into into a more more complex complex polyhedron, polyhedron, the the subdisubdism aller parts, vision being being carried carried out out in in such such aa way way as as to to be be compatible compatible on on that that portion portion vision of their boundaries shared by each pair of faces. In this way the whole polyof their boundaries shared by each pair of faces. In this way the who le polyhedron becomes transformed formally into a more complicated one with hedron becomes transformed formally into a more complicated one with aa larger number number of of faces faces of of each each dimension. dimension. The The various various topological topological properties, properties, larger or numerical or algebraic invariants, should be the same for the subdivided or numerical or algebraic invariants, should be the same for the subdivided complex as for the the original. original. complex as for The simplest simplest examples. examples. 1) 1) Everyone Everyone is is familiar familiar with with the the elementary elementary result result The called “Euler’s Theorem”, which, so we are told, was in fact known prior to called "Euler's Theorem" , which, so we are told, was in fact known prior to Euler: Euler: For any any closed, convex polyhedron in 3-dimensional 3-dimensional Euclidean Euclidean space space ]R3, IR3, the the For closed, convex polyhedron in number of less the number of of edges the number number of of (2-dimensional) (2-dimensional) number of vertices vertices less the number edges plus plus the faces, is 2. faces, is 2. Thus V -- E E ++ F is aa topological in that that it it is the Thus the the quantity quantity V F is topological invariant invariant in is the same for any subdivision of a convex polyhedron in lR3. same for any subdivision of a convex polyhedron in ]R3. 2) Another Another elementary observation of of aa topological topological nature, also dating dating back 2) elementary observation nature, also back to Euler, is the so-called “problem of the three pipelines and three to Euler, is the so-called "problem of the three pipelines and three wells” wells" .. Here one is given three points al, ~2, a3 in the plane lR2 (three “houses”) and Here one is given three points al, a2, a3 in the plane]R2 (three "houses") and three other points Al, A2, A3 (“wells”), and it turns out that it is not possible three other points Al, A 2 , A3 ("wells"), and it turns out that it is not possible to join each house ai to each well Ai by means of a non-self-intersecting path to join each house ai to each well Aj by means of a non-self-intersecting path (“pipeline”) in such a way that no two of the 9 paths intersect in the plane. ( "pipeline") in such a way that no two of the 9 paths intersect in the plane. (Of course, this is possible in IR3.) In topological language this conclusion may (Of course, this is possible in ]R3.) In topologie al language this conclusion may be rephrased as follows: Consider the one-dimensional complex (or graph) conbe rephrased as follows: Consider the one-dimensional complex (or graph) consisting of 6 vertices ai, Aj, and 9 edges xij, i, j = 1,2,3, where the “boundary” sisting of 6 vertices ai, Aj, and 9 edges Xij, i, j 1,2,3, where the "boundary" of each edge, denoted by dxij, is given by dxij = {ai, Aj}. The conclusion is of each edge, denoted by ÔXij, is given by ÔXij {ai,A j }. The conclusion is that this one-dimensional complex cannot be situated in the plane R2 without that this one-dimensional complex cannot be situated in the plane IR 2 without incurring self-intersections. This represents a topological property of the given incurring self-intersections. This represents a topological property of the given 0 complex. complex. 0 These two observations of Euler may be considered as the archetypes of the These two observations of Euler may be considered as the archetypes of the basic ideas of combinatorial topology, i.e. of the topological theory of polyhebasic ideas of combinatorial topology, i.e. of the topological theory of polyhedra and complexes established much later by Poincare. It is important to bear dra and complexes established much later by Poincaré. It is important to bear in mind that the use of combinatorial methods to define and investigate topoin mind that the use of combinatorial methods to define and investigate topological properties of geometrical figures represents just one interpretation of logical properties of geometrical figures represents just one interpretation of such properties, providing a convenient and rigorous approach to the formulasuchofproperties, providing convenient rigorous approach to theremaining formulation these concepts at thea first stage ofand topology, though of course tion of these concepts at the first stage of topology, though of course remaining useful for certain applications. However those same topological properties aduse fuI certain applications. same situations, topological for properties mit of for alternative formulations However in variousthose different instance adin mit of alternative formulations in various different situations, for instance in

Chapter Chapter 1. 1. The The Simplest Simplest Topological Topological Properties Properties

77

the contexts contexts ofof differential differential geometry geometry and and mathematical mathematical analysis. analysis. For For an an exexthe ample, ample, let let us us return return toto the the general general convex convex polyhedron polyhedron of of Example Example 11 above. above. By By smoothing smoothing off off its its corners corners and and edges edges aa little, little, we we obtain obtain aa general general smooth, smooth, closed, closed, convex convex surface surface inin R3, ]R3, the the boundary boundary of of aa convex convex solid. solid. Denote Denote this this sursurface face by by M2. M 2 . At At each each point point xx of of this this surface surface the the Gaussian Gaussian curvature curvature K(x) K (x) isis defined, defined, as as also also the the area-element area-element da(x), da(x), and and we we have have the the following following formula formula of of Gauss: Gauss:

JJK(x)da(x)

1 5.G/.I

2~

K(x)da(x)

== 2. 2.

(0.1) (0.1)

M2

In In the the sequel sequel itit will will emerge emerge that that this this formula formula reflects reflects the the same same topological topological property property as as does does Euler’s Euler's theorem theorem concerning concerning convex convex polyhedra. polyhedra. (Euler’s (Euler's thetheorem can be deduced quickly from the Gauss formula (0.1) by continuously orem can be deduced quickly from the Gauss formula (0.1) by continuously deforming deforming aa suitable suitable surface surface into into the the given given convex convex polyhedron polyhedron and and taking taking into account the relationship between the integral of the Gaussian curvature into account the relationship between the integral of the Gaussian curvature and also and the the solid soUd angles angles at at the the vertices.) vertices.) Note Note that that the the formula formula (0.1) (0.1) holds ho Ids also for nonconvex closed surfaces “without holes”. A third interpretation, as it for nonconvex closed surfaces "without holes". A third interpretation, as it turns out, of the same general topological property (which we have still not turns out, of the same general topological property (which we have still not formulated!) in the to Maxwell: Maxwell: formulated!) lies lies hidden hidden in the following following observation, observation, attributed attributed to Consider an island with shore sloping steeply away from the island’s Consider an island with shore sloping steeply away from the island's edge edge into into the surface has features; then the sea, sea, and and whose whose surface has no no perfectly perfectly planar planar or or linear linear features; then the the number of peaks plus the the number number of pits less less the passes is number of peaks plus of pits the number number of of passes is exactly exactly 1. This be easily into an closed surfaces in 1. This may may be easily transformed transformed into an assertion assertion about about closed surfaces in lR3 by formally extending the island’s surface underneath so that it is convex ]R3 by formally extending the island's surface underneath so that it is convex everywhere water (i.e. the island be "floating", “floating”, with everywhere under under the the water (i.e. by by imagining imagining the island to to be with aa convex convex underside satisfying the same assumption as the surface). The resultresultunderside satisfying the same assumption as the surface). The ing floating floating island island then then has has one further pit, the deepest deepest point point on on it. it. ing one further pit, namely namely the We conclude that for a closed surface in R3 satisfying the above assumption, We conclude that for a closed surface in ]R3 satisfying the above assumption, the number of of peaks peaks (points (points of of locally locally maximum maximum height) height) plus plus the the number number of of the number pits (local minimum points) less the number of passes (saddle points) is equal pits (local minimum points) less the number of passes (saddle points) is equal to 2, the same number as appears in both Euler’s theorem and the Gauss ta 2, the same number as appears in bath Euler's thearem and the Gauss formula (0.1) for surfaces without holes. formula (0.1) for surfaces without holes. What if the polyhedron or closed surface in lR3 or floating island is more What if the polyhedran or closed surface in ]R3 or floating island is more complicated? With an arbitrary closed surface M2 in R3 we may associate an complicated? With an arbitrary closed surface M2 in ]R3 we may associate an integer, its “genus” g > 0, naively interpreted as the “number of holes”. Here integer, its "genus" 9 ::?: 0, naively interpreted as the "number of holes". Here we have the Gauss-Bonnet formula

we have the Gauss-Bonnet formula

1

JJ

%SJK(x)da(x) ;7'( K(x)da(x)

= 2 - 29, = 2 - 2g,

(0.2) (0.2)

M2

and the theorems of Euler and Maxwell become modified in exactly the same and the theorems of Euler and Maxwell become modified in exactly the same way: the number 2 is replaced by 2 - 2g. Since Poincare it has become clear way: the number 2 is replaced by 2 2g. Since Poincaré it has become clear that these results prefigure general relationships holding for a very wide class that these results prefigure general dimension. relationships holding for a very wide class of geometrical figures of arbitrary

of geometrical figures of arbitrary dimension.

8

Chapter Chapter 1. The The Simplest Simplest Topological Topological Properties Properties

Gauss certain topological topological properties properties of Gauss also also discovered discovered certain of non-self-intersecting non-self-intersecting (i.e. in Iw 3. ItIt is well known known that simple, closed, closed, concon(i.e. simple) simple) closed closed curves curves in JR3. is weIl that a simple, tinuous (or ifif you piecewise smooth, or even piecewise linear) tinuous (or you like like smooth, smooth, or or piecewise smooth, or even piecewise linear) curve separates separates the into two curve the plane plane lR2 JR2 into two parts parts with with the the property property that that itit is imimpossible from one part to the other of a continuous path possible to to get get from one part to the other by by means means of continuous path avoiding curve. The of this intuitively avoiding the the given given curve. The ideally ideally rigorous rigorous formulation formulation of this intuitively obvious system ofaxioms of axioms for for geometry geometry and obvious fact fact in in the the context context of of an an explicit explicit system and analysis carries the title “The Jordan Curve Theorem” (although of course analysis carries the title "The Jordan Curve Theorem" (although of course in is, in simplified form, form, already in the axiom sysin fact fact itit is, in somewhat somewhat simplified already included included in the axiom system; if one is not concerned with economy in the axiom system, then it might might tem; if one is not concerned with economy in the axiom system, then it just one of conclusion (as (as for for just as well well be included included as one of the the axioms). axioms). The The same same conclusion a simple, simple, closed, continuous curve) holds also for any “complete” curve closed, continuous curve) holds also for any "complete" curve in in lR2, i.e. a simple, unboundedly extended, curve both both JR2, Le. a simple, continuous, continuous, unboundedly extended, non-closed non-closed curve of those ends limit points points in in the finite of those ends go go off off to to infinity, infinity, without without nontrivial nontrivial limit the finite plane. This principle generalizes in the obvious way to n-dimensional space: plane. This princip le generalizes in the obvious way to n-dimensional space: aa closed IR” separates separates itit into parts. In local version closed hypersurface hypersurface in in JRn into two two parts. In fact fact aa local version of this principle is basic to the general topological definition of dimension (by of this principle is basic to the general topological definition of dimension (by induction on n). induction on n). There is however less obvious obvious generalization generalization of of this this principle, principle, havThere is however another another less having its most familiar manifestation in 3-dimensional space iR3. ing its most familiar manifestation in 3-dimensional space JR3. Consider Consider two two continuous simple closed (loops) in in JR3 LR3 which which do do not incontinuous (or (or smooth) smooth) simple closed curves curves (loops) not intersect: tersect:

'l(t);= (xUt),xr(t),xy(t)),

,l(t + 21l") = 'l(t),

'2(7) = (X§(7),X~(7),X~(7)),

,2(7+21l") ='2(7).

Consider Di bounded bounded by Tiyi, i.e. continuous Consider aa “singular "singular disc” disc" Di by the the curve curve ,i, i.e. aa continuo us map of the unit disc into lR3: Z$ = $(T,$), i = 1,2, a = 1,2,3, where map of the unit disc into JR3: xi = xi(r,cp), i = 1,2,0: = 1,2,3, where 0 05 4 sending the of the the unit onto ,i: yi: o 5: : ; rr 5::::; 1,1, 0::::; cp 5::::; 21r, 21l", sending the boundary boundary of unit disc disc onto

xf(r, CP)lr=l

=

xf(cp),

0: = 1,2,3,

where 1, and = 2. where q5 cp = = tt for for ii = = 1, and 4 cp = = 7r for for ii = 2.

,1

,2

Definition 0.1 Two Two curves and 72 in in lR3 said to to be nontrivially Definition 0.1 curves yr and JR3 are are said be nontrivially linked if the curve 72 meets every singular disc D1 with boundary (or, linked if the curve '2 meets every singular disc Dl with boundary yi (or, equivalently, if the curve yi meets every singular disc Dz with boundary 72). equivalently, if the curve meets every singular disc D 2 with boundary '2).

,1

'1

Simple shown in in Figure 1.1. In In n-dimensional n-dimensional space space IRn certain Simple examples examples are are shown Figure 1.1. JRn certain pairs of closed surfaces may be linked, namely submanifolds of dimensions pairs of closed surfaces may be linked, namely submanifolds of dimensions pp and 1. In curve in in IR2 may be be linked and q q where where pp + + q q= = n n -- 1. In particular particular a a closed closed curve JR2 may linked with a pair of points ( a “zero-dimensional surface”) this is just the original with a pair of points ( a "zero-dimensional surface") - this is just the original principle simple closed closed curve curve separates principle that that a a simple separates the the plane. plane. Gauss introduced an invariant of a link consisting of two two simple simple closed closed Gauss introduced an invariant of a link consisting of 3 curves yi, 72 in lR3, namely the signed number of turns of one of the curves curves ,l, ,2 in JR , namely the signed number of turns of one of the curves around other, the linking coefficient coeficient hl, {n,n},2} of link. His formula for for around the the other, the linking of the the link. His formula this is this is

Chapter Chapter 1. 1. The The Simplest Simplest Topological Topological Properties Properties

(a) (a) Unlinked Unlinked curves curves

99

(c) (c) Linking Linking coefficient coefficient 44

(b) (b) Linking Linking coefficient coefficient 11 Fig. Fig. 1.1 1.1

@%w-+Y2(~)1 N

=

{n,rz>

=

;

!.f 71 72

Ir1(t>

,Yl -

-

72)

r2(t)13

’

(0.3) (0.3)

where ] denotes of vectors and (( ,, )) denotes the the vector vector (or (or cross) cross) product product of vectors in in lR3 ]R3 and where [[ ,, 1 the Euclidean scalar product. Thus this integral always has an integer the Euclidean scalar product. Thus this integral always has an integer value value N. of the the curves to be and the lie in in N. IfIf we we take take one one of curves to be the the z-axis z-axis in in lR3 ]R3 and the other other to to lie the (5, y)-plane, then the formula (0.3) gives the net number of turns of the the (x, y)-plane, then the formula (0.3) gives the net number of turns of the plane z-axis. plane curve curve around around the the z~axis. It is interesting to note that the linking coefficient (0.3) may be zero even It is interesting to note that the linking coefficient (0.3) may be zero even though are nontrivially nontrivially linked (see (see Figure 1.2). Thus Thus its its having having though the the curves curves are linked Figure 1.2). non-zero value represents only a sufficient condition for nontrivial linkage of non-zero value represents only a sufficient condition for nontrivial linkage of the loops. the loops. Elementary topological properties of paths and homotopies between them Elementary topological properties of paths and homotopies between thern played an important role in complex analysis right from the very beginning played an important role in complex analysis right from the very beginning of that subject in the 19th century. They without doubt represent one of of that subject in the 19th century. They without doubt represent one of

Fig. 1.2. The linking coefficient = 0, yet the curves

Fig. 1.2. The linking coefficient linked 0, yet the curves are non-trivially are non-trivially linked

10

Chapter The Simplest Properties Chapter 1. 1. The Simplest Topological Topological Properties

the important features features of of the the theory theory of of functions functions of of a complex complex variable, variable, the most most important instrumental to to the effectiveness and cess of of that theory in in aIl instrumental the effectiveness and suc success that theory all of of its its appliapplidefined and and single-valued single-valued cations. A A complex complex analytic function f(z) f(z) is often often defined cations. analytic function only the complex complex plane, in some region U U C c R2 free of of poles, poles, only in in a part part of of the plane, i.e. in sorne region JR2 free branch each closed closed contour CU branch points, points, etc. etc. The The Cauchy Cauchy integral integral around around each contour y'Y C yields a “topological” of the the contour: contour: yields "topological" functional functional of

f

lIb) = f(zkk f(z)dz, MY) f Y ""Y

(0.4) (0.4

in the sense the integral integral remains remains unchanged continuous homotopies homotopies in the sense that that the unchanged under under continuous (deformations) of the the region U, i.e. by deformations of of 'Y y (deformations) of the curve curve y'Y within within the region U, by deformations avoiding the points of of the the function. function. It It is this this very latitude - the the avoiding the singular singular points very latitude possibility the closed without affecting the integral integral possibility of of deforming deforming the closed contour contour without affecting the which enormous opportunities which opens opens up up enormous opportunities for for varied varied application. application. More complicated topological phenomena in the 19th century century More complicated topological phenomena appeared appeared in the 19th in essence essence beginning beginning with and Riemann Riemann - in with the the invesinvesin with Abel Abel and in connexion connexion with tigation complex variable, only implicitly by an tigation of of functions functions f(z) f(z) of of a complex variable, given given only implicitly by equation equation F(z,w) 0, W w = = f(z), (0.5) (0.5) F(z, w) == D, f(z), or else of analytic continuation throughout the plane, plane, of of aa function function or else by by means means of analytic continuation throughout the originally given as analytic and single-valued only in some portion of the originaIly given as analytic and single-valued only in sorne portion of the plane. The former situation arises in especially sharp form, as became clear plane. The former situation arises in especiaIly sharp form, as became clear after in the context of of Abel's Abel’s resolution of the the wellwellRiemann and and Poincare, Poincaré, in the context resolution of after Riemann known problem of the insolubility of general algebraic equations by radicals, known problem of the insolubility of general algebraic equations by radicals, where the the function F(z, w) w) is in two where function F(z, is aa polynomial polynomial in two variables: variables: F(z, w) = wn + al(Z)wn-l

+ . . . + a,(z)

= 0.

(0.6) (0.6)

Such polynomial equation equation has, general, finitely many isolated isolated branch branch Such aa polynomial has, in in general, finitely many points zi,“.,zm in the plane, away from which it has exactly distinct roots roots points Zl, ... , Zm in the plane, away from which it has exactly nn distinct wj(z) , z # zk (k = 1,. . . , m). Here the region U is just the plane Iw2 with Wj(z) , z i- Zk (k = l, ... , m). Here the region U is just the plane JR2 with the the m branch m branch points points removed: removed:

u = It2\ {Zl,. .. )z,}. It turns general the cannot be the It turns out out that that in in general the branch branch points points cannot be merely merely ignored, ignored, for for the following reason. In some neighborhood of each point zs that is not a branch foIlowing reason. In sorne neighborhood of each point Zo that is not a branch point, (0.6) determines determines exactly functions Wj(z) wj(z) such such the equation equation (0.6) exactly nn distinct distinct functions point, the that F(z, wj(z)) = 0. If, however, we attempt to continue any one wj of these that F(z, Wj(z)) = O. If, however, we attempt to continue any one Wj of these functions that neighborhood, neighborhood, we difficulty of of functions analytically analyticaIly outside outside that we encounter encounter aa difficulty the following sort: if we continue Wj along a path which goes round some of the following sort: if we continue Wj along a path which goes round sorne of the and back back to may happen we obtain obtain the branch branch points points and to the the point point zc, zo, itit may happen that that we

Chapter 1. The Simplest Topological Topological Properties Chapter The Simplest Properties

11 11

Fig. 1.3 1.3 Fig. nontrivial i.e. that we arrive arrive at at one other solutions solutions at at nontrivial “monodromy” "monodromy",, i.e. that we one of of the the other 20: Zo: w,(zo) # q(zo), s # j. Proceeding consider aIl all possible loops 'Y(t), r(t), aa S 2 tt S 2 b, b, Proceeding more more systematically, systematically, consider possible loops in the region U = lR2 \ { zi, ... ,zm}, with $a) = y(b) = za. Each such loop in the region U = lR 2 \ {ZI,"" zm}, with 'Y(a) ::=: ')'(b) = Zoo Each such loop determines aa permutation of the the branches of the the function function w(z): w(z): if if we we st start at determines permutation of branches of art at the branch given by wj(z) and continue around the loop from a to b, then we the branch given by Wj(z) and continue around the loop from a to b, then we arrive when at the defined by by w,, y(t) determines determines arrive when tt = bbat the branch branch defined ws , so so that that the the loop loop ')'(t) aa permutation j -+ s of the branches (or sheets) above ZO: permutation j -> s of the branches (or sheets) above Zo:

Y + o-r> u,(j) The inverse path y-l

= s.

(i.e. the path traced backwards

from b to a) yields the

The inverse path ')'-1 (Le. the path traced backwards from b to a) yields the inverse permutation cry-l : s + j, and the superposition yi .ys of two paths inverse permutation (j'YI : S -> j, and the superposition ')'1 . ')'2 of two paths yi (traced out from time a to time b) and 72 (from b to c), i.e. the path ob')'1 (traced out from time a to time b) and ')'2 (from b to c), i.e. the path obtained by following yi by 79, corresponds to the product of the corresponding tained by following ')'1 by ')'2, corresponds to the product of the corresponding permutations: permutations: ~71% = ~-72Ou-Y17

uy-1 = (up.

(0.7) (0.7)

In the general, non-degenerate, situation the permutations of the form CJ?

In the general, non-degenerate, situation the permutations of the form (j'Y generate the full symmetric group of permutations of n symbols. (This is the generate the full symmetric group of permutations of n symbols. (This is the underlying reason for the general insolubility by radicals of the algebraic equaunderlying reason for the general insolubility by radicals of the algebraic equation (0.6) for n 25.) To seethis, note that the “basic” path 35, j = 1,. . . , m, tion for from n 2':5.) see this,thenote thatbranch the "basic" ')'j, j = 1,·", m, which(0.6) starts za,To encircles single point path zj, and then proceeds which st arts from zo, encircIes the single branch point Zj, and then proceeds back to zo along the same initial segment (see Figure 1.3) corresponds, in backtypical to Zo situation along theofsame initial non-degenerate segment (see Figure corresponds, in the maximally branch 1.3) points, to the interthe typical situation of ma:ximally non-degenerate branch points, to the interchange of two sheets (i.e. (T-,, is just a transposition of two indices). The claim change of twofrom sheets just transpositions a transpositiongenerate of two indices). The daim then follows the(i.e. fact(j'Yj thatis the all permutations. then from thethat fact the thatpermutation the transpositions generate aIlif permutations. It follows is noteworthy cry is unaffected the loop y is It is noteworthy that the permutation (j'Y is unaffected if the loop ')' is subjected to a continuous homotopy within U, throughout which its beginsubjected to a remain continuous within U, throughout its beginning and end fixedhomotopy at za. This is analogous to the which preservation of at Zo. This is analogous to the preservation of ning and end remain fixed the Cauchy integral under homotopies (see (0.4) above), but is algebraically the Cauchy integral under homotopies (see (0.4) above), but is algebraically

12

Chapter 1. 1. The Simplest Topological Topological Properties Chapter The Simplest Properties

more dependence of the permutation gy on path "( y is more complicated: complicated: the the dependence of the permutation 0", on the the path non-commutative, contrast with with the integral: non-commutative, in in contrast the Cauchy Cauchy integral:

This consideration leads naturally to to a group with elements the hohoThis sort sort of of consideration leads naturally group with elements the motopy continuous loops ending at a particular particular motopy classes classes of of continuous loops y(t) ,(t) beginning beginning and and ending point U, for for any any region, region, or indeed any any manifold, manifold, complex complex or or topological point .zo Zo EE U, or indeed topological space U. This U (with (with base point ze) spa ce U. This group group is called called the the fundamental fundamental group group of of U base point zo) and (U, zo). 20). The The Riemann Riemann surface surface defined defined by w) = 0 thus and is denoted denoted by by ~1 7fl (U, by F(z, F(z, w) thus gives homomorphism monodromy - from from the the fundamental fundamental group group gives rise rise to to a homomorphism - monodromy to of permutations permutations of its "sheets", “sheets”, i.e. the branches of of the function to the the group group of of its the branches the function w(z) w(z) in in a neighborhood neighborhood of of z == ~0: Zo:

°

f7 : m(U,

zo)

+

&,

(0.9) (0.9)

where denote the the symmetric group on on n n symbols, symbols, and U is is as as before before -- aa where S, Sn denote symmetric group and U region of E2. region of]R2. For F, on other hand, hand, the F(z, w) 20) == transcendent al functions functions F, on the the other the equation equation F(z, For transcendental 0 may many-valued function with infinitely may determine determine aa many-valued function w(z) w(z) with infinitely many many sheets sheets (n = = 00). oo). Here simplest example (n Here the the simplest example is is

°

F(z,w)=expw-z=O, F(z,w)=expw-z=o,

U=R2\0, w=lnz. U=]R2\0, w=lnz.

In in aa natural way by means of of the In this this example example the the sheets sheets are are numbered numbered in natural way by means the integers: taking zo = 1, we have wk = lnzs = 2rik, where k ranges over integers: taking Zo = 1, we have Wk = ln Zo = 27fik, where k ranges over the the integers. y(t) with with hl \yJ == 1, 1, "(0) y(O) = = y(27r) 1, going going round round the the point point integers. The The path path "(t) "(27f) = = 1, zz == 00 in clockwise direction direction exactly exactly once, once, yields yields the y --> -+ 0"" cry, in the the clockwise the monodromy monodromy "( a,(k) = k 1. O",(k) = k-l. An theory where the non-abelianness non-abelianness of of the the fundafundaAn interesting interesting topological topological theory where the mental group r(U, ze) plays an important role is that of knots, i.e. smooth mental group 7f(U, zo) plays an important role is that of knots, i.e. smooth (or, if preferred, closed curves curves (or, if preferred, piecewise piecewise smooth, smooth, or or piecewise piecewise linear) linear) simple, simple, closed or, more generally, the theory of links, as intro-Y(t) C c ]R3, R3, Y(t + 27f) 27r) = = "(t), 7(t), or, more generally, the theory of links, as intro"(t) "(t + duced above, a link being a finite collection of simple, closed, non-intersecting duced ab ove , a link being a finite collection of simple, closed, non-intersecting curves yi, ... . ,,"(k yk C C Iw3. For kk > > 1, 1, one the matrix linking ]R 3. For one has has the matrix with with entries entries the the linking curves "(l, coefficients {ri, rj}, i # j, given by the formula (0.3), which however does not coefficients {ri, 'j}, i =1- j, given by the formula (0.3), which however does not determine all of the topological invariants of the link. In the case k = 1, that determine all of the topological invariants of the link. In the case k = 1, that of knot, there is no such coefficient coefficient available. available. Let Let "( y be be aa knot knot and and U U the of aa knot, there is no such the complementary region of Iw3: complementary region of ]R3:

U=R3\y. U =]R3 \ T

(0.10) (0.10)

It turns out that the fundamental fundamental group rr(U, za), where where Zo za is is any any point point of of It turns out that the group 7f1(U,ZO), U, is abelian precisely when the given knot y can be deformed by means of U, is abelian precisely when the given knot "( can be deformed by means of aa smooth (i.e. byan by an "isotopy", “isotopy” , as it is called) into into the the trivial trivial smooth homotopy-of-knots homotopy-of-knots (i.e. as it is called) knot, i.e. into the unknotted circle S1 c IK2 c Iw3, where the circle S1 lies in in knot, i.e. into the unknotted circle SI C ]R2 C ]R3, where the circle SI lies

Chapter Chapter 1. The The Simplest Simplest Topological Topological Properties Properties

13

for for instance instance the the (z, (x, y)-plane y)~plane (see (see Chapter Chapter 4, 4, $5). §5). Elementary Elementary knot knot theory theory (see (see Figure Figure 1.5) 1.5) shows shows that that the the abelianized abelianized fundamental fundamental group group

where where [xi, [KI, 7ri] Kt] denotes den otes the the commutator commutator subgroup subgroup of of rri, KI, and and U U is, is, as as before, before, the the knot-complement in lR 3, is for every knot infinite cyclic. More generally, knot-complement in ]R3, is for every knot infinite cyclic. More generally, for for a link . . ,,1'd ok} the {'YI, ... the group group Hi(U) Hl (U) is is the the direct direct sum sum of of kk infinite infinite cyclic cyclic groups. groups. link (71,. For any topological space U, the abelianized fundamental group For any topological space U, the abelianized fundamental group 7rr/[nr,~i] KI/[KI' KI] is is called called the the one-dimensional one-dimensional integral integral homology homology group group,, denoted denoted by by HI(U) Hl (U) as above, or by HI (U; Z). The group operation in HI is always written additively. above, or by Hl (U; Z). The group operation in HIis always written additively.

(a) Unknotted curve (a) Unknotted curve (the (the “trivial” "trivial" knot) knot)

(b) (b) The The simplest simplest nontrivial nontrivial knot “trefoil” knot) knot (the (the "trefoil" knot)

(c) The “figure-eight” (c) The "figure-eight" knot knot

Fig. 1.4

Fig. lA

We earlier that that in We saw saw earlier in planar planar valued function analytic U valued function analytic in in U in U, in U,

regions c R2 the Cauchy integral of a singleregions U UC ]R2 the Cauchy integral of a single(without poles!) around closed closed contours contours l' y lying lying (without poles!) around

f

In If b) == f f(z)dz, f( z )dz, 7 'Y

determines a complex-valued linear form on the one-dimensional homology determines a complex-valued linear form on the one-dimensional homology group Hl(U; Z) (see (0.8) in particular). group HI(U; Z) (see (0.8) in particular). It is appropriate to round off our collection of elementary topological obIt is appropriate to round off our collection of elementary topological observations with a more modern example, dating from the 1930s namely the servations with a more modern example, dating from the 1930s, namely the theory of singular integral equations on a circle or arc, which originated as one theory of singular integral equations on a circle or arc, which originated as one of several important boundary-value problems in the 2-dimensional theory of of several important boundary-value problems in the 2-dimensional theory of elasticity (Noether, Muskhelishvili). Subsequently this theory came to have elasticity (Noether, Muskhelishvili). to have much greater significance by virtue of Subsequently its considerablethisroletheory in the came development much greater significance by virtue of its considerable role in the development of the theory of elliptic linear differential operators and pseudo-differential of the theory operators and4 pseudo-differential operators. Let ofHI,elliptic Hz belinear Hilbertdifferential spaces, and let A : HI Hz be a NoetheHl, H 2 bein Hilbert and let A ---> H 2 be a Noetheoperators. rian operatorLet(Fredholm modern spaces, terminology), i.e.: Hl a closed (and bounded) in modern terminology), bounded) rian operator linear operator (Fredholm with finite-dimensional kernel Ker Le. A =a {hclosed 1 A(h)(and = 0) (not to linear operator with finite-dimensional kernel Ker A = {h 1 A(h) = O} (not to

14

Chapter 1. The The Simplest Simplest Topological Topological Properties Chapter Properties

-

Fig. 1.5 1.5 Fig.

be the kernel kernel of and finite-dimensional finite-dimensional be confused confused with with the of an an integral integral operator!), operator!), and “cokernel”, i.e. kernel of the adjoint operator A* : Hz --+ HI. It It turns turns out "cokernel", i.e. kernel of the adjoint operator A* : H 2 ~ Hl. out that the index i(A) of such an operator A, defined by that the index i(A) of such an operator A, defined by i(A) = dim(Ker dim(Ker A*), A*), i(A) = dim(Ker A) A) -- dim(Ker

(0.11) (0.11)

i.e. difference in numbers of of linearly linearly independent solutions of of the difference in the the respective respective numbers independent solutions i.e. the the equations A(h) = 0 and A*(h*) = 0, is a homotopy invariant. This means the equations A(h) = 0 and A*(h*) = 0, is a homotopy invariant. This means simply i(A) remains simply that that the the index index i(A) remains unchanged unchanged under under continuous continuo us deformations deformations of the operator A, although the individual dimensions on the right-hand side of the operator A, although the individual dimensions on the right-hand side of of the the equation equation (0.11) (0.11) may may change. change. In operators K, .6?, In the the simplest simplest case case of of nonsingular nonsingular kernels kernels K(z, K(x, y) y) of of integral integral operators the “Fredholm alternative” was discovered at the beginning of this century. the "Fredholm alternative" was discovered at the beginning of this cent ury. In egect just t&e assertion assertion that that In the the language language of of functional functional analysis analysis this this is is in in effect just the i(A) = 0 for operators A of the form A = 7 + K, where K is a “compact i(A) = 0 for operators A of the form A = î + K, where K is a "compact operator”, g(M) is every bounded subset M operator", i.e. i.e. K(M) is a a compact compact subset subset of of Hz H 2 for for every bounded subset M of HI, and the operator 7 is an isomorphism of the Hilbert spaces and of Hl, and the operator î is an isomorphism of the Hilbert spaces HI Hl and Hz. compact operator operator to operator H 2 . In In fact fact the the addition addition of of a a compact to any any Noetherian Noetherian operator A preserves the Noetherian property, so that the simple$ deformation A preserves the Noetherian property, so that the simplest deformation in in the the class of Noetherian operators has the form At = A0 + tK (with A0 = 7 in class of Noetherian operators has the form At = Ao + tf{ (with Ao = î in the the classical classical Fredholm Fredholm situatian). situation). For is a For singular singular integral integral operators, operators, on on the the other other hand, hand, the the index index is a rather rather more complicated topological characteristic. Much classical work of the 1920s more complicated topological characteristic. Much classical work of the 1920s and 1930s was devoted to explicit calculation of the index via the kernel and 1930s was devoted to explicit calculation of the index via the kernel of of the the operator. operator. Far-reaching Far-reaching generalizations generalizations of of this this theory theory to to higher-dimensional higher-dimensional manifolds, manifolds, culminating culminating in in the the Atiyah-Singer Atiyah-Singer Index Index Theorem, Theorem, have have come come to to be be of exceptional significance for topology and its applications. of exceptional significance for topology and its applications.

§1. Observations from general topology. Terminology

15

This example shows how topological properties arise not only in connexion with geometrical figures in the naive sense, but also in mathematical contexts of a quite different nature.

Chapter 2 Topological Spaces. Fibrations. Homotopies §l. Observations from general topology. Terminology Although topological properties are sometimes hidden behind a combinatorial - algebraic mask, they nonetheless all partake organically of the concept of continuity. The most general definition of a continuous map or function between sets req uires very little in the way of structure on the sets. As introduced by Fréchet, this structure or topologyon a set X, making it a topological space, consists merely in the designation of certain subsets U of X (among aU subsets of X) as the open sets of X, subject only ta to the requirements that the empty set and the who le space X be open, and that the collection of aIl the open sets be closed under the following operations: the intersection of any finite subcollection of open sets should again be open, and the union of any collection, infinite or finite, of open sets should likewise be open. The complement X \ U of any open set U is called a closed set. The closure of any subset V c X, denoted by V, is the smallest closed set containing V. A continuous map (or, briefly, map) f : X ---'; Y between topological spaces X and Y, is then one for which the complete inverse image f-l(U) of every open set U of Y is open in X. (The complete inverse image f-l(D) of any all x E X such that f(x) E D.) A compact subset DeY, is just the set of aU space is a topological space X with the pro perty that, given any covering of X by open sets Ua, i.e. Ua = X, there always exist a finite set of indices ŒI, ... ,ŒN such that the open sets Ual' ... , UaN already COyer X, i.e. there is always a finite subcover. It can be shown that in a compact space X every sequence of points Xi, i = 1,2, ... , has a limit point X oo in X, i.e. a point such that every open set containing it contains also terms Xi of the sequence for infinitely many i. A Hausdorff topological space is one with the pro perty that for every two distinct points Xl, X2 there are disjoint open sets U1 , U2 containing them: UI n U2 = 0, Xl E UI , X2 E U2 . A topological space X is called a metric space if there is a real-valued "distance" p(XI, X2) defined for each pair of points XI,X2 E X, continuo us in XI,X2 (with respect to the "product topology" on X x X - see below), satisfying

U

. 16 16

Chapter Topological spaces. Fibrations. Homotopies Chapter 2. Topological spaces. Fibrations. Homotopies

P(XI,X2) > 0 p(X,X) = 0,

if

Xl

f.

X2,

P(XI,X2) = P(X2,XI),

Metric space X is is one one in which Metric spaces spaces are always Hausdorff. A path-connected space each pair path in X, pair of of points x1, Xl, x2 X2 can be joined joined by a continuous path X, aa path being a map Il ---+ X. Every space dedet X of of an interval interval Il to to X. Every topological topological space composes into composes into pairwise disjoint, disjoint, maximal maximal path-connected path-connected components (pathcomponents). The set of path-components is denoted by 7ro(X) TO(X) and is is components). path-components of of X is called the “zero-dimensional analogue analogue of the homotopy groups” of X X; i in gengencaIled the "zero-dimensional homotopy groups" eral this this set does except in certain certain does not not come come with with aa natural natural group structure, structure, except cases(see (seebelow). special (and important) important) cases below). Given spaces X, Y, we we can form their Y, Given topological topological spaces X, Y, their direct product product X x Y, with ordered pairs (x, y), X, Y y E and as as aa "basis" “basis” for its with points points the the ordered pairs (x, y), xX E EX, E Y, Y, and for its topology is open X and V V in Y. Y. topology the the subsets subsets of the form U x V where U is (An obtained by taking taking the union of any (An arbitrary arbitrary open set of X x Y is then obtained collection collection of of basic basic ones.) ones.) Given topological spacesX, Y, one usually endows the set set Yx Yx of aIl all conX, Y, usuaIly endows Given topological spaces tinuous -+t Y the compact-open compact-open topology. tinuous maps maps f :: X X Y with with aa topology topology called caIled the topology. This follows: take any compact compact set set K K ceX, X, and and any any open open set set V V This is is defined defined as as follows: take any of Y; the set of all maps f sending K to V, f(K) c V, is then a typical basic of Yi the set of aIl maps f sending K to V, f(K) c V, is then a typical basic open set If-the space X and Y Y is is aa metric metric space space with with open set of of Yx. Y x. If the space X itself itself is is compact compact and metric p, then Yx is a metric space with metric j? given by metric p, then Y x is a metric space with metric fi given by

fi(f,g)

= max X p(f(x),g(x)), xE

f, g:

X

-t

Y.

There is from aa pair pair of of There is yet yet another another simple simple but but important important construction construction from topological X, Y, Y, namely bouquet X Y. Strictly Strictly speaking speaking the namely their their bouquet X VV Y. the topological spaces spaces X, bouquet spacesX, Y, i.e. i.e. spaces spaceswith specified points bouquet is is defined defined for for pointed pointed spaces X, Y, with specified points x0 E X, Yo yo E E Y; Y; their their bouquet VY is the space resulting resulting from from identification identification Xo EX, bouquet X X V Y is the space of ~0 and Yo, yo, ~0 in the union of of X and Y: Y: Xo and Xo z ~ yo, Yo, in the formal formaI disjoint disjoint union X and of

( (S' \

"

Fig. 2.1. bouquet X Y Fig. 2.1. The The bouquet X VV Y

5§l. 1. Observations Observations from general general topology. topology. Terminology Terminology xx V vY x y(). Y = = XOY/xo XÜY/xo ::::::! Yo.

17 17

(1.1)

More More generally, given any closed closed subset subset A cc X and map ff :: A ---+ - - t Y, Y, one may identification form form the analogue of the following following analogue of the the bouquet: bouquet: may by by means means of of identification Xv(A,f)Y=XtiY X V (A, f)Y

X Ü y /x=f(z), / x::::::! f(x), zcA. xEA .

An the mapping Y. An important important case case of of this this is is the mapping cylinder cylinder Cf Cf of of a a map map ff :: Z Z -+ - - t Y. Consider the product of 2 with an interval I = [a, b], and form the identificaConsider the product of Z with an interval l [a, bJ, and form the identification space tion space Cf = (2 xx I)LJY/(z,b) M f(z), E 2. (1.2) Cf = (Z I)ÜY/(z, b) :::::3 f(z), zz E Z. (Here 2Z xx Il plays role of and 2 topology on (Here plays the the role of X X and Z xx {b} {b} that that of of A.) A.) The The topology on the space Cf Cf is subset of Cf is the space is defined defined in in the the natural natural way: way: a a subset of Cf is taken taken as as open open precisely natural maps Cf precisely ifif its its complete complete inverse inverse images images under under the the natural maps 2Z xx lI + ----> Cf and Y and Y --+- t Cf Cf,, are are both both open. open. On induced topology On any any subset subset A A of of a a topological topological space space X X the the subspace subspace or or induced topology is defined by taking the open sets of A to be simply the intersections is defined by taking the open sets of A to be simply the intersections with with A A of the open sets of X. of the open sets of X. A of a topological space X is said to converge in X, A sequence sequence of of points points xi Xi of a topological space X is said to converge in X, ifif itit has has a limit in X , i.e. of X with the property that every a limit in X , i.e. aa point point xW Xoa of X with the property that every open set U containing 5, contains the xi all but for almost almost all an ii (that (that is is for for aIl but open set U containing XCX) contains the Xi for finitely many i). The topology on X may be recovered from the knowledge finitely many i). The topology on X may be recovered from the knowledge of of its convergent sequences. its convergent sequences. A homeomorphism between topological spaces X and Y is a continuous, A homeomorphism between topological spaces X and Y is a continuous, one-to-one surjection f : X + Y, such that the inverse f-l : Y --+ X is one-to-one surjection f : X Y, such that the inverse f-1 : y ----> X is also continuous. Here the continuity of the inverse function f-l 1 does not in also continuous. Here the continuity of the inverse function f- does not in general follow from the other defining conditions; it does follow, however, if general follow from the other defining conditions; it do es follow, however, if X is compact and Y Hausdorff. X is compact and Y Hausdorff. Functional analysis provides many examples of continuous bijections with Functional analysis provides many examples of continuous bijections with discontinuous inverses. In particular, for spacesof real-valued smooth funcdiscontinuous inverses. In particular, for spaces of real-valued smooth functions there are various natural kinds of convergence definable in terms of tions there are various natural kinds of convergence definable in terms of different numbers of derivatives, so that the existence of continuous’bijections different numbers of derivatives, so that the existence of continuo us 'bijections with discontinuous inverses is to be expected even in such relatively concrete with discontinuous inverses is to be expected even in such relatively concrete contexts. contexts. One often encounters topological spaceswhich carry at the same time some One often encounters topological whic:h carry at the in same algebraic structure, compatible with spaces the topological structure the time sensesorne that algebraic structure, compatible with the topological structure in the sense that the various algebraic operations are continuous when considered as maps; thus the various algebraic operations are continuous when considered as maps; thus one has topological groups, topological vector spaces, topological rings, etc. oneFrom has topological vector' spaces, topological rings, etc. the purely groups, abstract topological point of view, it is very natural to consider topoFrom the purely abstract point of view, it is very natural to consider topological spaces which have the property of being locally Euclidean, although logical which have the property of being locally Euclidean, although in fact spaces most naturally occurring examples of such entities come with some in fact most naturally occurring examples of such entities come with sorne additional smooth or piecewise linear structure (PL-structure). additional smooth or piecewise linear structure (PL-structure). Definition 1.1 A topological manifold (of dimension n) is a Hausdorff Definition 1.1 XA with topological manifoldthat (of each dimension is a Hausdorff topological space the property of its n)points x has an topological space X with the property that each of its points an open neighbourhood U (i.e. open set, or “region”, containing xx) has which open neighbourhood U (Le. open set, or "region", containing x) which

18 18

Chapter Homotopies Chapter 2. Topological Topological spaces. spaces. Fibrations. Fibrations. Homotopies

is homeomorphic W (for homeomorphic to to an open set of n-dimensional Euclidean space space ]Rn some sorne fixed fixed n). Thus n-manifold is covered by open sets U, each each homeomorphic to ]Rn, II%“, Thus an n-manifold sets Ua and therefore therefore each having via some specific homeomorphism having induced on itit via sorne specifie cp U, --+ + RF, . . , x”, each region of overlap U U,Ct n Uo 'P : Ua ]Rn, local co-ordinates x:, x;, ... x~. On each nU fJ there will then there will then be defined two systems systems (or more) of local co-ordinates, and hence each of these these to the hence a co-ordinate co-ordinate transformation transformation from from each the other: other:

XCtj

-

fj ( Ct

x 1fJ '· .. ,xfJn) '

x

(1.3)

X

fog=1.

$2. Homotopies. Homotopy §2. Homotopies. Homotopy type type A briefly homotopy A continuous homotopy homotopy (or briefly homotopy or deformation) deformation) of a map ff :: X ---t Y, is a continuous map of the cylinder X x Il to --+ Y, a eylinder to Y: F=F(z,t):XxI-+Y, F = F(x, t) : X x l

--+

Y,

XEX, x E X,

aa
(I (I an an interval interval [a, [a, b]) b)) for for which which F(x, for all F(x, a) a) = = f(x) f(x) for aIl xx E E X. X. Two ---+ Y Y are is aa continuous Two maps maps f,f, gg :: X X --+ are homotopic homotopie ifif there there is continuous homotopy homotopy F F such sueh that that F(x, E X. X. F(x, a) a) = = f(x), f(x), F(x, F(x, b) b) = = g(x), g(x), xII: E One Y, i.e. One often often needs needs to to consider consider in in this this context context pointed pointed spaces spaces X, X, Y, i.e. with with particular points x0 E X, yo E Y specified. For such spaces maps f : X -+ Y particular points Xo EX, Yo E Y specified. For such spaces maps f : X --+ Y are to be “pointed”, i.e. to satisfy yj~, and and are usually usually also also required required to be "pointed", i.e. to satisfy f(xo) f(xo) == Yo, homotopies “pointed” maps are then homotopies between between "pointed" maps are then also also normally normally “pointed”, "pointed", in in the the sense that one requires F(xo, t) = yo for all t. sense that one requires F(xo, t) = Yo for aIl t. Each equivalence class Y constitutes Each equivalence class of of homotopic homotopie maps maps ff :: X X --+ --+ Y constitutes a a pathpathcomponent of the function space Yx, and is called a homotopy class of maps component of the function space Y x, and is called a homotopy class of maps X --) Y (or of pointed maps, as the case may be). Thus the set ro(Yx) X --+ Y (or of pointed maps, as the case may be). Thus the set ?rD (Y X) is is comprised classes. comprised of of homotopy homotopy classes. Sometimes one has deal with of spaces spacesA A cc X, B c Y, where where the Sometimes one has to to deal with pairs pairs of X, BeY, the appropriate maps f : X --+ Y are those for which f(A) c B. Such appropriate maps f : X --+ Y are those for which f(A) c B. Such a a map map of of pairs pairs is is denoted denoted by by

53. §3. Covering Covering homotopies. homotopies. Fibrations Fibrations

19

f : (X,4 (X, A) ----> B) ,, - (Y, (Y,B) the space space of all aIl such such maps maps of of pairs has as its path-components and the has as path-components the analogous homotopy analogous homotopy classes classesof maps maps of pairs. There are are several important important reasons, as see below, for considering also category of triples reasons, as we we shall see also the category triples (xo E which the appropriate (20 E A c X) X) for which appropriate maps maps f : X ----> + Y are are those those for f(A) cc B Band f(xo) = yYo, BeY) another such which both which both f(A) and f(zrc) 0, where (Yo (ye E B c Y) is another such triple. Here one homotopy classes pointed maps maps of pairs. triple. one has has homotopy classesof pointed Definition 2.1 A continuo continuous Y is is called aa homotopy equiveqvivDefinition us map f ;: X + ----> Y alence if if there inverse” map 9 g:Y Y ----> --+ X, X, i.e. such such that that the there is a “homotopy "homotopy inverse" alence two composites 9 go ---+ Y are are homotopie homotopic to the two composites 0 f : X X + X and f 0o 9g : Y ---t respective identity maps identity maps lx :x---+x : X ----> X lx

(lx(x) = x), (lx(x) == x),

ly : Y ---t ---+ Y ly:

(ly(y) y). (lY(Y> == = Y>.

yield (For pointed pointed spaces spaces one modifies this definition definition in the obvious way to yield of a pointed homotopy the concept of homotopy equivalence.) The spaces spaces X, X, Y Y are are then said homotopy equivalent, X X N '" Y, said to be homotopy Y, or to have have the same same homotopy type. Suppose now that following conditions hold: X is Suppose that the following is aa subspace subspaceof Y Y (or embedded in Y); Y); f : X --f --f X restricted restricted ----> Y is the inclusion map; g 9 : Y ----> emhedded to X; and, finally, finally, throughout homotopy F to X is the the identity identity map on X; throughout the homotopy deforming f o g : Y -+ Y to the identity map ly, we have F(z, all deforming 0 9 identity ly, we F(x, t) = xICfor aU x E X cC Y. In this situation situation the subspace subspace X is is called aa deformation deformation retract retract Iw” has deformation retract retract of lower of Y. Y. For instance any open region Y Y of of]Rn has a deformation space ]Rn Rn has has any point point as as aa deformation The whole Euclidean space deformation dimension. The retract. Spaces Y property are said to be be contractible contractible (over retract. Spaces y with with the latter latter property o. themselves) or homotopically homotopically trivial: trivial: Y '" w 0. retraction of space Y onto c Y is is aa map ff :: Y -+ X A retraction of aa space onto aa subspace subspace X C y ----> with the property pro perty that with that the restriction restriction of f to X is is the identity identity map on X: X: e s ace X is then called a retract Y. fi x 1 x· The space is then a retract of Y. fix =1x. Th P

$3. Covering Fibrations §3. Covering homotopies. homotopies. Fibrations Consider a an arbitrary arbitrary mapa (continuous) (continuous) map p : X -+ ----> Y. Y. We say that that an ping ff : Z 2 ---t --f Y (via p) if if there is aa mapping 9g : Z 2 ----> + X such that y is covered covered (via such that pog. f ==pog. Suppose now that that we have aa homotopy homotopy F : Z x lI ----> Suppose -+ Y, Y, where 1 I = [a, [a, b], b], and that initial time t = a the map (2) = F(z, some that at the initial rnap ffez) F(z, a) is covered by sorne mapg:Z--+X. map g: Z - ? X. Definition map Definition 3.1 The The rnap space any homotopy space Z and any homotopy F 2 g(z), Z -+ ----> Y is covered (by (by g(z), Y is covered “up "up above" Y above” in

p : X ----> --+ Y is is called a fibration fibration ifif given any --+ Y whose whose initial initial rnap map f(z) : Z xxlI ----> fez) = = F(z, F(z, a) : homotopy F "down “down below" below” in say), the whole homotopy X by some homotopy G : Z x Il ----> some homotopy --+ X, X, i.e.

20 20

Chapter spaces. Fibrations. Chapter 2. Topological Topological spaces. Fibrations. Homotopies Homotopies

ppoo G(z, G(z, t) t). The G is t) == F(z, F(z, t). The homotopy homotopy G is called called a covering homotopy homotopy for for F with initial initial map map g. with For various technical definition is is often often technical reasons reasons a a weakened weakened form of this definition employed in situations situations where the condition imthe space space Z has has one or another condition posed on itit (for example, cellularity Chapter 3). However the essential essential posed cellularity - see see Chapter character of such changes. changes. character of the the concept of fibration fibration is unaffected by such Usually the is imposed Usually the following folIowing additional additional condition condition is imposed in the above definition, that each zr EE Z 2 remaining homotopy tion, namely that each point point Zl remaining fixed under the homotopy F(z, subinterval of [a, [a, b], likewise remain fixed on F(z, t) t) for all aIl t in any subinterval bl, should likewise that G(z, t). t). that subinterval subinterval under G(z, In situations the construction homotopy In the the most important important situations construction of a covering homotopy is carried “homotopy connexion". connexion”. Roughly Roughly speaking aa carried out out by by means means of a a "homotopy homotopy connexion path in Y beginning homotopy connexion is is aa recipe for obtaining obtaining from a a given path at ye E Y and any prescribed point x0 E X above ye, Yo E y point Xo E Yo, aa unique covering path in X beginning at xc. Furthermore this covering path should depend depend path beginning xo. Furthermore covering path continuously on both the given path in Y and the initial point x0 E X at which continuously both the path initial point Xo E the covering path is to begin; this secures the covering-homotopy property for the covering path is to begin; this sec ures the covering-homotopy property for all reasonably well-behaved spaces 2. aIl reasonably well-behaved spaces Z.

M= M=

M=

I~

f

N= & N=~ a

N=

N=S’VS’

b

N=S’VSZ

Fig. 2.2 Fig. 2.2

Some more Some more terminology: terminology: given given aa fibration fibration p p :: X X --f ---> Y, Y, we we call calI p p aa projecprojection, Y the each space space Fy (y), Y y E E Y, Y, aa the total total space, space, Y the base, base, and and each Fy = = pm1 p-l(y), tion, X X the fiber of the fiber of the fibration. fibration. Take Z fiber FyO of aa fibration Y, 9g :: Z Z ---> + X X the the Take Z to to be be a a fiber Fyo of fibration pp :: X X -+ ---> Y, inclusion Z ---> --+ Y Y to to be the single single point point inclusion and and ff :: Z be the the projection projection of of Z Z = = Fv,, FyO to to the

$3. Covering Covering homotopies. Fibrations §3. homotopies. Fibrations

21

in Y joining joining Yo ys to other point yi. Using Using the yo y(t) bbee a path Yo E Y. Y. Let Let "f(t) path in to any any other point YI. the covering-homotopy property (in the form of the above-mentioned “homotopy covering~homotopy property (in the form of the above-mentioned "homotopy connexion”, we always always presuppose) presuppose) it it is straightforward to establish connexion" , which which we straightforward to establish the the following important fact: following important fact: All of Y are homotopy homotopy equivalent. equivalent. AU fibers fibers FV Fy over over aa path-connected path-connected component component of Y are We shall in fact assume henceforth that the base Y of a fibration is pathWe shaIl in fact assume henceforth that the base Y of a fibration is pathconnected. connected. Here simplest examples. examples. Here are are the the simplest 1. Covering A map map pp :: X Y is is aa covering and X X is is aa 1. Covering spaces. spaces. A X --+ -+ Y covering map, map, and covering if it it is with aa discrete F, i.e i.e aa space space aIl all of of covering space space of of Y, Y, if is aa fibration fibration with discrete fiber fiber F, those subsets are (so that its points, which may may be infinite in those subsets are open open (so that its points, which be infinite in number, number, are all isolated isolated from one another), another), such for each Y there there is is are aIl from one such that that for each point point Yy EE Y an U of of Y y (y (y E U c Y) Y) whose complete inverse inverse image image an open open neighbourhood neighbourhood U EUe whose complete p-‘(U) direct product product U U xx FF with F = U, {z~}: p-l(U) is is homeomorphic homeomorphic to to the the direct with F Ua {x a }: p-‘(~) = UU u& 2 U x F, p-l(U) = a ~ U x F, where U, cC X X is is aa homeomorphic copy of The sets U, are are open open in where each each Ua homeomorphic copy of U. U. The sets Ua in XX and the restriction plu, : U, --+ U of p to each of them is a homeomorand the restriction plua : Ua -+ U of p to each of them is a homeomorphism U. The The existence connexion is easily established established phism with with U. existence of of aa homotopy homotopy connexion is easily for covering spaces. for covering spaces.

M

Here the covering space is

ô

~f

d.

Here the covering an infinite tree of space “crosses”is an infinite tree of "crosses" (without cycles and therefore (without cyclesEach and vertex therefore contractible). has Each vertex contractible). four edges incident with it has four edges incident with it

cx:JN Fig. 2.3

Fig. 2.3

Concrete examples of covering spaces over regions of W2 were discussed crete 1examples of covering spaces over regionsInofthose JR2 were discussed in Con Chapter in connexion with Riemann surfaces. examples the

in Chapter 1 in connexion with Riemann surfaces. In those examples the

22

Chapter spaces. Fibrations. Chapter 2. Topological Topological spaces. Fibrations. Homotopies Homotopies

homotopy in the examples are are homotopy connexion connexion is constructed constructed in the obvious obvious way. way. Other Other examples depicted in in Figures Figures 2.2 and depicted and 2.3. 2. Let BBeY c Y be any pair spaces,and denote by X 2. Serre Serre fibrations. fibrations. Let pair of spaces, X the all paths y, : [a, + Y, B: the space space consisting of aIl [a, b] bJ ----> Y, in Y beginning in B:

,(b) y(b) E Y, ,E Y E X X .

,(a) EE B, r(a) B,

Consider the evaluation map p : X --+ the evaluation ----> Y defined by

ph) == y(b). ,(b). P(Y) This + Y. Y. This defines defines a Serre Jibration fibration pp : XX ----> To see ---+ Y Y is fibration, consider any path path y( Y(T), see that that pp :: X ----> is indeed a fibration, T), bb <~ TT <~ c, path ,y EX, E X, c, in the base base Y beginning at the end of aa given path y(b) associating with each point of the the curve y(7) the the path path ‘yr y(b) == y(b). ,(b). By By associating with each point yy of curve Y(T) obtained by from y(b) to y(7) = y, we triviaIly trivially obtained by adjoining adjoining to to ,y the the segment segment from y(b) to Y(T) = y, we obtain a i.e. recipe recipe for Y/(T) in obtain a homotopy homotopy connexion, connexion, i.e. for covering covering each each path path Y(T) in the the base Y by a path yr in the total space X with Yb = y (see Figure 2.4). In base Y by a path in the total space X with = , (see Figure 2.4). In

'T

,b

'T

"......

/.

l l

1

r(a) ,(a)

----

........ ...... ,

Y(T) = Y -----__ -0 h--------------=-~----------~ y(b) Y(C) ,(b) = y(b) y(c) ....

"-

'--

b
altlb

Fig. 2.4 Fig. 2.4 this total space embedded copy of the space B, B, this example, example, the the total space X X contains contains an an embedded copy of the space const., which we may therefore identify consisting of the constant paths y(t) G consisting of the constant paths ,( t) == const., which we may therefore identify with that this is aa deformation deformation retract of X. with B. B. ItIt is is easy easy to to see see that this subspace subspace BB is retract of X. Of especial importance is the situation where B is a single point {yo}, yo Y; Of especial importance is the situation where B is a single point {Yo}, Yo E E Yj the space X is then contractible. In this case X is usually denoted by EYo, the space X is then contractible. In this case X is usuaIly denoted by Eyo, and over yy E Y by (A s noted earlier in in the the more more general and the the fiber fiber F3/ Fy over E Y by n(yo, D(yo, y). y). (As noted earlier general context, the fibers Q(yo, y) are all of the same homotopy type provided the context, the fibers D(yo, y) are aIl of the same homotopy type provided the base Y is yo the Q(yo,yo) R is is the the loop loop base Y is path-connected.) path-connected.) For For yy = = Yo the space space D(yo, yo) == D space of Y yo. Y based based at at Yo. space of 3. (or fiber covering-homotopy 3. For For locally locally trivial trivial fibrations fibrations (or fiber bundles) bundles) the the covering-homotopy property somewhat more difficult to to establish. establish. For For these these fibrations X --f property is is somewhat more difficult fibrations pp :: X ----> Y aIl all the are actuaIly actually homeomorphic homeomorphic to space F. The defining defining Y the fibers fibers Fy Fy are to aa space F. The conditions are as follows: follows: it it is is required to covering covering spaces, spaces, conditions are as required that, that, analogously analogously to each of the the base Y should should have have a U, yyEU EU C C Y, Y, whose whose each point point yy of base Y a neighbourhood neighbourhood U,

$4. §4. Homotopy Homotopy groups groups and and fibrations. fibrations. Exact Exact sequences sequences

23 23

inverse X, is inverse image p-‘(U) p-l(U) ceX, is homeomorphic homeomorphic to to the the direct direct product product U xx F via via aa homeomorphism qq :: p-l(U) p-l(U) --f --t U U x FF “compatible” "compatible" with with the the projection projection p. p. (Here (Here the the fiber fiber F F need need not not be be discrete, discrete, as as in the the case case of of covering covering spaces.) spaces.) Let Let {{Ua,} Ua} be be aa covering covering of of Y by by such such open sets sets U,. Ua. Then Then given given any two two sets sets U,, Ua, Uo U(3 of of the the covering, covering, we have on the the complete complete inverse image of of their their intersection intersection UC2 Ua. nn u, U(3 = ua,a Ua.,(3 two two homeomorphisms homeomorphisms defined: defined: +a : p-l

(Ua,p)

-

Ua,p Ua.,(3 xx FF ,,

40 : p-l

(Ua,a)

-

Ua,p x F

The Ua,p x F then respects fibers and The map map X,,p )..a.,{3 = 4a cPa. o0 4;’ cP 73 1 :: U,,p U0,(3 xx F F + - - t Ua.,(3 x F then respects fibers and induces the identity map of the base UQ,p. induces the identity map of the base Ua.,{3. ItIt therefore therefore has has the the form form

where F is a homeomorphism of the fiber over w, varying >:a.,(3(w) : F F + - - t F is a homeomorphism of the fiber over w, varying where ia,o(w) continuously with w. On Up n we require require intersections Ua,o,-, U ,{3,"Y == U, Ua n n U(3 n U, U"y we continuously with w. On intersections

a

x%PoXfl,?IoXr,a = 1. The are called glue&g or transition The maps maps A,,0 )..0,(3 are called glueing or transition covering of Y by regions U, open regions Ua satisfying satisfying the the covering of Y by open that for each Q there exists a homeomorphism that for each 0: there exists a homeomorphism 4w,, : p-l

(ua) -

(3.1) (3.1)

functions. there is is such such aa functions. IfIf there above requirements, notably above requirements, notably

ua x F,

compatible with p, and these homeomorphisms collectively satisfy (3.1), then compatible with p, and these homeomorphisms collectively satisfy (3.1), then the covering-homotopy property of a fibration follows, and moreover these the covering-homotopy property of a fibration follows, and moreover these data characterize the fiber bundle uniquely. data characterize the fiber bundle uniquely. The concept of a fiber bundle, as just described, turns out to be fundaThe concept of a fiber bundle, as just described, turns out to be fundamental in the theory of manifolds, in differential topology and geometry, and mental in the theory of manifolds, in difIerential topology and geometry, and in the major applications of these theories. We shall encounter the concept in the major applications of these theories. We shaH encounter the concept repeatedly in the sequel. In the most important examples of fiber bundles repeatedly in the sequel. In the most important examples of fiber bundles the homotopy connexions will be determined by “differential-geometric conthe homotopy connexions will be determined by "difIerential-geometric connexions” on the total spaces of the bundles. It is pertinent to note that nexions" on the total spaces of the bundles. It is pertinent to note that for certain bundles such “differential-geometric connexions”, when expressed for certain bundles such "difIerential-geometric connexions", when expressed in term of local co-ordinates, turn out to be what are termed by physicists in term of local co-ordinates, turn out to be what are termed by physicists “Maxwell-Yang-Mills fields”. "Maxwell-Yang-Mills fields" .

$4. Homotopy groups and fibrations. Exact sequences. Examples §4. Homotopy groups and fibrations. Exact sequences. Examples The homotopy groups of a topological space, which we shall now define, are homotopy groups of a topological which we shaH define, are theThe most important invariants, and playspace, a fundamental role now in topology. It the most important invariants, and play a fundamental role in topology. It

. 24 24

Chapter 2. 2. Topological Topological spaces. spaces. Fibrations. Fibrations. Homotopies Homotopies Chapter

has has turned turned out out that that they they are are of of crucial crucial importance importance also also in in the the applications applications of of topological topological methods methods to to modern modern physics, physics, determining determining for for instance instance the the structure structure of of singularities singularities (“declinations”) ("declinations") in in liquid liquid crystals. crystals. However However we we shall shaH not not in in this this section section be be in in aa position position to to embark embark on on a description description of of the the more more serious serious methods methods of of computation computation of of homotopy homotopy groups; groups; this this will will have have to to be be postponed postponed until until we we consider consider manifolds manifolds and and homology homology theory. theory. Let Let S” sn denote denote the the n-dimensional n-dimensional sphere sphere in in IP+‘, JRn+l, i.e. i.e. the the subspace subspace consistconsisting . . ,,P) ing of of the the points points (x0,. (x o, ... xn) satisfying satisfying 2 (xy2 a=0

5 1.

Definition (P, so) SO)-+ Definition 4.1 4.1 The The set of pointed pointed homotopy homotopy classes classes of maps maps (sn, ----t (X, e n-dimensional (X, x0), xo), is called th the n-dimensional homotopy homotopy group of (X, (X, x0), xo), and is denoted by ~n(x, 1l'n(X, ~0). xo). by

(a) (a) n= n = 11

Os,

so -

0

S1 S’

f

Y '1'

/-+ (x Io) 2)0 50----(X, 0

3--' 9

8 S1 V 51 SlVS'

’

(b) n=2

(b) n = 2

Fig. 2.5. 2.5. ff + -I-9g == (f (f vVg) 9) 00'IjJ @ Fig. We have have yet yet to to specify specify the the group group operation operation on on the the elements elements of of this this set, set, Le. i.e. We on the the homotopy homotopy classes. classes.Let Let J, f, gg be be two two maps maps (sn, (P, so) SO)----t + (X, (X, xo). x0). Their Their sum sum on is then then defined defined as as the the composite composite first first of of the the map map 'I11, from sn Sn onto onto the the bouquet bouquet is /J from S” V V sn Sn which which identifies identifies the the equator equator of of sn 5’” to to the the point point 80 SOon on it, it, followed followed by by sn the map map of of the the bouquet bouquet to to the the space space (X,xo) (X,x0) which which coincides coincides with with Jf on on the the the first sphere sphere of of the the bouquet bouquet and and with with 9g on on the the second second (see (seeFigure Figure 2.5). 2.5). first

54. and fibrations. fibrations. Exact §4. Homotopy Homotopy groups groups and Exact sequences sequences

25

Extended classes of of maps maps this sum is well-defined, and yields yields Extended to to homotopy homotopy classes this sum well-defined, and a group operation on on 7l"n(X,XO) ~,(X,ze) ( see Figures > 1 this (see Figures 2.6, 2.6, 2.7). 2.7). For For n > this group group group operation operation is abelian; consequence of fact that for n > > 1 the the operation abelian; this this is a consequence of the the fact that for n-sphere rotated, while the point point sa se fixed, fixed, so as n-sphere may may be continuously continuously rotated, while keeping keeping the

x0 D+ /’ /--- ‘TN \ ‘-.

5’ f-----o---\---...j X'

/I / 653 1; 5 Fig. 2.6 Fig. 2.6

sn ,VSn 0: SJ

Sn

so 50

( x , Xo) x0) (X,

(-a) 8 Fig. 2.7

Fig. 2.7

f

f Fig. 2.8

Fig. 2.8

to interchange its upper and lower hemispheres its upper hemispheres nto >interchange 1 the additive notationandis lower used for the group n> 1 the additive notation is used for the group

(see Figure 2.8). Hence for (see Figureon2.8). operation T~(X,Hence ze). for operation on 7l"n(X,XO).

26 26

Chapter Chapter 2. Topological Topological spaces. spaces. Fibrations. Fibrations. Homotopies Homotopies

Observe Observe that that the the elements elements of of ~,(X,za) 7l'n(X,XO) may may also also be be realized realized as homotopy homotopy classes of of maps maps ff of of the the disc disc Dn Dn sending sending the the boundary boundary dD” aDn == Snel sn-1 toto the the classes point point 50: Xo: (Dn,sn-l) ----+ (X,x0). (X,xo). ff:: (Dn, 9-l) --f Any Any path path r(t), ,(t), 1 5::; tt <::; 2, in in X, X, beginning beginning at at y(l) ,(1) = x0 Xo and and ending en ding at at y(2) ,(2) == ~1, Xl, yields yields a map map of of the the cylinder cyl in der S”-l sn-l xX 11 to to X: X:

sn-l

X

1

----+

1 ~ X,

which which depends depends only only on on the the coordinate coordinate t.t. IfIf we we take take the the union union of of this this map map with with any map f : (D”, S’+‘) + (X, ~a), then we obtain an extension of any map f : (Dn, sn-l) ----+ (X, xo), then we obtain an extension of the the latter latter along along the the path path y: T (f,-y)

: (D” u (9-l

x I), 9-l

x (2))

---+ (X,x1).

By By means means of of this this construction construction we arrive arrive at a a natural natural isomorphism between the (see Figures 2.9, the n-th n-th homotopy homotopy groups with with different different base base points x0, xo, x1 Xl (see 2.10, 2.11): 2.10,2.11): -yp

: 7&(X,

x0)

-

%(X,X1)

i

L

D; Dl :: 2 2

D~ :

Sb sb

i

L

(X j )2 ::; 2

j=1 80

= (1,0, ... ,0)

8~

= (2,0, ... ,0)

Fig. 2.9 Fig. 2.9

Fig. 2.10 2.10 Fig.

(xj)’j )2 ::; 2 11 (X

j=l j=l

Fig. 2.11 2.11 Fig.

54. §4. Homotopy Homotopy groups groups and and fibrations. fibrations. Exact Exact sequences sequences

27 27

The The isomorphism isomorphism -y!“’ 'Y~n) depends depends only only on on the the homotopy homotopy class class of of the the path path y'Y in in the the class class of of all aU paths paths joining joining 50 Xo to to 21. Xl. In In the the case case 50 Xo = xiXl each each such such homotopy homotopy class class is is a homotopy homotopy class class of of loops based based at at xc, Xo, and and is is therefore therefore the the element element of of the first homotopy group ri(X, 50) (the fundamental group of (X,x0)). We the first homotopy group 71'1(X,XO) (the fundamental group of (X,xo)). We conclude conclude that that the the fundamental fundamental group group xi71'1 (X, (X, xc) xo) acts acts naturally naturally as as aa group group of of operators on each of the groups 7rr,(X,zo). For n = 1 this action is operators on each of the groups 71'n(X,Xo). For Tb = 1 this action is just just the the conjugating conjugating action action inducing inducing inner inner automorphisms automorphisms of of the the group group ~1 71'1 (see (see Figures Figures 2.12, 2.13): 2.12, 2.13): y!l)(u)=ywy-l, UE7r1, YE7r1.

X0

=0 f""- -

~-1

~

-

--,

1

1

1

1

,

t

1

1

1

! "

",'

2l~)

(Ll)

Fig. 2.13

Fig. Fig. 2.12 2.12

Fig. 2.13

A very important class of topological spaces is that of simply-connected A very important class of topo1ogica1 spaces is that of simply-connected ones, i.e. those with trivial fundamental group 7ri (X, ~0). In this case it folones, Le. those with trivial fundamental group 71'1 (X, xo). In this case it f01lows from the preceding discussion that the homotopy groups n,(X,xc) are lows from the preceding discussion that the homotopy groups 71'n(X, xo) are essentially independent of the choice of base point zc (for path-connected essentially independent of the choice of base point Xa (for path-connected spaces X); 7rn(X, 50) may simply be defined as consisting of the free homospaces X); 71'n(X, xo) may simply be defined as consisting of the free homotopy classesof maps S” + X, without specifying any base point. In the topy classes of maps X, without specifying any base point. In the general case (of connected but not necessary simply-connected spaces) the general case (of connected but not necessary simply-connected spaces) the free homotopy classesof maps S” --+ X (i.e. unpointed) are determined alfree homotopy classes of maps X (i.e. unpointed) are determinecl a1gebraically as the orbits under the action of ~1 = ~i(X,zc) on the group gebraically as the orbits under the action of 71'1 71'1(X,Xa) on the group r,(X, ~0). In the casen = 1, these are just the conjugacy classesof the group 71'n (X, xo). In the case n = 1, these are just the conjugacy classes of the group

sn

sn _

Xl. 71'1·

It follows easily from its lt follows easily from its product of two spacesis just product two spaces: spaces is just of those of factor of those factor spaces:

definition, that the n-th definition, that the n-th the direct product of the the direct product of the

homotopy group of a homotopy group of a n-th homotopy groups n-th homotopy groups

Tn (X x y,zo x Yo) = rn (X,x:,) x nn (Y,Yo) X Y,xa x Ya) = 71'n (X,xo) X 71''1 (Y,ya)·

71'n (X

There is also the smash (or tensor) product of spaces: There is also the smash (or tensor) product of spaces:

x A Y = x x Y/(X x yo) v (zo x Y) ) XAY XxY/(XxYo)V(xaxY),

1

28 28

Chapter 2. 2. Topological Topological spaces. spaces. Fibrations. Fibrations. Homotopies Homotopies Chapter

where where the the relationship relationship between between the the homotopy homotopy groups groups of of X, X, YY and and XX A/\ YY is is given bilinear pairing pairing of of homotopy homotopy groups: gro7tpS: given by by aa bilinear

@: n(X) @Tn(Y) -

m+,(X A Y) ,

where where @ if> has has the the form form @=(f,g), if> = (i, g) ,

ff::9-x, Si

---+

X,

g:S”-Y, g: sm

---+

Y,

sl+““slAs”. sl+m ~ Si /\ Sm .

Given Given any any group group r, 7r, we we define define the the integral integral group group ring ring Z Z [n] [7r] of of x7r to to have have as the formal formaI finite finite linear linear combinations combinations of of the the elements elements of of 7r 7r with with as elements elements the integer integer coefficients: coefficients: N

a= a

L 5 ÀiUi, but7

XiÀi EE Z, Z,

IfdiET.E 7r . Ui

i=l i=l

Addition is defined defined Addition is is as as usual usuai for for linear linear combinations, combinations, and and the the multiplication multiplication is as C Xiu,, we set ÀiUi, bb = CE pjvj, jLjVj, we set as follows: follows: given given aa = E a.b= a·b

(~Xzu~).(~~~lij) (LÀiUi)' (LIJ,jVj) =~&P~(G.v,). = LÀil),j(Ui·Vj). i,j iJj

There an alternative, equivalent, definition of the the There is is an alternative, equivalent, definition of usual in the context of analysis: the elements are usual in the context of analysis: the elements are + Xi) with finite finite support, added in in aa :: 7r 7r + Z Z (ui (Ui À i ) with support, added valued functions, and multiplied via “convolution” valued functions, and multiplied via "convolution" functions a, a, bb their their product product is is given given by by functions

L

L

v·w=u

vE'/T

a. b(u) = c a· b(u) =

(4.1) (4.1)

(integral) group group ring ring more more (integral) taken to be the functions taken to be the functions the usual way for integerintegerthe usual way for as follows: given two such as follows: given two such

a(w)b(w) = c a(v)b(v-‘u) a(v)b(w) = a(v)b(v-1u) . V~W=‘u. VET7

((4.2) 4.2)

As already noted, this is the same ring Z[n]. This concept can be generalized As already noted, this is the same ring Z[7r]. This concept can be generalized to the situation of a continuous (i.e. topological) group by replacing the sums to the situation of a continuo us (Le. topological) group by replacing the sums by integrals. by integrals. The point for us here is that, to put it in algebraic language, every hoThe point for us here is that, to put it in algebraic language, every homotopy group 7r,(X,za), n > 1, is in a natural way a Z[r]-module, with motopy group 7rn (X,xo), n > 1, is in a natural way a Z[7r]-module, with 7r = ~1(X, zs), i.e has the structure of a linear space with ring of “scalars” 7r 7rl(X,XO), i.e has the structure of a linear space with ring of "scalars" Od. Z[7r]. The fundamental group r1 was introduced by Poincark. It was only in the The fundamental group 7rl was introduced by Poincaré. It was only in the 1930sthat the higher homotopy groups rr,, n > 1, were defined (by Hurewicz). 1930s that the higher homotopy groups 7rn , n > 1, were defined (by Hurewicz). Initially the Z[n]-module structure of the latter groups went unnoticed, and the Z[7r]-module structure of the latter groups went unnoticed, and Initially the abelianness of the 7r, for n > 1 created a false impression of their formal the abelianness of the 7rn with for n~1. > It1 created false impression of their formaI simplicity in comparison was onlya somewhat later (beginning with 7rl. It was only somewhat later (beginning with simplicity in comparison with Whitehead sometime in the 1940s) that their module structure came to be Whitehead sometime in the 1940s) that their module structure came to be used - although rather unsystematically. The active and systematic exploitarather unsystematically. active and systematic used although tion of the Z[r]-module structure of the The higher homotopy groups exploitaoccurred tion of the Z[7r]-rnodule structure of the higher homotopy groups occurred

§4. Homotopy groups Exact sequences $4. Homotopy groups and fibrations. fibrations. Exact sequences

29

from 1960s on, on, in course of of intensive intensive development development of of the theory of of from the the 1960s in the the course the theory non-simply-connected manifolds. It is curious that in certain particular probnon-simply-connected manifolds. It curious that in certain particular problems module-theoretic considerations played played a significant lems of of homology homology theory, theory, module-theoretic considerations significant role already in the 1930s. raie already in the 1930s. We the category triples (za c X). turn to to the category of of triples (xo EE A A eX). We now now turn Definition 4.2 The The n-dimensional relative homotopy group 1r r,(X, q,), Definition 4.2 n-dimensional relative homotopy group n (X, A, xo), n > 1, has as its elements the homotopy classes of maps n > has as elements homotopy classes of maps

ff:: (W (D n , S--l, sn-l, so) so) -- - t (X, A, 20)) xo) , where structure is defined on on this this set set in in where f(P) f(sn) cc A, A, f(sa) f(so) = = ~0. xo. The The group group structure is defined a that of of the the corresponding homotopy group group (see (see to that corresponding absolute absolute homotopy a similar similar manner manner to Figure Figure 2.14). 2.14). It out that that for n > > 22 the the groups 7rn(X, 50) are are abelian. abelian. Here it is is It turns turns out for n groups 1r n (X, A, A, xo) Here it the TT~(A,zO) which acts as as aa group on the the 1r n,(X, A,zo), the group group 1rl(A,xo) which acts group of of operators operators on (X,A,xo), n n > 2, they may as Z[r]-modules with 1r 7r = = 1rl 7rr(A, (A, ~0). n > 2, so so that that they may be be considered considered as Z[1r]-modules with xo).

X’bO

x1=0

x1;: 0

-

Y

'f

Fig. 2.14

Fig. 2.14

In the “relative” context there arise three natural homomorphisms: In the "relative" context there arise three natural homomorphisms:

A : ~(x, 50) 8 : rn(X, A, xo) G : ~,(A,xo) The definition of the The definition of the map f : (P, S-l) 4 map f: (Dn,sn-l) - - t

~(x, -4x0), P-l(A,

xo),

(4.3) (4.3)

nn(X,xo).

homomorphism j, depends on the observation that a homomorphism j* depends on the observation that a (X, 50) may be regarded as a map g of triples: (X,xo) may be regarded as a map 9 of triples: f = g : (II”, S-l, so) ---+ (X, A, x0), f g: (Dn,sn-l,so) --t(X,A,xo), in view of the fact that x0 E A. in view of the fact that Xo E A. The homomorphism 8 is defined by restricting maps f : (P, P-l) + The homomorphism [) is defined by restricting maps f : (Dn, sn-l) - - t (X, A, x0) to the boundary dD”n = P-l of D”: (X, A, xo) to the boundary [)D = sn-lof Dn:

... 30 30

Chapter 2. Topological Topological spaces. Chapter spaces. Fibrations. Fibrations. Homotopies Homotopies g == &-I (Sn-l,~o) +---> (A,xo). (A,x,,). f18n-1 : (sn-l,so) The “inclusion i, comes comes directly directly from the the inclusion inclusion A The "inclusion homomorphism” homomorphism" i. from A

cc

X. X. ItIt is is almost almost obvious obvious homomorphisms i,, j,, homomorphisms i., j*,

also composition of any any two two "neighbouring" “neighbouring” also that that the the composition of d8 yields the zero homomorphism: yields the zero homomorphism:

j*oi* j. 0 i. =o,0,

aoj, j. =o,0, 80

i*od=O, i. 08

= 0,

whence we whence we infer infer that that Im lm i,i.

Ker j,, c Ker j.,

Im j,j. lm

c

8 cc Ker Ker ii,.•. Ker 8, d, Im cc Ker lm 8

It turns out (after (after aa little that these subgroup inclusions are in It turns out little calculation) calculation) that these subgroup inclusions are in fact equalities (called “exactness conditions”): fact equalities (caUed "exactness conditions"): Im i, = lm i.

Ker Ker j,, j.,

Im lm j,j. =

This expressed by This is is normally normaUy expressed by groups and homomorphisms is groups and homomorphisms is

Ker 8, Ker 8,

Im 8d = Ker i,. lm

((4.4) 4.4)

the the following sequence of of the statement statement that that the following sequence “exact”: "exact":

+ . k T,(X,A,Q) 5 T+~(A,Q,) -3 % TT,(X,Z~) ~~ r,(A,xo) 7r n (A,xo) ~ 7r n (X,xo) ~ 7r n (X,A,xo) ~ 7r n -l(A,xo) ---> ... (4.5) (4.5) This is is called the exact exact homotopy sequence of the pair (X, A). This caUed the homotopy sequence of the pair (X, A). The construction of the homotopy absolute ones 7rn(X, 50) The construction of the homotopy groups groups -~ the the absolu te ones 7rn (X, xo) on the one hand and the relative ones rrn(X, A,q) on the other - may on the one hand and the relative ones 7r n (X, A, xo) on the other may be regarded covariant functors the category of pointed be regarded as as determining determining covariant functors from from the category of pointed topological (or the category of triples, as the case may be) to the topological spaces spaces (or the category of triples, as the case may be) to the category of (abelian) groups. This means simply that maps ...

category of (abelian) groups. This me ans simply that maps f :x

f :X

+

--->

Y,

x0 + Yo, Xo - t Yo,

Y,

(A ---> B) (A B)

of pointed spaces (X, 50) (or triples (X, A, 50)) determine homomorphisms of of pointed spaces (X, xo) (or triples (X, A, xo)) determine homomorphisms of the corresponding homotopy groups

the corresponding homotopy groups f* : %(X,X0)

-

%(Y,

f* : rm(X, A, xo) These homomorphisms These homornorphisms map g : (D”,n S-l) + map g : (D , sn-l) ---> ciate the map

ciate the map and similarly

Yo),

nn(Y, B, YO).

are obtained essentially in the following way: to each are obtained essentially in the following way: to each (X, ~0) representing an element of 7rn(X, 20) we asso-

(X, xo) representing an element of 7rn (X, xo) we asso-

f 0 g : (D”,

s-l> (Y, Yo) , fog: (Dn,Sn-l) --->(Y,Yo),

for triples.

andIn similarly for triples. the particular examples above we had ln the particular examples above we had i :A +

i :A

X,

X,

i(xo)

i(xo)

i, : nn(A,xo) -

= x0,

xo,

rn(X,xo),

54. Homotopy Homotopy groups and fibrations. fibrations. Exact Exact sequences sequences §4. groups and

31 31

(the inclusion inclusion homomorphism), and (the homomorphism), and

j : j:

{

x-x X~X Xo ~ A x0-A x0 - Xo x0 Xo

,

The above above collect.ion collection of elementary elementary properties of the the homotopy homotopy (and likewise likewise The of properties of (and homology) groups ~ especially their functoriality and the exact sequence homology) groups especially their functoriality and the exact sequence of aa pair pair (X, (X, A) A) ~ leads, leads, after after very very substantial substantial further development, development, to an an of further to elaborate algebra,ic apparatus for topology, as we shall see below. elaborate algebn\Ïc apparatus for topology, as we shal! see below. In one case, of of extreme extreme importance importance in connexion connexion with algebraic algebraic methods In one case, in with methods for computing the homotopy groups, the relative homotopy groups reduce for computing the homotopy groups, the relative homotopy groups reduce to the absolute ones. Consider an arbitrary fibration p : X -----) Y, with the to the absolute ones. Consider an arbitrary fibration p : X Y, with the covering-homotopy property (as usual with respect to any prescribed initial covering-homotopy property (as usual with respect to any prescribed initial condition 0). Let Let Yo ye E E Y, Y, and and zo E p-‘(ye) = Fo. An important, important, albeit condition at at t == 0). Xo E p-l(yO) = Fo. An albeit not especially difficult, theorem asserts that there is an isomorphism not especial!y difficult, theorem asserts that there is an isomorphism

t

((4.6) 4.6) This is established projection homomorphism p* (arising from from the the This is established using using the the projection homomorphism p* (arising projection p of X to Y sending the fiber FO to the point ye). Starting with projection p of X to Y sending the fiber Fo to the point Yo). Starting with the we may isomorphism intuitively as the covering-homotopy covering-homotopy property, property, we may see see this this isomorphism intuitively as follows. Each map f : D” + Y, representing an element of x,(Y, yn), may

fol!ows. Each map

f : Dn

~

Y, representing an element of 7l'n(Y, Yo), may

-4 (Di)

Dt”

aDi

so

@ Fig.

2.15

Fig. 2.15

be lifted to X in view of the contractibility of the disc D” to the point so on be lifted to X S”-’ in view of the contractibility of the disc Dn to the point So on its boundary (see Figure 2.15). By lifting this map to X we obtain a its boundary sn-l (see Figure 2.15). Dy lifting this map to X we obtain a covering map D” -+ X which maps the boundary S’“-l not necessarily to a covering map Dn ~ X which maps the boundary sn-l not necessarily to a point, but to the fiber FO over yo. A straightforward argument now yields the point, but to the fiber(4.6). Fo over Yo. A straightforward argument now yields the desired isomorphism desired (4.6). As a isomorphism consequence of this isomorphism, i.e. ultimately of the coveringAs a consequence this isomorphism, ultimately homotopy property, oneof obtains the following i.e.exact sequence ofofthe the coveringfibration homotopy property, one obtains the following exact sequence of the fibration (rather than of the pair (X, Fo)):

(rather than of the pair (X, Fo) ):

32 32

Chapter 2. Topological Chapter 2. Topological spaces. spaces. Fibrations. Fibrations. Homotopies Homotopies

( 4.7) Important 1. Let Important cases. cases. 1. Let pp :: X X --) Y Y be be aa covering-space covering-space projection projection (see (see above) with (discrete) fiber Fo. Then since ni(Fo, ~0) = 0 for i > (0 denoting denoting above) with (discrete) fiber Fo. Then since 7l'i(Fo, xo) 0 for i 2': 11 (0 the the exact exact sequence in this the trivial trivial group), group), the sequence (4.7) (4.7) becomes becomes in this situation: situation: for for for for

nn >> 12 n ==

1, 1, 1, 1,

0-

%(X,Zo)

3

Tn(Y,Yo)

0 -

?(X,Zo)

-

w(Y,

Yo)

-5

0; ~o(Fo,Zo)

-

0.

Thus Thus from from exactness exactness we we obtain: obtain:

(4.8) and ~0) is isomorphic to a subgroup of rl(Y, yo), and in in the the case case nn == 11 that that 7rl(X, 7l'1 (X, xo) is isomorphic ta a subgroup of 7l'1 (Y, Yo), furthermore in such way that the furthermore in such way that the set set of of cosets cosets is is “isomorphic” "isomorphic" to to (i.e. (Le. in in oneoneto-one correspondence with) the number of components of the fiber; in other ta-one correspondence with) the number of components of the fiber; in other 0 words each coset sheet of words each coset corresponds corresponds to to aa sheet of the the covering. covering. D 2. Consider the Serre fibration over any space Y; thus here (as before) the 2. Consider the Serre fibration over any space Y; thus here (as before) the total of all total space space XX = EVO EyO is is the the space space of aU paths paths y(t), 1'(t), aa 5::::: tt <::::: b, b, beginning beginning at at the point yo = r(a). Clearly X is contractible. The fiber PO = p-‘(~0) the point Yo = 1'(a). Clearly X is contractible. The fiber Fo = p-1(yO) over over the space R(yo), consisting of of all the chosen chosen point point Y/O Yo isis the the loop loop space [J(yo), consisting aU closed closed paths paths beginning and ending at yo. For this fibration the exact sequence (4.7) yields: beginning and ending at Yo. For this fibration the exact sequence (4.7) yields: 0+

7r,(Y,y/o)

5

q-l(.n(YO),~O)

-

0,

n 2 1,

whence we infer the isomorphism; whence we infer the isomorphism; %(Y,Yo)

"

%-l(%/O)r~O),

(4.9) (4.9)

where zo denotes the trivial loop with image the point yo for all t. This where xo denotes the trivial loop with image the point Yo for aIl t. This isomorphism was in fact used by Hurewicz to define the homotopy groups isomorphism was in fact used by Hurewicz to define the homotopy groups 0 recursively. D recursively. Returning now to an arbitrary fibration p : X -+ Y, we recall the basic Returning now to an arbitrary fibration p : X ~ Y, we recall the basic assumption that the fibration comes with a “homotopy connexion”. In parthat the fibration cornes with a "homotopy connexion". In each parassumption ticular this allows us to translate a fiber along a path in the base; i.e. to ticular this allows us ta translate a fiber along a path in the base; Le. to each path y(t), a 5 t 5 b, in the base there corresponds a map of fibers path 1'(t), a::::: t ::::: b, in the base there corresponds a map of fibers r: F. 4

;Y: Fo

~

Fl,

Fo =

FI =

P-‘(Yo),

P-‘(YI),

F1 , Fo p-l(yO), Fl = p-l(yt}, YO = r(a), ~1 = y(b), Yo = 1'(a), Yi = 1'(b),

which depends continuously on the path y. A closed path y beginning and which depends continuously path T equivalence A closed path l' beginning and ending at yo yields in this wayona the homotopy ending at Yo yields in this way a homotopy equivalence 7 : F. --+ Fo,

;Y: Fo

~

Fo,

$4. Homotopy Homotopy groups groups and and fibrations. fibrations. Exact Exact sequences sequences §4.

33 33

which in turn turn induces induces "monodromy" “monodromy” isomorphisms between the the n-th n-th homohomowhich isomorphisms between topy groups: groups: topy

:r* : 7rn (Fo,xo) ~ 7rn (Fo,:r(xo)).

We met met with with a particular particular case of of this this in in the the discussion discussion of of Riemann Riemann surfaces surfaces We in Chapter Chapter 1, 1, where where we we had had nn = = 0, 0, the the base base was was aa region region of of ]R2, Iw2, and and the the in monodromy, denoted by uy, was a permutation of the discrete fiber. monodromy, denoted by (J'Y' was permutation the discrete fiber. It is is worthwhile worthwhile distinguishing, from among among the the classes classes of of aIl all covering covering It distinguishing, from spaces, the regular ones, namely those where the image p,ni(X) c ,1(Y) (Y) spaces, the regular ones, namely those where the image p* 7r1 (X) C 7r1 is a normal subgroup of rr(Y). For such a covering space the quotient group is a normal subgroup of 7rl (Y). For such a covering space the quotient group acts naturaIly naturally as as aa discrete discrete group group of of transformations transformations m(Y)/p*n(X) r 7r1 (Y)/P*7r1 (X) = r acts (homeomorphisms) of the space X, the action being defined via the transport transport (homeomorphisms) of the space X, the action being defined via the of fibers along closed paths y in the base Y, i.e. via monodromy; the paths of fibers along closed paths "( in the base Y, Le. via monodromy; the paths belonging to those homotopy classes contained in the image p,rr(X) yield belonging to those homotopy classes contained in the image p* 7r1 (X) yield trivial monodromy. trivial monodromy. The largest largest covering covering space space X X of of aa given given space space Y Y is is called called aa universal universal covcovThe ering space of Y. It can be shown to exist uniquely for a large class of spaces ering space of Y. It can be shown to exist uniquely for a large class of spaces Y. The The space space X X is is simply-connected: simply-connected: 7r1(X) ri(X) is is trivial, so that, that, in in view view of of the the trivial, so Y. above, it is a regular covering, and r = ri(Y) has a natural discrete action on above, it is a regular covering, and r = 7r1 (Y) has a natural dis crete action on it, with orbits the fibers p-‘(y). Since X is simply-connected, its homotopy it, with orbits the fibers p-l(y). Binee X is simply-connected, its homotopy groups are independent of the the base point xo. ~0. The The group group rr (acting (acting discretely discretely base point groups are independent of and freely on X, i.e. only the identity element fixes any point) induces homoand freely on X, i.e. only the identity element fixes any point) induces homomorphisms morphisms y E r. y"( :: 7rn(X) 7rn(X), n > 1, 1, "( 7r (X) +~ 7r (X), n> E r. n

n

Taking into account that r Taking into account that with the natural geometric with the natural geometric

r

= ,i(Y, ys), we thus have actions of ~1 on all nn = 7r1(Y' Yo), we thus have actions of ?Tl on all 7rn interpretation. interpretation.

Example 1. Let U c Iw3 be the region obtained by removing from lR3 a line Example 1. Let U C ]R3 be the region obtained by removing from ]R3 a line and any point off that line. The region U then has as deformation retract the and any point off that line. The region U then has as deformation retract the bouquet of the circle and 2-sphere: bouquet of the circle and 2-sphere:

u N s2 v s1 = Y. One easily deduces that ri(U) = ni(S’ V 5”) ” Z 2 nl(S1), where One easily deduees that 7r1(U) = ?TI(S2 V SI) ~ Z ~ ?TI(Sl), where infinite cyclic group, with generator t, say. Consider the universal infinite cyclic group, with generator t, say. Consider the universal U, or, rather, the homotopically equivalent universal cover X of the U, or, rather, the homotopically equivalent universal coyer X of the s2 v s1 = Y: S2 V SI = Y: X+Y=S2VS1.

Z is the Z is the cover of coyer of bouquet bouquet

The space X turns out to be representable as the line Iw (coordinatized by A, The space X turns out to be representable as the line ]R (coordinatized by À, say) with 2-spheres ST attached at the integer points j (see Figure 2.16). The say) with 2-spheres at the action of ,1(Y) on X isattached here given by integer points j (see Figure 2.16). The action of ?Tl (Y) on X is here given by t(X) = x + 1, t (sj2) = s;+1. (4.10)

SJ

(4.10)

The space X has as its 2-dimensional homotopy group 7rz(X) the direct sum The space X has as its collection 2-dimensional homotopy groupbasis 7r2(X) direct sum of a countably infinite of copies of Z, with {dj} thesay, each dj of a countably infinite collection of copies of Z, with basis {d j } say, each dj

34 34

Chapter 2. Topological Topological spaces. Fibrations. Homotopies Chapter 2. spaces. Fibrations. Homotopies

2

52

52

dad -2

QQ

a

-1

z.

t(X) = x + 1, t (sjg = s;+l Fig. 2.16 2.16 Fig.

defined geometrically geometrically by the obvious obvious map map 8S22 ---+ 5’:. The The action action of of 7fl rrr(Y) on defined by the - 8J. (Y) on 7rz(X) is is then by 7f2(X) then obviously obviously given given by t@J

= dj+l.

(4.11) (4.11)

In view isomorphism (4.8), (4.8), this describes also also the the structure structure of of 7f2( 7r2(S2 V In view of of the the isomorphism this describes 82 V 9) ” 7@,(U), mc 1 usive of the action on it of .iri(S2 V S”) ” Z. In module 1 8 ) ~ 7f2(U), inclusive of the action on it of 7fl(.<,2 V S1) ~ Z. In module language, 7r2(U) ~2 (U) is is thus Z[r]-module (K = 1I'1(U)) nr(U)) on on the the one one generator generator language, thus aa free free Z[7f]-module (7r = dd == do, da, since since each element a of 7rz(U) has the unique form each element a of 7f2(U) has the unique form +M E Aid(d), j Àjt (d), j=-N

a =

a

2:

=

joc-N

where the Xj are integers, C Xjti E Z[r], and 7r = ni(Y, ~0).

o0

Example 2. The real projective plane lRP2 has as its points the equivalence Exarnple 2. The real projective plane lRP2 has as its points the equivalence classesof non-zero real vectors (A’,0 X1, A”) w here two triples are equivalent if classes of non-zero real vectors (À ,'\ l , ,\2) where two triples are equivalent if one is a nonzero scalar multiple of the other: one is a nonzero scalar multiple of the other:

(MO,xxl, xX2) N (x0,xl, x2) ) x # 0; in other words the points of IRP2 are the straight lines through the origin in in other words the points of lRP2 are the straight lines through the origin in IR3, with the origin removed. One obtains exactly two representatives of each lR 3 , with the origin removed. One obtains exactly two representatives of each equivalence class by imposing the requirement of unit length: equivalence class by imposing the requirement of unit length: 2

-&A)2

2:(,\1)2

j=O

= 1. 1.

j=O

From this it is clear that lRP2 may be considered as the orbit space of the From this it is2-sphere clear that be considered orbit action on the S2 lRP2 of themay discrete group Z/2 asof the order 2: space of the action on the 2-sphere 8 2 of the discrete group Z/2 of arder 2: (x0, xl, x2) N (-x0, -xl, --x2) Since the group 7ri(S2) is trivial this covering is universal for RP2. From (4.8) Since the group 11'1(8 2) is trivial this covering is univers al for lRP2. From (4.8) we have we have 7r&!P) cz 7r2(W2) 2 z. (4.12) (4.12)

35 35

$4. Exact sequences sequences §4. Homotopy Homotopy groups groups and and fibrations. fibrations. Exact

Denote by d aa generator 7r2(lRP2). 7t-i(IRP2) G+Z/2 Z/2 in view of generator of 11' 2 (lRp2). We also also have 11' 1 (lRP2) G:! the above description as the orbit orbit space cover under description of lRP2 lRP2 as space of its universal universal coyer the action ri(RP2), which can be be action of of Z/2; Z/2; write write t for the generator of of 11' 1 (lRP2), tt22 = 1, l, which regarded as acting appropriately appropriately on the sphere structure of 11' n2(RP2) as acting sphere S2. S2. The structure 2 (lRp2) as is then then given by as Z[7ri]-module Z[11'1]-module is t(d) = = -d, -d, t(d)

et2 == 1.1.

It. is established that 7ri(Sn) = It is readily readily established that 11'i(sn) = It follows much as in the case n 2 just It follows much as in the case n = = 2 just real has real projective projective space spa ce RP”, lRpn, one one has

and that r,(P) = Z. Z. 00 for for ii < < n, n, and that 11'n(sn) = treated that for the n-dimensional treated that for the n-dimensional

i > 1;

7ri(RPn) ” 7ri(S”),

(4.13) (4.13)

7rl(RPn) g z/a.

(4.14) (4.14)

Moreover the generator generator tt of of the group 11' rl(IRPn) acts on on the basis element element dd Moreover the the group 1 (lRpn) acts the basis of rn(IRPn) (regarded as Z[ni]-module) according to the formula: of 11'n(lRpn) (regarded as Z[11'1]-module) according to the formula: t(d) = (-l)“+‘d,

(4.15) (4.15)

where l)n+l arises from the reflection 5 --z of of lRn+l II@’ restricted where the the factor factor (- l)n+l arises from the reflection x4 ___ -x restricted to the sphere 5’” c IP+‘, as it affects orientation. to the sphere sn c lRn+l, as it affects orientation. We for the space lRPC>O, IRPoo, defined as the the direct direct We remark remark in in conclusion conclusion that that for the space defined as limit of the sequence of spaces RPn, n = 1,2, ., each embedded naturally in limit of the sequence of spaces lRpn, n 1,2, ... , each embedded naturally in its successor, the fundamental group is again Z/2, while the groups ri(iRP”), its successor, the fundamental group is again Z/2, while the groups 11'i(lRPOO), 0 i > 1, are all trivial. i > 1) are aU trivial. 0 Example (or, more more generally, generally, every Example 3. 3. Every Every connected connected planar planar region region (or, every open open 22dimensional manifold) has as deformation retract a one-dimensional complex. dimensional manifold) has as deformation retract a one-dimensional complex. Every one-dimensional complex is homotopy equivalent to a bouquet of finitely Every one-dimensional complex is homotopy equivalent to a bouquet of finitely or countably infinitely many circles. (The latter case is conveniently realized or countably infinitely many circles. (The latter case is conveniently realized as the complex depicted in Figure 2.17 (c).) as the complex depicted in Figure 2.17 (c).)

o 8 (a) Circle (a) Circle

~ -1 a 1 2 :A

(b) Bouquet (b) Bouquet

(c) Circles (c) Circles points X= points À

attached at the integer attached at the integer n of the X-line n of the À-line

Fig. 2.17 Fig. 2.17

It is easy to construct covering spacesfor these l-complexes that are trees, is easy to construct covering for these graph l-complexes are trees, andIt so simply-connected. (A tree spaces is a connected without that cycles.) The and so simply-connected. (A tree is a connected graph without cycles.) The

36

Chapter 2. Topological Topological spaces. Fibrations. Fibrations. Homotopies Homotopies Chapter

bouquet S1 V VS’, has the tree shown shown in Figure 2.18 as as its univers universal bouquet SI SI, for instance, has Figure 2.18 al cover. To see see this, observe that that every every vertex lies above vertex in the covering graph lies the vertex of the bouquet and therefore therefore must have aa neighbourhood the single vertex the bouquet homeomorphic vertex of the bouquet, bouquet, i.e. i.e. to aa homeomorphic to some sorne neighbourhood neighbourhood of the vertex “cross”. On this tree the free group on two generators acts freely in a natural "cross" . this acts a natural way, with orbit space the bouquet S1 V S’. It follows by this sort of argument with orbit space SI SI. It follows 0 that ni(K) is a free group for every 1-complex l-complex (or graph) K. K. that 7r 1 (K) is 0

/""

/

-

...-

""'"'-

./

1

"- \

\ 1

1

\

1 '\.

......

...... " ' - - _ . - - ' / '

./

/

Fig. 2.18. The univers universal tree of of s’ v SI s’ Fig. 2.18. The al covering covering tree SI V Example 4. The n-dimensional torus torus T T” n is is obtained obtained from from ]Rn IF??by means of of Example The n-dimensional by means the identification the identification n )N (-1 -n) . , ,À ,A”) ÀI , ... rv (Xl,. À , ....)X”) ,À ((xl,.

whenever (Xi1 -- x1, . . ,, À A”n -- x”) all components components integers. integers. We We infer infer immeimmewhenever (.À 5. 1 , ... 5. n) has has all diately discrete, free action of the integer integer lattice lattice Zn on ]Rn, BP, and and the the orbit orbit diately a a discrete, free action of the zn on space is the quotient group: space is the quotient group: Tn = IFP/Zn,

(4.16) (4.16)

T1 = S1 = lR1/Z.

It that It follows follows that nl(Tn)

g Z”,

ri(T”)

= 0

for i > 1.

oq

x

o -2 -2

o -1 -1

o 0 o

o 11

x -+ x + 1, 7ri(Si) = z Fig. 2.19 2.19 Fig.

o 2 2

o 33

c

$4. §4. Homotopy Homotopy groups groups and and fibrations. fibrations. Exact Exact sequences sequences

37 37

Example Example 5. 5. Up Up toto homeomorphism homeomorphism every every closed, closed, orientable orientable surface surface isis aa sphere-with-handles sphere-with-handles (see (see Figure Figure 2.20). 2.20). This This fact fact was was established established by by Mobius. Mübius. We shall denote the sphere with g handles attached by Mi; the integer We shan denote the sphere with 9 handles attached by Mi; the integer g9 isis called called the the genus. genus. For For g9 >2:: 22 each each of of the the surfaces surfaces Mi Mi may may be be obtained obtained as as an an under the discrete, free action orbit space of the Lobachevskian plane L2 orbit space of the Lobachevskian plane under the discrete, free action of of aa group group rr ofof (metric-preserving) (metric-preserving) transformations transformations (see (see Figure Figure 2.21, 2.21, where where L2 L2 isis represented as the interior of the unit disc). For the upper-half plane model, represented as the interior of the unit dise). For the upper-half plane model,

(a) (a) 99 = 11

(b) (b) g9 = 2 2

(cl (c) 99 = 33

Fig. Fig. 2.20 2.20

r

Im the group by the the following lm .zz >> 0, 0, of of L2, L 2 , the group r may may be be taken taken as as that that generated generated by foUowing

P=$ (3

4g

The transformations A1 and Az The and A2 move transformations the center of theAloctagon move the center of the octagon to the points indicated by the to the points indicated by the arrows arrows

Fig. 2.21 Fig. 2.21 2g linear-fractional transformations Al,. . , AQ : L2 + 2g linear-fractional transformations Al," ., A 2g : L 2 the half-plane) :

L2 (all preserving L 2 (aU preserving

the half-plane):

(~

Al = ( $

~l

.!, e

), ),

Al, = B,“+lA&l, k A k = B-k+l A 1B 9 9

cos cy sin f f

B, _= ( , B -sincos acy sin cos aa ) 9 - sin a cos a > ' 2g - 1 2g 1 Πf-a=-irT= 74g ,

l

,

cos p + vGFs-2p

ll=ln = ln cos (3 sin + y'c.OS2f3 /3 ’ sin (3 p=$.

(3= ~. 4g

(4.17)

(4.17)

38

Chapter 2. Topological Topological spaces. Fibrations. Fibrations. Homotopies Homotopies Chapter spaces. b db) d )

a

(Here a matrix matrix a (Here (( c

represents the the linear-fractional linear-fractional transformation represents transformation

zz+----+

az +b + b az czfd’+d cz

--.

2 .) Figure 2.21 2.21 shows shows some some of of this this transferred transferred over to to the the unit-dise unit-disc model of of L L2.) Figure over model ItIt can can be be checked that checked that

Al .

A2gA;’

. A;;

= 1,

(4.18) (4.18)

and shown that all relations relations among AI, . . ,A Aag are consequences of (4.18). and shown that aU among Al,"" 2g are consequences of (4.18). Thus 7ri(i$) is isomorphic isomorphic to the the group group defined defined by by the the generators generators Al,. , Aa9 Thus 11'1 (lH;) is to Al,' .. ,A 2g with the single single defining defining relation (4.18). One One may may find find other other generators generators with the relation (4.18). al,. , a,, bl, , ... ,, b b, for for the group 11'1 .rrr(Mi) with the the single single defining defining relation: relation: al,.·., a , b the group (M;) with 1

g

g

albla,1b~1a2b2a~1b,1

a,b,a,‘b,’

= 1.

((4.19) 4.19)

The effect effect of the orbit under the the action action of of this group of transforThe of taking taking the orbit space space under this group of transformations is, essentially, to identify identify appropriate appropriate sides of of aa 4g-gon 4g-gon in in L L22 (in IR2 mations is, essentially, to sides (in JR2 0 for g9 = = 1) see Figure 2.22. for 1) -- see Figure 2.22. 0

g=l

9

g=2

=1

g=2

Fig. 2.22 Fig. 2.22

Example 6. The closed non-orientable surfaces may be constructed from Example 6. The closed non-orientable surfaces may be constructed from the orientable ones by taking the orbit spaces of the latter under a suitable the orientable ones by taking the orbit spaces of the latter under a suitable discrete, free action of Z/2, analogously to the construction of IRP2 from S2, discrete, free action of 7l/2, analogously to the construction of JRp2 from S2, described earlier. There arise two families of nonorientable closed surfaces:

described earlier. There arise two families of nonorientable closed surfaces:

N,2,1 and q,2. N;,l and may N;,2'also be obtained from polygons of 4g and 4g+2 sides respectively These These may also be obtained from polygons of 4g and 4g+2 sides respectively by suitable identification of edges - see Figure 2.23. One can find generators by suit able identification of edges - see Figure 2.23. One can find generators al, . , a,, bl, , b, for .iri(Ni,,) with the single defining relation: al, ... , a g , bl , ... , bg for 7rl (N;,l) with the single defining relation:

Ng”,, : 2 .

Ng,l

alb&b,’

. . . a,_lb,_la,=l,b,=l,a,bga,lb,

.

= 1.

1.

(4.20)

(4.20)

2) is given by generators For Ni,2, on the other hand, the group ni(Ni For N;,2'bl, on the the group , b,, other c, and hand, the single defining 7rl (N;,2) relation is given by generators al,...,ag, al, ... , a g ,

bl , ... , bg , c, and the single defining relation

54. §4. Homotopy Homotopy groups groups and and fibrations. fibrations. Exact Exact sequences sequences

39 39

(4.21) (4.21 ) That That these these two two families exhaust exhaust the the closed, closed, non-orientable non-orientable surfaces surfaces was was first first 00 established by by Poincare. Poincaré.

.[} ô

g=1 N& N6,2 (the (the projective projective plane) plane) (b) N,2,,, (b) N;,2, g9 22: 00

N:,, Nf,l (the (the Klein Klein bottle) bottle) (a) 1 (a) N&, N;,l, g9 Z2: 1 Fig. Fig. 2.23 2.23

ItIt can shown that that for all 2-dimensional (i.e. surfaces), surfaces), open open or or can be be shawn for a11 2-dimensional manifolds manifolds (i.e. closed, with the exception of IwP2 and S2, all of the higher homotopy groups closed, with the exception of IRp2 and S2, an of the higher homotopy groups are tri vial: are trivial: (4.22) 7Ti(M2) = 0 for i > 1, M2 # RP2, s2. (4.22) A the property that an all but its homotopy groups are are A space space X X with with the property that but one one of of its homotopy groups trivial is called an Eilenberg-MacLane space, denoted by K(n,n) if trivial is called an Eilenberg-MacLane space, denoted by K(1r, n) if (4.23)

g T,Ti(X) = 0 for i # 72. 1rTn(X) n (X) 2! 1r, 1ri(X) = 0 for i =f. n.

(4.23)

(In fact, K(r, n) may b e realized as a “CW-complex” - seebelow.) We have (In fact, K(1r,n) may be realized as a "CW-complex" see below.) We have already encountered spaces of the type K(n, 1). Note in particular that the aIre ad y encountered spaces of the type K(1r, 1). Note in particular that the space RP” mentioned at the end of the Example 2 is of type K(Z/2,1). space IRpoo mentioned at the end of the Example 2 is of type K(7l../2, 1). 00 Example 7. The Hopf fibration. The complex projective space CP” of n Example 7. The Hopf fibration. The complex projective space cpn of n complex dimensions is defined, analogously to IRPn, by identifying non-zero complex dimensions is defined, analogously to IRpn, by identifying non-zero vectors in Cnfl if they are non-zero multiples of one another by complex vectors in C n +! if they are non-zero multiples of one another by complex scalars: scalars: (x0,. . . (A”) n ) x E @, x # 0. n - (MO, . ) XX”) (ÀO, ... , À )

rv

(ÀÀo, ... , ÀÀ ),

À E

C,

À

=f. O.

Restricting to just those points of cn+’ satisfying Restricting to just those points of C n +1 satisfying

we infer that CP” may be realized as the orbit space under the free action of we cpn be realized as the orbit space the infer circlethat group S1may ” U(1) ” 5’02 on S2n+1 given by under the free action of the circle group SI 2! U(l) 2! S02 on s2n+1 given by

40 40

Chapter Chapter 3.3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Homology \0 (A’,. (A ,

\n) --+ (eio: A\0 , .. .. .. ,,eeiaXn) io: A\n) .• , A”) --lo (eiaXo,

. . . ,A

This This defines defines the the Hopf Hopf fibration: fibration: pp :: S2%+l s2n+l ---+ - - f @P”, cpn, with with fiber fiber S1 SI ?~ U(1) U(l) g~ or ii >> 11 (see SO2. S02' Since Since 7r1(S1) 7rl(Sl) ”~ Z, Z, 7ri(S1) 7ri(Sl) = 0 ffor (see above), above), we we obtain obtain from from the the exact exact sequence sequence of of the the fibration fibration (see (see (4.7)) (4.7)) the the following following isomorphisms: isomorphisms:

°

7ri (s2”+1) (S2n+l) E~ 7ri (CP”) (cpn),, x2,+1

P

2n+l)

7r2 (CP2)

”

7Tzn+1(CP”)

2 Z”

ii #i- 2, Ei z,

(4.24) (4.24)

7r1(s1).

ItIt follows 03, follows by by taking taking the the direct direct limit limit as as nn --+ --f 00, that that the the infiniteinfinitedimensional complex projective space CPm has the type of the dimensional complex projective space Cpoo has the type of the EilenbergEilenbergMacLane MacLane space space K(Z, K(Z, 2). 2). From From (4.24) (4.24) and and the the fact fact that that @P1 CP! ”~ S2 S2 we we deduce deduce also Hopf’s theorem: also Hopf's theorem: 7r3 (s2) ” z. 0 (4.25) (4.25) Example 8. Consider Example 8. Consider aa knot, knot, i.e. Le. an an embedded embedded circle circle S1 SI in in the the 3-sphere 3-sphere S3, S3, and U, say. and its its complement complement S3 S3 \\ S1 SI = U, say. (Earlier (Earlier we we considered considered knots knots in in iR3; JR3; here here we we have have adjoined adjoined aa point point at at infinity infinity to to obtain obtain the the compact compact space space S3.) S3.) AA theorem of Papakyriakopoulos asserts that 7rz(U) = 0, whence theorem of Papakyriakopoulos asserts that 7r2(U) = 0, whence itit can can be be shown without difficulty difficulty that that nj(U) 7rj(U) = 00 for for jj >> 2. 2. Hence Renee knot-complements knot-complements shown without 0 are all Eilenberg-MacLane spaces of type K(n, 1). 0 are all Eilenberg-MacLane spaces of type K(7r, 1). In conclusion we note that of the simple, well-known examples of spaces In conclusion we note that of the simple, well-known examples of spaces of type K(x, n), there is only one with n > 1, namely K (Z, 2) = CP”. As of type K(7r,n), there is only one with n > 1, namely K(Z,2) = CPoo. As far as the apparatus of algebraic topology is concerned, for the most part the far as the apparatus of algebraic topology is concerned, for the most part the existence of the spaces K(n, n) suffices; in only one important paper (by E. existence of the spaces K(7r, n) suffices; in only one important paper (by E. Cartan) is knowledge of a concrete algebraic model made significant use of. Cartan) is knowledge of a concrete algebraic model made significant use of.

Chapter Chapter 33 Simplicial Complexes and CW-complexes. Simplicial Complexes and CW -complexes. Homology and Cohomology. Their Relation to Homology and Cohomology. Their Relation to Homotopy Theory. Obstructions

Homotopy Theory. Obstructions 31. Simplicial complexes §l. Simplicial complexes

Simplicial complexes (first introduced by Poincark) furnish the most eleSimplicial complexes (first introduced Poincaré) furnish the e1ementary, convenient, most accessible to abyrigorous treatment, and most cleanest mentary, convenient, most accessible to a rigorous treatment, and cleanest

51. §1. Simplicial Simplicial Complexes Complexes

41 41

means means for for defining defining the the homology homology and and cohomology cohomology groups groups and and investigating investigating their their formal formaI properties, properties, as as we we shall shan shortly shortly see; see; however however they they are are inconvenient inconvenient when when itit comes cornes to to particular particular concrete concrete calculation calculation of of these these groups. groups. An is defined as An n-dimensional n- dimensional simplex simplex on anis as the the convex convex hull hull in W’ ]Rn of of any n+l n +1 points . . ,, on points ~0,. 0:0, ... O:n not not contained contained in in any any (n - 1)-dimensional 1)-dimensional hyperplane hyperplane (see (see Figure Thus the the points points of of cn an = (~0, (0:0, .... ,, (1~~) O:n) are the the linear linear combinations combinations Figure 3.1). 3.1). Thus of . . ,, (Y, of the the vertices vertices (~0,. 0:0, ... O:n (regarded (regarded as as n-vectors) n-vectors) of of the the following following form:

nn nn X~ctj, cLx.j j ZT11,> xix j >2> 0. xX E E 8, an, xX = cLxjO:j, o. j=o j=o j=O j=O

(1.1 ) (1.1)

A determined by A face face of of a simplex simplex on an of any dimension n is the the simplex simplex determined by any

a0 !:X 0 .

•


•

0: 1

•

(a) O-dimensional (b) (a) O-dimensional (b) l-dimensional 1-dimensional

(c) 2-dimensional 2-dimensional (c)

(d) 3-dimensional (d) 3-dimensional

Fig. 3.1 Fig. 3.1 proper of the i.e. the the convex convex hull in ]Rn R” of of aa proper proper subset subset of of proper subset subset of the vertices, vertices, Le. hull in ((1~0,. , a,}. In particular the faces of dimension n 1 are just the simplexes {o:o, ... , O:n}. In particular the faces of dimension n 1 are just the simplexes ai n-1 = (cl&. . ,G!^i,.. . ,c&),

(1.2) (1.2)

(where the hat indicates that a symbol is to be considered omitted). Thus cP (where the hat indicates that a symbol is to be considered omitted). Thus an has exactly n + 1 faces of dimension n - 1. has exactly n + 1 faces of dimension n - 1. A simplicial complex K is then an arbitrary (finite or countably infinite) A simplicial complex K is then an arbitrary (finite or countably infinite) collection of simplexes, with the following properties: collection of simplexes, with the following properties: 1. Together with each simplex in the collection, all of its faces of all dimen1. Together with each simplex in the collection, all of its faces of all dimensions should also be in the collection. sions should also be in the collection. 2. Any two simplexes in the collection that intersect should either coincide 2. Any two simplexes in the collection that intersect should either coincide (i.e. be the same simplex) or intersect precisely in a common face. (Le. be the same simplex) or intersect precisely in a common face. A simplicial complex is most conveniently A simplicial complex is most conveniently cl!0)...) a, )“‘) together with those (finite) 0:0, ... , O:n, ... , together with those (finite) simplexes of the complex. simplexes of the eomplex.

given by indicating its vertices given by indicating its vertices subsets of these that determine subsets of these that determine

42 42

Chapter 3. CW-complexes. Homology Chapter 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. Homology

One might consider more general complexes composed composed not simplexes One might consider more general complexes not just just of of simplexes but also of polyhedra (or rather polytopes), polytopes), still still satisbut also of other other general general convex convex polyhedra (or rather satisfying the the conditions in fact fact often often be considering fying conditions 1 and and 2 above; above; and and we we shall shaH in be considering such complexes. show that by means means of of subdivision subdivision such such complexes. However However itit is easy easy to to show that by such complexes may simplicial ones. into the complexes may be refined refined to to simplicial ones. In In investigations investigations into the homolhomology of concrete complexes, of "consolidation" “consolidation” is ogy of concrete complexes, an operation operation of is often often employed, employed, involving simplexes into into convex convex polytopes. polytopes. involving the the union union of of appropriate appropriate sets sets of of simplexes Remark. simplicial complexes, cubic are of interest. Such Remark. Besides Besides simplicial complexes, cu bic complexes complexes are of interest. Such a complex is made from cubes In of all dimensions (see Figure complex made from cubes In of aH dimensions (see Figure 3.2), 3.2), fitting fitting together of the together into into a cubic cu bic complex complex under under the the analogues analogues of the conditions conditions 1, 2 above. Every simplicial complex may, by means of appropriate subdivision, above. Every simplicial complex may, by means of appropriate subdivision, be given the form of a cubic complex, and vice versa. The convenience of be given the form of cu bic complex, and vice versa. The convenience of cubic in the that a product cubic complexes complexes consists consists in the fact fact that that In+m I n+ m = I”In Xx I”, lm, so that product of cubic complexes is complex, and and they derive interest interest also of cubic complexes is again again a cubic cubic complex, they derive also from from the fact that R” has regular tesselations into cubes (regular cubic the fact that ]Rn has regular tesselations into cubes (regular cubic lattices). lattices). In statistical mechanics questions arise the theory of lattice In modern modern statistical mechanics several several questions arise in in the theory of lattice models concerning cubic subcomplexes occurring as images of submanifolds models concerning cu bic subcomplexes occurring images of submanifolds of of Iw” with prescribed In the the case in particular, particular, ]Rn with prescribed regular regular cubic cubic tesselations. tesselations. In case n == 3, in itit turns necessary to description of of the the maps maps of of surfaces surfaces turns out out to to be be necessary to find find aa description 2 into the regular cubic lattice in Iw3, where M22 is subdivided M2 into M into the regular cubic lattice in ]R3, where M is subdivided into a-cubes 2-cubes (squares), structure. Refinements of the the respective (squares), and and the the map map preserves preserves this this structure. Refinements of respective subdivisions the regularity regularity of must be be such such as as to to preserve preserve the of the the lattice lattice in in Iw3. ]R3. 1’ subdivisions must

i”rD•

II•

rl

0

•

I

(a) (a)

(b) (b)

(cl (c)

dIJ r3

(4 (d)

Fig. 3.2 Fig. 3.2

Returning simplicial (or (or convex complex K, Returning to to our our arbitrary arbitrary simplicial convex polyhedral) polyhedral) complex K, we note that if K is finite, i.e. consists of only finitely many simplexes (or we note that if K is finite, i.e. consists of only finitely many simplexes (or convex polytopes), characteristic, already mentioned then the the Euler-Poincare’ Euler-Poincaré characteristic, aIready mentioned convex polytopes), then in Chapter 1, may be defined for it: in Chapter 1, may be defined for it:

X(K) =

L(-IF,j,

(1.3) (1.3)

j~O

11 “Novikov’s considered recently by N.P. Dolbilin, M.A. Stank0 and M.I. "Novikov's problem”, problem", considered recently by N.P. Dolbilin, M.A. Stanko and M.!. Shtogrin. Shtogrin.

§§l. 1. Simplicial Simplicial Complexes Complexes

43 43

where number of of j-dimensional (or j-dimensional convex where rjIj is is the the number j-dimensional simplexes simplexes (or j-dimensional convex polytopes) If every every simplex dimension < n and some K. If simplex of of K K has has dimension:::; and there there are some polytopes) of of K. simplexes n, then then we complex K n-dimensional. simplexes of of dimension dimension n, we say say that that the the complex K is n-dimensional. The m-dimensional skeleton K is the made up up of of aU all simplexes simplexes The m-dimensional skeleton of of K the subcomplex subcomplex made of m. of K K of of dimension dimension <:::; m. Given two simplicial complexes complexes K we say say that K’ is a subdivisubdiviK and and K’, K', we that K' Given two simplicial sion simplex of finitely many simplexes of of K' K’ sion of of KK ifif each each simplex of K K is a union union of of finitely many simplexes and the simplexes simplexes of contained linearly in the simplexes of of K. There and the of K’ K' are are contained linearly in the simplexes K. There is subdivision of any any simplicial simplicial complex, complex, called called the barycentric is aa standard standard subdivision of the barycentric subdivision, defined inductively as follows: A point (or zero-dimensional simsubdivision, defined inductively as follows: A point (or zero-dimensional simplex) does not subdivide. The barycentric subdivision of a one-dimensional plex) does not subdivide. The barycentric subdivision of a one-dimensional simplex (era, cri) is carried out out by introducing aa new new vertex at the the cencensimplex crl (J'1 = (0'0,0'1) is carried by introducing vertex at so that a1 becomes the complex made up 3 vertices tre (midpoint) of a’, tre (midpoint) of (J'l, so that (J'1 becomes the complex made up 3 vertices and one-simplexes (the edges). To barycentrically subdivide an an n-simplex n-simplex and 22 one-simplexes (the edges). To barycentrically subdivide

.• dtl

d’ cL' •

. • 4 (b)

(a) Fig.

3.3.

The

barycentric

subdivision

of a simplex

Fig. 3.3. The barycentric subdivision of a simplex

CF c IfP, one first barycentrically subdivides the faces of on, then introduces (J'rt C ]R,rt, one first barycentrically subdivides the faces of (J'rt, then introduces aa further new vertex cr’ E on at the 9, and and admits admits as as new new centre (centroid) (centroid) of of (J'n, further new vertex 0" E (J'rt at the centre simplexes (of the barycentric subdivision) all those of the form (PO, . . . , ,&, cu’) simplexes (of the barycentric subdivision) aIl those of the form ({30, ... , (3k, 0") where (PO,. , &) is an arbitrary simplex of the barycentric subdivision of a where ({30, ... l (3k) is an arbitrary simplex of the barycentric subdivision of a face of on. The totality of new simplexes then comprises the barycentric subface of (J'n. The totality of new simplexes then comprises the barycentric subdivision of cm (see Figure 3.3). The barycentric subdivision of a simplicial division of (J'n (see Figure 3.3). The barycentric subdivision of a simplicial complex is then obtained by barycentrically subdividing all of the simplexes complex is then obtained by barycentrically subdividing aU of the simplexes of the complex, ensuring that on faces common to two or more simplexes the of the complex, ensuring that on faces common to two or more simplexes the subdivisions coincide. subdivisions coincide. Definition 1.1 Two simplicial complexes K1, K2 are said to be combiDefinition 1.1 Two simplicial complexes KI, K 2 are said to be combinatorially equivalent if there exists a simplicial complex K isomorphic to a natorially equivalent if there exists a simplicial complex K isomorphic to a subdivision of each of them. subdivision of each of them. At a certain stage in the development of topology, when all of the known At a certain stage inwere the development of topology, when of the known topological invariants defined in combinatorial terms, aIlthe conjecture the conjecture topological invariants were defined in combinatorial terms, known as the Hauptuermutung der Topologie, regarded then as of the very regardedthat thenanyas two of the very known as the Hauptvermtdung first importance, was proposed. der This Topologie, is the conjecture homeofirst importance, was proposed. This is the conjecture that any two homeomorphic complexes are combinatorially equivalent. It was shown to be valid It was shown to beinvalid morphic complexes combinatorially equivalent. methods in dimensions < 3 byaremeans of direct, elementary (by Moise the in dimensions :::; 3 by means of direct, elementary methods (by Moise in the

t 44 44

Chapter 3.3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Homology Chapter

1950s - at at least least for for manifolds), manifolds), but but turned turned out out to to be be false false in in dimensions?: dimensions > 6 1950s (Milnor in the early 1960s). Then in the late 1970s it was shown (by (by EdEd(Milnor the early 1960s). Then the late 1970s it was shown wards) that the double suspension of any 3-dimensional manifold that is a wards) that the double suspension of any 3-dimensional manifold that “homology 3-sphere” (i.e. has the same homology as the S-sphere) is actually the 3-sphere) is actually "homology 3-sphere" (Le. has the same homology homeomorphic to to the the 5-sphere, 5-sphere, and and itit is is certainly certainly the the case case that that the the associated associated homeomorphic 5 triangulation of S5 is combinatorially inequivalent to the trivial one, namely triangulation of 8 is combinatorially inequivalent to the trivial one, namely as the boundary of a simplex 06. 2 as the boundary of a simplex (76. 2 Notwithstanding the truth truth or or otherwise otherwise of of the the Hauptvermutung Hauptvermutung for particparticNotwithstanding the for ular spaces, spaces, the the simplest simplest topological topological invariants invariants depending depending for for their their definition definition ular on aa combinatorial combinatorial structure imposed imposed on on aa spaee space -- for for example example the the EulerEuleron structure Poincarg characteristic characteristic and and the the homology homology groups groups (see (see below) below) -- turn turn out out to to be be Poincaré invariant not not only only under under homeomorphisms homeomorphisms but even even under under homotopy homotopy equivequivinvariant but alences (Alexander (Alexander in in the the 1920s). 1920s). (It (It would would appear appear that that this this was was in in fact fact the the alences source of of the concept of of homotopy homotopy equivalence.) equivalence.) Deeper Deeper invariants invariants - the the inteintesource the concept grals of "Pontryagin “Pontryagin classes" classes” over over cycles, cycles, which which admit admit of of aa combinatorial combinatorial defidefigr aIs of nition by by the the Thom-Rohlin-Schwarz Thorn-Rohlin-Schwarz theorem, and and "Reidemeister-Whitehead “Reidemeister-Whitehead theorem, nition torsion” also be defined initially initially combinatorially. combinatorially. While it it has has long long torsion" -- may may also be defined While been known (since the middle 1950s for the Pontryagin classes, and since the been known (sinee the middle 1950s for the Pontryagin classes, and since the 1930s for Reidemeister-Whitehead torsion) that these quantities are not ho1930s for Reidemeister-Whitehead torsion) that these quantities are not homotopy it turns turns out out that that they they are are at at least least topological topological invariants, invariants, Le. i.e. motopy invariants, invariants, it invariant under homeomorphisms (Novikov in the mid-1960s for the Pontryainvariant under homeomorphisms (Novikov in the mid-1960s for the Pontryagin in the 1970s for for Reidemeister-Whitehead Reidemeister-Whitehead Edwards-Chapman in the 1970s gin classes, classes, and and Edwards-Chapman torsion; note that for n = 3 this follows from Moise’s theorem of of the 195Os, torsion; note that for n = 3 this follows from Moise's theorem the 1950s, mentioned above). Reidemeister-Whitehead torsion constitutes an essentially mentioned ab ove ) . Reidemeister-Whitehead torsion constitutes an essentially self-contained of topology. shall subsequently return to to later later deself-contained chapter chapter of topology. We We shaH subsequently return developments in connexion with this invariant (occuring in the 1960s and 7Os), velopments in connexion with this invariant (occuring in the 1960s and 70s), which have come to comprise a unique and highly complex theory. In the mid which have come to comprise a unique and highly complex theory. In the mid 1960s Sullivan outlined a beautiful theory having as a consequence that the 1960s Sullivan outlined a beautiful theory having as a consequence that the Hauptvermutung holds for simply connected PL-manifolds M provided that Hauptvermutung holds for simply connected PL-manifolds M provided that the group Hs(M; i%) does not have 2-torsion.3 3 Kirby and Siebenmann have the group H3(M; Z) does not have 2-torsion. Kirby and Siebenmann have developed a theory clarifying completely the relationship between PL- and developed a theory clarifying completely the relationship between P L- and topological structures on manifolds in dimensions n # 4.4 In any case for topological structures on manifolds in dimensions n oF 4. 4 In any case for *The results of Freedman and Donaldson (early 1980s) show that the Hauptvermutung 2Thealso results of Freedman andmanifolds. Donaldson (early 1980s) show that the Hauptvermutung is false for four-dimensional is false alsofirst for four-c\imensional 31n the version of his workmanifolds. Sullivan placed no restrictions on the group H3(M); of his workrequirement Sullivan placec\ the 3In needthefor first the version above-mentioned emergedno restrictions in a discussionon the with group BrowderH3(M); and the need for the above-mentionec\ requirement emerged in a discussion with Browder Novikov (Spring, 1967). An exposition of Sullivan’s theory has not yet appeared in and the Novikov (Spring, An exposition Sullivan's theory has not yet appeared in the literature (at least 1967). as of 1994, the time ofofwriting). literature (at least 1994, the writing). 4The book by R.asC.of Kirby and time L. C.of Siebenmann, Fozlndational essays on topological 4The book by R. C. and Kirbytriangulations, and L. C. Siebenmann, Foundational essays manifolds, smoothings, Pranceton University Press, 1977,on topological Annals of 1977, Annals manifolds, smoothings, Princeton University mathematics studies, no. and 88, triangulations, is devoted to this theory. In this book Press, the authors have ap-of mathematics studies,to no. devoted to this theory.ofInSullivan’s this booktheory. the authors apparently attempted give 88, an isindependent validation However have there parently to give an inindependent validation Sullivan's theory.results. However are several attempted crucial statements the book whose proofs ofrequire Sullivan’s For there exare several in the book whose proofs Sul!ivan's ample, in thecrucial present statements author’s opinion the proofs on the require pages 268-296 do results. not seem For to exbe ample, in independently the present author's opinionto the proofs the pages 268-296 seemimporto be complete of references certain of on Sullivan’s results. It isdoof not course complete independently certain ofinSullivan's It is of course important that these fundamental of references results be to established full rigour results. and published. It should be tant that results ofbe the established in full rigour and published. noted here these that fundamental some of difficulties theory outlined by Sullivan have beenIt should overcome be

noted here that sorne of difficulties of the theory outIined by Sullivan have been overcome

5§l. 1. Simplicial Simplicial Complexes Complexes

45 45

each each nn 2~ 55 there there exist exist altogether altogether only only finitely finitely many many homeomorphism homeomorphism classes classes of (P L-) manifolds manifolds (Novikov, (Novikov, in in the the midmidof closed, closed, smooth smooth or or piecewise piec:ewise linear linear (PL-) 196Os, 1960s, in in the the simply-connected simply-connected case). case). More More detailed detailed results, results, dating dating from from the the late 1960s and the 197Os, will be indicated at the conclusion of 55 of Chapter late 1960s and the 1970s, will be indieated at the conclusion of §5 of Chapter 4. 4. Returning Returning to to our our exposition exposition of of the the elementary elementary theory theory of of simplicial simplicial comcomplexes, we define a map of simplexes cr; --+ UT to be any mapping of the the plexes, we define a map of simplexes 0"1 --+ 0"2 to be any mapping of vertices vertices of of a? 0"1 toto those of of ap, 0"2' extended extended linearly linearly to to the the whole who le of of cry. 0"1' AA simsimplicial plicial map map ff :: K1 KI + K2 K2 of of complexes complexes is is then then aa map map whose whose restriction restriction to to each each simplex simplex is is a a map map of of simplexes; simplexes; thus thus aa simplicial simplicial map map between between complexes complexes is f(Œj) == ,0j /3j of of the the vertices vertices cyj Œj of of K1, KI, where where is also also determined determined by by the the images images f(c~j) the is one the /3j (3j are are vertices vertices of of Kz. K 2. A A piecewise piecewise linear linear (PL-)map (PL~)map K1 Kl ---+ --+ K2 K 2 is one that that is K1 and is simplicial simplicial between between some sorne suitable suitable subdivisions subdivisions I?l, KI, I?2 K 2 of of KI and K2 K 2 (not (not necessarily nec:essarily barycentric). barycentric). The asserts The fairly fairly straightforward straightforward “Theorem "Theorem on on Simplicial Simplicial Approximation” Approximation" asserts that every continuous map f : K1 --+ Kz of complexes, can be abitrarily that every continuous map f : KI --+ K 2 of complexes, can be abitrarily of suitable subdivisions of closely closely approximated approximated by by aa PL-map P L~map g9 :: I?1 KI 4 --+ I?2 K2 of suitable subdivisions of K1 and Ka, which is homotopic to f. KI and K2' which is homotopie to f. Sometimes is used, to which, given as Sometimes a a variant variant of of this this theorem theorem is used, according according to which, given as before any map f : K1 --+ Kz, there is a simplicial map g : l?l + Kz of nowof before any map f : KI --.; K2' there is a simplicial map 9 : Kl --+ K 2 now some subdivision of K1 only, no longer necessarily close to f, but homotopic some subdivision of KI only, no longer necessarily close to f, but homotopie to homotopy throughout throughout whieh which the motion of of each each image image to ff by by means means of of a a homotopy the motion point is confined to a single simplex of Kz. point is confined to a single simplex of K • 2

Definition 1.2 A A simplicial simplicial complex complex KK is is an n-dimensional P PL-manifold Definition 1.2 an n-dimensional L~manifold if, after application of a sequence of barycentric subdivisions, the combinatoif, after application of a sequence of barycentric subdivisions, the combinatorial neighbourhood of each simplex of K - i.e. the complex made up of the rial neighbourhood of each simplex of K - i.e. the complex made up of the simplexes, together with their having that that simplex simplex as as aa face face -- is is simplexes, together with their boundaries, boundaries, having a complex combinatorially equivalent to on. a complex combinatorially equivalent to O"n.

A PL-manifold of dimension n is said to be orientable if its n-simplexes A P L~manifold of dimension n is said to be orientable if its n~simplexes may all be so oriented that the orientations induced on each (n - 1)-simplex may aU be so oriented that the orientations induced on each (n - l)-simplex from the orientations of the two n-simplexes of which it is a common face, are from the orientations of the two n~simplexes of which it is a common face, are opposite. Each particular way of orienting the n-simplexes of an (orientable) opposite. Each particular way of orienting the n~simplexes of an (orientable) PL-manifold so as to satisfy this criterion, is called an orientation of the P L~manifold so as to satisfy this criterion, is called an orientation of the PL-manifold. P L~manifold. Every PL-manifold is, of course, a topological manifold. However not every Every PL-manifold is, of course, a topological manifold. However not every triangulation of a topological manifold yields a simplicial complex which is a triangulation of a topological manifold yields a simplicial complex which is a PL-manifold: we have already mentioned Edwards’ theorem (from the 1970s) PL-manifold: we have already mentioned Edwards' theorem (from the 1970s) to the effect that the triangulation of the sphere S5 arising from the double to the effect that the triangulation of the sphere S5 arising from the double suspension CCM3 of any homology 3-sphere M3 (i.e. a 3%manifold with HI = suspension EEM 3 of any homology 3~sphere M 3 (i.e. a 3 manifold with Hl = by other

topologists;

for instance

all required

results

of homotopy

theory

have

been

estab-

by other instancein ail established by topologists; Milgram and forMadsen, theirrequired book results Classifyingof homotopy Spaces in theory Surgery have and been Cobordism lished by Milgram and Madsen, their Annals book Classifying Spa ces studies, in Surgery of mathematics no. and 92. Cobordism This work of man~jolds, University Press, in1979, of manifolds, University 1919,notAnnals of mathematics uses several technical tools Press, and ideas yet developed in 1967. studies, no. 92. This work uses severa! technical tools and ideas not yet deve!oped in 1967.

l 46 46

Chapter 3. 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Chapter Homology

ci \

CK

\

Lf3)~7 (a) The The cone cone on K (a) on K

(b) The The suspension suspension EK CK = = (CKh (CK)l U U (CKh (cTK)~ (b) Fig. 3.4 3.4 Fig.

Hz = = 00 but with 11"1 ~1 =J # 0) 0) is combinatorially equivalent to ag55 yet 2 H but with is not not combinatorially equivalent to yet CCM3 EEM3 ~ 2 S5 (where E means “is homeomorphic to”). 8 5 (where ~ means "is homeomorphic to"). We define sorne some of of these these terms. The cone cone on on a complex We shall shaH now now define terms. The a simplicial simplicial complex K, denoted by CK, is the complex formed by joining every point of by K, denoted by CK, is the complex formed by joining every point of K K by means of an interval to a new vertex CY outside K; the triangulation of CK me ans of an interval to a new vertex CI: outside K; the triangulation of CK making it a simplicial complex is carried out in the obvious manner (seeFigure making it a simplicial complex is carried out in the obvious manner (see Figure 3.4(a)). The suspension CK of K is then the union of two cones on K with 3.4(a)). The suspension EK of K is then the union of two cones on K with their common base K identified: their corn mon base K identified: ZK = (CK)l

EK

UK (CK)2.

(1.4) (1.4)

(CKh UK (CKh.

The suspensioncan then be triangulated in the obvious way (seeFigure 3.4(b)) The suspension can then be triangulated in the obvious way (see Figure 3.4(b))

mdl do

\\\\ ’ dl /-El do d2 ciO

Fig. 3.5. Subdivisions of cylinders on simplexes Fig. 3.5. Subdivisions of cylinders on simplexes

The basic operations on topological spaces introduced in 31 of Chapter 2,

The the basicbouquet operations spaces mapping introduced in §1 of namely K1 VonKz,topological the (simplicial) cylinder Cf,Chapter f : A ---f2, namely the bouquet KI V K2' the (simplicial) mapping cylinder Cf, : A ---> Kz, A c K1, and the product K1 x Ka, all go over to the class off simpliK2' A C KI, and the product KI x K2' aU go over to the class of simplicial complexes upon following each operation by some (standard) subdivision. cial complexes upon following each operation by sorne (standard) subdivision.

47

§2. The homology and cohomology groups. Poincaré duality

(Thus although the product of two simplexes is not a simplex, it is readily triangulated see Figure 3.5.)

§2. The homology and cohomology groups. Poincaré duality We now turn to the definition of the homology and cohomology groups of simplicial complexes. The algebmic boundary of any simplex an = (ao, . .. , an) is defined as the following formallinear combination of its faces of dimension n 1 n

aa n

I)-l)iar-l,

(2.1)

i=O

where ai n - l = (ao, ... , â i , . .. , an), the hat over ai indicating its omission. Observe that aoa = o. For any complex K the group Cn(K) ofn-dimensional integml chains of K is the free abelian group consisting of all finite formaI integral linear combinations of the n-simplexes of K: cE

Cn(K),

c

L .\.a~,

a~ E K;

Cl<

here a indexes the n-simplexes of K, and the coefficients .xCl< are integers. From the defining formula (2.1) for the algebraic boundary of a simplex we obtain the boundary opemtor a on the integral n-chains of K:

and thenee the following chain complex C*(K):

Co(K)

0,

(2.2)

where a 0 a = O. We now define for each n ;:::: 0 the n-dimensional integml homology group of K by

Hn(K; Z)

Ker 8/Im

a,

(2.3)

noting that this makes sense, i.e. Ker a ::J lm a, sinee a0 a = O. The elements of the subgroup Ker a c Cn(K) are called cycles of K, and of the subgroup lm a c C n boundaries. The operations of forming Cl 0 C 2 and Hom( Cl, C 2 ) from arbitrary pairs Cl, C 2 of (additively written) abelian groups, will be important for us; they are defined as follows: 1. The abelian group Cl 0 C 2 is obtained from the group of all finite formaI sums of expressions gl 0 g2, gl E Cl, g2 E C 2, by imposing the bilinearity

relations:

i Chapter Chapter 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Homology

48 48

(9: (g~ +d) + gn 8’2® g2

=

s;

@g2

f&@g2,

91 091 @® (9;(g~ +

=

91

@d?

+

(2.4)

(2.4)

+ 99092)

91

@ 9;.

The The resulting resulting group group Gi GI @ ® G2 is is called called the the tensor tensor product product of of Gr G I with with G2. Gz. 2. The as as its The abelian abelian group group Hom(Gr, Hom(G I , G2) G2) hhas its elements elements the the homomorphisms homomorphisms hh :: Gr + G2 from the abelian group Gi to the abelian group Gz, two two such G I -) G 2 from the abelian group G I to the abelian group Gz, such homomorphisms being added in the natural way: homomorphisms being added in the natural way:

(hl + b)(g) = h(g) + b(g).

(2.5) (2.5)

In of aa field, one denotes denotes Hom(Gr, Gs)2 ) In the the case case where where G2 G 2 isis the the additive additive group group of field, one Hom( G I , G by GT. (Cf. the notation V* for the dual space of a vector space V, consisting of by Gi. (Cf. the notation V* for the dual space of a vector space V, consisting of all vector-space homomorphisms from V to the one-dimensional vector space, aIl vector-space homomorphisms from V to the one~·dimensional vector space, i.e. V.) Note Z 2~ G G Note in in particular particular that that G G@ ®Z i.e. scalar-valued scalar-valued linear linear functions functions on on V.) and Hom(Z, G) z G. and Hom(Z, G) ~ G. Equipped can construct from aa chain chain complex Equipped with with these these two two operations operations we we can construct from complex C(K) and any abelian group G the following chain and cochain complexes C(K) and any abelian group G the following chain and cochain complexes over over G: G: (9 (i)

C(K)@G: C(K)®G:

(ii) (ii)

Hom(C(K), Hom(C(K), G) G) ::

... 5

C,(K)

@G 2

C,-1(K)

8 G 5

... ,

ca* . ..) , . . . ziY P(K; G) + za*- cn-l(K; C+l(K; G) + ‘* - ... cn(K; G) G)

... + -

and a* is the formal dual operator of where = Hom(C,(K),G) where Ci(K; Ci(K; G) G) = Hom(Ci(K), G) and 0* is the formaI dual operator of 6’. We call the groups Cn(K;G) = C,(K) G the chain groups of K with 8. We call the groups Cn(K; G) = Cn(K) @ ® G the chain groups of K with coeficients from G, and the C”(K; G) the cochain groups of of K over G. The coefficients from G, and the cn(K; G) the cocha in groups K over G. The homology groups H, and cohomology groups Hn with coeficients G are homology groups Hn and cohomology grmtps Hn with coefficients from from Gare defined in terms of the above two algebraic complexes by dcfined in terms of the above two algebraic complexes by H, (K; G) = Ker d/Im d, Ker d c Cn(K) @G; Ker oC Cn(K) ® G; Hn(K;G) Ker a/lm a, Ker d* c Cn(K;G). Hn(K;G) = Ker d*/Im a*, Hn(K; G) Ker 8* C Cn(K; G). Ker 8* /Im 8*, The elements of the group Ker a* are called cocycles, and of the group Im a* The elements of the group Ker 8* are called cocycles, and of the group lm 0* coboundaries.

coboundaries.

Remark. Chains and cochains may of course be defined also with coeffiRemark. Chains and cochains may of course be defined also with coefficients from a non-abelian group (written multiplicatively in this case). Howcients from a non-abelian group (written multiplicatively in this case). However then the boundaries, coboundaries and cohomology groups admit of natcver then the boundaries, coboundaries and cohomology groups admit of natural definitions only in dimensions 0 and 1. The zero-dimensional cocycles ural definitions only in dimensions 0 and 1. The zero-dimensioBal cocycles (which are actually also the elements of the zero-dimensional cohomology (which are actually also the elements of the zero-dimensional cohomology group H’(K; G)) are just functions g defined on the 0-simplexes of K, the group HO(K; G)) are just fnnctions 9 defincd on the O-simplexes of K, the vertices, taking the same value at the two ends of each edge: vertices, taking the same value at the two ends of each edge: a*g(d, = d&Mdi,o)-l> 8*g(0';) = g(O'~,l)g(O'~,O)-l, (2.6) a*!9

=

1 *

d~&l)

= d4,o).

(2.6)

§2. The The homology homology and and cohomology cohomology groups. groups. Poincak Poincaré duality duality 52.

49 49

Hence H’(K;G) HO(K; C) 2s:: GC xx ...... xx G, C, where where the the number number of of direct direct factors factors GC isis the the Hence number number of of connected connected components components of of K. K. InIn dimension dimension one one we we have: have: a*gb&

= g(a~,2)g(a~,l)-lg(a~,o),

(2.7) (2.7) 9 = a*.f -

dd,l)

= dd,oM~:,o).

For For nonabelian nonabelian G, C, H’(K; Hl (K; G) C) will will in in general general be be simply sim ply a set set without without aa natural natural group structure, since the coboundaries need not constitute a group structure, since the coboundaries need not constitute normal normal subgroup subgroup of the group group of of cocycles. cocycles. of the Generalizations to the the case case nn == 22 have have been been suggested suggested although although there there Generalizations to isis some to the the naturalness naturalness of of the the definition definition in in this this case, case, and and they they some doubt doubt as as to have have so so far far not not found found significant significant use. use. One One may may for for now now at at least least consider consider such such generalizations generalizations merely merely arbitrary. arbitrary. Later Later on, on, when when we we come come to to the the theory theory of of sheaves sheaves and and fiber fiber spaces, spaces, we we shall shaH see (K; F) HO(K; F) and and H1 H 1(K; (K; F) F) with with coefficients coefficients see definitions definitions of of cohomology cohomology groups groups Ho in of non-abelian non-abelian groups, groups, whose whose naturality naturality will will be be very very clear. clear. 00 in aa “sheaf” "sheaf" FF of Returning Returning to to the the standard standard situation situation of of additive additive abelian abelian groups groups G, C, of an n-cochain introduce next the scalar product introduce next the scalar product (f,c) (1, c) EE G C of an n-cochain ff and and n-chain n-chain c: c: a

cy

we we an an

(2.8) (2.8)

This then induces scalar product the eleeleinduces aa natural natural scalar product between between the This scalar scalar product product then ments of the n-th cohomology and homology groups. ments of the n-th cohomology and homology groups. Example 1. If If C G is is the the additive additive group group of of aa field field (for (for instance instance
G) ” (H,(K;

G))* ,

(2.9) (2.9)

and the scalar product (2.8) is non-degenerate. and the scalar product (2.8) is non-degenerate. Example 2. Suppose G = Z and the complex K is finite. For appropriate Example 2. Suppose C Z and the complex K is finite. For appropriate maximal free abelian direct factors @ and H, of Hn(K; Z) and H,( K; Z) maximal free abelian direct factors * fin and fin of Hn(K; Z) and Hn(K; Z) respectively, one has fin = g,, . The scalar product (2.8) between fin and respectively, one has fin = (fin) ( > *. The scalar product (2.8) between fin and fi, is non-degenerate and, relative to naturally corresponding free bases, has fin is non-degenerate and, relative to naturally corresponding free bases, has determinant 3~1. For the torsion subgroups there is an isomorphism (in the determinant ±1. For the torsion subgroups there is an isomorphism (in the case K finite) case K finite) (2.10) Tor Hn(K; Z) S Tor H,-l(K; Z). (2.10) Example 3. Again suppose G is the additive group of any field. Then it E.xample 3. Again suppose C is the additive group of any field. Then it follows from the definitions that follows from the definitions that

l1 50

Chapter Simplicial Complexes Complexes and CW-complexes. CW-complexes. Homology Homology Chapter 3. Simplicial

Hn(K;G) H”(K;G)

=

Horn (Hn(K; (H,(K; Z), Z), G), G) , Hom

H,(K; G) Hn(K;G)

=

H,(K; Z) @ G. Hn(K;Z)@G.

If the (for instance instance G = Q) Q) then then If the field field G G has has characteristic characteristic 0 (for H”(K;Q) ?EHn@QNHom g,,Q Hn(KiQ) ~ H n @Q ~ Hom (Hn,Q). (

>

(2.11) (2.11)

If the characteristic pp < < 00 03 (for = Zjp) Z/p) then then for for If the field field G has has finite finite characteristic (for instance instance G = finite complexes one has has finite complexes K one H”(K; Z/p) Hn(K; Zjp)

=

Horn (Hn(K; (k(K; z), Zjp) Z/P) @ K-I(K; Q, Z/P), Hom Z), EB Horn Hom (Tar (Tor Hn-l(K; Z), Zjp) , (2.12) (2.12)

H,(K; Z/p) == (Hn(K; (k(K; Z) @Z/P) 69 ((Tar z)> @ @ Zjp). Z/P). Hn(Ki Zjp) Z) @Zjp) EB ((Tor ff,-l(K; Hn-I(K; Z)) From (2.11) and (2.12) we we see see that the ranks ranks of of the the homology groups over over Z/p From (2.11) and (2.12) that the homology groups Zjp are R, Cc. In standard the rank rank are at at least least equal equal to to their their ranks ranks over over Q, lR, C. In standard notation, notation, the dimension as vector space space over over Q) Q) is denoted of ( i.e. dimension of the the group group H,(K; Hn(K; Q) Q) (i.e. as aa vector is denoted by Betti number) and the rank (dimension) H,(K;Z/p)Zjp) by by by b, bn (the (the n-th n-th Betti number) and the rank (dimension) of of Hn(Ki bp) b~) 22': b,. bn . Example Example

4 (Poincaré's (Poincare”s Theorem). Theorem). For For any any finite finite complex complex K one has has K one

2)

x(K) = c(-l)jbj. X(K) = -l)jbj . j2°

(2.13) (2.13)

The be valid for the the bi,“‘: The analogue analogue of of this this turns turns out out to to be valid for b;p):

2)

x(K) (2.14) (2.14) X(K) == x(-1)jbl”‘. -l)1bJP). j>O j2° As noted earlier, the Euler-Poincare characteristic x(K) is is defined as the the alteralterAs noted earlier, the Euler-Poincaré characteristic X(K) defined as nating sum Cj20(-l)jrj, where “/j is the number of j-dimensional simplexes nating sum 2: j2 0 ( -1 )1-Yj, where 'Yj is the number of j-dimensional simplexes of The name “Betti numbers” was introduced Poincare, however however not as of K. K. The name "Betti numbers" was introduced by by Poincaré, not as the torsion-free rank of the groups Hj (K; Z). (It is easy to see that the group the torsion-free rank of the groups Hj(K;Z). (It is easy to see that the group Ho(K; abelian of the number number of of connected connected Ho(K; Z) Z) is is always always free free abelian of rank rank equal equal to to the components of K.) A many-sided investigation of the homology groups over components of K.) A many-sided investigation of the homology groups over various coefficient groups began in the late 1920s after E. Noether gave the forvarious coefficient groups began in the late 1920s after E. Noether gave the formal definition of them, thereby bringing algebraic order into the non-algebraic maI definition of them, thereby bringing algebraic order into the non-algebraic 0 homology created by by the topologists. homology theory theory created the topologists. D ItIt is not difficult to calculate the homology and cohomology of a product is not difficult to calculate the homology and cohomology of a product K1 to aa simplicial simplicial complex: complex: KI xx K2 K 2 refined refined to H,(K1

x Kz;Z)

”

L

c

i+j=m

&(h’l; @), Hi(K I ; Hj(K2; Hj(K2; Z)),

i+j=m

H”(K1 Hm(Kl xx K:!;Z) K2iZ)

z

C><

L

c H”(K1; Z)). Hi(Kli Hj(Kz; Hj(K 2i Z)). i+j=m i+j=m

(2.15) (2.15)

52. §2.

The The homology homology and and cohomology cohomology groups. groups. Poincare Poincaré duality duality

51 51

Over Over aa field field GG these these formulae formulae simplify simplify toto the the following: following: Hm(K1

x K2; G)

L

Z

c

H,(Kl; H i (K 1 ; G)G) @® H,(K2; H j (K 2 ; G), G),

ifj=m i+j=m

H”(K1

x K2; G)

L

c

”

(2.16) (2.16) i Hi(K1;

G) @ Hj(K,; 2 ;G). G). H (K I ;G)®Hj(K

i+j=m i+j=m

Note Note that that aa simplicial simplicial map map between between complexes complexes induces induces homomorphisms homomorphisms between between the the respective respective homology homology groups groups (and, (and, likewise, likewise, though though in in the the opposite opposite direction, between the cohomology groups), in view of the fact that such maps direction, between the cohomology groups), in view of the fact that such maps commute commute with with the the boundary boundary operator operator d: â: f

:Kl

-K2,

f*

f* : Hn(K2;

: fL(Kl;

G)

G) +

Hn(K1;

-

fL(K2;

G),

(2.17) (2.17)

G).

Thus, groups, the and Thus, as as for for the the homotopy homotopy groups, the operations operations of of forming forming the the homology homology and cohomology groups are, as they say, functorial (covariantly and contravaricohomology groups are, as they say, functorial (covariantly and contravariantly simplicial complexes and simplicial simplicial antly respectively) respectively) on on the the category category of of simplicial complexes and maps. maps. AA simplicial simplicial map K2, where simplicial homotopy homotopy isis aa simplicial map FF :: K1 KI xX I1 ---f -+ K2' where K1 x 1 is subdivided (barycentrically or a refinement thereof see Figure KI x 1 is subdivided (barycentrieally or a refinement thereof see Figure 3.5) so as to turn it into a simplicial complex. To each simplex on of K1 there 3.5) so as to turn it into a simplicial complex. To each simplex (Jn of KI there corresponds chain representing representing cn xx 1 I in K1 xl, x I, and corresponds the the (n (n ++ 1)-dimensional 1)-dimensional chain (Jn in KI and the chain F(o” 1) in in K K2. . Denote Denote by fa and fb respectively the maps maps the image image chain F((Jn xx 1) by fa and .th respectively the 2 of the base and lid of the cylinder K1 x I (I = [a,b]):

[a, b]):

of the base and lid of the cylinder KI xl (I

F(x, a),

fa(x) fb

and denote

: K1

the chain

x {b}

F(an

-

K2,

fb(z)

fb(x)

x 1) by D(an).

=

=

J’(T~),

F(x, b),

The following

formula:

and denote the chain F((Jn x 1) by D((Jn). The following formula: qao”) which

has a simple

f aq?)

geometric

= fa(an)

interpetation,

- fb(a”),

is a consequence

(2.18)

(2.18) of the obvious

which has a simple geometric interpetation, is a consequence of the obvious decomposition a(an x I) = ((aa”) x I) u (on x {a}) U (an x {b}), and the decomposition â((Jn xl) ((â(Jn) x 1) U ((Jn X {a}) U ((Jn X {b}), and the appropriate attaching of signs. The formula (2.18) extends by linearity to appropriate attaching of signs. The formula (2.18) extends by linearity to arbitrary n-chains. In particular if z is a cycle, dz = 0, then the formula gives arbitrary n-chains. In particular if z is a cycle, âz = 0, then the formula gives f&)

from which

it follows

that

-

homotopic

.fb(Z)

=

*wz),

maps induce

(2.19)

(2.19) the same homomorphisms

from whieh it follows that homotopie maps induce the same homomorphisms of the homology groups. The analogous conclusion is similarly valid for cohoof the homology groups. The analogous conclusion is similarly valid for cohomology. mology. The homotopy invariance (i.e. invariance under homotopy equivalences) homotopy and invariance (Le. invariance homotopy equivalences) of The the homology cohomology groups, is under thus ultimately a consequence of the homology and cohomology groups, is thus ultimately a consequence

.,. 1

52 52

Chapter 3.3. Simplicial Simplicial Complexes Complexes and and CW-complexes. Homology Chapter CW-complexes. Homology

of barycentric subdivision, would of their their invariance invariance under under barycentric subdivision, which which was was known, known, itit would appear, to the homology of finite finite appear, to Poincare. Poincaré. The The invariance invariance of of the homology and and cohomology cohomology of complexes under follows by invoking the the additional additional complexes under homeomorphisms homeomorphisms then then follows by invoking fact that any self-map of a finite complex, having fact that any self-map ff :: K +-+ K of finite complex, having the the property property that close to to its its image f(z), is homotopic to the the that every every point point zx is sufficiently sufficiently close image f(x), homotopie to identity map: For, given homeomorphism K1 -+ --f Kz of identity map: f N rv 1~. 1K. For, given any any homeomorphism h : Kl Kz of finite complexes, sufficiently fine fine subdivisions of the the finite complexes, then then after after carrying carrying out out sufficiently subdivisions of complexes, can (by the Simplicial Approximation Theorem complexes, we we can (by the Simplicial Approximation Theorem - see above) ab ove ) 1 respectively find simplicial maps 7, 9 approximating h and h-l such that the fi approximating and h- respectively such that the find simplicial maps composites 70 9 and 90 fare as close as desired to the identity maps 1~~ and composites 10 fi and fi 01 are close desired to the identity maps 1K2 and 1~~) homotopic to to them. them. 1K 1 , and and therefore therefore homotopie In the infers via simplicial approximation that any any (con(conIn the same same way way one one in fers via simplicial approximation that tinuous) simplicial complexes complexes induces induces homomorphisms homomorphisms of the the homoltinuous) map map of of simplicial of homology and groups, that are unchanged (continuous) homotopies. homotopies. ogy and cohomology cohomology groups, that are unchanged by by (continuous) Thus the and cohomology of a simsimThus the operation operation of of taking taking the the homology homology and cohomology groups groups of plicial complex is a functor with values depending only on the homotopy type plicial complex is a functor with values depending only on the homotopy type of the and the homotopy classes classes of them. of the complexes complexes and the homotopy of maps maps between between them.

1,

Consider now the map Consider now the diagonal diagonal map A(x) (z,s),x), .1(x) == (x,

A :K K --+ .1: -+ K K xx K, K,

K complex. K aa simplicial simplicial complex. Definition 2.1 Given Given any any ring ring C, G, we define the of two cohomology cohomology Definition 2.1 we define the product product oftwo classes aa E G) and and bb EE Hq(K; G) to be the quantity classes E Hj(K; Hj (K; C) Hq (K; C) to be the quantity ab=A*(a@b) ab = .1 * (a

(9

E Hm(K;G), b) E Hm (K; C),

+ q,

m=j+q, m =j

(2.20) (2.20)

where where A* : Hm(K

x K;G)

+

Hm(K;G)

is map A. is the the homomorphism homomorphism induced induced by by the the diagonal diagonal map Ll. Here denotes the cohomology class the product product K naturally Here aa @I (9 b b denotes the cohomology class of of the K xx K K naturally determined by a and b as follows: The chain complex C, (K1 x K2) of a determined by a and b as follows: The chain complex C*(K1 x Kz) of a product product may appropriate sum of tensor may be be identified identified naturally naturally with with the the appropriate sum of tensor products: products: Cm(K1 xx Kz) Cm(Kl Kz) ==

L

c Ci(W 1 ) @Cq(Kz), Cj(K (9 C q (K 2 ), j+q=m j+q=m

a(a @ b) = (da) @ b + (-l)ja

(2.21) ) (2.21

%Jab,

where basis for for C,(Kl K2)z) the of sorne some simplicial simplicial where we we take take as as aa basis C* (K 1 xX K the simplexes simplexes of subdivision of the products al x o-i of simplexes ofI uz of K1, K2. (For CWsubdivision of the products x of simplexes of K 1, K 2 . (For CWcomplexes (see may sim simply take the basis elements.) elements.) complexes (see below) below) one one may ply take the ~“1 ai xx 02” as as basis ItIt follows that a pair zi, .zz of cocycles of K1, K2 respectively, of dimensions follows that a pair Zl, Zz of cocycles of Kl' Kz respectively, of dimensions

ui ag

aL ag

ag

§2. The homology homology and cohomology Poincaré duality $2. The cohomology groups. groups. Poincare duality

53 53

m cocycle z1 zl xx 22 Z2 say, (K l x and itit is this j,j, q, yields yields a cocycle say, in H Hm(K1 x K K2), + q, and this 2 ), m = j + "product cocycles" that that defines the monomorphism “product of of cocycles” defines the monomorphism

Hj(K l ) @ Q9 Hq(K + Hj+q(K Hi+q(Kl l x K K2)2 ) H(K1) HP(K2) 2 ) -+ used (2.20). used in in the the definition definition (2.20). Equipped with multiplication, the the direct of the the cohomology cohomology groups Equipped with this this multiplication, direct sum sum of groups becomes skew-commutative associative ring55 with with identity identity element element becomes a graded graded skew-commutative associative ring 1 E H’(K; G), called ring of K (with coefficients from from an HO(K; G), called the the cohomology cohomology ring of K (with coefficients arbitrary commutative ring G with multiplicative identity): arbitrary commutative ring with a multiplicative identity):

2:.

H*(K; G) H*(K; G) == xHi(K;Hj(K; G). G).

(2.22)

j'20 PO

Here "skew-commutativity" “skew-commutativity” signifies signifies that that for a, b as as in the preceding definition definition has one has jq ba, jj=dima, ab = (-l)jqba, = dima, q = dim b. (2.23) ab=(-l) q=dimb.

the cohomology ring is functorial sense that The formation formation of the functorial in the (usual) sense that --+ K, K, the mapping mapping 1* -+ H*(L; H*(L; G) induced by any map f : L -+ the f” :: H*(K; H*(K; G) -+ is a ring ring homomorphism. is Kolmogorov and Alexander course of ItIt was was Kolmogorov Alexander who, in the mid-1930s, in the course carrying certain topological work (of (of Van Kampen carrying out out an algebraic analysis of certain topological work Kampen and Pontryagin, Pontryagin, for instance) where cohomological ideas ideas were were being used used in embryo, groups explicitly explicitly and introduced introduced the multiplithe cohomology groups multipliembryo, defined the cation of cohomology classes. classes. It seems that that they they were led cation It seems led to the cohomological product partly partly by the analogy with analysis where cohomology groups product with tensor analysis with R, defined in terms differential forms and so so having having with coefficients in lR, ter ms of differential multiplication, had already been (by E. Cartan), an obvious multiplication, been discovered (by Cartan), and and partly by the analogy analogy with Lefschetz' ring of intersections of cycles partly with Lefschetz’ cycles on closed closed the cohomology ring ring (see (see below). manifolds, dual to the In is pertinent “Pontryagin duality", duality”, which In this this connexion connexion itit is pertinent to mention mention "Pontryagin for finite complexes K is the isomorphism finite complexes Hj(K; Char G) ~ g Char Hj(K; G), Hj(K;Char Hj(K;G), where G is any abelian group and Char G is the group of of characters of G, i.e. of arbitrary arbitrary homomorphisms homomorphisms G-+S1%Jl”S02.

For finite finite abelian groups itit is not difficult For seethat that G 2 difficult to ta see ~ Char G, for G = Z that Char Char Z ”~ SI, ~ Z. In general we clearly clearly have that S1, and for G = SI, S1, Char SI S1 g one has has (Pontryagin) (Pontryagin) Char Char G g ~ G. SIn modern terminology, terminology, a “super-ring”. "super-ring". 51n modern

l1 54 54

Chapter 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Chapter Homology

In result of of investigating investigating the algebraic algebraic aspects of of earlier earlier In the the early early 1930s 1930s, as as a result the aspects work and Alexander on "duality “duality laws" laws”, , Pontryagin established the of Poincare Poincaré and Alexander on Pontryagin established the work of following following isomorphisms: isomorphisms: Hj(M”;

Char

G) 2 Char

H,-j(iF;

G),

for and any orientable orientable PL-manifold PL-manifold M”, Mn, and for any W,(S”\K;CharG)rCharH+-i(K;G), Hj(sn \ K; Char G) ~ Char Hn-j-1(K; G), 0O<j
H3(Mn; G) Hj(Mn;G)

r::o!

Z

H”+(M”; G) Hn-j(Mn;G)

H,(S” G) Hj(sn \ K; K; G)

r::o!

2

Hn-j-l(K; G), Hn-j-l(K; G),

(2.24) < j < < n - 1. 0 <

The these isomorphisms isomorphisms is to as "Poincaré “Poincare duality”, and the the The first first of of these is referred referred to duality", and second as “Alexander duality”. second "Alexander duality". The multiplication of integral cohomology classes may be defined defined alternaalternaThe multiplication of integral cohomology classes may be tively in terms of multiplication of cochains. Let the vertices of K be 00, ~1 ... ·,., tively in terms of multiplication of cochains. Let the vertices of K be 0'0, 0'1 and let f and g be cochains of dimensions j and q respectively, in other words and let f and g be cochains of dimensions j and q respectively, in other words integer-valued functions on the sets of simplexes of these dimensions. For each integer-valued functions on the sets of simplexes of these dimensions. For each oriented simplex cm = (oci”, . , oltnb) of K with m = j + 4, we set oriented simplex (Jm = (O'i o , ... ,Qi m ) of K with m = j + q, we set fg (urn)

= fk&7(~;)>

(2.25) (2.25) gf = (Q...>Qi,),

CT;= ((Yi,). . ) a&,).

(Note ai and meet in single vertex.) vertex.) The The multiplication multiplication (2.25) is is (Note that that (Ji and ~2” (Jg meet in aa single (2.25) not skew-commutative (in fact fact not even well-defined) well-defined) at the level of of cochains, cochains, not skew-commutative (in not even at the level but does does determine determine aa well-defined well-defined skew-commutative product of elements elements in in but skew-commutative product of the cohomology groups, in view of the equations the cohomology groups, in view of the equations a*(fg) = (8* (~*f)gf)g + -t (-I)j (-l)Wa*gL 8*(fg) = f(8*g),

(2.26) (2.26)

fg (-l)jqgf = 8*cjJ. a*& fg -- (-l)jqgf = The equation 2.26 is valid valid for cocycles only. only. The second second equation 2.26 is for cocycles Multiplication of cochains yields another important operation: the cap cup prodprodMultiplication of cochains yields another important operation: the uct. Let c be a chain of dimension m and a, b cochains of dimensions j, reuct. Let c be a chain of dimension m and a, b cochains of dimensions j, qq respectively, where m = j+q. The cap product anb is then the chain determined spectively, where m = j+q. The cap product anb is then the chain determined via scalar product chains with with cochains cochains by by via the the scalar product of of chains

(anc,b) (c,ab). (a n c, b) == (c, ab).

(2.27) (2.27)

The cap product product of aa cochain cochain with chain then then determines determines the cap cap product product The cap of with aa chain the

between homology classes: classes: between cohomology cohomology and and homology

§2. §2. The The homology homology and and cohomology cohomology groups. groups. Poincar6 Poincaré duality duality

55 55

u n ccE E Hm-j(K; an Hm_j(K; Z), Z), (2.28) (2.28) aa EE Hj(K; Hj(K; Z), Z),

ccE E H,(K; Hm(K; Z). Z).

Under Under maps maps ff :: Ki Kl +-+ K2, K2' the the cap cap product product obeys obeys the the following following rule: rule:

f*(f*b) 4 = an f* (f* (a) n ne) an f&9, f*(c), (2.29) (2.29) aa E Z), E P(K; Hj(K;Z), Each Each uct, uct, the the

ccE E H,(K; Z). Hm(K;Z).

integral integral homology homology class class cc EE H,(K; Hm(K; Z) Z) determines, determines, via via the the cap cap prodprodfollowing duality operator: following duality operator: P(K;

Z) 4

H&K;

Z), (2.30) (2.30)

a

f-t

an c.

ItIt turns m-dimensional orientable turns out out that that inin the the case case where where KK isis aa closed closed m-dimensional orientable manifold the fundamental fundamental class [Mm] (i.e. the the manifold M”, Mm, and and cc is is the class [Mm] EE H,(Mm;Z) Hm(Mm; Z) (Le. class [Mm] represented by the manifold A4 itself with a chosen orientation; class [Mm] represented by the manifold M itself with a chosen orientation; ifif M the connected then then H”(M”;Z) Hm(Mm; Z) 2~ ZZ with with generator generator [Mm]), [Mm]), then then the M isis connected operation (2.30) is just the isomorphism of Poincare duality: operation (2.30) is just the isomorphism of Poincaré duality: D : Hj(Mm;

Z) %

Hm-j(Mm;

Z),

D(U)

= u n [Mm].

(2.31) (2.31)

The operation to cohomological multiplicaThe operation DD determines determines aa dual dual operation operation to cohomological multiplication, called the intersection of cycles (for orientable manifolds M):

tion, called the intersection of cycles (for orientable manifolds M):

(D(a)) 00 (D(b)) (D(b)) = D(ab), D(ab)> (D(a)) H,+(M”;

Z) 0 &&Mm;

Z) c II,-,-JM”;

(2.32) 25).

(2.32)

We shall give below a simple geometric interpretation of this operation. For We shaH give below a simple geometric interpretation of this operation. For non-orientable manifolds, Poincare duality and the intersection of cycles are non-orientable manifolds, Poincaré duality and the intersection of cycles are defined in mod 2 homology, i.e. with G = Z/2. For piecewise linear (PL-) Z/2. For piecewise linear (P L-) defined in mod 2 homology, i.e. with G manifolds K = Mm, the Poincari! duality isomorphism has a simple geomanifolds K Mm, the Poincaré duality isomorphism has a simple geometrical interpretation. This involves the Poincare’ dual complex DK of the metrical interpretation. This involves the Poincaré dual complex DK of the complex K, originating in the idea of the dual of a graph in the plane or on a complex K, originating in the idea of the dual of a graph in the plane or on a two-dimensional surface. Let K’ be the barycentric subdivision of K. For each barycentric subdivision of K. For each two-dimensional surface. Let K' be the we simplex om of K of largest dimension, define its dual Dam to be the point simplex (Jm of K of largest dimension, wc dcfine its dual D(Jm to be the point of K’ at the center of am; the points {Dam} are then to be the vertices of of K' at the center of (Jm; the points {D(Jm} are then to be the vertices of the dual complex DK. At the other extreme, the dual Da0 of a vertex 0’ of the dual complex DK. At the other extreme, the dual D(Jo of asa vertex (Jo of K is taken to be the convex m-dimensional polytope obtained the union of K is taken to be the convex m-dimensional polytope obtained as the union of all m-simplexes of the barycentric subdivision K’ of K, having o” as a vertex the barycentric subdivision K' of K, having (Jo as a vertex aIl rn-simplexes of (see Figure 3.6). Each simplex & of K, 1 5 k < m, has at its center a point of K, 1 (m ~ k- ~k)-dimensional m, has at its center a point (see barycentric Figure 3.6). subdivision Each simplex the K’; (Jktheof dual “simplex” Dukof the baryccntric subdivision K'; the dual (m k)-dimensional "simplex" D(Jk of DK is then obtained by consolidating the (m - k)-simplexes of K’ that

of DK is then obtained by consolidating the (m - k)-simplexes of K' that

56 56

Chapter 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Chapter Homology

are transverse transverse to the the k-simplex k-simplex uk, and and have have the the center center of of (J'k uk as a vertex vertex (see (see are to (J'k, Figure 3.6). 3.6). Thus Thus the the complex complex DK DK is obtained obtained by consolidation consolidation from K', K’, and and by from Figure topological space it it coincides coincides with with the the complexes complexes K and and K' K’ which which are are K as a topological spaee just different different subdivisions subdivisions of the the same same PL-manifold PL-manifold M”. just of Mm.

Here m m = 2. 2. The The simplexes simplexes of of Here the complex complex K K are are indicated indicated the by the the solid solid line-segments, line-segments, those by those of the the dual dual complex complex DK DK by by of dotted tine line segments segments dotted

Fig. 3.6 Fig. 3.6 We the intersection simplex u& with We now now define define the intersection index index of of each each simplex (J'~ with Da; of DK (where cr& and CT; are simplexes of K of the same D(J'b of DK (where (J'~ and (J'b are simplexes of K of the same

each polytope polytope each dimension) dimension) to to

be 1 if ui meets DO;, and 0 otherwise: be 1 if (J'~ meets D(J'b, and 0 otherwise: d a o Duj,

= SaB.

(2.33)

(2.33)

This extends by linearity to yield the intersection This extends by linearity to yield the intersection index index of of arbitrary arbitrary pairs pairs of of cycles of complementary dimensions:

cycles of complementary dimensions:

(2.34)

(2.34) c o DC’ = &&“p@ (where on the right-hand sides the Einstein summation convention is being (where on the right-hand sides the Einstein summation convention is being used). used). In the orientable case the boundary operator commutes with the dual in In the orientable case the boundary operator commutes with the dual in the sense that the sense that (2.35) Da(c) = h’*D(c).

D8(c) = 8* D(c).

(2.35)

(In the case of non-orientable PL-manifolds this formula remains valid pro(In the case of non-orientable PL-manifolds this formula remains valid provided the coefficients are from Z/2.) The Poincare-duality isomorphism (2.31) isomorphism (2.31) vided the coefficients are from 7l../2.) The Poincaré-duality now follows from (2.35) since the homology of the given manifold M” coinnow follows from (2.35) sinee the homology of the given manifold Mm coincides on the one hand with that of K, a simplicial subdivision of it, and on the one hand with that of K, a simplicial subdivision of it, and on cides on the other hand with that of DK, a subdivision of M” into convex polytopes.

the other hand with that of DK, a subdivision of Mm into convex polytopes.

$3. §3. Relative Relative homology. homology. The The exact exact sequence sequence of of aa pair pair

57 57

The The intersection intersection of of cycles cycles of of noncomplementary noncomplementary dimensions dimensions of of KK and and DK DK respectively, respectively, also also clearly clearly makes makes geometrical geometrical sense. sense. Thus Thus the the operation operation of of ‘
$3. §3. Relative Relative homology. homology. The The exact exact sequence sequence of of aa pair. pair. Axioms CW-complexes Axioms for for homology homology theory. theory. CW-complexes As in aa natural As in in the the case case of of the the homotopy homotopy groups, groups, one one can can define define in natural way way the the relative relative homology homology and and cohomology cohomology groups groups of of aa pair pair (K, (K, L), L), where where KK is is aa simplicial simplicial complex complex and and LL aa subcomplex. subcomplex. AA relative class of Telative chain chain of of the the pair pair (K, (K, L) L) isis simply simply an an equivalence equivalence class of chains chains of K whose pairwise differences are chains of L; i.e. two chains are now to be be of K whose pairwise differenees are chains of L; Le. two ehains are now to considered the same if they are congruent modulo the chains of L. Hence the considered the same if they are congruent modulo the chains of L. Henee the relative Telative chain chain complexes complexes C,(K,

L) = C,(K)/C,(L).

(3.1) (3.1)

The operator 8a on C(K, L) The boundary boundary operator on C(K, L) isis that that induced induced by by operator on K-chains; since X’,(L) c Cj-i (L) this does operator on K -chains; sinee aCj (L) c Cj -1 (L) this do es defined operator: defined operator: .. +

Cj(K,

L) 5

Cj-l(K,

the usual the usual boundary boundary indeed yield aa wellwellindeed yield

L) ----f a .

The jth homology gmup group of of the the pair pair (K, (K, L) L) is is then then defined defined by by The jth relative Telative integral integral homology Hj(K,

L; Z) = Ker a/Im d. Ker a/lm o.

(3.2) (3.2)

Hj(K,L;7L)

The elements of Ker 8 and Im 8 are here called relative cycles and relative The clements of Ker a and lm a are here called relative cycles and relative boundaries respectively. b01mdaries respectively. The relative homology groups Hj(K, L; G) with coeficients from an arThe Telative homology groups Hj(K, L; C) with coefficients fmm an arbitrary abelian group G are defined, analogously to the absolute homology bitraTY abelian gmup C are defined, analogously to the absolute homology groups over G, using the complex C(K, L) 8 G; and the relative cohomology groups over C, using the complex C(K, L) 0 C; and the Telative cohomology groups Hj (K, L; G) over G are defined via the complex Hom(C( K, L), G) and gmups Hj (K, L; C) oveT C are defined via the complex Hom( C(K, L), C) and the relative coboundary operator a*. A relative G-cochain may be thought of the relative coboundary operator 0*. A relative C-cochain may be thought of geometrically as a function with values in G, defined on the set of simplexes geometrically as a function with values in C, defined on the set of simplexes of K and vanishing on those of L. of K and vanishing on those of L. Much as in the situation of the homotopy groups, one obtains the exact Much as in the situation of the homotopy groups, one obtains the exact homology and cohomology sequences of the pair (K, L): homology and cohomology sequences of the paiT (K, L): . -% H,(K) .

z

Hj(K)

3

H,(K,

L) L

HjeI(L)

5

Hj-I(K)

+

.. ,

(3.3) (3.3)

I’

Hj(K,

L) +f?mHj-l(L)

z

H?‘(K)

-

. .. .

(3.4)

58

Chapter Chapter 3. 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Homology

The The constructions constructions of of the the relative relative homology homology and and cohomology cohomology groups groups and and of of the the respective respective exact exact sequences sequences (3.3), (3.3), (3.4) (3.4) are are all all functorial functorial on on the the category category of of pairs (K, L) L) and and simplicial simplicial maps maps of of pairs pairs (Ki, (KI, Li) LI) L~ (Kz, (K2 , L2). L 2 ). These These objects objects pairs (K, are are all aIl homotopy homotopy invariants; invariants; the the induced induced homomorphisms homomorphisms f*, f*, f*f* are are unaffected unaffected by by homotopies homotopies of of maps maps of of pairs. pairs. In In a similar similar fashion fashion one one obtains, obtains, yet yet more more generally, generally, exact exact sequences sequences of of triples > LL:J > M: triples (K, (K, L, L, M), M), KK:J M: ... k H,(K, L) 5 Hj-l(L, - ' M) M) ... % ~ Hj(K,A4) H(K M) ++ ~ f&(K, H·JI(K M) ---+4 .. . . ,, J ' M) .J..:.. H(K J ' L) ~ H JI(L -' (3.5) (3.5) .• .• 8* .• .... ..~Elj(I(.hl)~Hi(K,L)~H~-‘(L,M)~Hj-l(K,~)t ..:- Hj(K,M) J- Hj(K,L) f - Hl-I(L,M)":- Hj-l(K,M) f - .. .. ..•. (3.6) (3.6) Noting that the quotient space k = K/L of a complex K by a subcomplex Noting that the quotient space K = KIL of complex K by subcomplex LL admits admits a natural natural triangulation, triangulation, one one has has the the following following result result (the (the Excision Excision Theorem): Theorem): f$(K,

L)

=

H,(K/L),

jj >> 0, 0,

”

0, 0,

(if K connected and (if K is is connected and LL #f- 8). 0).

(3.7) (3.7)

&(K,L)

The from the space The Excision Excision Theorem Theorem follows follows from the observation observation that that the the quotient quotient space K/L is clearly homotopically equivalent to the complex obtained by attaching KIL is clearly homotopically equivalent to the complex obtained byattaching the along L: the cone cone on on LL to to K K along L: K/L N K uL CL. KILrvKuLCL. For For the the suspension suspension CK EK homology sequence the homology sequence of of the tractibility of the complex tractibility of the complex phism: phism: H,(K,xo)

= (C (CK)l (CK)z, infers from the exact exact = Kh UK UK (C Kh, one one infers from the pair (CK, (CK)z) in conjunction with the conpair (EK, (CKh) in conjunction with the conCK, the following important suspension isomorCK, the following important suspension isomor” Hj+l(CK,xo),

j 2 0.

(3.8) (3.8)

Starting of the two-point space K K == SO So = = Starting from from the the homology homology of the discrete discrete two-point space (~0, xi}, and and iterating isomorphism, one one obtains obtains by induction {xo,xI}, iterating the the suspension suspension isomorphism, by induction the of the the homology homology groups groups of the spheres: spheres:

H,(P,

so; Z) E fP(l!P,

so; Z) = z,

&(P,

so; G) ” Hn(Sn,

so; G) Z G.

(3.9) (3.9)

According to to aa theorem theorem of Eilenberg-Steenrod, homology (and (and cohomolcohomolAccording of Eilenberg-Steenrod, homology ogy) theory is uniquely determined (i.e. up to isomorphism) by the conditions ogy) theory is uniquely determined (Le. up to isomorphism) by the conditions (Eilenberg-Steenrod axioms) of of functoriality, functoriality, homotopy invariance, invariance, existence existence (Eilenberg-Steenrod axioms) homotopy of the exact sequence of a pair (3.3), and the normalization axiom (essentially of the exact sequence of a pair (3.3), and the normalization axiom (essentially

53. sequence of §3. Relative Relative homology. homology. The The exact exact sequence of a pair pair

59

the excision axiom holds; itit is the property property (3.9)), (3.9», provided provided that that the the excision axiom (3.7) (3.7) also also holds; is the distinguishes homology the latter latter axiom axiom that that distinguishes homology theory theory from from homotopy homotopy theory. theory. In the 1960s, 1960s what what are or extraordinary, homology are known known as generalized, generalized, or extraordinary, homology In the (and cohomology) theories to be be intensively exploited as (and cohomology) theories began began to intensively exploited as part part of of the the apparatus are the as the ordinary theory apparatus of of topology. topology. These These theories theories are the same same as the ordinary theory except the generalized except for for the the normalization normalization axiom: axiom: the generalized homology homology groups groups of of the the sphere (Sn, se) may be non-trivial in dimensions other than n, or, equivalently, sphere (sn, so) may be non-trivial in dimensions other than n, or, equivalently, the of aa contractible contractible space the (absolute) (absolute) generalized generalized homology homology groups groups of space (e.g. (e.g. aa disc disc or a point) may be non-trivial in non-zero (including negative) dimensions. or a point) may be non-trivial in non-zero (including negative) dimensions. We We shall shaH be be considering considering questions questions related related to to this this below. below. Poincari: duality can be extended to the relative context (without (without changing changing Poincaré duality can be extended to the relative context the underlying geometric assumptions). Let M” be a closed, orientable PLthe underlying geometric assumptions). Let Mm be a closed, orientable P Lmanifold, c M” manifold, and and K K c Mm aa subcomplex. subcomplex. There There always always exists exists aa finite finite subcomplex subcomplex LL of the open U = of M, M, which which is is aa deformation deformation retract retract of of the open complementary complementary set set U Mm\K, L c U. Wesh a 11 understand the homology of the pair (IvIm, U) to Mm \ K, LeU. We shaH understand the homology of the pair (Mm, U) to be be that following generalized that of of (Mm, (Mm, L). L). We We then then have have the the following generalized version version of of Poincare Poincaré duality: duality: Hj(K) “= H”+(Mm, U). (3.10) (3.10) Example the m-sphere. m-sphere. Provided Example 1. 1. Take Take M” Mm = S”, sm, the Provided S” sm \\ the exact cohomology sequence of the triple (S”, U, *) the exact cohomology sequence of the tri pIe (sm, U, *) (see (see isomorphism isomorphism H”-q!?“, U) z~ H”-+‘(u, *), jj #f=. O. 0. Hm-j(sm,U) H m - j l(U,*),

U is U is non-empty, non-empty, (3.5)) (3.5)) yields yields the the

From this we infer the isomorphism - see From this we in fer the isomorphism of of Alexander Alexander duality duality-see (2.24) (2.24) (valid (valid also for j = 0): also for j = 0): H,(K,xo)

” Hm-j-l(U,uo),

x0 E K,

ug E U. 0

(3.11)

(3.11)

Example 2. Let M” be any orientable manifold and let K = W” m be the Example 2. Let Mm be any orientable manifold and let K w be the closure of a non-empty open set in Mm, so that K is a submanifold W” m (with closure of a non-empty open set in Mm) so that K is a submanifold W (with boundary aWm)m of the same dimension m. From (3.11) and the Excision boundary 8W ) of the same dimension m. From (3.11) and the Excision Theorem (3.7) one obtains readily the following isomorphisms: Theorem (3.7) one obtains readily the following isomorphisms:

Hj(Wm,dWm)

=

H”-qwy

H+F,

”

H,pj(Wm).

aWm)

0 These isomorphisms constitute what is known as Lefschetz duality. These isomorphisms constitute what is known as Lefschetz duality. 0 Homotopy-invariant homology theories are most conveniently constructed homology are most conveniently andHomotopy-invariant investigated for the categorytheories of countable cell complexesconstructed (or CWthe category of countable cell complexe8 (or CWand investigated for complexes). We have already encountered - for instance in the process of complexes). We have already encountered for instance in the process of constructing the Poincare dual of a simplicial complex - the necessity, or at the necessity, or at constructing the Poincaré dual of a simplicial complex

60 60

Chapter Chapter 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Homology

least least convenience, convenience, of of considering considering consolidations consolidations of of simplicial simplicial complexes, complexes, where where the structural units are convex polyhedra or polytopes, for the structural units are convex polyhedra or polytopes, for instance, instance, rather rather than than simplexes. simplexes. CW-complexes CW-complexes are are the the most most general general and and convenient convenient objects objects of this type. To begin with, consider a simplicial complex of this type. To begin with, consider a simplicial complex K. K. We We shall shaU say say that that KK isis decomposed, decornposed, or or subdivided, subdivided, into into cells cells ifif we we are are given given aa family farnily of of cells ceUs (discs) (discs) 0: D~ of of various various dimensions dimensions nn and and a collection collection of of simplicial simplicial maps maps

1Jr'; : D~

-+

K,

with with the the following foUowing properties: properties: (i) (i) Each Each map map !Pg 1Jr;; isis one-to-one one-to-one on on the the interior interior Int Int 0: D~ of of the the cell ceU Dz; D~; (ii) (ii) The The union union of of the the cells ceUs (i.e. (Le. of of the the images images !Pz(Dz)) 1Jr;;(D~)) is is all all of of the the complex complex K, K, and and their their interiors interiors !Pz(Int 1Jr;;(Int DE) D~) are are pairwise pairwise disjoint; disjoint;

a

(iii) !&Q(aDt) of the (iii) For For each each n, n, the the image image 1Jr;; ( D~) (the (the boundary boundary of the cell ceU P,Q(DE) ift'; (D~) )) lies lies in the union of the cells of dimension < n. in the union of the ceUs of dimension < n. Under we say been endowed the structure structure Under these these conditions conditions we say that that KK has has been endowed with with the of a CW-complex K. It is clear from this definition that, given an orientation of a CW-cornplex K. It is clear from this definition that, given an orientation of cell Pz(Dz) of such celK (assuming (assuming orientability) orientability) each each n-dimensional n-dimensional ceU 1Jr;;(D~) of such aa celof K lular subdivision is a linear combination of the (oriented) n-simplexes of K lular subdivision is a linear combination of the (oriented) n-simplexes of K contained in the cell: contained in the cell:

ift';(D~)

L

Àœ,/(J~,

Àcq

= 0,

"'1

where the c$ are n-dimensional simplexes of K. (Note that since distinct cells where the (J~ are n-dimensional simplexes of K. (Note that since distinct ceUs intersect only along their their boundaries boundaries if if at at all, all, the the interior interior of of each each n-simplex n-simplex intersect only along ay” is contained in the interior of exactly one cell.) Denote the cell !?c(Dz) (J~ is contained in the interior of exactly one cell.) Denote the ceU ift;; (D~) by K”,. Since the boundary 8DE of each disc 0: is mapped simplicially to the by ~~. Since the boundary aD~ of each disc D~ is mapped simplicially to the union of the simplexes of K of dimension < n (the (n - 1)-skeleton of K), union of the simplexes of K of dimension < n (the (n - l)-skeleton of K), the boundary operator 8 defined already (in $2) for s@plicial chains, yields the boundary operator a defined already (in §2) for simplicial chains, yields an expression for the boundary 8~: of each n-cell of K as an integral linear an expression for the boundary a~~ of each n-cell of K as an integral linear combination of cells of dimension n - 1: combination of cells of dimension n - 1: tk”,

= c

[&“a : d-l] 6

P$-l,

aoa

= 0.

(3.12) (3.12)

The integers [K: : K:-‘] are called the incidence numbers of the CW-complex The integers [~~ : ~~-1 ] are called the incidence nurnbers of the CW -complex i?. K. The cellular homology and cohomology groups of the CW-complex g are The cellular hornology and cohornology groups of the CW-complex K are now defined, analogously to the simplicial versions, in terms of integral cellterrns integml cellnow defined, analogously C,(K))to the andsimplicial cellular versions, cochains, inand, overof an arbitrary chains (comprising chains (comprising C*(K)) and cellular cochains, and, over an arbitrary abelian group G of coefficients, in terms of C, (2) @ G and Hom(C, (g), G). abelian grouphomology G of coefficients, in terms of C*(K)of®aGand Hom(C*(K), G). The cellular and cohomology groups CW-complex coincide The cellular homology and cohomology groups of a CW-complex coincide

53. Relative Relative homology. homology. The The exact exact sequence of a pair pair §3.

61 61

with (i.e. (i.e. are are naturally naturally isomorphic isomorphic to) to) their their simplicial simplicial analogues analogues as defined defined with above (in (in §2). $2). above The n-skeleton n-skeleton Kn Kn of of a CW-complex CW-complex K is the the union union of of the the ceUs cells of of dimendimenThe K sion S 5 n. Clearly Clearly each quotient quotient space space Kn K”/K”-l bouquet of of spheres spheres sion / Kn-l is a bouquet K”/Kn-l

” S;” v.. . v S; v .

,

representing the respective n-cells /"\;~, IE~, 0: (Y = 1,2, 1,2,. ... . . ,m, , m, ...... Rence Hence the the group group representing the respective n-cells H,(Kn, ’ free is f ree abelian abelian with free generators represented by by the the cells cells Hn(Kn, Kn-‘; Kn-l; Z) Z) is with free generators represented n:, and the boundary operator 8 may be defined alternatively as the com/"\;~, and the boundary operator may be defined alternatively as the composite of the the maps figuring in in the exact homology sequence of of the the posite of the the maps a,, j,j* figuring the exact homology sequence pairs (Kn, Knpl), (Kn-l,Kn-2): (K+‘, Kn-‘): pairs (Kn,Kn-l),

a.,

n n H,(K”,Kn-‘;Z) H n (K , K -

1 . Z)

,

a

-% H,-l(K”-l;Z)

% H,-1(Kn-1,Kn-2;Z)

The definition of CW-complex can be somewhat somewhat broadened broadened by by dispensing dispensing The definition of aa CW-complex can be with the dependence on an initially given simplicial decomposition. Thus aa with the dependence on an initially given simplicial decomposition. Thus topological space X is called a CW-complex if there is given a collection of topological space X is called a CW-complex if there is given a collection of maps Pz : 02 --+ X of discs (balls) 0: of various dimensions (whose images, maps 1JtJ; : D~ ---) X of discs (balls) D~ of various dimensions (whose images, or, the maps themselves, are are called called cells) cells) with with the same or, strictly strictly speaking speaking the maps themselves, the same properties (i), (ii), ( iii ) as before: each !PE should be one-to-one on IntDz; properties (i), (ii), (iii) as before: each 1JtJ; should be one-to-one on IntD~; the cells should all of !P~(IntD~) of the cells the union union of of the the cells should be be aIl of X; X; the the interiors interiors !JrJ; (IntD~) of the ceUs should be pairwise disjoint; and for each n the set !Pz(aDz) = P,Q(F-‘) should be pairwise disjoint; and for each n the set 1JtJ;(aD~) = 1JtJ;(sn-l) should be contained in the union of the cells of dimension 5 n 1. It is is also also should be contained in the union of the ceUs of dimension S n 1. It required that the topology on X be the appropriate identification topology, in required that the topology on X be the appropriate identification topology, in -+ 2 to be continuous (2 any topological the sense that for a mapping f : X the sense that for a mapping f : X ---> Z to be continuo us (Z any topological 0 !Pz all be continuous (or in other words f space) space) itit suffices suffices that that the the maps maps ff 0 1JtJ; all be continuous (or in other words f should be continuous if it is the union of maps of the cells, each pair of which should be continuous if it is the union of maps of the ceUs, each pair of which agrees on the shared portion of the cells’ boundaries). agrees on the shared portion of the cells' boundaries). In this general case the n-skeleton of a CW-complex X is the subcomIn this general case the n-skeleton of a CW-complex X is the subcomplex Xn made up of the cells of dimension < n. Clearly each quotient space plex xn made up of the ceUs of dimension S n. Clearly each quotient space Xj1Xj-l is again a bouquet of j-dimensional spheres: Xi /Xj-l is again a bouquet of j-dimensional spheres: xj/xj-1

” s:’ v . v SA v .

)

each sphere Si arising from exactly one j-cell ~j, (= ‘=%%)L each sphere S~ arising from exactly one j-cell /"\;~ (= !Jr&.(D~)), Q = 1,. , m, The boundary operator d has the same form as before (see 0: 1, ... ,m, .... The boundary operator has the same form as before (see (3.12)), where now the incidence number [ &+’ 3 of a pair of (oriented) a G] /"\;1] of a pair of (oriented) (3.12)), where now the incidence number [/"\;~+l as the degree of the composite map: cells K;+‘, 6; is defined (equivalently) ceUs /"\;~+1, /"\;1 is defined (equivalently) as the degree of the composite map:

a

:

apj&tl) + acKjdtl)- si determined by !Pi+l 1 and the above identification, i.e. essentially as the determined by of !Jrl+ above of identification, as the (signed) number timesand the the composite !Pi+lI acoi+,+l)Le. with essentially the identification (signed) number oftimes the composite of !Jrl+1la(D~+l) with the identification xi/xi-l + v S$ “wraps” the j-sphere i3(Di+‘) around the corresponding xj / X j - l ---> V S~, "wraps" the j-sphere a(D~+1) around the corresponding

62 62

Chapter Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Chapter 3. Simplicial Homology

°°

j-sphere j-sphere of of the the bouquet. bouquet. ItIt follows follows that that d 0o 8 = 0, 0, and and one one then then defines defines the the homology and cohomology groups of the CW-complex X in the homology and cohomology groups of the CW-complex X in the usual usual way, way, obtaining groups defined above. obtaining groups naturally naturally isomorphic isomorphic to to those those defined above. Any CW-complex may be constructed by iterating the operation: following operation: Any CW -complex may be constructed by iterating the following Given a space X and countable collection of maps fa : Sj-’ + Given a space X and countable collection of maps fa : Sj-1 ----4 X, X, one one constructs a new space Y by attaching discs Dj, to X by identifying constructs a new space Y by attaching discs D~ to X by identifying the the boundary Dj, with (Sj-‘) via fa: boundary of of each each D~ with the the corresponding corresponding fol fa (Sj-1) via fa: Y=X Y=X

U

0; u . Dj, u u ((DiU ... D~U"')' > (.fl>...JCX>...) (h, .. ,'!,,,, ,)

(3.13) (3.13)

The topology Thus starting The topology on on YY is is defined defined in in the the natural natural way. way. Thus starting with with aa countcountable discrete set K” of points (to be eventually the O-skeleton), able dis crete set KO of points (to be eventually the O-skeleton), one one first first atattaches the l-cells l), then l-skeleton the 1-cells (j(j == 1), then to to the the resulting resulting 1-skeleton the 2-cells 2-cells are are taches the attached, and so attached, and so on. on, We note in conclusion the CW-complex is We note in conclusion the fact fact that that every every CW-complex is homotopically homotopically equivalent to some simplicial complex. equivalent to some simplicial complex. Example 1. The Example 1, The simplest simplest cellular cellular decomposition decomposition of of the the sphere sphere S” S" isis as as follows: to 0-dimesional cell (vertex) one attaches to aa single single O-dimesional cell (vertex) K’ ",0 one attaches follows: dimensional cell kn by means of the obvious map P S” dimensional cell ",n by means of the obvious map if! :: D” Dn +----4 sn boundary of D” to the point KO: !P(aDn) = {KO}. Clearly 86’ = boundary of D" to the point ",0: if!(oD") {"'O}, Clearly 0",° that the homology that the homology groups groups are are as as follows: follows:

aa single single nnsending sending the the 0 = dKn, so

0"''', so

0

Ho(Y)

z ” Hn(Sn), Ho(S") ”~ Z ~ Hn(S"),

Hi(S”)

Hj(sn) == 00

0

for j # 0,n. for j =1- 0, n,

o

Example 2. a) The real projective spaces have the following cell decompoExample 2. a) The real projective spaces have the following ceU decompowith boundary operators as indicated: sitions, with boundary operators as indicated:

sitions,

RP”

= K0 u d u . u P, ",OU",lU'··U",",

IRP"

~2 = RPj a,$

0",0

= aKl

=

0",1

\ &W-l = a,$

=

= {(x0

:

= aK5 = .

0",3 = 0",5 = ...

: xn) = a,2k+l

=

(3.14)

1 xi # 0, xk = 0 for Ic > j}

O",2k+1

=

= 0,

= ... = 0,

,

aK2k = 2K2k-1,

O",2k = 2",2k-1,

(3.14)

k > 0. k > O.

b) For the complex projective spaces we have: b) For the complex projective spaces we have: CP”

= KO u 2 u . u fP, /'i,0 U ",2 U . , . U /'i,2n,

CP" =

p$=

(y3 \cpi-1

= {( 2’ : a& O/'i,2j

= 0, 0,

: z”)

1 24 # 0, zk = 0 for k > j}

for j > 0.

for j > O.

,

(3.15) cl

o

Example 3. The construction of the closed orientable and non-orientable Example The construction of the closed non-orientable surfaces by 3. identification of appropriate pairs orientable of edges ofand convex polygons

surfaces by identification of appropriate pairs of edges of convex polygons

§3. Relative Relative homology. homology. The The exact exact sequence sequence of of a pair pair §3.

63 63

(see (see Figures Figures 2.22, 2.22, 2.23) 2.23) yields yields the the following following minimal minimal cellular cellular decompositions decompositions of those those surfaces: surfaces: of

Mg:

(i) For For the the orientable orientable surfaces surfaces I$: (i) K 0 u K: u ‘.

u K$ u K2,

8~; = 0,

for j = 1,. . ,2g,

8~’

= 0

amily of (ii) For For the the first first ffamily of non-orientable non-orientable surfaces surfaces Nf,s, Nl,g, g9 >~ 1: (ii) K

0

u K:

u . .

u KQl

UK2,

t&i

= 0,

for j = 1,. . . (29,

d&2 = 24.

(iii) (Hi) For For the the second second family family of of non-orientable non-orientable surfaces surfaces N;,,,, N:},g' g9 2~ 0: n”UK~lJ.4JK;,UK2,

dr;:

= 0,

for j = 0,. . (29,

8~

= 2~4~

The homology homology groups groups of of the the closed closed surfaces surfaces may may be easily computed from The be easily computed from these formulae. formulae. these The intersection intersection indices indices of of one-dimensional one-dimensional cycles these surfaces The cycles on on these surfaces (taken (taken modulo 2 for the non-orientable ones) are determined by the intersection inmodulo 2 for the non-orientable ones) are determined by the intersection indices of of pairs pairs of of the the above above basic basic l-cycles, 1-cycles, and and these these are are as follows: Writing dices follows: Writing U, NF,9, Ni,,9, ai = = ~ii, K:§.j, bii = ~i~-lr K:§i-l' one one has, has, for for all aU surfaces surfaces I$, Nl,g, N:},g'

b

Mg,

aioaj=biobj=O, ai 0 aj = bi 0

bj = 0,

(3.16) (3.16)

a,obj =Sij,bij, ai 0 b j =

where 6,j bij is is interpreted interpreted mod the non-orientable surfaces. For the surfaces where mod 22 for for the non-orientable surfaces. For the surfaces Ni. g there there is is the the additional additional basic 1-cycle K:6 N&, basic l-cycle K; = Cc say; say; the the addition additional al basic basic this 1-cycle intersection indices indices arising arising from from this intersection l-cycle are are as follows: follows: cot= co c

= 1,

coai=cobi=O co ai = co bi

=0

(mod2). (mod 2).

(3.17) (3.17)

Any of cycles cycles ai, ai, bi, . . ,g , g (and (and c in the case case of of Ni.,g) IV;,,), , satisfying satisfying Any collection collection of bi , i = 1, ... in the (3.16) called a canonical canonical basis for for the the cycles cycles on on (and (3.17) (3.17) ifif appropriate) appropriate) is is called basis (3.16) (and the surface. the surface. Via the isomorphism of Poincaré Poincare duality, which in each each case case sends sends the the Via the isomorphism of duality, which in basic cohomology class of H2, H2, basic element element of of the the group group Ho Ho to to the the fundamental fundamental cohomology class of we can case of the of we can infer infer from from (3.16) (3.16) (and (and (3.17 (3.17 in in the the case of Ni,g)) N&)) the structure structure of the the cohomology rings cohomology rings

H*(M;; 7:.), H*W;; @,

H*(Nf,g; 7:./2), H*(%,,; 7W),

H*(Ni,g; 7:./2). H*(Ni’,,; V4.

o0

We have have already Hopf's‘s Theorem: We already mentioned mentioned Hopf Theorem:

HICK;4 c n(K)/ [“l(K), m(K)1>

(3.18) (3.18)

64 64

Chapter Chapter 3. 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW -complexes. Homology Homology

valid valid for for any any CW-complex CW-complex K. K. (Here (Here [7ri, [7l'1' “11 7l'1] denotes denotes the the commutator commutator subgroup subgroup of of the the group group ~1, 7l'1, i.e. i.e. the the subgroup subgroup consisting consisting of of those those elements elements of of ~1 7l'1 which which become become trivial trivial ifif commutativity commutativity isis imposed imposed on on ~1.) 7l'1.) As As an an extension extension of of this this we we have have Hurewicz’ Hurewicz' Theorem: IjE7ri(K)=OforOl and ifn > 1

there is a natural natural isomorphism H,(K;

Z) 2 Q(K).

(3.19) (3.19)

(Note (Note that that since since for for aa path-connected path-connected space space the the homotopy homotopy groups groups defined defined with with reference to different base points are canonically isomorphic, we shall future reference to different base points are canonically isomorphic, we shaH in in future not relative homotopy not indicate indicate base base points points explicitly, explicitly, and and likewise likewise for for the the relative homotopy groups ri(K, L) where L is path-connected.) groups 7l'i(K, L) where L is path-connected.) The The relative relative version version of of Hurewicz’ Hurewicz' theorem theorem is is as as follows: follows:

If7ri(K,L)=OforO 1 there is a natural isomorphism n > 1 there is a natural isomorphism

H,(K, L;Z)”~7l'n(K,L)/{a r,(K, L)l {a= t(a)), Hn(K,L;Z) t(a)},

(3.20) (3.20)

over rl(L) considered considered as where 7rn(K, L), and where aa ranges ranges over over7l'n(K,L), andtt ranges ranges over7l'1(L) as aa group group of operators acting as defined earlier on T~( K, L) of operators acting as defined earlier on 7l'n(K, L).

This of Hurewicz' Hurewicz’ theorem more complete in This relativized relativized version version of theorem is is actually actually more complete in that it applies in the non-simply-connected case also, and so embraces Hopf’s that it applies in the non-simply-connected case also, and so embraces Hopf's theorem. theorem.

Example. If L is a path-connected of the CW-complex K, such Example. If Lis a path-connected subcomplex subcomplex of the CW-complex K, such K/L is a bouquet of k n-spheres, n > 1: that KIL is a bouquet of k n-spheres, n > 1:

that

K/L=S:v...vS;, KIL SrV .. ·VSk, then rn(K, L) is free as a Z [n]-module (rr = ~1 (L)) with Ic natural free genthen 7l'n(K, L) is free as a Z [7l']-module (7l' = 7l'1(L)) with k natural free genII erators corresponding to the spheres of the bouquet.

erators corresponding to the spheres of the bouquet.

0

$4. Simplicial complexes and other homology theories. §4. Simplicial complexes and other homology theories. Singular homology. Coverings and sheaves. The exact sequence Singular homology. Coverings and sheaves. The exact sequence of sheaves and cohomology. of sheaves and cohomology. In this section we shall use simplexes as a means for constructing a hoIn this section we shall usetopological simplexes spaces as a means for constructing a homology theory for arbitrary X. We shall also define the mology thcory for arbitrary topological spaces X. We important shall also define the cohomology groups with coefficients from a “sheaf”, in various cohomology groups with coefficients from a "sheaf", important in various questions touching on complex-manifold theory and complex analysis. theory and complex analysis. questions touching on complex-manifold We begin with the (‘singular” homology theory of a general space X. We with the "singular" homology theory general X. We We begin define an n-dimensional singular simplex of X to beof aa pair (c?, space f) where I?’ define an n-dimensional singular simplex of X to be a pair ((Jn, 1) where (Jn is a standard n-simplex and f is a map of the simplex to X:

is a standard n-simplex and

f is a map of the simplex to X:

54. Simplicial Simplicial complexes complexes and other other homology homology theories theories §4.

65 65

f :: (Jn un + X. x. --'t

The singular singular n-simplexes n-simplexes of of X X thus thus form form a continuum, continuum, as as it it were, were, of of maps maps ff :: The on + X. A singular n-chain in X is then a formal integral linear combination (Jn --t X. A singular n-chain X then formaI integrallinear combination of finitely finitely many many singular singular simplexes. simplexes. We We denote denote the the corresponding corresponding singular of singular chain complex by C,(X), with boundary operator d : Cn(X) + C,-r defined chain complex by C*(X), with boundary operator â: Cn(X) - - ' t C n - 1 defined on a singular simplex (on, f) by the formula: on a singular simplex ((Jn, f) by the formula:

a((Jn, cc 1) f) â

n

= I) C(-i)j (ql, f) f) ,, -IF ((Jr;-l, j=o j=O

(4.1) (4.1 )

where n-1 is is the the j-th (oriented) (n (n -- 1)-dimensional 1)-d imensional face face of of (Jn. 17~. In In ter terms of where aj uj-l j-th (oriented) ms of the singular chain complex C,(X), one defines in the usual way the singular the singular chain complex C* (X), one defines in the usual way the singular homology and cohomology cohomology groups X. These These are are easily verified as as being hohomology and groups of of X. easily verified being homotopy invariants of X, and functorial. It is also easy to see that the singular motopy invariants of X, and functorial. It is also easy to see that the singular (co)homology of aa one-point one-point space space are in non nonzero (co)homology groups groups of are zero zero in zero dimensions. dimensions. The extension to arbitrary abelian coefficient group G is essentially beThe extension to arbitrary abelian coefficient group G is essentially as as before. The singular relative homology and cohomology groups Hj(X, A; G) and fore. The singular relative homology and cohomology groups Hj(X, A; G) and Hj (X, A; G) are defined analogously analogously to to the earlier versions. versions. If If Gis G is aa ring then Hj (X, A; G) are defined the earlier ring then an analogous multiplication of singular cohomology classes can be defined an analogous multiplication of singular cohomology classes can be defined making direct sum sum H*(X; groups into making the the direct H*(X; G) G) of of the the singular singular cohomology cohomology groups into aa graded ring, the singular cohomology ring over G. Finally, it can be shown graded ring, the singular cohomology ring over G. Finally, it can be shown that CW-complex the singular singular homology homology and and cocothat ifif X X is is aa simplicial simplicial or or CW -complex then then the homology groups coincide (i.e. are canonically isomorphic to) those defined homology groups coincide (Le. are canonically isomorphic to) those defined earlier. earlier. Singular homology is particularly Singular homology is particularly convenient convenient for for investigating investigating spaces spaces of of maps from one complex to another, or, more precisely, their homotopy inmaps from one complex to another, or, more precisely, their homotopy invariants, which are crucial for building up the technical apparatus used in variants, which are crucial for building up the technical apparatus used in computing the homotopy groups of spheres and other important complexes. computing the homotopy groups of spheres and other important complexes. As mentioned above, for certain kinds of topological spaces, in particular As mentioned ab ove , for certain kinds of topological spaces, in particular those arising in complex geometry and complex analysis, cohomology theories those arising in complex geometry and complex analysis, cohomology theories with coefficients in ‘(sheaves” are useful. with coefficients in "sheaves" are useful. A presheaf F (of groups, rings, etc.) over a topological space X is a correA presheaf:F (of groups, rings, etc.) over a topological space X is a correspondence associating with each open set U of X a group (or ring, etc.) &,, spondence associating with each open set U of X a group (or ring, etc.) :Fu, and to each inclusion V c U of open sets a homomorphism and to each inclusion V c U of open sets a homomorphism hJ

:&I

4>vu : :Fu

called the restriction called the restriction further required that further required that

-3v, --'t

:Fv,

(4.2) (4.2)

homomorphism corresponding to the pair V c U. It is corresponding to the pair V cU. It is ifhomomorphisrn W c V c U are open sets then if W c V c U are open sets then

4wv 0 hw = 4wlJ. 4>wv

0

4>vu = 4>wu.

A presheaf is called a sheaf if it has the following further properties: A presheaf is called a sheaf if it has the following further properties:

(4.3)

(4.3)

66 66

Chapter Chapter 3. 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Homology

U

(i) (i) For For any any open open cover coyer of of UU by by open open sets sets V, Va cC U, U, UVtiVa = U, U, the the vanishvanisha

ing E 3~, ing of of all aU the the restriction restriction homomorphisms homomorphisms 4v-v CPvo u & on an element f E Fu, q5v,~(f) cpvau(f) = 0, 0, should should entail entail ff = 0. O. (ii) (ii) Given Given open open sets sets V, Va as as in in (i), (i), and and elements elements fa fa E E 3~~) Fv", one one from from each each 3~~, Fv"" at at which which for for every every cy, a, p(3 the the two restriction restriction homomorphisms corresponding V, n VP cC V, the sense to the the pairs pairs Wap Wa,e = = Va n V,e Va and and Wao Wa,e cC Vi, V,e, agree agree in in the sense ing to that that 4K,K(fa) ((4.4) 4.4) CPW",fjV", (fa) = hL,V,(fPL CPw",fjVfj(f,e), there of 3~ all (1~ there should should exist exist an an element element ff of Fu such such that that for for aU a dw,u(f)

= fa,

The The elements elements of of the the group group (or (or ring ring etc.) etc.) 3~ Fu over over a a region region U U cC X X are are called ealled sections sections of of the the sheaf sheaf 3:F above above U. U. Example sheaf has set U, U, where where G G Example 1. 1. The The constant constant sheaf has 3~ :Fu == G G for for every every open open set 0 is some fixed group (or ring, etc.). is sorne fixed group (or ring, etc.). 0 Example of continuous continuous functions. each 3r~ Example 2. 2. The The sheaf sheaf of of germs germs of functions. Here Here eaeh :Fu is the ring of real-valued functions defined on U. One has, analogously, the is the ring of real-valued fun ct ions defined on U. One has, analogously, the sheaf allowing complex holomorsheaf of of germs germs of of smooth smooth functions, funetions, and, and, aUowing eomplex values, values, of of holomorphic, meromorphic or algebraic functions, over respectively a smooth, complex phie, meromorphie or algebraic funetions, over respeetively a smooth, eomplex manifold or algebraic variety X. One may consider more generally the sheaf manifold or algebraic variety X. One may consider more generally the sheaf of germs of vector fields, or general tensor fields on X of some smoothness of germs of veetor fields, or general tensor fields on X of sorne smoothness class, even the sections of X (see (see dass, or or even the sheaf sheaf of of germs germs of of sections of any any fiber fiber bundle bundle over over X 0 below). 0 below).

Fig. 3.7. Here the nerve is the tetrahedron ABCD Fig. 3.7. Here the nerve is the tetrahedron AB CD

In the 1920s Lefschetz initiated a program of applying the algebraic meth1920s Lefschetz initiated program of applying theearly algebraic methodsInofthe combinatorial topology to thea study of general spaces, advances in ods of combinatorial topology to the study of general spaces, early advances in

$4. other homology §4. Simplicial Simplicial complexes complexes and and other homology theories theories

67 67

which Alexandrov and and Tech. introduced which were were made made by by Vietoris, Vietoris, Alexandrov Cech. Alexandrov Alexandrov introduced the concept the nerve nerve of of a covering covering of of a space by countably countably many many open open the concept of of the space X by subsets V,, complex K K {VO!} { Va} defined defined as follows: subsets VO!, U V, VO! = X; X; this this is a simplicial simplicial complex a,.<; follows:

U O!

The indices ~1, . . of taken as K {VO!}' {Va}, and and aa The indices &,, ao, al,'" of the the {Vol} {VO!} are are taken as the the vertices vertices of of K finite subset {cy,, , . , ~i,~} of n + 1 indices is regarded as an n-simplex if the the finite subset {aio' ... , ai,.} of n + 1 indices is regarded as an n-simplex if corresponding open sets have non-empty intersection (see Figure 3.7): corresponding open sets have non-empty intersection (see Figure 3.7): va,, n . . n vai,

#

0.

The cohomology groups Hj(K; Hj(K; G) G) of of the the nerve K {VO!} {I&} are are then called The cohomology groups nerve K K = K then called the cohomology groups of the covering {I&}. With each cover { Wn} “inthe cohomology groups of the covering {VO!}' With each coyer {W,J "inscribed” in the covering {Va}, i.e. such that each W, is contained in some scribed" in the covering {VO!}' Le. such that each W/I', is contained in sorne V,, K {W/I',} {Wn} --f K {VO!} {Va} of of nerves, nerves, namely namely VO!' there there is is aa natural natural simplicial simplicial map map K ---> K that sending each vertex K of K {W6} to any vertex Q: of K {V,} satisfying that sending each vertex K- of K {W/I',} to any vertex a of K {VO!} satisfying W, cC VO!' V,. (If (If there is more more then then one one such choose any one.) This This map map of of W" there is such o, a, choose any one.) vertices does indeed determine a simplicial map, since if W,, n. . n WKrL # 8, vertices does indeed determine a simplicial map, since if W/I',Q n· .. n W/I',n 1- 0, then the intersection V,, n . . ’ n V,Tb is also non-empty if WKj c V+. Thus then the intersection VO!o n ... n VO!n is also non-empty if W/I',j C Vaj . Thus there map of nerves there is is aa simplicial simplicial map of nerves @WV :: K K {W/I',} {Wn) Wwv

K {Vi}.

Note {I&} and and {Uo} of the the space space X Note also also that that for for any any two two coverings coverings {VO!} {U(3} of X there there is is aa covering {Wn} inscribed in these coverings (for example with W, = U, n V,, covering {W/I',} inscribed in these coverings (for example with W/I', UO! n V(3, K = c@) with corresponding maps of nerves: K- = a(3) with corresponding maps of nerves:

K{W/I',}

whence there arises an inverse system (or “spectrum”) {K {I&}, @WV} of whence there arises an inverse system (or "spectrum") {K {VO!} , wwv} of complexes K {Va} and simplicial maps between them. The inverse system complexes K {Va} and simplicial maps between them. The inverse system e considered as providing a simplicial approximation mayb {K(b), {K {VO!} , @WV> wwv} may be considered as providing a simplicial approximation of the space X; for instance if X is a compact metric space, then the inverse of the space X; for instance if X is a compact metric space, then the inverse limit of {K {Va} , PWV} is homeomorphic to X. to X. limit of {K {VO!}' wwv} is homeomorphic Each simplicial map PWV : K {Wn} - K {Va} induces a homomorphism Each simplicial map wwv : K {W/I',} K {VO!} induces a homomorphism of the cohomology groups of the corresponding coverings: of the cohomology groups of the corresponding coverings: Sk,, : Hi (K{V,};G) + Hj(K{W,};G), and one can take the direct limit of the system { Hj (K {I&} ; G) , !Ph,} and one can take the direct limit of the system {Hi (K {VO!} ; G) , Wwv} of groups and homomorphisms, called the jth spectral (or 6ech) cohomolof calledlimit the jth spectralessentially (or Cech)ascohomologygroups grozlp and of thehomomorphisms, space X. The direct is defined follows: ogy group of the space X. The direct limit is defined essentially as follows:

68 68

Chapter Chapter 3. 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Homology

Starting Starting with with the the elements of of the the Hj Hj (K (K {Vo} {Va} ; G) G) for all coverings {Va} {Va} as as

generators, generators, take take elements 5x E Hj(K Hj (K {l&} {Va} ; G) G) and yy E Hj(K Hj (K {Up} {U,a} ; G), G), possibly sibly corresponding corresponding to to different different coverings, to to be equal (i.e. we impose this this as as an abelian-group abelian-group relation relation among the the elements) ifif there there exists exists a finer finer covering covering {IV&} (z) and {W,,} such that that the the images images S&,, !Jiwv(x) and @l;lv(y) !Jiwu(y) of of the the two elements coincide in the the group group Hj(K Hj (K {IV&} {W,,};; G): G):

Hj (K (K Na) {Va};; G) Hi

Â

Hj (K {W,,}; G)

~ Hi (K W,) ; G) Hj(K{U,a};G)

The similarly as as the The jth jth spectral spectral (or tech) Cech) homology group group of of X X is defined similarly the inverse limit of the jth homology groups of all coverings of X, where of course limit of the jth of aH X, course the the homomorphisms all go the other other way. The The modern formulation formulation of of spectral spectral cohomology theory is cohomology and and homology homology theory is due due to to Tech. Cech. Given any triangulated manifold, it is Given triangulated manifold, it is not not difficult difficult to to construct construct a covering covering whose homology and cohomology groups coincide with those whose homology and cohomology groups coincide with those of of the the original original triangulation. triangulation K triangulation. (This (This may may be be done done by by taking taking a a sufficiently sufficiently fine fine triangulation of observing that of the the manifold, manifold, and and observing that the the Poincare-dual Poincaré-dual complex complex DK DK yields yields a a covering by the open convex polytopes U, = Int Duz, where the uz the covering by the open convex polytopes Ua = Int D(J"~, where the (J"~ are are the vertices of One can can now seethat { Ua} of of this vertices of the the triangulation triangulation K. K. One now see that the the nerve nerve K K {Ua} this covering coincides with the barycentric subdivision of the complex K.) Hence covering coincides with the barycentric subdivision of the complex K.) Hence one infers, invariance of of simplicial homology under barycentric subsubone infers, via via the the invariance simplicial homology under barycentric division, the equivalence of Tech homology with the simplicial version. Since division, the equivalence of Cech homology with the simplicial version. Since the Tech groups are clearly invariant homeomorphisms, one one the Cech homology homology groups are clearly invariant under under homeomorphisms, obtains in this manner the invariance under homeomorphisms of the simplicial obtains in this manner the invariance under homeomorphisms of the simplicial homology (but not not their their homotopy homotopy invariance). invariance). homology groups groups (but Using nerves of coverings of a give a natural Using nerves of coverings of a topological topologie al space space X X one one can can give a natural definition of of X coeficients from sheaf 3 definition of the the cohomology cohomology groups groups of X with with coefficients fram aa sheaf F over over X, introduced originally, itit would would appear, by Leray. K {Va} {I&} be the nerve nerve X, introduced originally, appear, by Leray. Let Let K be the of aa covering Va} of of X, X, as as defined defined above. above. To To each each simplex simplex (0:0, (era,... . . . ,O:n) , cy,) of of the the of covering {{Va} nerve one associates the obvious group (or ring, etc.): nerve one associates the obvious group (or ring, etc.): ” ((Qo,. 0:0, ...

- Fv JSu,n...nV,, 74 ,O:n) --+ .. 0'0 n···nv Un

The cochains values in in the the sheaf sheaf3 (corresponding to to the the covering covering {Va}) {Va}) The cochains ff with with values F (corresponding are then defined as those functions on the simplexes of K {Va} which at an an are then defined as those functions on the simplexes of K {Va} which at arbitrary simplex (NO,. . , a,) have value in 3r/m0n.,.nv,,, . The coboundary arbitrary simplex (0:0, ... , O:n) have value in Fvaon ... nvun . The coboundary operator is naturally defined by operator a* 8* is naturally defined by

54. §4. Simplicial Simplicial complexes complexes and and other other homology homology theories theories

69 69

n+l n+l

2)

(8’.f, ” 1 Qn+l)) (8*!, (00, (ŒO"", Œn+I)) == C(-l)%kf(cYo, -l)j4>j!(Œo, '.... ,&j,. , Cij , ... , Œ n +1), . . ,%fl)r j=O j=O

where where q5j 4>j isis the the restriction restriction homomorphism homomorphism

4>j : F(n i,#j v..) ~'t

--t

F(n. v".), 't

'1.

From From this this one one obtains obtains the the sheaf sheaf cohomology cohomology group group of of dimension dimension nn correspondcorresponding ing to to the the particular partieular covering covering {Va}, {Va,}, and and the the cohomology cohomology group group H”(X; Hn(x; 3) F) (nth (nth cohomology cohomology group group of of XX with with coefficients coefficients in in the the sheaf sheaf 3) F) isis then then the the limit limit of of the the resulting resulting spectrum spectrum of of such such groups groups arising arising from from all aH coverings coverings of of X. X. Example Example 1. 1. For For the the sheaf sheaf 3F of of germs germs of of continuous continuous functions functions k (1 < lc < co) on a smooth of smoothness class C” of smoothness class C (1 :::: k :::: 00) on a smooth manifold manifold Mn, Mn, difficult difficult to to show show that that Hj(W;3) Hj(Mn;F) == 0, 0, jj >> 0. O. The The zero-th zero-th cohomology cohomology group group H”(Mn; HO(M n ; 3) F) space of (admissible) functions spaee of (admissible) functions defined defined on on

or functions functions or itit is is not not too too

is is here here just just the the infinite-dimensional infinite-dimensional 0 the whole manifold Mn. the whole manifold Mn. 0

Example Example 2. 2. For For the the sheaf sheaf 3F of of germs germs of of holomorphic holomorphie functions functions on on aa closed closed complex manifold X, one has: complex manifold X, one has: (i) (i) H”(X;3) HO(X; F) constant constant

Z @ (since the only globally globally holomorphie holomorphic functions functions are are the the ~ te (sinee the only functions); fun ct ions ); 0 finite-dimensional vector space. (ii) for jj >> 0, (ii) for 0, that that Hj(X;3) Hj(X;F) isis aa finite-dimensional vector space. 0

Subsheaves 31 c 32, and quotient sheaves 33 = 32131 can be defined in Sllbsheaves FI C F 2 , and qllotient sheaves F3 = F 2 /Ft can be defined in a natural fashion, although difficulties may arise in defining the quotient, so a natural fashion, although difficulties may arise in defining the quotient, so that in fact it is generally defined only as a presheaf. Thus we have a short that in fact it is generally defined only as a presheaf. Thus we have a short exact sequence of sheaves: exact sequence of sheaves:

o This yields the following exact sheaf-cohomology sequence having as particular This yields the following exact sheaf-cohomology sequence having as partieular cases all of the various versions of exact cohomology sequences: cases all of the various versions of exact cohomology sequences: +

H4(X;

31) 2

zP(X;

32) L

Hq(X;

Fs) -2

Hq+l(x;

31) +

... (4.5)

Example 1. Let 32 be a constant sheaf over a CW-complex K, i.e. a fixed Example 1. Let F be a constant sheaf over a CW-complex K, i.e. a fixed group G is associated 2 with each open set of K. As the subsheaf 31 of 32 we group G is associated with each open set of K. As the subsheaf FI of F2 we take the sheaf over K which is constant (= G) over a given subcomplex L take the sheaf over K whieh is constant (= G) over a given subcomplex L of K, and zero outside L. The q-th cohomology group of K with values in of K, and zero outsidc The values q-th cohomology group K exact with values in 31 is Hq(L; G). ofThe sequencein 32 is then Hq(K; G), andL. with F2 is then Hq(K; G), and with values in FI is Hq(L; G). The exact sequence

70

Chapter 3. Simplicial Simplicial Complexes Complexes and CW-complexes. CW-complexes. Homology Homology Chapter

(4.5) cohomology sequence sequence of of the the pair pair (K, (K,L)L) (see §3, 53, (4.5) reduces reduces to to the the exact exact cohomology (3.4)). 00 (3.4)). Example 2. sheaves of of abelian abelian groups groups over over K, K, Example 2. Let Let 31 FI CC 32 F2 be constant constant sheaves 3s (4.5) yields yields a natural natural relationship F3 = 3i/3~. F1/F2. In In this this situation situation (4.5) relationship between between the the cohomology of K with with different of coefficients. coefficients. Thus Thus for instance cohomology groups groups of different groups groups of for instance ifif the group for for 31 Z/p and for F 322 is Z/p2, that for for 3s the constant constant group FI is Zlp and for Zlp2, then then that F3 is the homomorphism the exact exact sequence sequence (4.5) (4.5) takes takes ZI p2 IZlp ”~ UP, Zlp, and and the homomorphism d* fj* of of the ~lP”lUP the form: the form: a* : Hq(K; z/p) + Hq+‘(K; Z/p). (4.6) (4.6) This p, and and is is called called the This homomorphism homomorphism is is usually usually denoted denoted by by (3, the Bockstein Bockstein homomorphism. Here is the direct definition of it: Let Z be an integral qqhomomorphism. Here is the direct definition of it: Let be an integral cochain in C*(K; Z) which on being reduced modulo p yields a cocycle z E cochain in C*(K; Z) which on being reduced modulo p yields a cocycle z E C*(K; Z/p), z = Z mod p. Taking coboundaries gives C*(K; Zlp), z mod p. Taking coboundaries gives

z

z

a*z 0, a*z 0, fj*z == 0, fj*z == pu, pu, a*u fj*u == 0, so the expression so that that the expression Ifj*~ pcz) -ia*, z = (3(z)

p

defines an an element element /3(z) H q+l(K;Z/p)Zlp) for E HQ(K;Z/p). defines (3(z) of of Hq+l(K; for zz E Hq(K; Zlp).

oq

Remark. It is quite often often there sheaves of of nonnonRemark. It is aa worth worth noting noting that that quite there arise arise sheaves abelian groups. For these one can naturally, and usefully, define objects abelian groups. For these one can naturally, and usefully, define objects H’(X; structure, and and H’(X; 3) which are merely merely HO (X; 3) F) which which come come with with aa group group structure, Hl (X; F) which are sets. For constant sheaves these objects were, in effect, introduced above (see sets. For constant sheaves these objects were, in effect, introduced above (see $2). Their d e fi m ‘t’ ion in the general case is completely analogous. We shall §2). Their definition in the general case is completely analogous. We shaH give specific examples below in the context of principal G-bundles. The group give specific examples below in the context of principal G-bundles. The group H”(X; 3) is called the group of sections of the sheaf 3, and the 3~ sections HO (X; F) is called the group of sections of the sheaf F, and the Fu sections over the U (or of sections). over the regions regions U (or germs germs of sections).

55. of non~simply-connected non-simply-connected spaces. §5. Homology Homology theory theory of spaces. Complexes of modules. Reidemeister torsion. Complexes of modules. Reidemeister torsion. Simple homotopy type Simple homotopy type The homology and cohomology theory of non-simply-connected complexes The homology and cohomology theory of non-simply-connected complexes presents curious and isolated algebraic features leading in certain situations presents curious and isolated algebraic features leading in certain situations to important consequences. As has already been mentioned, for non-simplyto important consequences. As has already been mentioned, for non-simplyconnected complexes K a relative analogue of Hurewicz’ theorem for a pair K a relative of Hurewicz' connected (K, L) can complexes be formulated, concerninganalogue the structure of thetheorem module for T,(K,a pair L) (K, L) can be formulated, concerning the structure of the module 1r L) over Z[r], where 7r = ni(L). If we regard Z as a trivial Z[r]-module,n CK, i.e. over Z[1r], where 1r = 1rl(L). If we regard Z as a trivial Z[1r]-module, Le. consider the elements of 7r as acting identically on Z, then this result may be consider theaselements 1r asisomorphism acting identically on Z, then this result may be considered asserting ofthe (cf. (3.18)) considered a..c:; asserting the isomorphism (cf. (3.18))

55. §5. Homology Homology theory theory of of non-simply-connected non-simply-connected spaces spaces

ffn(K,L;4 g G(K,L) %[a]z,

71 71

(5.1) (5.1)

under under the the conditions conditions nj(K, n j (K, L)L) == 00 for for 0 < < jj << n, n, nn > > 1. l. Given over aa ring Given two two modules M, M, N Nover ring R, R, we we define define their their tensor tensor product product M @R over RR to E M, ®R N Nover to be the the module generated by by all aIl symbols m m@ ® n, m mE M, nnE E N, N, subject subject to to the the usual bilinearity bilinearity relations relations (cf. (2.3)): (2.3)): X(m @n) = (Am) @n = m 8 (An), À(m®n) (Àm)®n m®(Àn),

X E R. ÀER.

(5.2) (5.2)

In In the the case case RR = Z Z this this reduces reduces to to the the tensor tensor product product of of abelian abelian groups groups (con(considered the natural natural way way as as Z-modules) Z-modules) introduced introduced in in $2. §2. sidered in in the With With any any non-simply-connected non-simply-connected simplicial simplicial (or (or CW-) CW -) complex complex K K there there is is naturally associated an algebraic complex C,(k) (where k is the universal naturally associated an algebraic complex C*(K) (where K is the universal cover chains in K) of of free free Z[n]-modules Z[11']-modules with with basis basis consisting consisting of of chains in natural natural coyer of of K) one-to-one correspondence with the simplexes (or cells) of K. This one-to-one correspondence with the simplexes (or ceUs) of K. This complex complex C,(K) C*(K) is is defined defined as as follows: follows: Let Let pp:: KK +

K, K,

(7rl(K) = O), (11'1(K) 0),

be group 11' 7r = 11' xi(K) be the the universal universal covering covering projection. projection. As As we we saw saw earlier earlier the the group 1 (K) acts freely on K, and in fact sends simplexes to simplexes (or cells to acts freely on K, and in fact sends simplexes to simplexes (or ceUs to cells); ceUs); this from the the fact over aa contractible subspacethe universal cover this follows follows from fact that that over contractible subspace the univers al coyer is trivial, i.e. is the product of that subspace with the discrete fiber F, which is trivial, i.e. is the product of that subspace with the dis crete fiber F, which is the group group IT ri(K) case. In our present context we we have as conconis the 11' = = 11' 1 (K) in in this this case. In our present context have as tractible subspaces the simplexes (or cells) of K, whence tractible subspaces the simplexes (or ceIls) of K, whence

U

p-1 (0,“) = /J “&. p-l (Œ~) = Œ~,. -VET ,E71'

Choose an "initial" “initial” n-simplex (or cell) c pm1 (~2). The remaining n-cells Choose an n-simplex (or ceU) a:r Œ~l C p-1 (Œ~). The remaining n-ceUs of p-l p-l (02) c k are then obtained by applying the elements of 7r to this one: of (Œ~) c K are then obtained by applying the elements of 11' to this one: Y(G)n ) = 4&, n "Y ( Œ"'1 = Œ"",

Y EE 71. "Y n.

Thus C,(g) is just the usual n-chain complex of i?, endowed with this Z[n]Thus Cn(K) is just the usual n-chain complex of K, endowed with this Z[11']action. It is clearly a free module. Note that the action of Z[K] commutes with action. It is clearly a free module. Note that the action of Z[n] commutes with the boundary operator d: the boundary operator â: ax(c) = M(c), â>.(c) = Àâ(c),

x E Qr]. Z[n].

À E

Thus we obtain C,(k) as a chain complex of free Z[n]-modules. Thus we obtain C*(K) as a chain complex of free Z[11']-modules. If the covering g is not necessa_rilyuniversal, but-still regular, then the If the covering K is not necessarily universal, but still regular, then the acts freely on K and sends simplexes quotient F = rl(K)/N, N ” TI(K), quotient = 11'1 (K)jN, N ~ 11'1 (K), acts freely on K and sends simplexes (cells) to simplexes (cells) in such a way that the orbit space k/r 2 K. Then (ceUs) to simplexes (ceUs) in such a way that the orbit space Kj r ~ K. Then as before the chain complex C,(K) comes with the natural r-action, turning as before the Z[F]-module. chain complex C*(K) co mes with the natural r-action, turning it into a free it into a free Z[r]-module.

r

72

Chapter 3. Simplicial Simplicial Complexes Complexes and CW-complexes. CW-complexes. Homology Homology Chapter

Any group r, r, i.e. any homomorphism homomorphism p : r r ---> + Any representation representation p of of a group Le. any Aut or abelian abelian Aut V, V, from from r to to the the group group of of automorphisms automorphisms of of some sorne vector vector space space or group V, can with the structure of (V, p). p). group V, can be be used used to to endow endow V with the structure of a ZJjr]-module Z[rj-module (V, Tensoring such a module module (V, (V, p) C,(K) just defined, we we Tensoring such p) with with the the Z[r]-module Z[r]-module C*(K) just defined, obtain the complex of respect to pp obtain the chain chain complex of K K with with respect

r

G(K;

P) = Cd@

From one constructs, From this this one constructs, as cohomology groups of K cohomology groups of K with with

(5.3) (5.3)

@‘;zpy (v, P).

appropriate factor factor modules, modules, appropriate respect to the representation respect to representation

the homology the homology and and p.

Example If pp is taken to trivial representation all of of r Example 1. If is taken to be be the the trivial representation sending sen ding aIl to the the identity identity automorphism of V, V, then then the usual homology cohomology to automorphism of the usual homology and and cohomology groups Hq(K; V) are Hq(K; V), V), H’J(K; Hq(K; V) are obtained. obtained. •I 0 groups Example manifold, dim dim K = n. For For each each Example 2. 2. Let Let K be be aa non-orientable non-orientable manifold, K = path class [r] E ~1 (K), define p[y] = sgn y, where sgn y = 1 if transport path class b] E 7rl(K), define pb] = sgn 'Y, where sgn 'Y = 1 if transport of aa co-ordinate frame around around y'Y preserves preserves orientation, orientation, and -1 otherwise. We of co-ordinate frame and -1 otherwise. We take V = Z, so that Aut Z Z Z/2 Z {*l}. For the resulting homology take = Z, so that Aut Z ~ Z/2 ~ {±1}. For the resulting homology and Hj (K; p) and Hj (K; p) there are are Poincare-duality and cohomology cohomology groups groups Hj(K; p) and Hj (K; p) there Poincaré-duality isomorphisms determined as earlier by the Poincare dual complex DK DK (see (see isomorphisms determined as earlier by the Poincaré dual complex §2): . §2): D: HJK;p) g H”-j(K;p), (5.4) (5.4) D Hn+(K;p). p). o q Hj(K;p) ” Hn_j(K; D:: Hj(K;p) and associate Example V == Z[7r], Z[T], and Example 3. 3. Take Take V associate action on Z[r] via multiplication on the action on Z[7r] via multiplication on the right. right. the identity p, yielding yielding the identity representation representation p, H,(K;p)”

H#;Z),

Hj(K;p)

with with This This

each element each element gives is gives what what is

G Hj(&Z).

its gg of of K 7r its essentially essentially 0 o

Of V, for for which which Of particular particular interest interest are are those those representations representations pp:: x7r +---> Aut Aut V, are trivial; in this situation the corresponding homology groups Hq(K; p) the corresponding homology groups Hq(K; p) are trivial; in this situation we we say that complex. IfIf K of dimension n, the the chain chain say that K K is is aa p-acyclic p-acyclic complex. K is is p-acyclic p-acyclic of dimension n, complex C, (K; p) exact sequence: sequence: complex C*(K; p) yields yields an an exact

a C,-,(K;p) a ... 5a C1(K;p) 3a C,,(K;p) + 0. 00---> + C,(K;p) % Cn(K;p) 5---> Cn-1(K;p) ---> ... ---> C1(K;p) ---> Co(K;p) ---> O. (5.5) (5.5) ItIt is is important for what follows to note that if V is a free abelian group with important for what follows to note that if V is a free abelian group with (or aa vector vector space free abelian abelian aa given given free free basis basis (or space with with given given basis), basis), then then the the free groups C, (K; p) (vector spaces if V is a vector space) will likewise each have groups C j (K; p) (vector spaces if V is a vector space) will likewise each have aa distinguished basis, whose members correspond to the cells of K. distinguished basis, whose members correspond to the cells of K. Remark. V free vector space) space) with basis. Remark. Assume Assume V free abelian abelian (or (or aa vector with prescribed prescribed basis. The "distinguished “distinguished basis” of Cj(K; C,(K; p) determined The basis" of p) just just described described is is uniquely uniquely determined

$5. Homology theory theory of non-simply-connected non-simply-connected spaces spaces §5. Homology

73 73

only one or more of of the the following elementary geometric geometric only up up to to application application of of one or more following elementary changes of basis: changes of basis: that arising of some or all cells: O'~ a: ---+ ---f -O'~; -a:; 1) that arising from from a change change of of orientation orientation of sorne or aU cells: 2) choice of of "initial “initial cell" cell” 0'~1 a$ from from the the complete complete 2) that that arising arising from from a different different choice inverse image p~l p-l (O'~) (0,“) = = U 0&, 'y"Y E E 11' 7r;; here here the the change change of has the form UO'~1" of basis basis has the form inverse image a; to aa representation representation p, a: --+ --+ ±ph)O'~. kfrp(y)aE. O'~ --+ ---+ f-p;, ±'"YO'~, or, or, with with respect respect to p, O'~ In what follows follows we we assume for cach each Cj(K; Cj(K; p) fixed "distinguished “distinguished basis”. 00 In what assume for p) aa fixed basis". Suppose now that V is finite-dimensional vector space space over over sorne some field field is aa finite-dimensional vector Suppose now that k, that K is p-acyclic, so that that we we have have the the exact exact k, with with prescribed prescribed basis, basis, and and that K is p-aeyclic, so sequence (5.5). If If that has but but two two terms terms sequence (5.5). that sequence sequence has o --f C,(K;

p) -% Cn-l(K;

P) -

0,

then simply an an isomorphism, and relative relative to to the distinguished bases for then d8 is is simply isomorphism, and the distinguished bases for is represented by a nonsingular matrix over k, with G(K; Cn (K; P) p) and and G-I(K; Cn~ 1 (K; p) p) is represented by a nonsingular matrix over k, with determinant det a8 = det C, (K; p). out that that with with every every finite finite p-acyclic determinant det det C*(K; p). ItIt turns turns out p-acyelic complex K one may associate a “determinant” relative to distinguished bases complex K one may associate a "determinant" relative to distinguished bases for the C,(K; p) in the corresponding exact sequence (5.5). In the case where for the Cj(K; p) in the corresponding exact sequence (5.5). In the ca.<;e where the sequence (5.5) has three the exact exact sequence (5.5) has three terms: terms: 0+

C,(K;

p) 5

C,-l(K;

p) z

Cnx.(K;

6’) -

‘1

this determinant is follows: Denote Denote the the distinguished bases this determinant is defined defined as as follows: distinguished bases of C,(K; p), Cn--l(K; p), Cn-2(K; p) by k, ~~-1, c,-2. The middle complex of Cn(K; p), Cn- 1 (K; p), Cn~2(K; p) by en, Cn~l, Cn~2. The middle complex as basis, where dP1(c,-2) denotes a set C,-l(K; Cn-1(I{j p) p) also also has has (ac,, (8c n , 6’-‘(c,-2)) 8~1(c-n~2») as basis, where 8~1(cn~2) denotes a set of arbitrary chosen inverse images of the members of c,-2, ordered correspondof arbitrary chosen inverse images of the members of Cn~2, ordered correspondingly. Since the latitude in the choice of a-‘(c,-2) is “triangular” in characcharacingly. Sinee the latitude in the choice of 8~1(cn~2) is "triangular" in ter, the determinant of the change of basis from c,-1 to (dc,, aP1(cn-2)) is ter, the determinant of the change of basis from Cn-l to (8c n , 8-1(Cn~2») is uniquely defined. The determinant det C,(K; p) is defined analogously in the uniquely defined. The determinant detC*(K;p) is defined analogously in the general case: Denote the distinguished basis of each Cj = Cj(K; p) by cj, general case: Denote the distinguished basis of eaeh C j Cj (K; p) by Cj, write Aj c t7-1 for the image of Cj under d, (j = 1, . . , n-l) and choose write Aj C Cj~l for the image of C j under 8, (j ::::: 1, ... , n-l) and choose bases aj for these A, arbitrarily. Set A0 = 0, A, = C, and a, = c,: bases aj for these Aj arbitrarily. Set Ao = 0, An = Cn and an = Cn : An = G,

Aj=aCjcCj-,,

0+

A, -+ Cn-l

0-

A,el

°

Al --f Co 5a A0 = o. Al ---+ Co ---> Ao = O.

0--->

+

5

j=l,...,

Cn-2 2

An-1

c Cn-2,

An-2 c G-3,

n-l,

Ao=O,

74

Chapter 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Homology Chapter

We We then then define define i : (u&- ‘u,-1)] .det [c,-s: (a,-,%‘a,-s)]. ' ) = det det [cn[cn-1:(amô-1an-1)]·det[cn_3:(an_z,ô-1an_3)]· .. det C, (K; = (K , p) d et C 1 * p [ . ( ô)] [ . ( -1 ) ] det[c,-:! : (a,-l,d-la,-2)].det[cn--4 : (u~-.&-~u~-~)]. . '’ det Cn-z. an-l, an -2 ·det Cn -4· an -3,ô an -4 ... for the for the arbitrary arbitrary pacyclic p-acyclic complex complex C,(K; C*(K; p). p). Definition 5.1 The The Definition 5.1 complexes K (i.e. with complexes K (i.e. with ter of K, K, and and is is ter torsion torsion of

above quantity quantity det (K; p), for p-acyclic p-acyclic above det C, C*(K; p), defined defined for Hj(K; p) = 0 for j > 0), is called the ReidemeisHj(K;p) for ~ 0), is called the Reidemeisdenoted by denoted by R(K, R(K, p). p).

It was 1930s (by Rham, Franz) that It was established established in in the the late late 1930s (by Reidemeister, Reidemeister, de Rham, Franz) that up and multiplication multiplication by of the the form form det 7r, the up to to sign sign and by any any number number of det p(y), p( 'Y), 'yY EE 7r, the invariant of of the the simpliquantity C,(K; p) == R(K,p) R(K, p) is quantity det detC*(K;p) is a combinatorial combinatorial invariant simplicial K, i.e. much later, in the cial complex complex K, Le. is unchanged unchanged by by subdivisions. subdivisions. Only Only much later, in the 1970s was was itit shown shown that fact aa topological topological invariant. invariant. (The (The varivari1970s, that R(K, R(K, p) is is in in fact ation from the above-mentioned latitude choosing the ation in in R(K, R( K, p) results results from the above-mentioned latitude in in choosing the distinguished bases of the Cj.) distinguished bases of the Cj .) Note L is any any simply-connected simply-connected complex, 7rlL ~1 L == 0, then one has has Note that that ifif Lis complex, then one

R(K R(K x x L,P) L, p) = R(K,p)x(L), R(K, p)x(L), where L. where x(L) X(L) is is the the Euler-Poincare Euler-Poincaré characteristic characteristic of of L. The quantity introduced in the first place (by Reidemeister) The quantity R(K, R(K, p) p) was was introduced in the first place (by Reidemeister) for defined as as follows: follows: Let Let A A for the the purpose purpose of of investigating investigating the the “lens "lens spaces”, spaces", defined be complex diagonal diagonal matrix with primitive roots of of unity be an an nn xx nn complex matrix with primitive mth mth roots unity as as its its diagonal diagonal entries: entries: A = (ajk) ,

ajk = e2Tiq”m6Jk,

A”

= I,,,,

(5.6) (5.6)

where (S,k) == I,,, x n matrix. mth where (Ojk) Inxn is is the the identity identity nnxn matrix. Since Since the the ajj ajj are are primitive primitive mth roots of 1, each qj must be relatively prime to m. The matrix A generates roots of l, each qj must be relatively prime to m. The matrix A generates acting on the sphere sphere S2+l according to the group isomorphic isomorphic to to Z/m Z/m acting on the s2n-l according to the aa group following rule: following rule: n ,i /+ e2riq71mzi -&y=Iz j l2 = l. 1. zj ~ e27riq;jmzj,, (5.7) (5.7)

L

j=l

j=1

Taking the space yields manifold known - 1)-dimensional Taking the orbit orbit space yields the the manifold known as as the the (2n (2n -l)-dimensional lens space: Lens space: L2n1 (5.8) (5.8) (ql,,..,qn) = s2”-1/A. It may be assumed q1 = = 11 by choosing in of A A the It may always always be assumed that that q1 by choosing in place place of the approappropriate generator of the cyclic group A generates. priate generator of the cyclic group A generates. When obtain the the 3-dimensional S-dimensional manifolds manifolds Lz qj, where where qq is When nn == 22 we we obtain L~ = = Lfl Lrl,q)' is relatively prime to m. Two such manifolds L$ and L$, (defined for same the same relatively prime to m. Two such manifolds L~, and L~" (defined for the m) are and only if m) are homotopy homotopy equivalent equivalent ifif and only if q’/q” q' /q" E== St2 ±.\2 mod mod m. m.

(5.9) (5.9)

§5. Homology Homology theory theory of of non-simply-connected non-simply-connected spaces spaces §5.

75 75

This follows follows in in aa straightforward straightforward manner using using the the structure structure of of the the cohomolcohomolThis manner ogy ring H* (Lz; Z/m) and the Bockstein homomorphism ,L? : H1(Li; Z/m) + ogy ring H*(L~; Z/m) the Bockstein homomorphism f3 : Hl(L~; Z/m) ---; H2(Li; Z/m) Z/m) (see (see (4.6)): (4.6)): For For the the fundamental fundamental homology homology class class [L~] [Li] of of the the lens lens H2(L~; space L~, Li, the the following following congruence congruence holds: holds: space (up(a),-+ ± [L~]) [Li]) == E ±q dq mod mod m, m, (af3(a), for for by by

any generator generator aa of of Hl(L~; H1(Li; Z/m) Z/m) ~ % Z/m. Z/m. Replacing Replacing aa in in this this congruence congruence any any other generator Xa then causes q to change to X2q. any other generator Àa then causes q to change to À2 q.

The lens lens spaces spaces L~;l~.~.,qn) L$-f, 1p7,) may may be be decomposed decomposed as as CW-complexes CW-complexes in aa manmanThe in ner independent of (91, . , qn), much as this was done for the projective space ner independent of (qi , ... , qn), much as this was do ne for the projective space IW2n-1. One One obtains obtains just one cell cell of of each each dimension: dimension: 0"0, (TO,0"1, cl,. .... ,, 0"2n-l, cr2n-1, with with IRp2n-l. just one boundary operator given by boundary operator given by @-

1= 0

aa2j = ma2j-1 > j > 0.

(5.10) (5.10)

However up on on the sphere S2+l, the induced cell-decomposition, However up the sphere S2n-1, the induced covering covering cell-decomposition, on which the cyclic group of order m generated by A acts freely and and so so as as on which the cyclic group of or der m generated by A acts freely to respect cells, does depend in an essential way on (91, . , qn). In the case to respect cells, does depend in an essential way on (ql, ... , qn)' In the case nn = = 22 we have the following cells induced on on s2n-1 S2+l = = S3 S3 from from those those of of L~: Lz: we have the following cells induced aS = Aja” , 3 The The cJs = O"g0

s = 0,1,2,3,

j = 0,l ,...,m-1.

(5.11) (5.11)

action cells is is as as follows follows (writing (writing action of of the the boundary boundary operator operator on on these these cells 0s ) s = 0, 1,2,3): O"s,s 0,1,2,3):

aa

=

(1 -A’J)02,

da2

=

(l+A+...+A”-‘)a’, (l+A+'''+Am-I)O"l,

ad BO"l

=

(1 -A)aO.

: Z : Z the the

+ (eaxilm) in the ---; (è1t'i/m) in the exact sequence (5.5) exact sequence (5.5)

For the representation p For the representation p space Lz is pacyclic, and space L~ is p-acyclic, and C,(L,3; p) is C*(L~; p) is

°---;

(1

o~c3~c,~c,~c,~o, a C 3 ---;

of = ~7~ (5.12) (5.12)

A) 0"0.

a

a

space V = space V = arising from arising from

C’,
the lens the lens complex complex

C 2 ---; Cl ---; Co ---; 0, where each Cj is one-dimensional, with distinguished generator ej say. In view where each Cj is one-dimensional, with distinguished generator ej say. In view of (5.12) the boundary operator is given by of (5.12) the boundary operator is given by de3 = (1 - e2?riqlm) e2, de2 = 0, (5.13) (5.13) &, = (1 - eariqlm) eo, Bel = (1 e 2 7l'i Q/m) eo, whence one computes the Reidemeister torsion, obtaining whence one computes the Reidemeister torsion, obtaining (5.14) R(L$ p) = (I- C”) (1 - C) mod (*Cal, R(L~, p) = (1 - (q) (1 - () mod (±(Q), (5.14)

76

Chapter Chapter 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Homology

21ri m where .. ItIt follows where [( = ee2rzlm / follows from from (5.14) (5.14) that that there there are are lens lens spaces spaces that that are are homotopically but not not combinatorially combinatorially equivalent. equivalent. Later Later results results (of (of homotopically equivalent equivalent but Moise, Moise, Chapman Chapman and and Edwards) Edwards) show show that that in in fact fact homotopically homotopically equivalent equivalent lens lens spaces spaces need need not not even even be be homeomorphic. homeomorphic. By By considering considering various various represenrepresentations tations pp for for nn >> 2, de Rham Rham and and Franz Franz have have shown shown that that in in fact fact only only in in trivial trivial cases spaces. cases does does one one have have combinatorial combinatorial equivalence equivalence of of lens lens spaces. A of the point point of of view view of A complete complete analysis analysis of of this this type type of of question question from from the homotopy out by by Whitehead. Whitehead. Suppose Suppose we we know know that that two two homotopy theory, theory, was was carried carried out finite are homotopically and LLare homotopically equivalent. equivalent. Can Can we we then then realize realize finite complexes complexes KK and aa homotopy them by by means means of of elementary elementary combinatorial combinatorial homotopy equivalence equivalence between between them operations? Rather than than embark embark on on aa detailed, detailed, exact exact description description of of these these operations? Rather operations the operations in in geometric geometric terms, terms, we we shall shall confine confine ourselves ourselves to to sketching sketching the essential essential algebraic algebraic picture. picture. Suppose contained in in KK as as aa simplicial simpli,.<::ial or or CW-subc_om_plex. CW-sub~rllPlex. Since Since Suppose LL isis contained KK N '"" L, L, the the complex complex of of relative relative chains chains C(K, C(K, z) L) = @, ESt C,(K, Ct(K, L) L) regarded regarded as is acyclic. acyclic. Consider Consider the the following following two two types types of of elementary elementary as aa Z[r]-module, Z[ll-]-module, is operation on = ESt $, Ct Ct over over any any associative arbitrary algebraic algebraic complex complex C C = associative operation on an an arbitrary ring R, where each C, has a distinguished basis (04): ring R, where each Ct has a distinguished basis {an:

(i) k, j,j, and and y'Y E R, (i) For For each each fixed fixed k, E R, 0: -+ cj a;-+ d

c73” aj -+cr,k+a~, ---> 'Yaj + aj,

ifif either or ii #"1- j.j. either tt #"1- kk or

(5.15) (5.15)

(ii) Stabilization: addition of direct summand Cc”) to to (ii) Stabilization: for for each each fixed fixed k, k, the the addition of aa direct summand C(k) C of the following form: C of the following form: 0-

°

c,--(k) 2.+ qy

+

0, 0,

(5.16) (5.16)

where Cp’, and Cp’k are cyclic, generated respectively by g&i, & with where Ckk:l and Ck ) are cyclic, generated respectively by ek-l, ek with %k = $1; thus the operation is defined as follows: oek = ek-l; thus the operation is defined as follows: cC ----> CEl) Cadk),C(k), (5.17) (5.17)

i.e.

Ci+Cj

forj#k-1,k.

If we define two acyclic algebraic complexes over R to be equivalent if one can If we define two acyclic algebraic complexes over R to be equivalent if one can be obtained from the other by means of a finite sucession of the operations be obtained from the other by means of a finite sucession of the operations (i), (ii) and th eir inverses, then under the operation of taking the direct sum (i), (ii) and their inverses, then under the operation of taking the direct sum the resulting equivalence classes form an abelian group K1 (R). This is the the resulting equivalence classes an abelian group This isquothe ‘Lgroup of determinants”; it may form be alternatively defined KIas(R). a certain "group of determinants"; it may be alternatively defined as a certain quotient group of the group of infinite matrices over R with only finitely-many tient group of the group of infinite matrices over R with only finitely-many non-zero off-diagonal entries and all but finitely many diagonal entries equal aIl but finitely many Kl(R) diagonal non-zero off-diagonal identity entries and to the multiplicative element of R. To obtain oneentries factors equal this to the multiplicative identity element of R. To obtain KI (R) one factors this

§5. §5. Homology Homology theory the ory ofof non-simply-connected non-simply-connected spaces spaces

77 77

group group by by the the normal normal subgroup subgroup generated generated by by all aIl commutators commutators and and all aIl elemenelementary tary matrices matrices with with diagonal diagonal entries entries 11 and and the the remaining remaining non-zero non-zero entry entry above above the the diagonal. diagonal. This This group group arises arises naturally naturally in in algebra algebra as as a result result of of the the axiomaxiomatization the fundamental fundamental properties properties of of the the determinant. determinant. atization of of the In In the the present present topological topological context context the the ring ring RR has has the the form form Z[r], Z[7l'] , and and in in view view of the aforementioned aforementioned latitude latitude in in the the choice choice of of distinguished distinguished bases bases - namely namely of the that that resulting resulting from from changes changes of of orientation orientation of of cells cells of of the the covering covering space space and and of the initial initial cell cell - itit is is not not so so much much Kr(Z[n]) KI (Z[7l']) but but rather rather its its factor factor group group of the Wh(r)

= Kl(%d)l(f~),

(the (the Whitehead Whitehead group), group), that that is is significant. significant. Thus for a pair L c K of homotopy we obtain Thus for a pair L c K of homotopy equivalent equivalent complexes, complexes, we obtain as as an an invariant the “determinant” of the complex C(K,z) of Z[n]-modules in the invariant the "determinant" of the c~mplex C(K, L) of Z[7l']-modules in the Whitehead Whitehead group group Wh(n). Wh(7l'). The The terminology terminology used used in in this this connexion connexion isis “simple "simple homotopy to another another by by means means homotopy type” type":: ifif one one such such complex complex can can be be changed changed to of complexes the above-defined above-defined elementary elementary combinatorial combinatorial operations operations then then the the complexes of the are Any representation representation pp :: 71' K +- 4 to be be of of the the same same simple simple homotopy homotopy type. type. Any are said said to GL(n,IW) to aa homomorphism to the the multiplicative GL(n, lR) gives gives rise rise to homomorphism from from K~@[TT]) K 1 (Z[7l']) to multiplicative group lR group of of Iw Kl(@,d) - II-X* yielding torsion discussed above. For simply-connected comyielding the the Reidemeister Reidemeister torsion discussed above. For simply-connected complexes the group K1 is trivial: Z[r] = Z, Kl(Z) = 0, and simple homotopy plexes the group KI is trivial: Z[7l'] = Z, KI (Z) = 0, and simple homotopy equivalence the ordinary ordinary kind. equivalence reduces reduces to to the kind. Already by the late late 1930s, 1930s beginning beginning with with the the work work of of Smith Smith and and RichardRichardAlready by the son, investigations of the homological properties of finite groups of transson, investigations of the homological properties of finite groups of transformations were being carried on. Suppose that a finite group G acts on aa formations were being carried on. Suppose that a finite group G acts on Cl&‘-complex K, sending cells to cells. From this action one obtains, much CW -complex K, sending cells to cells. From this action one obtains, much as above, a complex of Z[G]-modules - even if G does not act freely. The as ab ove , a complex of Z[G]-modules even if G does not act freely. The subcomplex of K consisting of the points of K fixed by every element of G will subcomplex of K consisting of the points of K fixed by every element of G will be denoted by KG, and the corresponding orbit complex by K/G. We have be denoted by KG, and the corresponding orbit complex by KjG. We have natural maps i : KG -+ K, 7r : K 4 K/G, and the induced homomorphisms natural maps i : KG - 4 K, 71' : K - 4 KjG, and the induced homomorphisms IA

5

H,(K),

H,(K)

-2

H,(K/G).

The finiteness of G allows one to define the transfer homomorphism The finiteness of G allows one ta define the transfer homomorphism

p* : K(K/G)

-

K(K),

(defined over any group of coefficients) as follows: Every j-chain c, in (defined over any group of coefficients) as follows: Every j-chain Cu in Cj(K/G) corresponds to an orbit {g(c) ] g E G} of j-chains of Cj(K), c E Cj(KjG) corresponds to an orbit {g(c) 1 9 E G} of j-chains of Cj(K), c E C,(K), under the action of the finite group G; the homomorphism pj is then Cj(K), under the action of the finite group G; the homomorphism ftj is then with c g(c) E C,(K), the induced by that associating each c, E Cj(K/G) induced by that associating each C" E Cj(KjG) with g(c) E Cj(K), the gEG

L

gEG

78

Chapter Homology Chapter 3. Simplicial Simplicial Complexes Complexes and CW-complexes. CW-complexes. Homology

(finite) of the the chains chains in orbit. It It follows that the the comcom(finite) sum sum of in the the corresponding corresponding orbit. follows that posite posite map map n*p* : H,(K/G) - H,(K/G) (5.18) (5.18) is just multiplication by ]G], and and just multiplication by IGI, pcL*r* : H,(K)

-

(5.19) (5.19)

K(K)

L

is (E Z[G] is determined determined by by the the action action of of c g9 (E Z[G] gEG gEG From (5.18) and and (5.19) (5.19) it it follows follows that that over over aa From (5.18) finite characteristic relatively prime to ]G], finite characteristic relatively prime to IGI, we we

if the the coefficient coefficient group group is is Z). Z). if field of characteristic zero or or field of characteristic zero have have

H,(K; k) H*(K; k)

E ~

p*H,(K/G; k) @ Ker Ker 7r*, 7r+, M*H*(K/G; k) EB

H,(K/G; k) H*(K/G; k)

” ~

H,(K; H*(K; k)G, k)G,

(5.20) (5.20)

where the G denotes denotes the the subgroup subgroup pointwise stabilized by by G. G. where the superscript superscript G pointwise stabilized Via the G-action each element p of Z[G] has associated with it the chain Via the G-action each element p of Z[G] has associated with it the chain complex pC(K) and the resulting homology groups Hf(K) = H,(pC(K)). complex pC(K) and the resulting homology groups Hf(K) = H*(pC(K)). The = Z/p is of of particular that here here The case case G G = = Z/p, Z/p, kk = Z/p is particular interest. interest. Note Note that H,P(K; Z/p) is to be understood as the homology of the complex pC(K; Z/p) Hf(K; 7l/p) is to be understood as the homology of the complex pC(K; 7l/p) and @Z/p. the following following elements elements of of the group ring ring and not not of of pC(K; pC(K; Z) Z) ® Z/p. Consider Consider the the group k[G], k = Z/p:

k[G], k

Z/p:

Cl= (]' = c Lg = g@ gEG

g = 1 + T + T22 + . + Tp-l 1 +T+T + ... +TP-l,

7=1-T,

T

1

(5.21) (5.21 )

T,

ffr = 7-c = 0,

O'T

TO

0,

where T is a generator of the cyclic group G of order p (written multiplicawhere T is a generator of the cyclic group G of order p (written multiplicatively). The exact sequencesdue to Smith are as follows: tively). The exact sequences due to Smith are as follows: ... ->

Hj(K; Z/p)~H'J(K;Z/p) -SHJ_l(K;Z/p) Hj-&Y;Z/p)

H j - 1 (K; Z/p)

2

If}

Hj_1(K G; Z/p) ~

.... (5.22)

(5.22)

Hj(K; Z/p)

HJ(K; Z/p)

H'J_l(K; Z/p)

Hj--l(K;

If}

Hj_l(KG; Z/p)

Z/p) I;

Hj_l(K; Z/p)

There are analogous exact cohomology sequences(with the arrows going the There way). are analogous exact cohomology sequences (with the arrows going the other other way). From these exact sequencesone infers the following result: If the group Z/p exact sequences one then infersthe theset following Z/p actsFrom on athese Z/p-homology n-sphere, of fixedresult: pointsIfofthe thegroup action is acts on a Z/p-homology n-sphere, then the set of fixed points of the action is

‘$6. Simplicial Simplicial and cell bundles bundles with with aa structure structure group group §6.

79 79

likewise a Zjp-homology Z/p-homology sphere of of some some dimension dimension r. r. Moreover Moreover if if p > 2, 2, then likewise sphere is even. even. Analogous results are are valid for homology n-discs n-discs in the category category n - r is (K, L) of complexes. complexes. (A (A Zjp-homology Z/p-homology n-sphere is is aa complex K with with of pairs (K, n-sphere I$(K; Zjp) Z/p) ~ ” Hj(sn; Hj(P; Zjp) Z/p) for for an all j.j. This definition definition may be be taken as as applying applying Hj(K; also in the case casen = -1, -1, where sn S” is is the empty set.) also Within the bounds of of the present exposition exposition it it is is not appropriate to to dwell dwell Within the bounds the present Ilot appropriate purely homological theory finite or compact transat greater greater length on the purely theory of finite formation its associated associated algebro-topological algebro-topological techniques, although formation groups groups with with its techniques, although several authors have investigated investigated this this topie. topic. We shall touch touch on on the the subject subject We shan several authors have of finite groups of of smooth smooth transformations below in in connexion connexion of finite and and compact compact groups transformations below with “K-theory”, and "bordism “bordism theOl·y". theory”. with "K-theory", “elliptic "elliptic operators”, operators", and

56. Simplicial and and cell cell bundles bundles with with aa structure structure group. group. §6. Simplicial Obstructions. Universal objects: universal fiber bundles and the the Obstructions. Universal objects: universal fiber bundles and universal of Eilenberg-MacLane universal property property of Eilenberg-MacLane complexes. complexes. Cohomology Cohomology operations. The Steenrod algebra. The Adams spectral sequence sequence operations. The Steenrod algebra. The Adams spectral We elaboration of of aa concept examined above above from from We now now turn tum to to the the elaboration concept already already examined the point of view of homotopy, namely that of a fiber bundle. Our present the point of view of homotopy, namely that of a fiber bundle. Our present definition include an an important new element element - the “structure group" group” the "structure definition will will include important new of the fiber bundle, occuring naturally as a subgroup of the group of selfof the fiber bundle, occuring naturally as a subgroup of the group of selfhomeomorphisms (transformations) of a fiber. Let homeomorphisms (transformations) of a fiber. Let pp :: E B E + ---> B

(with fiber F) (with fiber F)

be a fiber bundle (as defined in Chapter 2, §3), where E, F and B are CWbe a fiber bundle (as defined in Chapter 2, §3), where E, F and B are CWor simplicial complexes and the projection p respects cells (simplexes). In this or simplicial complexes and the projection p respects cells (simplexes). In this context the prescribed structural maps are assumed to be defined for some context the prescribed structural maps are assumed to be defined for sorne open neighbourhood U(oE) of each cell (or simplex) c: of the base B; hence open neighbourhood U(O'~) of each ceU (or simplex) O'~ of the base B; hence on p-‘(oz) we have on p-l(O'~) we have da : ~-‘(a;) --+ u,” x F, 4YCY. : p-l(O'~) ---> O'~ x F, where the restrictions to fibers are homeomorphisms of F. It is further rewhere the restrictions to fibers are horneornorphisms of F. It is further 1'equired that the transition function on some small open neighbourhood of the quired that the transition function on some smaU open neighbourhood of the boundary (or faces) common to two cells (simplexes): boundary (or faces) common to two ceUs (simplexes):

have the form have the form

Jkdd~7 Y) = (Z>L&)(Y))

ÀCY.{3(x, y)

(x, :\CY.{3(X) (y))

where &(z) is a transformation of the fiber F for each point II: E U,, = where :\CY.{3 (x) is a transformation of the fiber F for each point x E UCY.{3 U(az n a?). We assumethat all maps &p(z) belong to a prescribed group U(O'~ n 0'13)' We assume maps G:\CY.;3(X) prescribed G of transformations of F;that theallgroup is the belong structureto agroup of the group fiber G of transformations of F; the group G is the structure group of the fiber

80 80

Chapter Homology Chapter 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology

bundle (E, B, G). ItIt is also required required that that the composite of transformations bundle (E, B, F, F, p, P, G). is also the composite of transformations X,,(z), th e identity 1‘d en t’t1 y element element of of the the group :\afi(X) , &,(z) :\fi"r(X) and and &(zr) :\"ra(x) is is the group G: S&(X)

0 i&(X)

0 &(x)

- 1,

(6.1) (6.1)

for Uap nU Up, nU U,,"ra,, the the common common domain domain of of the the of the the intersection intersection U afi n fi"r n for each each point point xx of transition functions xap, ipy and &. Thus the fiber bundle is determined transition functions :\a,6, :\fi"r and :\"ra' Thus the fiber bundle is determined essentially the maps essentially by by the maps & :: U Uao - G, (6.2) (6.2) :\afi afi ---4 G, with condition (6.1) imposed. with the the condition (6.1) imposed. A cross-section cross-section of of the map ff :: B -+ E E such such that that ppoo ff = = 1lg.B. A the bundle bundle is is any any map B ---4 AA map of fiber bundles (or fiber bundle map) between fiber bundles with the map of fiber bundles (or fiber b~mdle map) between fiber bundles with the same fiber F and group G: same fiber F and group G: @:: (E,B,F,p,G) if> (E, B, F,p, G) -

(E’, B’, P’, G), G), (E', B', F, F,p',

is the natural as given given by @ :: E E ---4 + E' E’ and and cp 4 :: B B ---4 + B’ is defined defined in in the natural way way as by maps maps if> B' commuting projections in in the sense that that commuting with with the the projections the sense cpop=p’o cp 0 p p'

0

cp. if>.

ItIt is that if> @ respect fibers, i.e. that its its restrictions restrictions to fibers be be is further further required required that respect fibers, i.e. that to fibers homeomorphisms. Moreover these homeomorphisms have to come from the homeomorphisms. Moreover these homeomorphisms have to come from the structure in the precise sense: sense: over over each each point point of of the structure group group G G in the following folIowing precise the base base BB the the map map @oq!+F+F #cd 0O if> CP~t 0 cp -;;1 : F ---4 F (where &,, 4a are any appropriate structural maps of the bundles E’, E (where CP~t, CPa are any appropriate structural maps of the bundles E', E respectively) should be a homeomorphism from G. respectively) should be a homeomorphism from G. Given a fiber bundle (E’, B’, F, p’, G) and an arbitrary map 4 : B + B’, Given a fiber bundle (EI,BI,F,p',G) and an arbitrary map cp: B B', one obtains a fiber bundle over B, the induced fiber bundle (E, B, F, p, G) and one obtains a fiber bundle over B, the induced fiber bundle (E, B, F, p, G) and a fiber bundle map a fiber bundle map @ : E - E’, c$ : B + B’, if> : E ---4 E', cp: B ---4 B', by means of the following construction: The distinguished open sets of B on by me ans of the folIowing construction: The distinguished open sets of B on which the structural maps 4a are to be defined, are taken to be the complete which the structural maps CPa are to be defined, are taken to be the complete inverse images U, = c$-‘(U’~) of the distinguished open sets U’, of B’ (and inverse images Ua = cp-1(U ' a ) of the distinguished open sets U ' a of B' (and then if necessary these may be refined to open neighbourhoods of cells). Above then if necessary these may be refined to open neighbourhoods of celIs). Above each such U, we set p-‘(U,) = U, x F. The product spaces p-‘(U,) are then each such Ua we set p-1(U ) Ua X F. The product spaces p-1(Ua ) are then joined together to form E aby means of the transition functions X,0: joined together to form E by means of the transition functions ).,afi: L&,Y)

= b&&9(~)),

x E ua n 4

where the transformation ia is defined wherethe thefiber transformation from bundle (E’, :\a,6 B’, F,isp’,defined G) as from the fiber bundle (E', B', F,p', G) as &(x)

= &(4(x))

Y E F,

in terms of the transformation ?& in terms of the transformation :\~,6 follows: follows: E G.

$6. §6. Simplicial Simplicial and and cell cell bundles bundles with with aa structure structure group group

81 81

The The total total space space EE may may be be considered considered formally formally as the the subset subset of of BB xx E’ E' consistconsisting (x, y)y) such such that that 4(x) 4>(x) == p’(y). p'(y). Thus Thus the the induced induced fiber fiber bundle bundle ing of of those those pairs pairs (x, over over BB isis obtained obtained by by “pulling "pulling back” back" via via 44> the the fiber fiber bundle bundle over over B’. B'. Remark. Remark. ItIt is is easily easily inferred inferred from from the the condition condition (6.1) (6.1) that that the the maps maps &p :\0:/3 represent represent aa one-dimensional one-dimensional cocycle cocycle of of the the base base BB with with coefficients coefficients in in the the sheaf sheaf FG Fe of of germs germs of of functions functions with with values values in in the the group group G G (see (see $4). §4). Hence Hence the (B; 3~)) and the fiber fiber bundle bundle isis associated associated in in this this way way with with an an element element of of H1 H1(B;Fe), and this with structure structure this correspondence correspondence affords affords aa classification classification of of bundles bundles over over BB with group group G G in in terms terms of of one-dimensional one-dimensional sheaf sheaf cohomology. cohomology. This This classification classification is, is, however, is however, of of an an essentially essentially linguistic linguistic or or tautological tautological character, char acter , although although itit is 00 occasionally the group group G G is is abelian. abelian. occasionally useful, useful, for for instance instance ifif the As As observed observed above, above, the the collection collection of of maps maps xap :\0:,B in in (6.2) (6.2) uniquely uniquely determines determines the the structure group G G is structure of of the the bundle bundle (E, (E, B, B, F, F, p, p, G). G). Of Of course course aa given given topological topological group is in of aa in general general realizable realizable in in many many different different ways ways as as aa group group of of homeomorphisms homeomorphisms of topological same structure the same same b_ase ~ase B, B, with with the the same structure topological space. space. Fiber Fiber bundles bundles over over the group G, and with the same collection of maps X,p, but with possibly different group G, and with the same collection of maps ).0/3, but with possibly different fibers, are said to be associated with one another. Clearly any fiber bundle fibers, are said to be associated with one another. Clearly any fiber bundle determines the structure structure of of all aIl of of its its associated associated bundles. bundles. determines the There as aa canonical universal universal realization realization of of aa topological topological group group G G as There isis aa canonical group of homeomorphisms of a fiber F, namely that where F = G and G acts group of homeomorphisms of a fiber F, namely that where F = Gand G acts on itself by means of (right) translations (see Figure 3.8): on itself by me ans of (right) translations (see Figure 3.8): g:C G .......... ---+ C, G, g:

g(h) g(h) == hg, hg,

h,g E G. G. h,g E

(6.3) (6.3)

A fiber (E, B, G, C,p) G,p) with this G-action Such bunbunA fiber bundle bundle (E, B, C, with this C-action is is called called principal. principal. Such dles arise most naturally in the situation of a topological group acting freely dIes arise most naturally in the situation of a topological group acting freely (i.e. implies 9g == 1) 1) on on aa space space E; E; here the orbit orbit space space E/C E/G plays plays g(x) == 5x implies here the (i.e. g(x) the role of the base B, and the projection is the natural one p : E --+ E/G. the role of the base B, and the projection is the natural one p : E ---+ E / G. In fact it is not difficult to show that every principal fiber bundle arises in In fact it is not difficult to show that every principal fiber bundle arises in this way, i.e. that there is a free action of G on the total space E such that this way, Le. that there is a free action of G on the total space E such that the base B is naturally identifiable with the orbit space E/G, and the fibers the base B is naturally identifiable with the orbit space E/C, and the fibers p-‘(z) with G. p-l(x) with C. Of particular importance is the case where G is a compact Lie group acting Of particular importance is the case where C is a compact Lie group acting freely on a smooth manifold E by means of smooth transformations, and the freely on a smooth manifold E by means of smooth transformations, and the orbit space B = E/G is likewise a smooth manifold. This describes, in essence, orbit space B = E / G is likewise a smooth manifold. This describes, in essence, the concept of a smooth fiber bundle, to be considered at greater length in the the concept of a smooth fiber bundle, to be considered at greater length in the sequel. Another particular case of interest is that where G is a discrete group; sequel. Another particular case of interest is that where C is a discrete group; here a principal G-bundle is simply a regular covering space, as considered here a principal C-bundle is simply a regular covering space, as considered earlier. earlier. Definition 6.1 A fiber bundle (E, B, F, G,p) is called universal if the asDefinition 6.1 A fiber bundle (E,B,F,C,p) is called universal if the associated principal bundle (E’, B, G, G, p) has the property that its total space sociated principal bundle (E', B, C, G,p) has the property that its total space E’ is contractible. E' is contractible. The significance of this definition will appear below. The significance of this definition will appear below.

Chapter 3. Simplicial Simplicial Complexes Complexes and CW-complexes. Cl&‘-complexes. Homology Homology Chapter

82 82

G=fR Fig. 3.8 3.8 Fig. The concept of of an an "obstruction" “obstruction” is of fundamental fundamental importance importance in in algebraic algebraic The concept is of topology; originally it arose arose in in attempting attempting to to answer answer the the following following questions. questions. topology; originally it Question 1. Let L subcomplex of complex K, K, and and let let ff :: L L ----7 + X X Question 1. Let L be be aa subcomplex of aa complex be a map from L to an arbitrary topological space X. How may f be be a map from L to an arbitrary topological space X. How may f be extended, ifif itit can can be extended, to to all all of of K? K? extended, be extended, Letf,:K+X,cu=1,2,betwomaps,andletF:LxI+Xbea Let fa : K ----7 X, a = 1,2, be two maps, and let F : LxI ----7 X be a homotopy between the restrictions of of these these two maps to to the the subcomplex subcomplex homotopy between the restrictions two maps L. Can the homotopy F be extended to all of K, i.e. to one betweenil fi L. Can the homotopy F be extended to ail of K, i. e. to one between and fi? and h? Question E ----7 + I3 fiber bundle (with base base B simplicial or or Question 2. 2. Let Let pp :: E B be be aa fiber bundle (with B aa simplicial

CW-complex), and let 1c,: L + E be aa cross-section of the bundle over over L ----7 E be cross-section of the bundle CW-complex), and let 1/J: the subcomplex L c B: plc, = 1~. How may one extend the cross-section the subcomplex L c B: p1/J IL' How may one extend the cross-section over The analogous analogous problem for homotopies homotopies of over all all of of B? B? The problem for of two two crosscrosssections is as follows: Given two cross-sectionsover B and a homotopy sections is as follows: Given two cross-sections over B and a homotopy between them defined over L c B, how can one extend this homotopy between them defined over L c B, how can one extend this homotopy to one between the original cross-sections, i.e over all of B ? to one between the original cross-sections, i. e over all of B?

3. Let (E, B, F, G,p) and (E’, B’, F, G,p’) be fiber bundles with Question 3. Let (E,B,F,G,p) and (E',B',F,G,p') be fiber bundles with the same fiber and structure group, L c B be a subcomplex as abov_e, the same fiber and structure group, L c B be a subcomplex as above, and let f : L -----f B’ be a map inducing a-map of fiber bundles f : B' be a map inducing a map of fiber bundles and let f : L p-l(L) + E’. How can one extend the map f to a map of fiber bundles p-l(L) ----7 E'. How can one extend the map to a map of fiber bundles E -+ E’ ? There is also the analogous question for homotopies between E ----7 E' ? There is also the analogous question for homotopies between two maps of fiber bundles. two maps of fiber bundles.

Question

i

i :

We begin with the first of these questions. Thus we seek to extend a map We begin with the first of these questions. Thus we seek to extend a map : L + X from the subcomplex L c K to the whole CW-complex K. fSuppose : L ----7inductively X from the subcomplex L c K to the who le CW -complex K. that the map f has already been defined on all cells of Suppose inductively that the map f has already been defined on all cells of dimension 5 n - 1. Thus if a: is any cell of K not contained in L then f is dimension S n 1. Thus if (j~ is any cell of K not contained in L then f is assumed already defined on its boundary au:: assumed already defined on its boundary 8(j~: .

f

fl a02 : au; ---+ x.

56. §6. Simplicial Simplicial and and cell cell bundles bundles with with a structure structure group group

83 83

Clearly Clearly this this map map may may be be extended extended toto the the interior interior of of 0,” O"~ ifif and and only only ifif the the map map flat= of 8~2 XX is of the the sphere sphere Sz-’ S~-l 5 80"~ u\ is null-homotopic. null-homotopie. InIn any any case case the the map map S’z-l S~-l -+ ---> XX defines de fines an an element element x (0:)

Thus Thus in L) in L)

E %-l(X).

provided contained provided nn >2 22 we we obtain obtain (by (by considering eonsidering all aU n-cells n-ceUs ~2 O"~ not not contained aa relative cochain relative cochain X(f)

E C”

(K,

L;

%1(X)),

called ealled an an obstruction obstruction to ta the the extension extension off. of f. Clearly Clearly the the map map fJ may may be be extended extended from X(f) E 0. from the the (n (n - l)-skeleton l)-skeleton of of KK to to the the n-skeleton n-skeleton ifif and and only only ifif >..(J) O. ItIt may (n - 1)-skeleton may be be possible, possible, however, however, to to change change the the map map fJ on on the the (n l)-skeleton KnV1 K n- 1 without without changing changing itit on on the the (n-2)-skeleton (n - 2)-skeleton KnP2 Kn-2 or or on on L. L. Observe Observe that that the (n -- l)-cell the restriction restrietion f]JI Gz-l n~l of of the the map map fJ to to each eaeh (n l)-cell $-’ 0"(3n-1 may may be be replaced replaced a(J

by with fJ on on the of the by aa map map g9 agreeing agreeing with on KnP2 Kn-2 Uu LL but but differing differing on the interior interior of the cells 0;3n-l not contained in L. Since f and g agree on the boundary a$-’ ceUs O"~-l not contained in L. Sinee J and 9 agree on the boundary 80"~-1 of of each such cell, they determine a map SnP1 + X, representing a homotopy each such ce Il , they determine a map sn-1 ---> X, representing a homotopy class (X). Taking account aIl all J.l( O"~-l) EE 7r,-i 1Tn-1 (X). Taking into into aceount class (“distinguishing ("distinguishing element”) element") ~(a;-‘) relative (n (n -- 1)-cochain such 0"~-1 we we obtain obtain aa relative l)-cochain sueh cells eeIls a;-’

p(f) 6E C c-ln - 1 (K, (K,L;~n-l(X)). {L(J) L; 1Tn~l(X)). Via the induced homomorphism f* : rrl(L) ---+ 7rr (X), the homotopy groups Via the induced homomorphism J* : 1T1(L) ---> 1T1(X), the homotopy groups nj(X) acquire the structure of Z [rr(L)]-modules, whence the cochain com1Tj(X) acquire the structure of Z [7rl(L)l-modules, whence the eochain eomplex C* (K, L; r+r(X)) acquires a Z [rr(L)]-module structure. It will be asplex C* (K, L; 1Tn -1 (X)) acquires a Z [1Tl (L)]-module structure. It will be assumed in what follows that rrl(L) ” rrl(K) = 7r (via the homomorphism sumed in what follows that 7r1 (L) ~ 1TI (I<) = 7r (via the homomorphism that the map f is already defined on the (n - 2)f* n(K)) and J* :: m(L) 1Tl(L) - 1Tl(K)) and that the map J is already defined on the (n - 2)skeleton of K. The following assertions are valid:

skeleton of K. The following assertions are valid: (i) The obstruction cochain X(f) asa cocycle in the complex C*(K,L;rrn-l(X)) (i) The obstruction cochain >..(J) is a cocycle in the complex C*(K,L;1Tn -I(X)) considered as a complex of Z [r]-modules: 8*(x(f)) = 0 . considered as a complex ofZ [1Tl-modules: 8* (>"(J)) = 0 . (ii) Alteration of th e map f (to the map g say) on that portion of the (n - l)(ii) Alteration of the map f (ta the map 9 say) on that portion of the (n - 1)skeleton of K not contained in L, without changing it on the (n - 2)skeleton of K not contained in L, without changing it on the (n - 2)skeleton (so that f = g on KnP2 U L) causes the obstruction cocycle skeleton (so thatbyJthe addition 9 on K nof- 2an U L) causes the obstruction cocycle l(f) to change arbitrary coboundary: \ (J) ta change by the addition of an arbitrary coboundary: X(f)

-

J+(f) + a*(P),

and a* is the coboundary homomorwhere p E Cm1 (K, L;rr,-l(X)), where J.ld*E :cn-l n-1 (X)), and 8* is the coboundary homomorphism: CnP1(K, (K, L; L; 1T7rn-1(X)) - C”n (K, L; rr+l(X)). phism: 8* : cn-1 (K, L; 7rn -I(X)) ---> C (K, L; 1Tn -I(X)), We conclude

that

We conclude that For it to be possible to extend the given map f to the n-skeleton of K while For it toit be possible to (nextend the given(but mapallowing f to thef n-skeleton of Konwhile keeping fixed on the - 2)-skeleton to be changed that keeping it fixed on the (n 2)-skeleton (b1d allowing J ta be changed on that

84

Chapter 3. Simplicial Simplicial Complexes Complexes and CW-complexes. Cl&‘-complexes. Homology Homology Chapter

portion the (n (n- -1) l)-skeleton not contained in in L) L) it it is nessesar-y nessesary and and sufficient suficient por-tion of the -skeleton not that the the obstruction obstruction cohomology class [>,(f)] [X(f)] in in the the complex complex of of Z[1l}modules Z[r]-modules that cohomology class (K, L; nn-l(X)) C* (K,L;1l'n_l(X)) vanish:

o0 =

[X(f)] E E H II-ln - 1 (K, (K, L; L; 1l'n-dX)) 7&1(X)). [À(f)] .

Allowing to be be changed changed on on cells cells of of dimension dimension :S < nn -- 22 (not (not contained contained in in Allowing ff to L) leads to complicated rules rules for changing the obstruction obstruction cochain À(f). X(f). L) leads to more more complicated In particular, particular, changes in ÀX(f) arising from from changes changesof of ff on on the the (n (n-2)-skeleton In changes in (f) arising - 2)-skeleton may be described in in ter terms of the the "cohomology “cohomology operations" operations” (introduced (introduced be fully fully described ms of may in the 1940s and early 1950s 1950s by by Pontryagin, Pontryagin, Steenrod, Steenrod, Postnikov, Postnikov, Boltyanskii Boltyanskii in the 1940s and early and Liao); we shall consider consider these these cohomology cohomology operations operations below. below. and Liao); we shall Observe that ifif the the complex complex K K is is simply-connected simply-connected (or (or the the homo homomormorObserve that phism f* : ~1 (L) --+ ri(X) is trivial) then the obstruction class [X(f)] lies in in phism f* : 1l'1(L) -7 1l'1(X) is trivial) then the obstruction class [À(f)] lies the ordinary relative cohomology group over the coefficient group 7rn-i(X). the ordinary relative cohomology group over the coefficient group 1l'n-l(X), The problem of of extending extending a reduces to to the the ab above The associated associated problem a homotopy homotopy reduces ove problem of of extending taking KI Ki == K K xx [a, [a, b], b], and and L1 c KI K1 to to problem extending aa map, map, by by taking LI C consist consist of of the the lid lid and and the the base: base: K1 x [a, b], KI = KKx[a,b],

L1 = (K x {a})

LI

u (K x {b}). (Kx{a})U(Kx{b}).

The prescribed map fi :: L1 X is defined by the two given maps fa : K + The prescribed map fI LI + ---t X is defined by the two given maps fa : K ---t X, fb : K -----) X. Suppose we have already extended extended the the homotopy homotopy LxI Lx I + X X, fb : K X. Suppose we have already X to the (n-l)-skeleton of K, (n > 3), so that we have F : (KnP1 U L) XI + X. X. to the (n-l)-skeleton of K, (71. 2 3), so that we have F : (Kn-l U L) xl Arguing above one obstruction to extending the the homotopy Arguing as as above one distils distils out out the the obstruction to extending homotopy FF to Kn as an element to Kn as an element X E Cn+’

(Kl,

Ll;r,(X))

,

n > 3.

It follows that the obstruction to the extension of the homotopy F to the It follows that the obstruction to the extension of the homotopy F to the skeleton of K is represented by the cohomology class of X, or, equivalently, skeleton of K is represented by the cohomology class of À, or, equivalently, the corresponding element the corresponding element [A] E IIn+’

(Kl,

L1; m(X))

= IP

(K; m(X))

nnby by

,

where the cohomology groups are defined with respect to the complexes of where the cohomology groups are defined with respect to the complexes of Z[r]-modules Z[1l']-modules 7.r= rl(K). C’ (K; G(X)), C* (Kl,L,; s(X)), C* (K; 1l'n(X)), C* (KI, LI; 1l'n(X)) , 1l' 1l'1(K). The second of the above extension problems is treated analogously: AssumThe second of the above extension problems is treated analogously: Assuming that the cross-section $J over L has been defined on the (n - 1)-skeleton that the cross-section 'I/J over L has been defined on the (n - 1)-skeleton ing F-l of the base B, one seeks to extend it from the boundary agz of each Bn-I of basecell’s B, one seeksThe ta extend from themap boundary âO'~ of each n-cell a: the to the interior. known it restriction n-cell O'~ ta the cell's interior. The known restriction map

'l/JloO';;: : âO'~

---t

E,

p'I/J

=

1,

56. §6. Simplicial Simplicial and and cell cell bundles bundles with with aa structure structure group group

85 85

determines de termines a map of of the the (n - l)-sphere 1)-sphere g-l

-+ aa,” Ap

-’ (a;)

sz CT; x F,

which which determines determines in in turn turn an element element of of the the group group m-1(F). 7rn-l(F). Note Note that that the the action action of of xl(B) 7rl(B) on the the fibers (as (as determined determined by by the the “homo"homotopy topy connexion” connexion" on the the fiber bundle bundle - see see Chapter Chapter 2, 53), §3), induces induces an action action of (B) on the nj(F), [XI (B)]-modules. Hence there 7rl(B) 7rj(F), turning turning them them into into Z Z[7rl(B)]-modules. there of ~1 arises, arises, much much as as before, before, an an obstruction obstruction cohomology cohomology class class to to the the extension extension of of the the cross-section cross-section $1 1/J: [A] [À] EE Hn(B; Hn(B; nn-l(F)), 7r n-l(F)),

where the where the the cohomology cohomology groups groups Hj Hj (B; (B; 7r,7rn~ 1r (F)) (F)) are are defined defined with with respect respect to to the complex of Z[7rr(B)]-modules C*(B; r+l(F)). By allowing $ to change complex of Z[7rl(B)]-modules C*(Bj 7rn -l(F)). By allowing 1/J to change on on the the the (n (n -- l)-cells l )-cells of of B B not not contained contained in in L, L, but but keeping keeping itit fixed fixed on on Fe2 B n - 2 UU L, L, the obstruction from obstruction cocycle cocycle X À can, can, as as before, before, be be changed changed to to any any other other cocycle cocycle from the same cohomology class [Xl. the same cohomology class [À J. The cross-sections is simiThe obstruction obstruction theory theory for for homotopies homotopies between between two two cross-sections is simi1 to B” is an n lar: the obstruction class to extending the homotopy from BnP1 lar: the obstruction class to extending the homotopy from B - to Bn is an element n,-lF). the group group IP(B; Hn(Bj 7r element of of the n -lF). The similarly to two, with one difference. The third third problem problem is is treated treated similarly to the the first first two, with one difference. Suppose Qn-r :: p-l U L) L) --+ E' E’ defined Suppose we we already already have have aa bundle bundle map map Pn-l p-l (P-l (Bn-l U defined n 1 UL, L c B, determining a map 4 : B”-l on that portion of E above Bn-’ --->, on that portion of E above B - U L, L c B, determining a map
,n -

p-1

5n-

1

5

(1) 3 E’, aa(u;)((J"~) xx {I} ~ E',

P’ O @P,-lla(o~)x{l}

=

41a(c7;).

p' 0 Pn-lIB(a;;:)x{l} , E' so determined, defines an element of 7rn -l (E'), and taking into consideration all cells a: on which q5is not yet defined, one obtains in ing into consideration all cells (J"~ on which
86

Chapter Chapter 3. 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Homology

IfIf BB isis simply-connected sirnply-connected then then as as earlier earlier the the obstruction obstruction class class belongs belongs to to the the ordinary cohomology group with coefficients from the group n+r(E’). ordinary cohornology group with coefficients frorn the group 7rn-l (El). We We infer infer the the following following result result concerning concerning our our three three questions: questions: IfIf the the homotopy homotopy groups gmups of of the the space spa ce XX of of Question Question 11 (or (or of of the the fibre fibre FF inin Question Question 2, 2, or or the the total total space space E’ El inin Question Question 3) 3) are are trivial trivial in in dimensions dimensions 5~ n, n, then < nn (the then for for CW-complexes ClV -complexes KK of of dimension dimension::; (the base base BB inin Questions Questions 22 and and 3) 3) every every map map (cross-section (cmss-section inin Question Question 2, 2, bundle bundle map map in in Question Question 3) 3) defined defined on on any 5 nn of any skeleton skeleton of of dimension dimension::; of KK (or (or B) B) can can be be extended extended to ta the the skeleton skeleton of of next higher dimension, and likewise for homotopies (in the senses appropriate next higher dimension, and likewise for homotopies (in the senses appmpriate to 5 nn -- 1. to the the three three questions) questions) defined defined for for any any skeleton skeleton of of dimension dimension::; l.

We We obtain obtain immediately imrnediately as as aa particular particular case case the the consequence consequence that that for for any any fibre fibre ere is up to a homotopy just one bundle map bundle bundle (E, (E, B, B, F, F, G,p) G,p) th there is up to a hornotopy just one bundle map @: B’, F, G, P’) <{>: (E,B,F,G,p) (E,B,F,G,p) -----+ (E’, (EI,BI,F,G,pl) to to the the corresponding corresponding universal universal bundle, bundle, since since the the total total space space E’ El is is contractible. contractible. Writing B’ = BG in standard notation for the base of the universal bundle, we we Writing BI = Be in standard notation for the base of the univers al bundle, deduce that every fibre bundle (E, B, F, G,p) determines a map f : deduce that every fibre bundle (E, B, F, G,p) deterrnines a map f : BB + BG Be uniquely uniquely to to within within aa homotopy hornotopy (which, (which, conversely, conversely, induces induces the the given given fibre fibre bundle from the universal bundle). Such a homotopy is of course a map map ff :: bundle frorn the universal bundle). Such a hornotopy is of course a BB xx IJ + BG, Be, and and so so induces induces aa fibre fibre bundle bundle over over the the cylinder cylinder BB xx I.J. Since Since the cylinder B x I, I = [a, b], may be contracted onto its base B x {a} in such the cylinder B x J, J = [a, bl, rnay be contracted onto its base B x {a} in such a a way way that that the the lid lid BB xx {b} {b} goes goes “identically” "identically" onto onto BB xx {a} {a} (in (in the the sense sense that that we have that Hj(B x I, B x {a}) = 0 over any ----f (~,a) for all x E B), (Xl b) (x,b) (x,a) for an x B), we have that Hj(B x I,B x {a}) 0 over any coefficient group. It follows that any fibre bundle over B x I may be mapped coefficient group. It follows that any fibre bundle over B x 1 rnay be rnapped to the restricted bundle over the base B x {a} in such a way that the induced to the restricted bundle over the base B x {a} in such a way that the induced map of basessendsthe lid B x {b} of B x I “identically” to B x {a}. For the map of bases semis the lid B x {b} of B x l "identically" to B x {a}. For the conclusion we wish to draw we require a preliminary definition. conclusion we wish to draw we require a prelirninary definition. Definition 6.2 Two fibre bundles over the same base B, with the same Definition 6.2 Two fibre bundles over the sarne base B, with the same fibre F and the same structure group G, are said to be equivalent if there fibre F and the same structure group G, are said to be equivalent if there exists a bundle map between them: exists a bundle map between them: @ : (E, B, F, G,P) - (E’, B, F, GP’), <{> : (E, B, F, G,p) ----+ (El, B, F, G,p/), inducing the identity map on the common base. inducing the identity rnap on the cornrnon base. The foregoing discussion has as upshot the following important result. The foregoing discussion has as upshot the following important result. Theorem 6.1 The set of the equivalence classes of jibre bundles with the Theorem 6.1 The set of the equivalence classes of fibre bundles with the same base B, Jbre F and group G, is in natural one-to-one correspondence same base B, fibre F andclasses gmup of G, maps is in ofnatuml one-to-one correspondence with the set of homotopy the baseB to the base BG of the with the set of homotopy classes of maps of the base B ta the base Be of the corresponding universal bundle. corresponding univers al bundle. Example 1. From the exact homotopy sequenceof the universal principal Example(F 1.= From theChapter exact homotopy sequence of the universinto al principal obtains, on taking account G-bundle G) ( see 2, (4.7)) one (F = G) (see Chapter 2, (4.7)) one obtains, on taking into account G-bundle the contractibility of the total space, the following isomorphism: the contractibility of the total space, the following isornorphisrn:

$6. §6.

Simplicial Simplicial and and cell cell bundles bundles with with aa structure structure group group T,(BG)

”

T-I(G)

(=

%-l(F)),

n

>

1.

87 87

(6.4) (6.4)

This This isomorphism isomorphism and and the the preceding preceding theorem theorem together together imply imply that that the the equivequivalence classes of G-bundles with base S” are classified by the group are classified by the group 7r,-r 11"n-l (G). (G). alence classes of G- bundles with base Recalling Recalling that that for for a topological topological group group G G the the connected connected component component Gc Go containcontaining the identity identity isis a normal normal subgroup, subgroup, so so that that xc(G) 11"o(G) == G/Go G/G o has has a natural natural ing the group structure, we infer that the action of ?‘rr(&) (” TO(G)) on group structure, we infer that the action of 11"1 (Be) 11"o(G)) on the the groups groups nn(BG) 11" n (Be) (” x+1(G)) 11"n-l (G)) isis identifiable identifiable with with the the action action of of TO(G) 11"0 (G) on on 7rn-r(G): 11"n-l (G):

sn

q(a) q(a) = q@, qaq-\

qq E no(G), 11"o(G),

aa E rn-l(G). 11"n-l(G).

(6.5) (6.5)

Hence Hence the the free free homotopy homotopy classes classes of of maps maps S” Sn ----f ----t & Be (i.e. (i.e. without without distindistinguished natural one-to-one one-to-one correspondence correspondence with with the the orbits orbits guished base base point) point) are are in in natural of these orbits 11"n-l(G) under under the the action action of of no(G) 11"o(G) given given by by (6.5), (6.5), so sa that that these orbits may may of n,-r(G) be be taken taken as as determining determining the the inequivalent inequivalent G-bundles G-bundles over over S”. Geometrically Geometrically speaking speaking this this reduction reduction of of the the classification classification problem problem for for GGbundles of the the group group 7rn-r(G) natural: sn to to the the consideration consideration of 11"n-l (G) is is very very natural: bundles over over S” Consider as the with their the sphere sphere S” as the union union 0; D+ UU D”_ Dr:.. of of the the hemispheres hemispheres with their Consider the boundaries (the equator) identified. Any G-bundle over Sn is trivial over each each boundaries (the equator) identified. Any G-bundle over sn is trivial over of of their Hence there there are are structural structural the hemispheres, hemispheres, in in view view of their contractibility. contractibility. Hence of the homeomorphisms: homeomorphisms: c+hl: p-l (D;) --+ D; x G,

sn.

sn

q52: p-l

(D:) (Dr:..)

+

D” xxG, G, Dr:..

(where G-bundle to be principal), principal), and and the the transition (where we we are are assuming assuming the the G-bundle to be transition function function Xl2 = (h4>2 q5& 1 :: S”-1 x G ----f 9-l x G À 12 sn-1 X G ----t sn-l X G on the equator has the form on the equator has the form XlZ(T

9) À12(X, g)

=

(T X12(~)(9)), (x, :\12(X)(g)),

where Xl2 : S”-l + G represents the appropriate homotopy class of 7rn- r (G). where :\12 : sn-l ----t G represents the appropriate homotopy class of 11"n-1 (G). 0

o

The above classification theorem was given its final form by Steenrod, The above classification theorem was given its final form by Steenrod, it would seem, although investigations in this direction involving particular it would seem, although investigations in this direction involving particular universal bundles of importance had been carried out earlier by Pontryagin, universal bundles of importance had been carried out earlier by Pontryagin, Ehresmann, Chern, and others. As far as obstructions are concerned, a version Ehresmann, Chern, and others. As far as obstructions are concerned, a version first appeared in the early 1930s in connexion with Van Kampen’s construction first appeared in the early 1930s in connexion with Van Kampen's construction of n-dimensional complexes K”,n n 2 2, that are higher-dimensional analogues of , n ?: 2, that are higher-dimensional analogues of n-dimensional the l-complex complexes figuring inK Euler’s problem of the three houses and three of the 1-complex figuring in Euler's problem of the three hou ses and three wells, in the sense that they are not embeddable without self-intersections in sense they are not embeddable without self-intersections wells, in the Il%2n. This idea wasthat subsequently elaborated on by Whitney, in particular in in ]R2n. This idea was subsequently elaborated on by Whitney, in particular in relation to cross-sections of fiber bundles, and was actively exploited in conta cross-sections and was and actively exploited in conrelation crete problems of homotopyof fiber theorybundles, by Pontryagin Whitehead. Evidently crete problems of homotopy theory by Pontryagin and Whitehead. Evidently

88 88

Chapter Chapter 3. 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Homology

the the final final formulation formulation of of the the concept concept of of an an obstruction obstruction is due due to to Eilenberg Eilenberg in in the in the the case case of of maps, maps, and and Steenrod Steenrod in the context context of of fiber fiber bundles. bundles. We We note note in in passing passing that that fiber fiber bundles bundles with with contractible contractible fiber fiber possess possess crosscrosssections, situation sections, which which are, are, moreover, moreover, all aH homotopic homotopie to to one one another. another. This This situation will theory of the theory of smooth smooth manifolds, manifolds, to to which whieh Chapter Chapter 44 is is will arise arise frequently frequently in in the devoted. devoted. In In concrete concrete topological topological problems problems itit is is most most often often the the so-called so-called first first obstrucobstruction that is used. This is the obstruction arising in attempting to extend maps tion that is used. This is the obstruction arising in attempting to extend maps (Question (Question 1 1 above) above) or or bundle bundle cross-sections cross-sections (Question (Question 2) 2) from from the the (n (n -- l)1)skeleton the n-skeleton first diskeleton to to the n-skeleton of of the the appropriate appropriate space, space, where where n n is is the the first dimension for which mension for whieh the the homotopy homotopy group group of of the the codomain codomain X X (in (in Question Question 1) 1) is is non-trivial: non-trivial: 72, XT%(X) ?Tn(X) #=1= 0, 0, 7rj(X) ?Tj(X) = 0 0 for for jj < < n, or or of of the the fiber fiber FF (in (in Question Question 2): 2): rj(F)=O G(F) ?Tn(F) # =1= 0, 0, ?Tj(F)

= 0 forj
Example 1. Consider classification up Example 1. Consider Hopf’s Hopf's problem problem of of the the classification up to to homotopy, homotopy, of the maps from an n-dimensional CW-complex K” to the n-sphere of the maps from an n-dimensional CW-complex Kn to the n-sphere S”. sn. Here Here there are no obstructions to map-extensions. There is just one obstruction there are no obstructions to map-extensions. There is just one obstruction to to extending T,(P) 2~ Z. extending a a homotopy, homotopy, and and this this lies lies in in the the group group Hn(Kn; lln(Kn; Z) Il) since sin ce ?Tn(sn) Il. It S” are in natural one-toIt follows follows that that the the homotopy homotopy classes classes of of maps maps K” Kn + ----> sn are in natural one-toone correspondence with the elements of the group Hn(Kn; Z). Equivalently, one correspondence with the elements of the group lln(Kn; Il). Equivalently, the homotopy classescorrespond one-to-one to the homomorphisms between the homotopy classes correspond one-to-one to the homomorphisms between the n-th integral homology groups of Kn and S”, since the n-th integral homology groups of Kn and sn, since

H”(K”;

Z) = H om (H,(W;

Z), Z),

Z ” H,(Sn; Z).

0

o

Example 2. The previous example may be (mildly) generalized by replacing Example 2. The previous example may be (mildly) generalized by replacing S” by any (n-l)- connected space X (i.e. one for which nj(X) = 0 for j < n). sn by any (n - 1)-connected space X (i.e. one for which ?Tj (X) = 0 for j < n). Thus one seeksto classify the homotopy classesof maps of an n-dimensional Thus one seeks to classify the homotopy classes of maps of an n-dimensional CW-complex K” to X. As before, in view of the (n - l)-connectedness, only CW-complex Kn to X. As before, in view of the (n l)-connectedness, only the first obstruction arises, and the analogous result is obtained: each map the first obstruction arises, and the analogous result is obtained: each map Kn + X is to within a homotopy determined by a unique cohomology class Kn ----> X is to within a homotopy determined by a unique cohomology class in Hn(Kn; m(X)), or equivalently, a unique homomorphism between the nth in lln(Kn; ?Tn(X)),groups or equivalently, unique integral homology of Kn anda X, sincehomomorphism between the nth integral homology groups of K n and X, since H”(K”; m(X)) = Horn (H,(Kn; Z), 23,(X; Z)), H,(X; Z) TZ TV. 0

Example

3. Suppose X is an Eilenberg-MacLane spaceof type K(r,

n), i.e.

Examplc 3. Suppose X is an Eilenberg-MacLane space of type K(?T, n), Le. 7rn(X) ” 7r, nj(X)=O

?Tn(X)

~?T,

for

jfn.

?Tj(X) = 0 for j

=1=

n.

56. §6. Simplicial Simplicial and and cell ceU bundles bundles with with a structure structure group group

89

The The argument argument of of the the preceding preceding two two examples ex amples leads leads to ta the the same same concluconclusion, K: The The set set of of homotopy homotopy classes classes sion, though though here here for for any any CW-complex CW-complex K: of K; ‘rr), of maps maps KK +----4 XX isis in in natural natural one-to-one one-ta-one correspondence correspondence with with Hn( Hn(K; 11'), and and so, sa, provided provided n >> 1, acquires acquires the the structure structure of of an an abelian abelian group. group. Since Since H,(K(r, Hn(K(11', n); n); Z) 71,) g9:: 7rn(K(7r, 11'n(K(11', n)) n)) g9:: 7r 11' and and H”(K(n, 7r), Hn(K( 11', n); n); 7r) 11') ”9:: Horn Hom (H,(K(n, (Hn (K(11', n); n); Z), 71,), 7r) 11') % 9:: Horn Hom (7r, (11',11'), there there isis aa distinguished distinguished element element uu of of fP(K(x, Hn (K (11', n); n); 7r), 11'), corresponding corresponding to to the the identity identity isomorphism isomorphism 11 :: n11' --+ ----4 K, 11', called caUed the the fundamental fundamental class class of of Hn(K(7r, Hn (K (11', n); n) j 7r). 11'). In In terms terms of of the the fundamental fundamental class class u, u, the the above above correspondence correspondence between between hohomotopy of H”(K; motopy classes classes [f] [il ofof maps maps ff :: KK +----4 K(r, K(11', n) n) and and elements elements of Hn(K; 7r) 11') isis given given by by [fl ~1. (64 (6.6) [f] ++ +-+ f*(u) f*(u) E E H”(K; H n (K;11'). Note Note that that in in the the case case nn = 11 the the above above is is valid valid also also for for non-abelian non-abelian groups groups 1 7r since, as observed earlier, H1(K; n) is defined (as a set) such 7r. 11' since, as observed earlier, H (K; 11') is defined (as a set) over over such 11'. We We conclude from (6.6) that: conclude from (6.6) that: The is universal classiThe Eilenberg-MacLane Eilenberg-MacLane space space of of type type K(n, K(11', n) n) is universal for for the the classification of n-dimensional cohomology classes over n (much as principal unification of n-dimensional cohomology classes over 7r (much as principal universal G-bundles are universal for the classification of equivalence classes of versai C-bundles are universal for the classification of equivalence classes of G-bundles). C-bundles). As of examples examples of of Eilenberg-MacLane Eilenberg-MacLane comAs noted noted earlier, earlier, we we know know aa series series of comand K(Z, 2) = c.P” (= lim @P”), but of the remainplexes of type K(n, l), plexes of type K (11', 1), and K (71" 2) = CpcXJ lim cpn), but of the remainn--+w n ..... oo der we know only that they exist. By virture of (6.6) these complexes play an der we know only that they exist. By virture of (6.6) these complexes play an 0 important technical role in topology. important technical role in topology. 0 Definition 6.3 A cohomology operation is a “natural” function Definition 6.3 A cohomology operation is a "natural" function Bx(.zi,. . , zk) of Ic variables zj E Hn3 (X; Gj), taking its values in H*(X; G). eX(Zl,"" Zk) of k variables Zj E Hnj (X; Cj), taking its values in H*(X; C). Here “natural” means that the operation should be functorial in the sense Here "natural" me ans that the operation should be functorial in the sense that it commutes with homomorphisms induced by maps f : Y ---+X: that it commutes with homomorphisms induced by maps f : y ----4 X: if

f*(q) = .zj.,

then

&(zi,.

,z(c) =

f*BX(zl,.

, zk).

(6.7)

The simplest example is afforded by the product 19= zr zk of elements of The simplest example is afforded by the product e = Zl ... Zk of elements of any number k of factors, with all Gj = G. It is however important that there C. It is however important that there any number k of factors, with aU C exist cohomology operations O(Z) ofj even one variable that are not definable exist cohomology operations e(z) of even one variable that are not definable merely in terms of the ring structure of H*(X; G), especially in the case merely in terms of the ring structure of H*(X; C), especially in the case G = Z/p. The simplest such “non-trivial” example is afforded by the Bockstein C 71,/p. The simplest such "non-trivial" example is afforded by the Bockstein homomorphism (see $4) homomorphism (see §4) Z/p) P : H4(X; Z/P) + fPfl(X; In view of the above-noted universal property of the Eilenberg-MacLane In view of the above-noted universal pro y of the Eilenberg-MacLane spaces in connexion with the classification ofpert cohomology classes,the totality spaces in connexion with the classification of cohomology the totality of cohomology operations of the above form is in naturalclasses, one-to-one correof cohomology operations of the above form is in natural one-to-one correspondence with the elements spondence with the elements

90 90

Chapter CW-complexes. Homology Chapter 3.3. Simplicial Simplicial Complexes Complexes and and CW-complexes. Homology

19E H* (K(G1,

no) x .

x K(Gk,

n/J; G) .

This correspondence is easily defined in terms the fundamental fundamental classes classes (see This correspondence easily defined ter ms of of the (6.6)): (6.6) ): Hn3 (K(Gj, nj); Gj) E Horn (Gj, Gj), uj * lG,, so so that that 6 = qu1,. For aa For maps maps

given space given space XX and and elements elements zj Zj fjIj :: X + K(G,, nj) such that that X -+ K(Cj,nj) such f = il fl x··· x ‘. 1

x X

fk : X X + Ik:

. . ) Uk).

(6.8) (6.8)

E Hnj Hnj (X; (X; Cj), Gj), jj = = 1, 1, ... . . ,, k, Ic, there there exist exist E zj = f*(uj), whence we obtain the map Zj = f*(Uj), whence we obtain the map

K(G1, nl) x . . . x K(Gk,

nk),

and the E H* H*(X;(X; C) G) of of the the operation 6 is is then then given given by and the value value Qx(zr, ex (Z1, ... ,, zk) Zk) E operation e by Bx(a,.

. . rzk)

=

f*~(‘%,...,Uk).

(This was observed observed by Serre in in the the early early 1950s.) 1950s.) (This was by Serre In addition to the operation p other particular non-trivial cohomology cohomology opopIn addition to the operation (3 other particular non-trivial erations had been discovered earlier: erations had been discovered earlier: 1) Pontryagin and powers: 1) Pontryagin squares squares and powers:

Pp : Hzq(X;

Z/p)

+

Hzpq(X;

Z/p’),

where mod p, and p p is prime. where P,(z) Pp(z) E 3zP mod p, and is prime. 2) squares and 2) Steenrod Steenrod squares and powers: powers: a) Steenrod squares: a) Steenrod squares: Sq’ : Hq(X;

Z/2)

---+ Hqfi(X;

Z/2),

where, in particular, Sq” = 1 and Sq’ is the Bockstein homomorphism where, in particular, SqO 1 and Sq1 is the Bockstein homomorphism for p = 2; for p = 2; b) Steenrod powers: b) Steenrod powers: St; : Hq(X;

Z/p)

--t Hq+i(X;

Z/p),

where, in particular, St, ’ = 1 and St; coincides with the Bockstein 1 and St~ coincides with the Bockstein where, in particular, Stg operator p. operator (3. The Steenrod squares are homomorphisms and have the following further The Steenrod squares are homomorphisms and have the following further basic properties: basic properties: (i) Sq” = 1, Sq’ = 0, Sqn(z) = .z2, z E Hn(X;Z/2). (i) SqO = 1, Sq1 = (3, Sqn(z) = Z2, Z E Hn(x; '1l/2). (ii) The Steenrod squares are stable, i.e. they commute with the suspension (ii) The Steenrod squares are stable, Le. they commute with the suspension isomorphisms isomorphisms

$6. §6. Simplicial Simplicial and and cell cell bundles bundles with with aa structure structure group group

E : Hq(X; (iii) (iii) S$(.zi Sqj(ZI . z2) Z2) =

Z/2)

*

Hq+l( cx;

91

Z/2).

I: Sq”(zi) Sqk(Zl)' Sql(zz). Sql(Z2)'

C

k+l=j k+l=j

(iv) (iv) Sqj Sq} == 00 for for jj << 0; Sqn(z) Sqn(z) == 00 ifif zZ has has dimension dimension << n. n. (v) (v) The The Steenrod Steenrod squares squares are are additive additive (in (in contrast contrast with with the the Pontryagin Pontryagin powers): powers): Sqi(z1 + z2) = Sqi(z1) + Sq-yzy.).

The The Steenrod Steenrod powers powers have have analogous analogous properties: properties: (i) (i)

St: Stg == 1, 1, St; Stb == ,B, f3, St; Stt == 00 ifif jj << 00 or or jj f::f:- 0,l 0,1 mod mod (2p (2p -- 2). 2).

(ii) suspension (ii) The The Steenrod Steenrod powers powers are are stable, stable, i.e. Le. they they commute commute with with the the suspension isomorphisms. isomorphisms. (iii) Sty(P-l), one (Hi) Writing Writing Pi pj for for St;j(P-l), one has has Spj(P-l)+l = ppi St P2j (p-l)+l = f3pj > p ,

(6.9) (6.9)

and formulae: and also also the the following following formulae: Pi(Zl

z2)

=

c I:

PYZl) pi(zd' Pk(Z2), pk(Z2),

i+k=j

(6.10) (6.10)

Hk=j

,B(Zl

z2)

=

P(q)

22 +

(-l)W

P(zz),

where m = dim zi. where m = dim Zl. (iv) for zz E E H H2n(X;Z/p), for jj > > n, n, Pi(z) 0. 2n (X;7l,/p), and (iv) P”(z) pn(z) == zp zP for and for pj(z) == o. (v) The Steenrod powers are additive: (v) The Steenrod powers are additive: sqz1 + z2) = St;l(z1) + Sti(zz). St~(Zl + Z2) St~(Zl) + St~(Z2). By results of Serre (in the case p = 2) and Cartan (p > 2) from the early By results of Serre (in the case p = 2) and Cartan (p > 2) from the early 1950s the Z/p-algebra of all stable operations (the Steenrod algebra dp) is 1950s, the 7l,/p-algebra of aU stable operations (the Steenrod algebra A p ) is generated multiplicatively by the Steenrod operations (the Sqi when p = 2, generated multiplieatively by the Steenrod operations (the Sqi when p 2, and the St; when p is an odd prime); it follows that all such cohomology and the St~ when p is an odd prime); it follows that aU sueh cohomology operations are additive. operations are additive. Let d: denote the Zlpsubspace of the Steenrod algebra A, consisting of A; denote the 7l,/p-subspaee of the algebraInAthe p eonsisting of the Let operations of degree n, i.e. a E d; if a Steenrod : Hq + Hq+n. case p = 2, the operations of degree n, Le. a E A; if a : Hq Hq+n. In the case p = 2, a cohomology operation a E dz is the null operation precisely if the element aa(~) cohomology operation a E A~ is the null operation precisely if the element vanishes for the appropriate fundamental class u: a( u) vanishes for the appropriate fundamental class u: ui E H1 (lRP,~~Z/2) , u = Ul. . UN E HN (lkf~,oo x .‘. x IRF~,Z/2))

Chapter 3. Simplicial Complexes and CW-complexes. CW-complexes. Homology Chapter 3. Simplicial Complexes and Homology

92 92

for some N > n. that IRpoo IWP” = K (Z/2; 1), l), and and that ring for sorne N > n. (Recall (Recall that K (Zj2; that the the ring H* (IRPl' (llw,” x ... ‘. . x RP,“; Z/2) H* IRPN'; Zj2) is polynomial.) From the properties properties of of the the Steenrod Steenrod squares squares listed listed above, above, it it is polynomial.) From the follows that the image AZ(U) c H* (RP,oO x x BP,“; Z/2) is a Z/2-vector follows that the image A2(U) C H* (IRPr x ... x IRPN';Zj2) Zj2-vector space spanned by those symmetric in the the variables ~1, ... . . UN, UN, variables Ul, space sparmed by those symmetric polynomials polynomials Uu,w in of the form: of the form: U w=

C il#i2#..-+.9 (

wj Wj

. UC’

UT

21, 1

= 2 - 11,, W W=(Wl,...,Wk), (Wl, ... We denote Sqw the operation defined defined by by We denote by by SqW the operation SqW(u~~~~us) SqW(Ul" ,u s ) =uw, = Uw ,

wj ~ > O. 0. Wj

, Wk ) ,

(6.11) (6.11)

degSqW == Cwi. degSqW I:Wj.

with with

sqi = SqW, sqw ( sl =

< 72,

i

hj 2h7

Note that Note that

Cwj

w=(l,...,l). (1, ... ,1). "--v--' i times i tinles

W =

One following product formula: One has has the the following product formula: &Y(az2) SqW(ZI Z2)

I:

= =

sqw’(.4sQw2(~2). SqWl(zdSqW2(Z2).

c

(w,w)=W

(Wl,W2)=W For > 22 we copies X Xj of of the the infinite-dimensional infinite-dimensional For pp > we consider con si der copies j H K(Z/p; 1) = P/Z/p. ere the cohomology ring H*(X,; ; Zjp) Z/p) has K(Zjp; 1) = SOO jZjp. Here the cohomology ring H*(X has j

vi E H1(Xj;Z/p),

Vj E

uj = &.

uj E H2(Xj;Z/p), 2

HI(X j ; Zjp),

Uj E

H (X j ; Zjp) ,

In this this case E A~, AZ, a a :: Hq In case an an operation operation aa E Hq + where where

lens space lens space generators generators

Uj

= {3Vj

Hqfn, is null null precisely precisely if if a( CL(V) 0, Hq+n, is v) = 0,

E HN1+N2(X1 X...X~N,+N~;~/~), N N vV=V~"~~/N~UN,+~~~~UN,+N~ VI" 'VN1UN1+l" 'UN 1+N2 E H 1+ 2 (Xl x··· x X N1 + N2 ;Zjp) , where Ni, iV2 > n. The image dP(v) is spanned by the polynomials of the where NI, N 2 > n. The image Ap(v) is spanned by the polynomials of the form: form: uwr,w”

UW',w"

= u .

(++I..

c

== u,

.u~+luJwl;i+l..

.,,;;+1>

lVi, . .vui,,

il < ... -c i, 5 Nl il < ... < is :S NI N2 > j, > . > j, > N1 N 2 ~ jl > ... > jq > NI W'l

< ... < w's' 1 + w'~

= phi ' w" l

deg a = c (2~: + deg a i (2w~ + 1)

I:

< ... < w"q' 1 + w"J

= pl,

'

h·2 > 0 , 1j > 0, _

+ I: (2wj) j

a : HQ(X;

Z/p)

-

Hqfdega(X;

Z/p),

a = SC,,,,,,).

36. §6. Simplicial Simplicial and and cell bundles bundles with with aa structure structure group group

93 93

The The number number ss 22': 00 appearing appearing in in this this formula formula is called ca11ed the the T-Cartan T- Cartan type type of of the the corresponding corresponding operation operation a, denoted more more usually usua11y by by s(a) s(a) or or r(a). T(a). The The type type ss and and the the degree degree deg a (s 5<:::: deg deg a) a) determine determine aa bigrading bigrading of of the the algebra A, A p (for (for pp L2': 2): 2): aa H ~ [s(a), [s(a), deg deg aa - s(a)] s(a)].. Of Of special special interest interest are are the the operations operations aa = Sk+,, Bk,w, where where of of course course we we have have kt2xwj

< N1+2N2.

The The operations operations Sk,e Bk,O = ek Bk say, say, are are called called i%lnor Milnor exterior exterior algebra: algebra: eke, BkBj = -ejek, -BjBk, S(ek) S(Bk) =

operations. operation.s. They They define define an an 1, 1,

and and have have the the further further property property ek(Zl

’ ~2)

=

ek(h2

f

dk(Z2)r

a formula a special special case case of of the the general general formula SCw’,w”)(zl

. z2)

=

c

(~1)s(,:,,~)(zl)s(,;,,;,)(zz).

(w;,w:)=w’ (w;I,w;)=w”

ItIt turns out that that the the algebra algebra of stable) Z/p-cohomology Z/p-cohomology turns out of all a11 (not (not necessarily necessarily stable) operations is generated via composition by the Steenrod operations and the the operations is generated via composition by the Steenrod operations and Bockstein operator p (together with the product operation in the cohomology Bockstein operator 13 (together with the product operation in the cohomology ring), and that all relations among these operations follows from the properties ring), and that ail relations among these operations follows from the properties listed above. Over other coefficient groups one has for instance the Pontryagin listed above. Over other coefficient groups one has for instance the Pontryagin powers; a classification of all cohomology operations was given by Cartan powers; a classification of aIl cohomology operations was given by Cartan in the mid-1950s. In the early years of that decade Serre had shown that in the mid-1950s. In the early years of that decade Serre had shown that cohomology operations over a field Ic of characteristic 0 (for instance Q, R, @) cohomology operations over a field k of characteristic 0 (for instance Q, lR, q are all trivial, i.e. reduce essentially to the cohomological product operation. are a11 trivial, Le. reduce essentially to the cohomological product operation. There are in such casesno stable operations aside from multiplication by a There are in such cases no stable operations aside from multiplication by a scalar: z H Xz, X E lc. scalar: Z ~ ÀZ, À E k. As a generalization of single-valued, everywhere-defined cohomology operAs a generalization of single-valued, everywhere-defined cohomology operations one has multi-valued, partially defined operations, as in the following ations one has multi-valued, partially defined operations, as in the following two examples. two examples. Example 1. Consider the Bockstein homomorphism Example 1. Consider tbe Bockstein homomorphism P : H”(X; UPI + Hk+yX; Z/p). As in $4 above, ,D is defined initially on integral cochains Z E C”(X; Z) which As in §4modulo above, p13tois adefined on integral cochains E Ck(X; Z) which reduce cocycleinitially z over Z/p, and then ,0(z) is defined as ;6’*E rereduce modulo p to a cocycle Z over Z/p, and then j3(z) is defined as ~â*z reduced mod p. If z is in Ker p, i.e. if the cocycle ia*? (mod p) is cohomologous duced mod p. If z is in Ker 13, Le. if the cocycle ~â*z (mod p) is cohomologous to 0, then aa*? = pu + d*y, and we can define to 0, then ~â*z = pu + â*y, and we can define

z

94 94

Chapter Chapter 3.3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Homology

1 /32(z) -\8*2 (mod (mod p), p), /32 (z) = Td*Z Pp provided provided we we work work modulo modulo Im lm /3. /3. Proceeding Proceeding inductively inductively we we similarly similarly define define on Ker ok-i, k > 2, pi = p, an operation pk with values determined modulo on Ker fA-l, 2': 2, /31 /3, an operation /3k with values determined modulo Im lm Pk-1,

1, byby /3k(Z)

=

~8'2

P

Pk /3k :: Ker Ker @k-l

(mod p),

zz E== Z2 (mod (mod p), p),

Hqfl(X;Z/p)/Im

-

2~ 2 E Cq(X;Z), cq(X; Z),

(6.12) (6.12)

P&i,

Ker Ker flk-1 /3k-l CC HQ(X; Hq(X; z/p). Z/p). IfIf /&(z) /3k(Z) == 00 for for all aU kk >2': 1, 1, then then the the class class zz EE Hq(X;Z/p) Hq(X; Z/p) isis obtained obtained by by reducing reducing an an integral integral class. class. Classes Classes &(Z) /3k(Z) always always arise arise from from integral integral classes classes of of 00 order order pk. pk.

Example E.1:ample 2. 2. Let Let kk be be satisfy ziz2 = 0, = satisfy Z1Z2 = 0, ~2~3 Z2Z3 = Massey M assey bracket bracket

(zl, z2, z3)

aa ring ring 0. O. For For =

B(zl,

or or field field and and such such triples triples

let k), ii = 1,2,3, let zjZj EE H”J(X; HHj(X;k), = 1,2,3, there is an operation called there is an operation caUed the the

z2, z3) c Hn1+nz+n3-1(X;

k),

(6.13)

(6.13)

defined defined only only up up to to the the addition addition of of an an element element of of the the form form Wl Z3 *±ZlW2, ZlW2, W1Z3

The

The definition definition 6”“~ (X; k) it is CHi (X; k) it is

wl

k), w2 E H W+Wl(X; n 1 (X; k), HHl+H2-1(X;k), W2 E H n 2+ 3- (X;k).

g Hnl+%-1

Wl E

of the Massey bracket is very simple: on cochains of the Massey bracket is very simple: on cochains in in the the given by

given by

(Zl, Z2, Zg) = dp1(zlZ2)Z3 * (Z1, Z2, Z3) = 8-1(ZlZ2)Z3

± Zldp1(Z2z3). Z1 8 -1(Z2 2 3)'

In the case that k is a field of characteristic zero, the Massey brackets and In the case that k is a field of characteristic zero, the Massey brackets and their analogues constitute generators, with respect to composition, for all their analogues constitute generators, with respect to composition, for aIl 0 multivalued, partial, natural cohomology operations.

multivalued, partial, natural cohomology operations.

0

We now return to our discussion of Steenrod squares. From the classification We now return to our discussion of Steenrod squares. From the classification algebra A2 given above (see, in particular, (6.11)) we see that the Steenrod given above (see, in particular, (6.11)) we see that the Steenrod algebra A 2 i is generated multiplicatively by the Steenrod squares of the form Sq2*, i 2 0. is generated multiplicatively by the Steenrod squares of the form Sq2 , i 2': O. This fact has the following nice application.

This fact has the foIlowing nice application. Example. Let K be a CW-complex such that Example. Let K be a CW-complex such that for j = 0, n, 2n, Hj(K; Z/2) ” Z/2 Hj(K;Z/2) ~ Z/2 for j = 0, n, 2n, Hj(K;

Z/2) = 0 for j # O,n,2n. 0 for j:f 0, n, 2n. Denoting by 2cj the generator of Hj (K; Z/2) for j = n, 2n, we also assume Denoting the generator of Hj for jcomplex, = n,2n, we also assume that z$ = by ~2,.Uj Examples of such K (K; are 7l../2) the real, quaternionic and that u~ tl2n' Examples of such K are the real, complex, quaternionic and

Hj(K;Z/2)

$6. §6. Simplicial Simplicial and and cell cell bundles bundles with with a structure structure group group

95

Cayley Cayley projective projective planes. planes. ItIt follows follows that that Sqnu, Sqn un = ~2, U2n (by (by property property (i) (i) of of the the Steenrod squares). If n is not a power of 2, this is not possible, since then Sqn Steenrod squares). If n is not power of this is not possible, since then Sqn would would be be decomposable decomposable as as a composite composite of of the the Sqj Sqj with with j.i << n,n, and and for for such su ch j.i (with the exception of j = 0) we have Sqju, = 0 in view of the assumption (with the exception of.i = 0) we have Sqju n in view of the assumption that that Hnfj(K; Hn+j (K; Z/2) 2/2) = 0. O. Hence Hence we we must must have have nn = 2’. 28 • This This was was shown shown by by Hopf, without using cohomology operations, in the 1940s and by Steenrod in Hopf, without using cohomology operations, in the 1940s, and by Steenrod in the the late late 1940s. 1940s. ItIt was Steenrod squares was further further (shown (shown by by Adams Adams in in the the late late 1950s) 1950s) that that the the Steenrod squares i i > 3, decompose Sqn with n = 2i, non-trivially as composites Sqn with n 2 , i > 3, decompose non-trivially as composites of of partial partial operations. that Sq%, Sqn un == ~2, U2n only only ifif nn =::= 1,2,4,8. 1,2,4,8. This This operations. In In particular, particular, itit follows follows that result the following result has has aa series series of of fundamental fundamental consequences, consequences, in in particular particular the following ones: ones: 1. R1, FL’, 1. The The only only real real division division algebras algebras are are]RI, ]R2, R4, ]R4, IRS. ]RB. 2. 2. The The stable stable tangent tangent bundle bundle of of the the sphere sphere S”-l sn-l is is trivial trivial only only ifif nn = 1,2,4,8. 1,2,4,8. 3. elements with 7rzn-l (Sn), only 3. There There exist exist elements with Hopf Hopf invariant invariant 11 in in the the groups groups 1f"2n_l(sn), only ififn n == 2,4,8.6 6 2,4,8. 4. 5’” of Hopf type (i.e. where the base and 4. There There is is aa fibration fibration Szn-l s2n-l -+ ---'> sn of Hopf type (i.e. where the base and 0 total spaces are spheres) only if total spaces are spheres) only if nn == 2,4,8. 2,4,8. 0 The squares for for CWCWThe essential essential idea idea behind behind the the construction construction of of the the Steenrod Steenrod squares complexes K is as follows. Consider the diagonal complexes K is as follows. Consider the diagonal A(K) cc K Ll(K) K xx K, K, A(K) Ll(K) = {(x,x)> {(x, x)},> and denote by 2 the subcomplex of those cells of K x K containing A. (Note and denote by Ll the subcomplex of those cells of K x K containing Ll. (Note that A itself is not a subcomplex in general.) Let A, : K 4 2 C K x K that Ll itself is not a subcomplex in genera!.) Let Ll 1 : K Ll c K x K be a cellular approximation of the diagonal, i.e. a cellular map close to the be a cellular approximation of the diagonal, i.e. a cellular map close to the diagonal map K -+ A, with the image of each cell CT~of K approximated in diagonal map K ---'> Ll, with the image of each cell (}"k of K approximated in a(gk x ak). This yields a product operation on the cell cochains in Ll((}"k x (}"k). This yields a product operation on the cell cochains in C*(K x K) ” C*(K) @C*(K), C*(K x K) ~ C*(K) 0 C*(K), defined by defined by ZiZj Llr(4'i 0 Zj). This operation turns H*(K) into a ring; it is the cohomological product deThis operation turns H*(K) into a ring; it is the cohomological product defined earlier for simplicial complexes - see $2. fined earlier for simplicial complexes - see §2. Consider the involution u : K x K + K x K given by Consider the involution (}" : K x K --+ K x K given by a(x, y) = (y,x), CT2= 1. O"(x,y) (y,x), (}"2 l. Although OA = A, the map cr does not (except in trivial cases) preserve A,: Although the map (}"is does not (except trivial cases) preserve 1: crAl # A,. (}" Ll This Ll, asymmetry the source of the inSteenrod squares. Let wLlbe (}" Ll 1 f. Ll 1 · This asymmetry is the source of the Steenrod squares. Let w be ‘jFor

the definition

of the

“Hopf

invariant”,

see Chapter

4, 53.

6For the definition of the "Hopf invariant", see Chapter 4, §3.

96

Chapter 3. Simplicial Simplicial Complexes Complexes and and eW-complexes. CW-complexes. Homology Chapter Homology

any cocycle cocycle in in C* C*(Al(K)), and define define cocycles cocycles Wo, ws, Wl, wi, ... . . , in in terms terms of of W w as as any (.1 1 (K)), and follows (over (over 'lL/2 Z/2 for for simplicity): simplicity): follows

wo = W, w, Wo

PWl = wo + u*wo,

a*(w, + + u*wt} u*wl) = = 0, 0, â*(Wl

a*w2

= w1+

u*w1,

a*(wz

+ u*w2)

d*Wi

=

+

d*(Wi

+

Wi-1

U*Wi-1,

U*Wi)

= 0, =

0,

.... ,) .

,

with the support support of every every Wi wi in in .1 A,(K). For cocycles z E C”(K; Z/2) set with the of 1 (K). For cocycles z E cn(K; 'lL/2) set

(z”)i = Ay(Wi) E C2”-i(K;

Z/2),

with Wo wc taken taken to to be be L1i(z®z), A;(z @z), and and then then define define the the (n (n - i)th i)th Steenrod Steenrod square square with 2n i of z as the cohomology class of the cocycle (Z”)i E H2n-i(K; Z/2): of z as the cohomology class of the cocycle (Z2)i E H - (K; 'lL/2):

Sqn-i(z) = (z2);.

°

ItIt is is immediate immediate from definition that Sq”+“(z) = 0 for > 0, 0, and and from this this definition that Sqn+i(z) for ii > Sqn(z) = z2. The remaining properties are more difficult to establish; in Sqn(z) z2. The remaining properties are more difficult to establishj in particular it is not clear from the above construction how the axiomatparticular it is not clear from the above construction how the axiomatically simple property = z, Sq” == 1, 1, follows. follows. On the the other other ically simple property Sq’(z) SqO(z) = z, i.e. Le. SqO On hand this property follows easily from Serre’s classification of the stable Z/2hand this property follows easily from Serre's classification of the stable 'lL/2cohomology operations mentioned above (whereby the group of stable operacohomology operations mentioned above (whereby the group of stable operations of degree is isomorphic Hj+,(K(Z/2, n); Z/2)). For, For, the tions of degree jj is isomorphic to to Hj+n(K('lL/2, n); 'lL/2)). the cohomology cohomology of the Eilenberg-MacLane space K(Z/2, n) is trivial in dimensions < n, n, and and of the Eilenberg-MacLane space K('lL/2, n) is trivial in dimensions < lP(K(Z/2, n); Z/2) ” Z/2, so that there is only one cohomology operation Hn(K('lL/2, n); 'lL/2) ~ 'lL/2, so that there is only one cohomology operation of the scalar scalar 11 EE Z/2. 'lL/2. of degree degree zero, zero, namely namely multiplication multiplication by by the The above construction of the Steenrod squares has consequences the The above construction of the Steenrod squares has consequences in in the purely algebraic theory of graded Hopf algebras. Let A be a graded algebra purely algebraic theory of graded Hopf algebras. Let A be a graded algebra over Z/p, with an identity element: over 'lL/p, with an identity element:

A=$A”,

A

= ?I>0 EBA n ,

A’=Z/p,

AO

'lL/p,

n~O

with

multiplication

with multiplication #:A@A+A,

4(a@b)=a.b,

An.Am~An+m.

We have the augmentation homomorphism E : A + Z/p = A’, where Ker E = We have the augmentation homomorphism e : A ---t 'lL/p = AO, where Ker e @+i - A”. The tensor product (over Z/p) EBn~l An. The tensor product (over 'lL/p)

A&alpA, A®z/pA,

EB

(A@A)j = @ A4 @A”,m (A®A)j = q+m=j Aq®A , q+m=j

is an algebra,

where

the product

is defined

by

is an algebra, where the product is defined by (u @b)(a’ @b’) = aa’ @bb’.

(a ® b)(a' ® b') = aa' ® bb'.

$6. Simplicial structure group group §6. Simplicial and and cell bundles bundles with with a structure

97 97

The algebra algebra A has a “twisting” homomorphism The A@ ®A A has "twisting" homomorphism CT:: ABA C! A ® A 4---- A@A, A ® A,

a(a@b) (-l)desadesbb@a, ® a. C!(a ® b) = (_l)degadegbb

Such if there homomorphism, Such an an algebra algebra A A is is called called aa Hopf Hop! algebra algebra if there is is aa diagonal diagonal homomorphism, i.e. a degree-zero algebra homomorphism of the form Le. a degree-zero algebra homomorphism of the form A*:A+A@A, .1 * : A ---- A ® A, A*(u) = aa ® @ 11 ++ 11 ® @aa ++ cai @a:, Ll*(a) = Laj ® a'j,

degal > > 0, degaj 0,

degay > 0. dega'j > O.

@ 11 and and 118® A elementwise to to Here = (u 1) .. (1 @ b) Here aa CXI ® bb = (a 18 ® 1) (1 ® b) and and AA ® A commute commute elementwise within multiplication by 3~1. A Hopf algebra A is said to be symmetric if the within multiplication by ±1. A Hopf algebra A is said to be symmetric if the diagonal A* commutes depending on on deg deg a) u) diagonal homomorphism homomorphism .1* commutes (up (up to to aa sign sign 3~ ± depending with the involution operator 0 : A @ A -+ ABA defined by a(a@b) = (b@a): with the involution operator C! : A ® A ---- A ® A defined by 0'( CL ® b) (b ® a): A*0 = &A*. It was observed by Milnor in the late 1950s that the Steenrod algebra A, It was observed by Milnor in the late 1950s that the Steenrod algebra Ap can be naturally endowed with the structure of a Hopf algebra by means of can be naturally endowed with the structure of a Hopf algebra by means of the formula for the action on a product of cocycles (see (6.10)). For p = 2, for the formula for the action on a product of cocycles (see (6.10)). For p = 2, for instance, one sets instance, one sets

L

A*(Sq%) = Sq’ @ 1 + 18 Sq’ + c Ll*(8qi) = Sqi ® 1 + 1 ® Sqi +

jfk=t

Sqi 18 Sqk. Sq) ® sl.

j+k=i

For any symmetric Hopf algebra A over Z/p, analogues of the Steenrod operaFor any symmetric Hopf algebra A over Zjp, analogues of the Steenrod operations can be defined on the corresponding cohomology algebra Exti*(Z/p, Z/p) tions can be defined on the corresponding cohomology algebra Ext~* (Zjp, Zjp) (defined below) : (defined below): St; : ExtT’n(Z/p, Z/p) --+ Exty+i’pn(Z/p, Z/p). 8t~ : Ext~,n(Zjp, Zjp) ____ Ext~+i,pn(Zjp, Zjp).

(6.14) (6.14)

For example, the operation St: coincides with a nontrivial operator cy* deterFor example, the operation Stg coincides with a nontrivial operator a* determined by the Frobenius-Adams operator? 7 o : x +-+ xpP on the algebra A. The a: x r-7 x on the algebra A. The mined by the operations St; Frobenius-Adams have more-or-less operator the same properties as the original Steenrod operations St~ have more-or-Iess the same properties as the original Steenrod operations, while St; = 0 only for i $?! 0,l mod (p-l), p > 2. Thus here there operations, while St~ := 0 only for i of=. 0,1 mod (p-l), p > 2. Thus here there are more nonzero operations St;, than was the case for the original Steenrod are more nonzero operations St~, than was the case for the original Steenrod operations. operations. These operations were discovered in the process of developing the algepro cess of developing the algeoperations were for discovered braicThese techniques needed calculatingin the stable homotopy groups using the the braic techniques needed for calculating stable homotopy groups using Adams spectral sequence (Novikov in the late 1950s Liulevicius in the early Adams spectral sequence (Novikov in the late 1950s, Liulevicius in the early 7Well-defined

when

A is a Hopf algebra

over

7Well-defined when A Îs a Hopf algebra over

z/p.

Zlp.

98

Chapter 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Chapter Homology

1960s). ‘,’ In the Adams spectral sequence sequence for spheres, spheres, the structure structure of the 1960s).8,9 Steenrod agebra agebra Ap A, and of its cohomology algebra algebra (6.15) (6.15)

Ext:t:(Zjp, Zjp)

role. In accordance accordance with with Adams' Adams’ method, the the computation computation of play aa crucial role. the algebra (6.15) uses an an "approximation" “approximation” of of the the Steenrod Steenrod algebra algebra Ap A, by by the algebra (6.15) uses finitely-generated symmetric symmetric Hopf Hopf algebras; algebras; here here the the above above analogues analogues of of the the finitely-generated Steenrod operations operations turn turn out to be very useful. useful. In In the the case case of of the the Steenrod Steenrod Steenrod out to be very algebra Ap with pp > > 2, 2, one one must must take take into into account account the the bigrading bigrading (Cartan (Cartan Ap with algebra type) of the algebra as as weU well a'l as its its approximating approximating finite finite Hopf Hopf algebras. algebras. type) of the this this algebra In applying applying the the Adams Adams spectral spectral sequence sequenceto the computation computation of of the the stable stable to the In homotopy groups of of any CW-complex K, K, one one considers considers the the cohomology cohomology alalhomotopy groups any CW-complex gebra H* (K; Z/p) == M M as as aa graded graded left left module module over over the the Steenrod Steenrod algebra. algebra. gebra H*(K;Zjp) Consider more generally any any left left module module M over aa graded graded Hopf Hopf algebra algebra A A M over Consider more generally (over Z/p). The cohomology algebra (over Zjp). The cohomology algebra Ext>* (M, Zjp) Z/p) Ext~*(M, of M may may be defined of the the module module M be defined mid-1950s). denote by A+ the the mid-1950s). We We denote by A+

(6.16) (6.16)

as follows (after (after as follows kernel @A”, kernel EBA n , of of

Cartan-Eilenberg Cartan-Eilenberg the augmentation the augmentation

in the the in homohomo-

?Ql

morphism and consider the complex f:, and consider the complex morphism E, M t

A+@M

t

A+@A+@M

c

. .c

A+@A+@

. @A+@M

c

. . . , (6.17)

where an the form form an %-chain” "n-chain" has has the where g n = (al @a2 @ . @a, @LX)E A+ ~3 A+ %I.. ‘18 A+ 18 M, aj E An3,

nj > 0,

x E M.

The n-chain (or “simplex”) gn is taken to have dimension n, and is graded The n-chain (or "simplex") an is taken to have dimension n, and is graded also by its degree, defined by also by its degree, defined by sThese operations were introduced by Novikov in order to compute the cohomology of were by Novikov in order intoparticular compute that the cohomology the 8These Steenrod operations algebra -Ap andintroduced modules over J$,. It emerged for any prime of Steenrod modules It emerged in particular any prime pthe there exists algebra a nontrivialA p and torsion elementoverz Ap. of the ring of stable homotopy thatof for spheres such p there a nontrivial torsion of the stableusing homotopy of spheres such that xp exists # 0 (1959). This result waselement proved xalso by ring Toda of(1960) a different approach. P that result wastheorem: pl'Oved also (1960) using Nishidax =fhas0 (1959). proved This the following For byanyToda nontrivial torsion a different element approach. 2 of the Nishida has following For that any xn nontrivial torsion of the homotopy ring proved xi of the spheres there theorem: exists n such = 0. Hence the element ring K: isx locally homotopy ring 1f! of spheres there exists n such that xn O. Hence the ring 1f~ is locally nilpotent although not nilpotent in the strong sense. In the late 1960s P. May published an nilpotent although not nilpotentmore in the strongcategorical sense. In the late 1960s May published article establishing (in slightly general language) all P. foundational resultsan article establishing (in slightly more general categoricallanguage) ail foundational results needed for these operations. needed for these Note: operations. g Ill-anslator’s A general Nilpotence Theorem has been proved for any spectra (HopA general has been proved spectra (Hopkins,9 Translator's Devinatz, andNote: Smith, in the Nilpotence mid-1980s). Theorem The simplest version of for the any Nilpotence Thekins, Devinatz, in the mid-1980s). The simplest version of the Nilpotence Theorem states that and if a Smith, “ring spectrum” X (or “stable space” with a multiplicative structure, orem asstates that if a "ring spectrum" X (orspaces) "stablehas space" with a multiplicative such the above-mentioned stable Thorn torsion-free cohomology H*structure, (X; Z), such every as theelement above-mentioned Thom spaces) has torsion-free then of Tor xi(X) stable is nilpotent. (For details and further cohomology developments, H* (Xi see Z), D. then everyNilpotence element ofand TorPeriodicity 1f! (X) is nilpotent. details and further developments, see D. Ravenel, in Stable (For Homotopy Theory, Annals of Mathematics Ravenel,no.Nilpotence and Periodicity in Stable Homotopy Theor'y, Annals of Mathematics Studies 128, Princeton, 1992.)

Studies no. 128, Princeton, 1992.)

$6. structure group §6. Simplicial Simplicial and and cell cel! bundles bundles with with a structure group

99

n

degon&+n. deg an = L + nj

n.

j=l j=l

The is defined by The boundary boundary homomorphism is n-1 n-l

Dan

aan

==

[email protected] ....~uaj~aj+l~....~ua,~~)+(-l)~(a~~...~ua,~~~ua,.z), L(al 0aj' aj+10 ... 0a n 0x) + (-lt(a10 ... 0a n-10a n · x),

C( j=l j=1

and the degree degree of of chains. homology of (6.17) with with and preserves preserves the chains. The The homology of the the complex complex (6.17) respect the above furnishes the the desired cohomology algebra algebra respect to to the above differential differential furnishes desired cohomology (6.16). The The particular Z/p = H*(S’) yields the cohomology algebra algebra (6.16). particular case case M M = Z/p H*(SO) yields the cohomology

mentioned If the homomorphism mentioned above. above. If the module module M M possesses possesses a a diagonal diagonal homomorphism A*:M+M@M ,d*: M compatible with diagonal of A,,p , then algebra compatible with the the diagonal of A then the the cohomology cohomology algebra Ext;F(M, Z/p) acquires the structure of a bigraded Z/p-algebra. Ext::t;(.)\'1,Z/p) acquires the structure of a bigraded Z/p-algebra. One construct the the Adams of groups groups (actually (actually One can can now now construct Adams spectral spectral sequence sequence of Z/p-vector spaces) E, = c E$, and differentials (homomorphisms) E~l, and differentials (homomorphisms) Z/p-vector spaces) Em = q,l q,l

L

----4 Eq+m,l+m-l d'm'. Eq,l mm'

d2

m

= 0,

such that such that (i) Ez>l % Ez&(M,Z/p), = H(E,,d,); (i) E~,l 2:! Ext:.lp (M, Z/p), Em+1 E m +1 H(Em , dm);

L

(ii) c E$,’ ” Gn,“(K), where Grr is the “adjoined group”. (ii) E~l 2:! G7r;(K), where G7r is the "adjoined group". q-1=t q-l=t

,

Here 7$(K) = 7r,+t (PK), where YT~(PK) = 0 for j < n, 0 < t 5 n Here 7rt(K) = 7rn+t(En K), where 7rj(EnK) = 0 for j < n, 0 :S t :S n The adjoined group GT (relative to a filtration TT = T(O) > r(l) > . .) is, The adjoined group G7r (relative to a filtration 7r 7r(O)::J 7r(1) ::J ... ) is, for rings etc., the direct sum of the successivefactors: for rings etc" the direct sum of the successive factors: (q&+1) = G71.=@ Gj7T. G7r = EB7r(j) /7r(j+l) = CD EBG 7r. jl0 QO j j~O

1.

1. as

as

j~O

The intersection of all members of the filtration arising from the Adams specThe sequence intersection members of is theprecisely filtration the Adams spectral (forofa aIl given prime p) thearising set of from elements of the stable tral sequence (for a given prime p) is precisely the set of elements of the stable homotopy groups of K of finite order relatively prime to p. finiteinorder relatively prime ta p. cases) the Adams homotopy groups In the case K of = K5’”of(and several other important In the case K = sn (and in several other important cases) the Adams spectral sequencehas as its terms bigraded Zlpalgebras E, with differentials with differentials spectral sequence termsf bigraded d, satisfying d,(w)has as = its d,(u)w udm(u) Z/p-algebras for bigraded Em elements U,V E E,, dm satisfying dm (uv) dm (u)v ± ud m (v) for bigraded elements u, v E Em,

100 100

Chapter Chapter 3.3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Homology

and and the the algebra algebra E, Eco is adjoined adjoined to to the the ring ring of of stable stable homotopy homotopy groups groups of of the the sphere 5’“: sn: 7rz = @4> 7T; = 7rn+k(P), n > k + 1, k>O

with with the the multiplication multiplication induced by by the the composition composition Sn+‘+J sn+i+j -% Snfi sn+i -% ~ S”, sn, which, via the realization of the stable homotopy groups of spheres in terms which, via the realization of the stable homotopy groups of spheres in terms of the “cobordism "cobordism ring ring of of framed framed manifolds” manifolds" (see (see Chapter Chapter 4, 4, §3), §3), corresponds corresponds of the to the multiplication induced by the direct product of manifolds. to the multiplication induced by the direct product of manifolds. In In the the case case of of the the sphere sphere KK == S”, sn, the the second second term term of of the the Adams Adams spectral spectral sequence is the algebra sequence is the algebra

where where Z/p Z/p has has the the A,-module Ap-module structure structure determined determined by by the the identical identical action action of of 1 E d,, and the annihilating action of the elements of d, of positive 1 E A p , and the annihilating action of the elements of A p of positive degree. degree. Here Here there there is is an an element element ho E E;‘l

= Ext>; (Z/p,

Z/p),

with with the the property property that that multiplication multiplication by by he ho is is adjoint adjoint to to multiplication multiplication by by p. p. Example 1. From the definition of the cohomology of an A,-module A4 (see Example 1. From the definition of the cohomology of an Ap-module M (see (6.17)) it follows that (6.17)) it follows that E$k

= Extc$M,

= HomiP (M, Z/p),

Z/p)

(6.18)

(6.18)

where only those homomorphisms are admitted that commute with the action where only those homomorphisms are admitted that commute with the action of A,. The action of d, on Z/p is given by of Ap- The action of A p on Z/p is given by a(u)=0

if

dega>O,

a(u)=O if dega>O, X(u) = Au À(u)

=

Àu

if a= X E Z/p, if a = À E Z/p,

where u E Z/p. The superscript k in Horn5 (M, Z/p) signifies that we only where u E Z/p. The superscript k in Hom~ p (M, Z/p) signifies that we only admit Z/plinear forms that are zero off M Tc (and of course commute with admit Z/p-linear forms that are zero off M k (and of course commute with the action of dP). When M = H*(K; Z/p) the zero-line E$* has a clear the action of A p ). When M = H*(K; Z/p) the zero-line Eg,* has a clear topological meaning: any map Sk 2 K induces a homomorphism of dPtopological meaning: any map Sk L K induces a homomorphism of Apmodules M = H*(K; Z/p) Li"* Z/p, and, consequently determines an element modules M =(M,H*(K; ---> Z/p, and, consequently determines an element f* E Homjhp Z/p); Z/p) the non-vanishing of this element guarantees that the 1* E HomÀ p (M, Z/p); the non-vanishing of this element guarantees that the map f is essential (i.e. not homotopic to zero). However, an element II: E map f is essential (i.e. not homotopie to zero). However, an element x E HomLp CM, UP) is induced by some map 5’” + K only if all differentials in HomÀ (M, Z/p) is induced byact some map sn ---+ K only if aIl differentials in the Adams spectral sequence trivially on Z: l' the Adams spectral sequence act trivially on x: d,(x)

= 0,

m > 2.

$6. Simplicial Simplicial and and ceU cell bundles structure group §6. bundles with with aa structure group

101 101

Note are many maps Sk Sk zL K the trivial trivial homohomoNote that that there there are many essential essential maps K inducing inducing the morphism (K; Z/p) Z/p) 1: L * Z/p; Z/p; for instance none of of the nontrivial nontrivial elements elements H*(K; morphism H* of positive degree of the stable homotopy groups of spheres can be “detected” of positive degree of the stable homotopy groups of spheres can be "detected" by homomorphisms of 0 by the corresponding corresponding homomorphisms of the the cohomology ring. 0

Example As noted above, for for the sphere 5’” we have M = Z/p. Z/p. In In this this Example 2. 2. As noted above, the sphere sn we have M case Extiz = Z/p and Ext:: = 0 for Ic # 0. It is not difficult to identify the case Ext~Op = Z/p and ExtAO,kp = 0 for k cf:. O. It is not difficult to identify the groups We sketch sketch the assuming for for simplicity simplicity groups ExtfdE(Z/p, Ext~~(Z/p, Z/p). Z/p). We the argument, argument, assuming i that the algebra is generated generated by by the the elements elements Sq2 Sq2i, there are that pp = = 2. 2. Since Sinee the algebra AZ A2 is , there are non-zero elements non-zero elements 12 i (Z/2,2/2) hi E Ext Extf4f hi E (Z/2, Z/2)

A2

i dual in fact fact account account for of the groups dual to to the the Sq2’. Sq2 • These These in for all all non-zero non-zero elements elements of the groups Ext>:(Z/2,2/2). Note that d,hi = 0 for q 2 2 if and only if there exists an Ext~: (71../2,71../2). Note that dqh i 0 for q 2: 2 if and only if there exists an element invariant 11 in stable homotopy homotopy group group of of element with with Hopf Hopf invariant in the the appropriate appropriate stable spheres (see Chapter 4, §3). Adams proved in the late 1950s that spheres (see Chapter 4, §3). Adams proved in the late 1950s that

d2 (h i ) = hohLl, where hohf-, # 0 for i 2 4. It is not difficult to establish Adams’ formula in where hohLl cf:. 0 for i 2: 4. It is not difficult to establish Adams' formula in the case i = 4: d2(hd) = heh$. Since d, (hd) = 0 for q > 3 for “dimensional reathe case i = 4: d2 (h 4) = hoh~. Since dq (h4) = 0 for q 2: 3 for "dimensional reasons” (there is no nontrivial element to be a target), the element h3 represents sons" (there is no nontrivial element to be a target), the element h3 represents a non-zero element ‘ils = (T of Hopf invariant 1 in the group $ = 7rn+~(Sn), a non-zero element h3 = (J' of Hopf invariant 1--in the group 7r~ 7r n +7(sn), -- -2 = 0. Since n > 8. By skew-symmetry we have -h3h3 = -hshs, whence 2x: n 2: 8. By skew-symmetry we have h3h3 -h3h3, whence 2h3 O. Since multiplication by ho is adjoint to multiplication by 2, the element hohg must multiplication by ho is adjoint to multiplication by 2, the clement hoh~ must be trivial in the infinite term E, of the Adams spectral sequence, whence be trivial in the infinite term Eoo of the Adams spectral sequence, whence

hoh$ = dz hoh~ dz and since hq is the and since h4 is the lishes the formula. lishes the formula. the products

the products

where hihi+ = where h i hi+l = are of particular are of particular chapter. lo chapter. lO

for some z E Ext>i6(Z/2,

for sorne

Z/2),

z E Ext~!6(71../2, 71../2),

only nonzero element, we must have z = hq, which estabonly nonzero element, we must have z = h 4, which estabWe note that the 2-line Ext:z(Z/2,Z/2) is generated by We note that the 2-line Ext~: (71../2,71../2) is generated by

hihj E Extf4t’+2” (Z/2,2/2),

(6.19) (6.19) 0 and all other products are nontrivial. The products h; 0 and aU other products are nontrivial. The products h; importance in manifold theory, to be considered in the next importance in manifold theory, to be considered in the next 0 0

lo ‘71-anslator’s Note: The elements h? (which are related to the “Arf invariant problem”) Translator's The elements h 2 (which the "Arf invariant are 10 the only doubleNote:products hihj about whichare itrelated is not toknown whether they problem") support are the only double products hihj that about they support nontrivial differentials. It is known the which elementsit is h:notareknown cycles whether of all differentials for that and the M.elements are cycles of ail jnontrivial = 1,2,3,4,5 differentials. (M. Barrat, It is J. known D. S. Jones Mahowald, mid-1970s). The differentials elements hlh,for jfor = j 1,2,3,4,5 (M. Barrat, S. Jones and4M. mid-1970s). 1978). The elements hlhj 2 4 are nontrivial, and J.ofD.order at least for Mahowald, j 2 5 (M. Mahowald, For a general for j :::: 4ofarethenontrivial, of order at least j :::: 5spectral (M. Mahowald, For a general account results ofand computations in the4 for Adams sequence 1978). for spheres see D. account ofComplex the results of computations the Adamsof spectral spheres see D. Ravenel, Cobordism and Stable inHomotopy Spheres, sequence Academic forPress, 1986.

hJ

Ravenel, Complex Cobordism and Stable Homotopy of Spheres, Academie Press, 1986.

102 102

Chapter Simplicial Complexes Complexes and and CW-complexes. CT&‘-complexes. Homology Homology Chapter 3. Simplicial

Applications spectral sequence of bordism bordism Applications of of the the Adams Adams spectral sequence to to the the computation computation of and cobordism rings of smooth manifolds were made around 1960 1960 by Milnor Milnor and Novikov. The Adams spectral sequence admits a natural generalization Novikov. spectral sequence a natural generalization in terms of extraordinary extraordinary (generalized) cohomology theory (Novikov, in the late theory (Novikov, 1960s) revealing a deep connexion with stable homotopy 1960s) revealing a deep connexion with homotopy theory, theory, while the Cartan-Serre method of computing stable (and unstable) homotopy Cartan-Serre computing homotopy groups does not admit such a generalization. does not admit such a generalization. Over the rationals sequence converges converges to the stable Over rationals the Adams spectral sequence n rational r;(K) Q9 @Q = (CnK) enough n), rational homotopy homotopy groups 7rj(K) = 7r7r,+j K) (for large enough n +j(E reducing yielding the earlier result of Serre Serre (from (from the reducing to a trivial trivial procedure yielding early 1950s) that that the Hurewicz Hurewicz homomorphism is in fact an isomorphism: early 1950s)

7qK) @Q 5 Hj(K;Q).

(6.20)

As mentioned the algebra dQ A Q of operations of rational rational cohomology mentioned before, the theory theory consists consists only only of scalar multiplications. multiplications. As in the of aa prime prime pp application application of spectral As already already noted, noted, in the case case of of the the Adams Adams spectral sequence investigation of sequence to to particular particular problems of homotopy homotopy theory theory entails investigation various over Z/p: Z/p: Hopf algebras algebras over various Hopf B=@By B = EBB n ,

O BO=Z/p, B = Z/p,

7220 n;:::O

with A :: B (seeabove). of such such with a a symmetric symmetric diagonal diagonal Ll B --) ---. B B @B Q9 B (see ab ove ). The The category category of algebras (where the morphisms preserve grading and the Hopf structure), algebras (where the morphisms preserve grading and the Hopf structure), together Hopf algebras, algebras, constitutes constitutes an an inintogether with with the the category category of of modules modules over over Hopf teresting algebraic model” which imitates certain properties of the homotopy teresting algebraic modepl which imitates certain properties of the homotopy category of For instance instance aa fiber with simply-connected category of CW-complexes. CW-complexes. For fiber bundle bundle with simply-connected base is modelled algebraically by an appropriate epimorphism + B B of base is modelled algebraically by an appropriate epimorphism ff :: A A ---. of Hopf algebras (where A is the “total space”, B the “base”) with the following Hopf algebras (where A is the "total space", B the "base") with the following special properties: there should exist exist aa Hopf subalgebra C C C c A “fiber”): there should Hopf subalgebra A (the (the "fiber"): special properties:

with ideal with augmentation augmentation ideal C+=@CncA+=@An, 7Ql

Ql

such that such that (i) is a (ac = ca for all cc E E A), (i) C C is a central central subalgebra subalgebra of of A A (ac = ca for aB E C, C, aa E A), and and Ker Ker ff has the form Ker f = AC+ = @A; has the form Ker f = AC+ = C+ A; ‘lIn particular, in these categories cohomology algebras H*,*(A; R) are defined for a Hopf 11 In particular, in these categories cohomology algebras HO,o (A; R) are defined for a Hopf algebra A and an A-module R. algebra A and an A-module R.

$7. The The dassical classical apparatus apparatus of theory §7. of homotopy homotopy theory

103 103

(ii) free left C-module, i.e. of elements elements (ii) the the algebra algebra A A is a free left C-module, Le. there there is a basis basis of ai 1,2) ... . .,0 ) 0 = no < < nI nr ::; < n2 ::; < n3 ns ::; 5 ... . . .,, with ae = 1, snch such that ai E E A”*, Ani, i = 0, 0,1,2, with ao that each of A can be uniquely expressed each element element aa of A can be uniquely expressed in the the form form

aa=

wi, cLCiai, i>O i~O

ci E C. c. Ci E

Under the above conditions there is aa spectral spectral sequence sequence (a (a special case of of the the Under the above conditions there is special case Serre-Hochschild spectral sequence sequence discovered discovered in the mid of aa "fiber “fiber Serre-Hochschild spectral in the mid 1950s) 1950s) of bundle” + B with "fiber" “fiber” C C which which imitates imitates certain certain properties properties of of the the B with bundle" ff :: AA ---t Leray sequence for CW-complexes (see §7). $7). This This Leray spectral spectral sequence for aa fiber fiber bundle bundle of of CW-complexes (see takes the sequence of of algebras algebras takes the form form of of aa tri-graded tri-graded spectral spectral sequence q,l,r E m'

d

. Eq,l,r

m·

---t

m

Eq+m,l-m+l,r 111

d2 0 '1n'

where: where: 1. EL:;

= H*(E;*>*,d,),

2. E;>l,r =

EB

3. @ 3.

d,(w)

= d,(u)v

zt udm(u);

L

L

rI +r2=r

rl+r2=r

c Hq>‘l (B; Hl,rz(c)) H1l”(C)) = c Hql”’ Hq,r l (Bi Hq,r l (B) (B) cgzlp ®Zjp H1)‘2(C); H I ,r 2 (C)i ?-I+r* =r Tl+T*=T

Ek’>’ E~l,r ” GHnlT(A), GHn,r(A),

q+l=n

q+l=n

where (A) denotes the adjoint certain filtration filtration where GH”>’ GHn,r(A) denotes the adjoint ring ring with with respect respect to to aa certain generating the above spectral sequence. The gradings indicated by the indices generating the above spectral sequence. The gradings indicated by the indices ~1, 7-2, ~3 correspond to the gradings of A, B, C respectively. rI, rz, r3 correspond to the gradings of A, B, C respectively. The analogues of the Steenrod operations St: (the Steenrod squares S$ for The analogues of the Steenrod operations St~ (the Steenrod squares Sqi for p = 2) mentioned above (see (6.14)) are related to the above spectral sequence p = 2) mentioned above (see (6.14)) are related to the above spectral sequence in much the same way that the ordinary Steenrod operations are related to in much the same way that the ordinary Steenrod operations are related to the Leray spectral sequence for fiber bundles, which we shall consider in the the Leray spectral sequence for fiber bundles, which we shaH consider in the next section. next section.

$7. The classical apparatus of homotopy theory. The Leray §7. The classical apparat us of homotopy theory. The Leray spectral sequence. The homology theory of fiber bundles. The spectral sequence. The homology theory of fiber bundles. The Cartan-Serre method. The Postnikov tower. The Adams spectral Cartan-Serre method. The Postnikov tower. The Adams spectral sequence

sequence

The method of “spectral sequences”, first discovered by Leray in the midTheinmethod of "spectral discovered Leray in the 1940s connexion with mapssequences", of spaces, first and so applying by in particular to midfiber 1940s in connexion with maps of spaces, and so applying in particular to fiber bundles, is of fundamental importance as an effective tool in homological algeis ofpossible fundamental as anthe effective tool in homological algebundles, bra, making (amongimportance other things) computation of the homology br a, making things) the computation the homology groups of an possible extensive (among class ofother spaces by means which avoidofdirect detailed groups of an extensive dass of spaces by means which avoid direct detailed examination of their geometrical structure. examination of their geometrical structure.

104 104

Chapter Complexes and Homology Chapter 3. Simplicial Simplicial Complexes and CW-complexes. CW-complexes. Homology

Let map of For each each j ::::: > 0 the the map map p p Let pp : Y + XX be be a map of topological topological spaces. spaces. For determines a presheaf 3j :Tj on Y given by 3; = I.@(p-l(U);

II),

j 2 0,

(7.1) (7.1)

on each each open set U of X X (and for any fixed abelian group D). D). Hence Rence there arise in turn turn the cohomology groups E;J = W(X;

39

(defined to for qq << 00 or or 1l < < 0). theorem asserts asserts the the existence existence to be be zero zero for 0). Leray’s Leray's theorem (defined of aa spectral spectral sequence converging to Hm(Y; D): D): of sequence converging to Hm(y;

(7.2) where where

Ed

m+1

= H*(E$,d,)

N Ker d,/Im

d,,

Eg=E$’= Ei;/ for for m>q+l, Eg:} m > q + l,

such that such that c

Ezl ” GH”(Y,D)

m

= &““-‘,B”“-‘-‘.

q+l=m

(7.3)

I=0 1=0

q+l=m

Here GHm(Y, D) D) is to EP(Y, D) by by means of aa certain certain Rere GHm(y, is the the group group adjoined adjoined to Hm(y, D) means of filtration filtration

For certain indices q, 1 the group E$ is easily identified; for instance Ezm = For certain indices q, l the group Eg:,t is easily identified; for instance E~m = B” c Hm(Y; D) is image under projection homomorphism homomorphism p*: p*: B O c Hm(y; D) is just just the the image under the the projection Eo,” o.

==B” =Imp*,

p* : H”(X;

D) +

H”(Y;

D).

(7.4)

For a locally trivial fibration Y 5 X with fiber F, the above Leray spectral For a locally trivial fibration Y X with fiber F, the above Leray spectral may be described as follows. Let us assume the base X is a CWseq7œnce may be described as follows. Let us assume the base X i8 a CWcomplex with n-skeleton Xn, n = 0, 1,2,. . The filtration of H”(Y; D) is complex with n-skeleton X n , n = 0,1,2, .... The filtration of Hm(y; D) is determined by the following natural filtration of the space Y: determined by the following natural filtration of the space Y: sequence

Y, = p-yxy,

Y, J% Y,

Yo c Yl c Y2 c . . . c Y, c . . . ) Y = Y, .

YO C YI C Y2 C ... C Yn C . .. , Y

=Y

00 •

From the inclusion map $6k : Yk + Y, we obtain for each m the induced From the inclusion map rPk : Yk ---+ Y, we obtain for each m the induced homomorphism 4; : Hm(Y) -+ Hm(Yk); clearly 4; is an isomorphism for rP'k : Hm(y) ---+ Hm(yÙ clearly rP'k is an isomorphism for homomorphism k = m. The desired filtration k = m. The desired filtration

57. §7. The The classical classical apparatus apparatus ofof homotopy homotopy theory theory

105 105

of Hm(Y; Hm(y; D), D), isis obtained obtained by by setting setting of B” Bk == Ker Ker {4&k {cfJ:n-k :Hm(Y;D)+Hm(Ym-k;D)}, : Hm(y; D) ---> Hm(ym_k; D)},

=

k=m,m-l,...,O. k m, m

- 1, ... , a.

IfIf in in addition addition the the base base XX isis connected connected and and has has just just one one vertex vertex (O-cell) (a-ceU) -which then the the quotient quotient which may may without without loss loss of of generality generality always always be be assumed assumed - then B”/B”-’ Bm / Bm-l in in (7.3) (7.3) isis isomorphic isomorphic to to the the image image of of Hm(Y;D) Hm(y; D) in in Hm(F;D) Hm(F; D) under under the the homomorphism homomorphism induced induced from from the the inclusion inclusion of of the the fiber fiber FF in in the the total total space space Y: Y: i :F + Im

{i’

:

Hm(Y;

D) -

Y,

Hm(F;D)}

F = P-~(Q), ” B”/B”-l

c Hm(F;D).

(7.5) (7.5)

Furthermore in Furthermore for for locally locally trivial trivial fiber fiber bundles bundles (and (and also also for for Serre Serre fibrations fibrations - in fact fact for for any any fibration fibration with with the the covering covering homotopy homotopy property) property) Leray’s Leray's theorem theorem gives the explicit explicit form form of of the the term term Ez’l: E:]"l: gives the Eq31 = H4(X. 1(H’(F.D) 7 2

7p)).

(7.6) (7.6)

Here group ~1 on H’(F; D) is Here aa representation representation pp of of the the fundamental fundamental group 7r1 (X) (X) on H1(F; D) is dedetermined by the action of ri(X) on the fiber F. If ri(X) acts trivially (in termined by the action of 7rl(X) on the fiber F. If 7rdX) acts trivially (in particular if ri(X) = 0), then it follows that Ez’l = H”(X; Hi(F)), i.e. the particular if 7rl(X) = a), then it follows that E:],,l = Hq(X; Hl(F)), Le. the term the cohomology cohomology of of the the direct product X X xx F: F: term Eill E:]"l coincides coincides with with the direct product c E;” x F; F; II). L E:]"l ”~ H*(X H*(X x D). q,l

(7.7) (7.7)

q,l

For bundle Y Y = = X Xx x F product) we we have have that that E:]"l Ei’l == E~}, Eg’, For the the trivial trivial bundle F (the (the direct direct product) i.e. the differentials d, are all zero for m 2 2. Le. the differentials dm are aU zero for m ? 2. However for a non-trivial fiber bundle (or “skew product”) with simplyHowever for a non-trivial fiber bundle (or "skew product") with simplyconnected base, the cohomology of the total space becomes different from connected base, the cohomology of the total space becomes different from H*(X x F; D) because of the existence of non-zero differentials d,, m > H*(X x F; D) because of the existence of non-zero differentials dm, m ? 2. Thus for a skew product there is “less” cohomology than for the direct 2. Thus for a skew product there is "less" cohomology than for the direct product. This is not surprising, for the following reasons. Assuming that the product. This is not surprising, for the foUowing reasons. Assuming that the fiber F, as well as the base X, is a CW-complex, the cell decomposition of fiber F, as weU as the base X, is a CW -complex, the ecU decomposition of the total space is the same as that of the direct product X x F since above the total space is the same as that of the direct product X x F since above each cell grn of K the bundle is trivial: each ceU am of K the bundle is trivial: p-‘(P) m 2 CP x F. p-l(a ) ~ am x F. The “skewing” in the cohomology groups H*(Y; D) of the total space Y apThe "skewing" in the cohomology groups H*(Y; D) of the total space Y appears when we apply the boundary operator d: pears when we apply the boundary operator

a:

aa~+l a(a1 x a~,) a q x (aa~) + (al + a2 + .. .)a~+l, where ajo;‘;‘” I lies above the (q -j)-skeleton Xq-i of the base X. The operator where lies above the (q (asj)-skeleton baseofX. operator ai can ajaV be explicitly calculated shown by xqLeray)j ofinthe terms theThe cohomology al can be explicitly calculated (as shown by Leray) in ter ms of the cohomology of the base with coefficients from the nr(X)-module H*(F). of the base with coefficients from the 7rl (X)-module H*(F).

106 106

Chapter 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Chapter Homology

IfIf DD isis aa ring differenring then then each each E2* E:r;* becomes becomes aa bigraded ring and the differential d, satisfies d,(zl)v f ud U&(V) elements ‘~L,z, satisfies d,(uv) dm (uv) = = dm (u)v ± (v) for bigraded elements u, v (and tial dm m the resulting resulting ring is incorporated Leray’s theorem). theorem). There is is aa ring structure structure is incorporated into into Leray's completely analogous homological version of the theorem, and over aa field the completely analogous cohomology sequencesare are mutually mutually dual. cohomology and homology homology spectral sequences The transgression homomorphism T in the Leray spectral sequence sequence is is dedeThe transgression homomorphism T in the Leray spectral fined as follows: consider the pair (Y, F) and the homomorphisms fined as follows: consider the pair (Y, F) and the homomorphisms H4(F) L

H9f1(X,

Hq+‘(Y, F)

* x0) % Hq+l(Y,

(the coboundary coboundary homomorphism), (the homomorphism), homomorphism). ((the the projection projection homomorphism).

F)

The composition (p*)06 = 7 is the the desired desired transgression. that T7 need need The composition (p*)-l’ 08 T is transgression. Note Note that not everywhere defined on Hq(F), may be be many-valued. An clement element Hq(F), and and may many-valued. An not be be everywhere defined on zZ of of Hq(F) if T(Z) ~(2) is defined. In In terms terms of of the the Leray Leray Hq(F) is is called called transgressive transgressive if is defined. spectral sequence of the fiber bundle the transgression T may be equivalently spectral sequence of the fiber bundle the transgression T may be equivalently defined = d,, dq , where where defined as as rT = Hq(F)

> E4”lq 2

Ei+llo

= Hqf’(X)/Ker

p*.

(7.8) (7.8)

It follows that for for aa transgressive transgressive element E Hq(F), H’J(F), we have have It follows from from (7.8) (7.8) that element Zz E we dj(z) = 0 for j < q. dj(z) = 0 for j < q. In the case all cohomology are modules modules over over the algebra Ap, dP, In the case DD = = Z/p, Z/p, aIl cohomology rings rings are the algebra and it turns out that the differentials in the Leray spectral sequencecommute and it turns out that the differentials· in the Leray spectral sequence commute with the Steenrod squares and powers; indeed, since 6 and p* commute with with the Steenrod squares and powers; indeed, since {) and p* commute with the Steenrod squares and powers, it follows that the differential d, does also. the Steenrod squares and powers, it follows that the differential dq do es also. This of crucial crucial importance importance in certain difficult difficult computations. computations. In In particular, particular, is of in certain This is if z E H’J(F) is transgressive then so is St;(z), and if Z E Hq (F) is transgressive then so is St~ (z), and L%;(Z) = St;+). TSt~(Z)

St~T(Z).

Example 1. Borel’s theorem (early 1950s): If H* (F; Q) is an exterior algebra Example 1. Borel's theorem (early 1950s): If H*(F; IQ) is an exterior algebra over Q and H*(Y; Q) = 0, then H*(X;Q) is the polynomial algebra in the over IQ and H* (Y; IQ) = 0, then H* (X; IQ) is the polynomial algebra in the images under transgression of certain exterior generators. Over the field Z/2, images under transgression of certain exterior generators. Over the field Z/2, the hypothesis that H*(F; Z/2) b e an exterior algebra is replaced by the the hypothesis that H* (F; Z/2) be an exterior algebra is replaced by the requirement that there exist a finite collection of transgressive generators wj E requirement that there exist a finite collection of transgressive generators Vj E Hni (F; Z/2) forming an additive basis for H*(F; Z/2): Hn.i (Fi Z/2) forming an additive basis for H*(Fi Z/2): ~j,,‘ujZ,~“,VjUjk>

jl

<j,

< . . . <j,.

The same conclusion follows: the ring H*(X; Z/2) is polynomial. By means The same conclusion follows: the ring H*(X; Z/2) is polynomial. By meand ans of this theorem the cohomology over Z/2 of a space of type K(Z/2,n), the cohomology over Z/2 of a space of type K(Z/2, n), and of this theorem 0 the Steenrod algebra d2 were computed (Serre, early 1950s). the Steenrod algebra A2 were computed (Serre, early 1950s). 0 Example 2. Using the Leray spectral sequence the following cohomology Example Using the Leray difficulty: spectral sequence the following cohomology rings can be 2.calculated without rings can be calculated without difficulty:

$7. The apparatus of of homotopy theory §7. The classical classical apparatus homotopy theory H*('Rpn;Z/2); H* (RP; Z/2);

H*('Rpn;z); H*(RP”; Z);

H*(SOO /Z/m; Z); H*(LF/Z/m; Z);

H*(cpn;z); H* (CP”; Z);

H*(SOO /Z/m; Z/m); H*(S”/Z/m; Z/m);

H*(SOniQ); H*(SO,; Cl!); H*(G; H*(BSO n ; Q); H*(BSO,;

107 107

4;

H*(BUn,·Z)· H*(BU,; Z); ,

Z H*(Spni H*(Sx 4;);

H*(BSO,; n ; Z/2); Z/2); H*(BSO

H*(BO,; n ; Z/2). H*(BO Z/2).

The calculations calculations are carried carried out using following fibrations: using the following fibrations: 1. 1.

s2n+ l 4 --+ cpn S2n+1 CT”

(fiber SI); (fiber S’);

2.

'Rp2n+l Iw2n+1 4--+ cpn CP”

(fiber SI); (fiber S’);

3. 3.

S2n+l/Z/m --+ cpn S2”f1/Z/m -+ CT’”

(fiber S’); (fiber SI);

sn-l --+ 5-l 4. 4. SOn so, --+

(fiber SOn-l); (fiber SO,-1);

5. 5. Un u,

(fiber U,-1); (fiber Un-di

s2n-l --f 9-l

6. 6.

E --+ --+ BO, BOn

(fiber 0,), E E rv N *; (fiber On),

7. 7.

E --+ + Bun E BUn

(fiber V,), E ""' N *. (fiber Un),

00

Example 3. 3. As aa trivial trivial case caseof Borel's Borel’s theorem above, one one has has for the space space K(Z,n): xX == K(Z, 72): Hn(x) , ifif n is A(u) > , u EE H”(X) is odd, A(u) H*(X; Q) is even. Hn(X) , ifif n is H*(X’Q) = {1 Q[u] , u E E Hn(X)

For finite finite groups 7r 7r the ring H*(K(n,n);Q) is zero. For finitely finitely the cohomology cohornology ring H* (K (7r, n); Q) is generated abelian groups 7r 7r the cohomology ring H*(K(n, H*(K(7r, n); R) over any coefficient coefficient ring ring R is finitely finitely generated. 0cl convert, by means construction In Chapter Chapter 1 we described how to convert, means of a a construction preserving into a Serre Serre fibration fibration preserving homotopy homotopy types, any map fJ : X -+ --+ Y into p:x+Y, p: X --+

L-X) X, Y F-Y, Y, X""' ""' Y,

p-f, p ""'

J,

where Y is Y, X X is is the space space of is the mapping cylinder cylinder of f,J, contractible contractible to Y, paths in Y with X x {O} (0) C c Y) and ending Y beginning beginning at points of of X (identified (identified with point,s of Y, and p sends sends each each such path to its end-point end-point in Y. Y. The space space of Y, such path in points X is contractible X contractible to to X. X. Cartan-Serre method of computing We now describe the Cartan-Serre computing the homotopy homotopy R-~(X) of aa simply-connected space X. Suppose 7rj(X) rj(X) = 0 for j < < nn groups 7rj(X) simply-connected space X. Suppose

108 108

Chapter 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Chapter Homology

K(x,(X), is then aa map map J f : X --+ --+ Y, for which the induced induced and Y = = K(7r n (X), n). There is f* : 7r 7rn(X) r,(Y) is an isomorphism. Applying homomorphism J* (X) --+ 7r (Y) is an isomorphism. Applying the above n n construction ta to the map J, f, we we obtain obtain aa fiber bundle (Serre fibration): fibration): construction p:x +

P w K(7r,(X),

n).

It follows that fibration pp :: X X ---t + Y Y has has the same samehomotopy homotopy It that the fiber F of the fibration groups as as X in dimensions dimensions above n: n: rj(F)

2 nj+l(X)

for j > n,

xj(F)

= 0

for jj < < n. n. for

(7.9) Hence computing (F) is is as as good good as as computing 7r* (X). In In its its simplest simplest form form Hence computing 7r* 7r* (F) computing 7r * (X). the Cartan-Serre method operates as follows: the Cartan--Serre method operates as follows: Step 1. 1. The The homology homology groups groups H*(F) H,(F) of fiber Fare F are calculated calculated by by me means Step of the the fiber ans of the homological Leray spectral sequence applied to the fibration of the homological Leray spectral sequence applied to the fibration p:X+Y. p:X Y. Step 2. Hurewicz’ theorem is invoked invoked to to compute compute the Step 2. Hurewicz' theorem is the group group 7rr,+i(X): n+l(X): H,(F)

2 nn(F) ” T,+~(X).

Step 3. The entire is repeated. repeated. Step 3. The entire process pro cess is However is a more convenient procedure for for calculating calculating the homotopy a more convenient procedure the homotopy However there there is groups of a simply-connected space X. By attaching cells of dimensions > qq groups of a simply-connected space X. By attaching ceUs of dimensions ::::: to the space X so as to “kill” the homotopy groups of dimensions 2 q, one to the space X so as to "kill" the homotopy groups of dimensions? q, one obtains a space X, > X for which obtains a space X :::J X for which q

nj(X)

” r.j(X,)

for j < q, (7.10)

7r.j(X,) = 0

for j > q. for j::::: q.

(7.10)

Thus if X is such that r.j(X) = 0 for j < n, then TV = 0 for all j, so that Thus if X is such that 7rj(X) = 0 for j < n, then 7rj(Xn) = 0 for aIl j, sa that X, is contractible, while Xn+i is of type K(r,n), K = rrn(X). Consider the X n is contractible, while X n + 1 is of type K(7r, n), 7r 7rn (X). Consider the following sequenceof fibrations (obtained by conversion of the corresponding following sequence of fibrations (obtained by conversion of the corresponding inclusions Xl+1 c Xl into fibrations): inclusions Xl+! C Xl into fibrations): (n + 1) (n + 1)

EL+2 -

(n + 2)

xn+3 -

(n

+ 2)

(with fiber F,+l N K(7h+l(X), n + 1)); (with fiber Fn+! tv K(7r n+!(X), n + 1));

%,l %+2 .

.

.

(with fiber Fn+2 N K(s+2(X), n + 2)); (with fiber tv K(7r n+2(X), n + 2)); . . . Fn+2 .

(with fiber &+k - K(~+k(x), n + k)). (with fiber Fn+k '" K(7r n+k(X), n + k)). Now a fiber bundle with fiber of type K(r,m) and with simply-connected Nowa bundle with determined fiber of type m) cohomology and with simply-connected base B, fiber is homotopically by K(7r, a single class base B, is homotopically determined by a single cohomology class (n

+ k) (n+ k)

i+k+l

-

Xn+k

57. §7. The The classical classical apparatus apparatus of of homotopy homotopy theory theory

109 109

11, Um E E H”+yBJr). H m+1(B, 1l').

Thus Thus to to each of of the the above above fibrations fibrations &+k+i XnH+1 --f ----+ Xn+k Xn+ k with with fiber fiber &+k F n+ k K(n,+k, K(1l'n+k> nn ++ k), k), there there corresponds corresponds a particular particular cohomology cohomology class class

UnH EE Hn+k+l(&z+k> HnH+l(Xn+ k , %+k(x)). 1l'nH(X)). h+k The The cohomology cohomology classes classes u,+k Un+k are are known known as as the the Postnikov Postnikov invariants invariants of of the the space space X. X. The The sequence sequence of of fibrations fibrations with with fibers Fn+k FnH N rv K(?r,+k, K(1l'nH, n + + k): .

4

xn+3

-

x,+2

-

xn+1

-

*

*

(7.11) (7.11)

(where (where ** here here denotes denotes a a contractible contractible space) space) is is called called the the Postnikov Postnikov tower tower for for X. It follows that the collection of groups rj(X) and Postnikov invariants uj X. It follows that the collection of groups 1l'j(X) and Postnikov invariants Uj determine the homotopy type of the the simply-connected space X. determine the homotopy type of the the simply-connected space X. The The Cartan-Serre Cartan-Serre method method enables enables one one to to compute compute the the Postnikov Postnikov invariants invariants in the course of recursively constructing the spaces X,+j, in the course ofrecursively constructing the spaces X n +j , invoking invoking knowledge knowledge of spacesK(TT, For the the rational of the the cohomology cohomology of of the the Eilenberg-MacLane Eilenberg-MacLane spaces K (1l', m). m). For rational homotopy groups 7ri(X)@Q this computational procedure becomes very much homotopy groups 1l'i(X)0Q this computational procedure becomes very much simpler simpler by by virtue virtue of of the the simplicity simplicity of of the the structure structure of of the the cohomology cohomology rings rings ff*(K(n, H*(K(1l', ml; m); Q). Q). Example 1. For sphere S” 7ri(Sn) 0@Q Q are are given by: Example 1. For the the sphere sn the the groups groups 1l'i(sn) given by:

7r,(P)

g z,

7r4n-1

@Q

=

Q,

with all groups trivial trivial (whence (whence it it follows follows that that in in aH all with aIl other other rational rational homotopy homotopy groups other cases,i(Sn) is finite). other cases 1l'i(sn) is finite). Example 2. For For simply-connected, simply-connected, finite finite CW-complexes CW-complexes X X the the homotopy homotopy Example 2. groups rj(X) are all finitely generated. (These two results were obtained by groups 1l'j(X) are aIl finitely generated. (These two results were obtained by Serre in the early 1950s.) Serre in the early 1950s.) Example 3. The Cartan-Serre theorem (early 1950s). If a simply-connected Example 3. The Cartan-Serre theorem (early 1950s). If a simply-connected space X is such that the algebra H*(X; Q) is a free skew-symmetric algebra, space X Ül such that the algebra H* (X; Q) is a free skew-symmetric algebra, then rj(X) @Q is isomorphic to the vector space of those linear Q-forms on then 1l'j(X) 0 Q is isomorphic to the vector space of those linear Q-forms on Hj(X; Q) that vanish on the nontrivially decomposable elements of Hj (X; Q). Hj (X; Q) that vanish on the nontriviaIly decomposable elements of Hj(X; Q). This theorem applies to H-spaces X since by Hopf’s theorem H*(X; Q) is free This theorem applies to H-spaces X since by Hopf's theorem H*(X; Q) is free skew-symmetric for such spaces.Hence in particular the result applies to loop skew-symmetric for such spaces. Hence in particular the result applies to loop spacesQX, so that knowledge of H* (QX; Q) for any finite simply-connected spaces ft X, so that knowledge of H* (ft X; Q) for any finite simply-connected CW-complex X yields complete information about the groups nj(X) @ Q. CW-complex X yields complete information about the groups 1l'j(X) Qi) Q. However such cohomology algebras are not always easily computed in detail. However such cohomology algebras are not always easily computed in detail. (For the definition of H-spaces and for further details, see Chapter 4, $2.) the definition of H-spaces and for further details, see Chapter 4, §2.) (For Much later, in the early 1970s Sullivan gave precise form to a general the1970s, Sullivanapplying gave precise form simply-connected to a general theMuch later, homotopy in the carly ory of rational type (“Q-type”) to finite, ory of rational homotopy type ("Q-type") applying to finite, simply-connected CW-complexes, where all computations are carried out in the rational cateCW where all computations out in the category,-complexes, i.e. when all invariants are tensoredare by carried Q. He showed thatrational the rational Le. when ail invariants are tensored by Q. He showed that the rational gory, homotopy type of such a CW-complex is determined by an equivalence classof homotopy type of such a CW-complex is determined by an equivalence class of

110 110

Chapter Chapter 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Homology

certain differential differential skew-symmetric geometry this led, certain skew-symmetric algebras. algebras. Via Via Kahlerian Kahlerian geornetry in particular, that the Q-type to the the result that Q-type of of a simply-connected simply-connected “KShler "Kahler manparticular, to ifold” is completely ifold" completely determined by its rational rational cohomology cohomology algebra (Deligne, Sullivan, Sullivan, Griffiths, Griffiths, Morgan, Morgan, in the the mid-1970s). rnid-1970s). 0q In certain special cases Cartan-Serre method method allows allows the homotopy In certain special cases the the Cartan-Serre the homotopy groups to be computed exactly, rather than just up to tensoring by Q. Adams groups to be computed exactly, rather than just up to tensoring by Q. Adams reorganized the of a sequence the Cartan-Serre Cartan-Serre method method into into the the form form of a spectral spectral sequence reorganized known as He observed observed that for the computhe “Adams "Adams spectral spectral sequence”. sequence". He that for the compuknown as the tation of stable homotopy it is actually more convenient to use the so-called tation of stable homotopy it is actually more convenient to use the so-called “Adams rather than than Postnikov Postnikov towers. towers. "Adams resolution”, resolution", rather Let X be a finite CW-complex homotopic equivalent of diLet X be a finite CW -complex (or (or homotopie equivalent thereof) thereof) of dimension < 2n 1 and with rrj(X) = 0 for j 5 n 1. We now regard mension < 2n - 1 and with 7f'j(X) 0 for j :::; n 1. We now regard the the cohomology ring H* (X; Z/p) cohomology ring H*(X; Zjp) as as a a module module over over the the Steenrod Steenrod algebra algebra A,. A p . Let Let e a basis for this module, and consider ,mlc), mj E HnT(X;%), b {WT... {ml,'''' mk}, mj E Hnj (X; Zjp) , be a basis for this module, and consider maps maps k, fj:X+K(Z/p,nj), K(Zjp,nj), j=1,2 j 1,2,,..., ... ,k, fJ:X such that uj E Hnj (K(Z/p, nj); Z/p) is the fundamental such that f,*(uj) Ij(uj) = mj, mj, where where Uj E Hnj(K(Zjp,nj);Zjp) is the fundamental class of the complex K(Z/p, nj). We obtain a map class of the complex K(Zjp, nj). We obtain a map k

k

II

II

f = fi f3 : X -+ fi K(Z/p,nj). 1 = Il: X ~ j=l K(Zjp,nj)' j=l j=1 j=l We convert space X-i N X We convert the the map map ff into into aa fibration fibration with with total total space X- 1 "" X and and fiber fiber X0 = F c X-1: X o Fe X- 1 : k f&-i y3- K(Zjp, K(Z/p,nj). j: X- 1 ~ II yj, l'j, l'j nj)'

j=l

rv

j=l

We then repeat the entire procedure with Xa in place of X-i, and so on, to We then repeat the entire procedure with X o in place of X-l, and so on, to obtain a filtration of the space X: obtain a filtration of the space X: X-X-,3X~3X1>X~3~~~ ) (7.12) (7.12) with with for j for j

the property that the A,-modules Mj+l = H*(Xj,Xj+l;Z/p) are free the property that the Ap-modules Mj+l = H*(X j , Xj+l; Zjp) are free > 0. The boundary homomorphism 2': O. The boundary homomorphism + H’+‘(Xj-1, Xj; Z/P) s : H’(Xj, xj+l; z/P)

in the cohomology exact sequence of the triple (XJ--1,X3, Xj+i) yields the in the cohomology sequence of the triple (Xj_1,Xj,X j +!) yields the differential dj : Mj+l exact ---+ Mj, determining the following complex of free A,differential d : Mj+! Mj, determining the following complex of free Apj modules: modules: . --f Mj+, 2 Mj % Mj-, + . 2 MO -r, Mpl. (7.13) (7.13) Mo Here M-1 E H*(X-1; Z/p) and E is defined to be the homomorphism Here M_ 1 ~ H*(X-1i Zjp) and ê is defined to be the homomorphism

$7. §7. The The classical classical apparatus apparatus of of homotopy theory theory H*(X-l,Xo;z/P)

111 III

H*(X_ 1 ; Zjp)

-

ff*(X-l;qP)

induced 8) -+ Xc). induced by by the the inclusion inclusion of of pairs pairs (X-i, (X- 1 ,0) ---4 (X-i, (X_1,X In particular, particular, in O)' In “stable "stable dimensions” dimensions" we we have have k

Mo

H*(X_ 1 , X o; Zjp) ~ H*(II K(Zjp, nj); Zjp), j=l

so so that that in in the the algebraic algebraic resolution resolution (7.13) (7.13) all all iUj Mj with with jj >?: 00 are are free free (and (and Ker Ker dj dj = Im lm d,+i); dj+t); this this therefore therefore represents represents an an acyclic acyclic resolution resolution of of the the .A,A pmodule (X; Z/p). module M-i M_ 1 E H* H*(X; Zjp). The The complex complex (7.13) (7.13) is is known known as as the the Adams Adams resolution of the the A,-module Ap-module H*(X; H*(X; Z/p), Zjp), and and the the diagram diagram of of “stable "stable spaces” spaces" resolution of or or spectra spectra X-X-1 ,

t- ,x0/,x1<.

.7xj~,/j+~.

X-l/X0

.

x0/x1

.’

xj+l/xj

. . .

(7.14) (7.14) is of the spaceX or the the geometric realization of of is called called the the Adams Adam8 resolution re80lution of the space X or geometric realization the (7.13). By groups one one obtains obtains the Adams Adams resolution resolution (7.13). By taking taking stable stable homotopy homotopy groups from diagram (7.14) the Adams spectral sequence.We We note that there there is is the the the diagram (7.14) the Adam8 8pectral sequence. note that from the following of Eilenberg-MacLane K(Z/p, nj): nj): following isomorphism isomorphism for for aa product product of Eilenberg-MacLane spaces spaces K(Zjp, HomA p (H*(II

K(Zjp, nj); Zjp), Zjp)

J

~ n~ ( II K(Zjp, n j )), J

and this isomorphism allows the Adams spectral sequenceto be handled very and this isomorphism allows the Adams spectral sequence to be handled very efficiently by means of homological algebra. efficiently by means of homological algebra. By using the above methods of Cartan, Serre and Adams, often in comBy using the above methods of Cartan, Serre and Adams, often in combination with other more special tools and ideas, several topologists have bination with other more special tools and ideas, several topologists have managed to carry out far-reaching computations of the stable and non-stable managed to carry out far-reaching computations of the stable and non-stable homotopy groups of spheres (Serre, Toda and others, from the early 1950sto homotopy groups of spheres (Serre, Toda and others, from the early 1950s to the mid-1960s). In the late 1960s and into the 1970s these techniques were the mid-1960s). In the late 1960s and into the 1970s these techniques were supplemented by those of extraordinary homology theory (see 58, 9 below). supplemented by those of extraordinary homology theory (see §8, 9 below). The stable homotopy groups of spheres $ = 7r,+i(SN), N > i + 1, and The stable homotopy groups of spheres nt = nN+i(SN), N > i + 1, and the associated skew-symmetric ring rz: the associated skew-symmetric ring n!:

n!

'L nt,

nt n}

i~O

are important in the theory of smooth manifolds. The first few stable homoare the theory of smooth The first few stable homotopyimportant groups of inspheres are given in the manifolds. following table: topy groups of spheres are given in the following table:

112 112

Chapter 3.3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Chapter Homology

ii=O= 0

'TT%ri == 1

11

22

33456 45

6

88

77

99

10 10

11 11

12 13 13 12

Z/2 '1'.,/2 Z/2 '1'.,/24 Z/24 0 0 '1'.,/2 Z/2 '1'.,/240 Z/240 Z/2 œ'1'.,/2 @Z/2 Z/2 œ @ Z/2 Z/2 œ @ Z/2 Z/2 '1'.,/6 Z/6 Z/504 Z/504 Z/4 '1Z'., '1'.,/2 00 '1'.,/2 00 '1'.,/4 Z/2

'1)

'1)2

1/2

1/

17

W

'1)217, 'l)U, V

'1)17, U

y

x

The multiplicative multiplicative and and additive additive relations relations among among the the indicated indicated generators generators are are The as follows: follows: 12 l2 as 2rj = 0, 0, 21] q2a = Y3, 2 q2v V 1]

= 256x, 256x, =

v3 1]3

= 12v, 12V,

rp 1]V

= 0, 0,

240~ 2400"

= 0, 0,

Tj3a = 0,

u2u = 0,

3w = ?p,

= vw, VW, yy =

6w = 0, 0, 6w

504X = = O. 0. 504x

Note 0, 1]0", qcr, q2a, x and their multiples Note that that the the elements elements q, 1], V, v, 0", 1]20", X and their multiples by means of framed normal bundles on the sphere S”, Le. i.e. are are by means of framed normal bundles on the sphere the homomorphism the homomorphism

sn,

(7.15) (7.15)

are realizable are realizable images under images under

(Some of the stable groups nj(SO) whose whose definition definition is is given given in in Chapter Chapter 4, 4, $3. §3. (Sorne of the stable groups 1Tj (SO) are tabulated in Chapter 4, $2.) Nontrivial p-components xj(P) CC 1Tj ‘/rj5appear are tabulated in Chapter 4, §2.) Nontrivial p-components 1T;P) appear first for j = 2p - 1; for p > 2, j < 2p(p - 1) - 1, they are as follows: first for j = 2p - 1; for p > 2, j ::; 2p(p - 1) - 1, they are as follows: ,w 1T(P) 3 ” z/p, Z/p, J 3 dp)

= z/p,

1T]P) ~ Z/p,

j = 2k(p - l), j 2k(p 1) ,

k=

k

j = 2p(p - 1) - 2, j 2p(p 1) 2,

l,...,p-1,

1, . . ,p

1, (7.16)

(7.16)

2p(p - 1) - 1. 1T]P) ”~ z/p2, Z/p2, jj == 2p(p - 1) 1.

7rF 3

For j # 2p(p- 1) -2, these 7ry’ are contained in Im J, i.e. T:) c Jnj(SO). It For j ::f 2p(p 1) 2, these 1TJP) are contained in lm J, Le. 1TJP) C J1Tj(SO). It turns out that the image under J contains only a small part of @; in particular, turns out that the image under J contains only a small part of 1T!; in particular, the group 7r,$p-1J-2 contains nontrivial elements that do not belong to that the group 1T~~~P_l)_2 contains nontrivial elements that do not belong to that image. image. 12We recall that in the algebra Extdz (z/2, z/2) the elements 2, 7, V, o are represented recall that ExtA2 (Z/2, '1'.,/2) the elements 2, '1), 1/, 0' are represented by 12We the elements ho, in hl, the hz, a.lgebra h3, respectively.

by the elements ho, hl, h2, h3, respectively.

§8. Definition Definition and and properties properties of of K-theory K-theory 58.

113 113

§8. Definition Definition and and properties properties of of K-theory. K-theory. $8. The The Atiyah-Hirzebruch Atiyah-Hirzebruch spectral spectral sequence. sequence. Adams Adams operations. operations. Analogues Analogues of of the the Thorn Thom isomorphism isomorphism and and the the Riemann-Roth Riemann-Roch theorem. theorem. Elliptic Elliptic operators operators and and K-theory. K-theory. Transformation Transformation groups. groups. Four-dimensional Four-dimensional manifolds manifolds The The first first and and the the most most important important generalizations generalizations of of cohomology cohomology theory theory - K-theory - were theory· were introduced introduced (or (or more more accurately accurately K -theory and and cobordism cobordism theory first first considered considered from from the the appropriate appropriate point point of of view) view) in in the the late late 1950s 1950s and and 1960s 1960s by by Atiyah. Atiyah. Subsequent Subsequent development development of of the the associated associated methodology methodology very very significantly significantly augmented augmented the the algebraic algebraic apparatus apparatus of of topology. topology. Moreover Moreover for for the the investigation investigation of of many many topological topological problems problems either either K-theory K-theory or or cobordism cobordism (bordism) (bordism) theory theory has has turned turned out out to to provide provide the the most most appropriate appropriate context. context. The The general general axiomatics axiomatics of of extraordinary extraordinary homology homology (and (and cohomology) cohomology) theory theory were out by by G. G. W. W. Whitehead Whitehead in in the the early early 1960s. 1960s. were worked worked out We fithe basic basic concepts concepts of of K-theory. K-theory. For For aa pair pair (K, (K, L) L) of of fiWe begin begin with with the nite CW-complexes the groups Ki(K, L) and Kg(K, L) are defined as the nite CW-complexes the groups K~(K,L) and Kg(K,L) are defined as the Grothendieck Grothendieck groups groups of of classes classes of of stably stably equivalent equivalent vector vector bundles bundles (real (real and and where “stable equivalence” of two two complex respectively) with base B = K/L, complex respectively) with base B = K / L, where "stable equivalence" of vector bundles ~1 and ~2 means that vector bundles /11 and /12 means that Vl CBEN1 = vz c3 EN2

(8.1) (8.1)

where e &Na t rivial vector fiber IRNi RN% or or C (CNi. Every Ni . Every Ni (i=1,2)isth where (i 1,2) is thee trivial vector bundle bundle with with fiber vector bundle Y over K/L has a stable inverse. This is explained in Chapter vector bundle /1 over K / L has a stable inverse. This is explained in Chapter 4, first realizes K/L/ L as as aa deformation deformation retract of aa manifold manifold U U :J > K K/L/ L 4, §$1: 1: one one first realizes K retract of over which v can be extended to a bundle stably equivalent to the tangent over which /1 can be extended to a bundle stably equivalent to the tangent bundle U; the the inverse inverse -/1 --v is is then realized as as the the normal bundle over over U U with bundle of of U; then realized normal bundle with respect to an embedding U c EV, q sufficiently large: respect to an embedding U c IRq, q sufficiently large: Y@(--v) =EN -0. (8.2) (8.2) /1 EB (-/1) = eN '" O. It is clear that under the direct sum operation on vector bundles, Ki(K, L) It is clear that under the direct sum operation on vector bundles, K~ (K, L) (A = IR, C) becomes a group, and, via the tensor product, in fact a commuta(A = IR, C) becomes a group, and, via the tensor product, in fact a commutative ring. tive ring. For a finite Cl&‘-complex K we set For a finite CW-complex K we set K;(K) = K;(K u *, *), A = lR,@, (8.3) (8.3) where * denotes where * denotes over K. K. overMotivated by Motivated by where where versal versaI

the one-point space, i.e. in effect we consider vector bundles the one-point space, Le. in effect we consider vector bundles the suspension isomorphism (see (3.6)) we define the suspension isomorphism (see (3.6)) we define K;j(K, L) = K;(CjK, CjL), j > 0,

CK, CL denot,e the suspensions of K, L. From EK, EL denote the suspensions of K, L. From G-bundles with G = lim O,, lim U,, one obtains G-bundles with G = lim On, Hm Un, one obtains

(8.4) (8.4)

properties of the uniproperties of the uninatural isomorphisms natural isomorphisms

Chapter Chapter 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Homology

114 114

Kij(K, L) s~ [23(K/L), K;j(K,L) [L'j(KjL), BO], (8.5) (8.5)

K&j(K,L) Ki?(K,L) ”~ [lP(K/L), [L'j(KjL), BU], BU],

where Yj denotes denotes the the set set of of homotopy homotopy classes classes of of maps maps XX *~ Y. Y. Bott Bott where [X, [X, Y] periodicity periodicity (see (see Chapter Chapter 4, $2) §2) then then yields yields

K;j-s(K,

L) = Kij(K,

L), (8.6) (8.6)

K- j - 2 (K L) K-j (K, (K L), L) K$“(K, ”C>< c K;j c 'L)"

which which allows aUows the the groups groups Ki(K, K~ (K, L) L) (A (A == IF%, lR, UZ) q toto be be defined, defined, via via periodicity, periodicity, for for all aU j,j, -oc -00 < < jj << 03. 00. The The functor functor K:(., K~ (-, .).) satisfies satisfies the the axioms axioms of of generalized generalized cohomology cohomology thetheory (since by ory (see (see $3): §3): homotopy homotopy invariance, invariance, functoriality, functoriality, the the excision excision axiom axiom (since by definition (K/L, *) (K, L)), K~(KjL, *) == K: K~(K, L)), and and existence existence of of exact exact sequences sequences of of pairs: pairs: definition Ki .

-

Kj(K,L)

+

Kj(K)

--) Kj(L)

3

--+...

Kj+‘(K,L)

(8.7)

The category of of finite finite comcomThe dual dual homology homology theory theory K,/‘(., KjC,·).) is is defined defined for for the the category plexes plexes by by K,/‘(K)

= K,‘(K,

*) = K;+(S”\K,

*‘),

n +

03,

(8.8) (8.8) Kf(K,L) *), Kj(K,L) == Kj/(K/L, Kj(KjL,*), i.e. CW-complex KK embedded i.e. as an an appropriate appropriate limit, limit, with with the the finite finite CW-complex embedded in in S” sn (for n). There There is simple geometric description of of the groups (for sufficiently sufficiently large large n). is no no simple geometric description the groups K./(K), they are used in theory. Kj(K), and and they are not not generally generaUy used in homotopy homotopy theory. Since bundle over the suspension suspension CK is determined determined by hoSince each each vector vector bundle over the EK is by aa homotopy motopy class class of of maps maps K K-U,, K +~ 0, On or or K ~ Un, for some n, r~, we for sorne we infer infer natural natural isomorphisms isomorphisms K&K)

Z [K, 01,

K&K)

z [K, U].

Hence one-point space) space) we Hence for for aa contractible contractible CW-complex CW-complex (in (in particular particular aa one-point we obtain obtain via via Bott Bott periodicity periodicity the the isomorphisms: isomorphisms: K;j(*) K;j(*)

C><

E

nj(BO) 7rj(BO)

C><

K;j(*) Ki;J(*)

C><

”

T+~(O), (j > > 0), 0), 7rj_l(O), (j

zs 7rj(BU) rj(SU)

C><

”

T+~(U), (j > > 0). 0). 7rj_l(U), (j

Note general KÂ K$(K, L) is is aa graded graded skew-commutative skew-commutative ring. The The above above Note that that in in general (K, L) ring. observations lead to the following description of the coefficient rings KF(*), observations le ad to the foUowing description of the coefficient rings Kf( *), Kj@(*): Kr(*):

58. Definition K-theory §8. Definition and and properties properties of of K-theory

115 115

1. Ki(*) Ki(*), Ki4(*), KJi(*) has has as as generators generators 1 E K~(*), h EE I-C;‘(*), K;l(*), u E K;4(*), v21 EE Ki’(*), K;8 (*), v-l v- 1 E Ki(*), K~ (*), with with relations

2h = 0,

h3 = 0,

u2 = 4v,

vu-l = l' 1;,

(8.9) (8.9)

w-l 1 E Ki(*), wE K~(*),

(8.10) (8.10)

2. 2. Kc(*) Kè (*) has has as as generators generators 1 E a*), Kg(*),

w W

c

E E Ke2(*), K 2 (*),

with the single ww-l 1 = 1, 1, so that with the single relation relation wwso that KC(*) 2 qw, w-l].

(8.11) (8.11)

Note multiplication by the case Ki(*)(*) (and multiplication by by Note that that multiplication by vv in in the case of of Ki (and multiplication w in the case of Kc(*)) corresponds to application of the Bott periodicity w in the case of Kê (*)) corresponds to application of the Bott periodicity operator. operator. The by means of the the Atiyah-Hirzebruch The ring ring K;(K) KÂ (K) may may be be computed computed by means of Atiyah-Hirzebruch spectral sequence, which exists for any generalized cohomology theory. In In the the spectral sequence, which exists for any generalized cohomology theory. present case the Atiyah-Hirzebruch spectral sequence has as its terms rings present case the Atiyah-Hirzebruch spectral sequence has as its terms rings Ek* d, satisfying E;n* with with differentials differentials dm satisfying *>* = H(Ez*, E m+l

d,),

d, : Egq ----t Eg+m+-m+l,

dk = 0, 0,

E2pyq= Hp(K; K;(*)),

(8.12) (8.12)

c Egq = GK;(K), GKÎJK) , p+q=s p+q=s

is adjoined to K;(K). The images of the differL Ekq E~q is adjoined to KÂ (K). The images of the differ-

i.e. the ring E& = c Le. the ring E');.;* =

P,q

entials d, are finite p,q groups, and for a finite CW-complex K one has d, = 0 entials dm are finite groups, and for a finite CW -complex K one has dm = 0 for m sufficiently large. The orders of the groups Im d,, and the relationship for m sufficiently large. The orders of the groups lm dm, and the relationship of the differentials d, to cohomology operations, were studied by Buchstaber of the differentials dm to cohomology operations, were studied by Buchstaber in the late 1960s. in the late 19608. For CW-complexes K with torsion-free integral cohomology groups, the For CW-complexes K with torsion-free integral Atiyah-Hirzebruch spectral sequence collapses in the cohomology case n = (E ,groups, i.e. d, =the0 Atiyah-Hirzebruch spectral sequence collapses in the case A = te , Le. dm 0 for m 2 3, yielding the isomorphism: for m 2': 3, yielding the isomorphism: H*(K; Kc(*)) ” K;(K). With n = lR the same condition on K implies that all groups Im d, have With at A =most lR the order 2. same condition on K implies that aU groups lm dm have order at most 2. Example 1. Let K = RP”. The ring K~(IRP”) is generated by the element 1. Let K = the lRpk.canonical The ringline K~ (lRpk) generated by thenontrivial element u =Example n - 1, where 11is bundle isover ID’” (with u 17 1, where 77 is the canonical line bundle over lRpk (with nontrivial

116 116

Chapter Chapter 3.3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Homology

Stiefel-Whitney Stiefel-Whitney class class WI(~); Wl(TJ); see Chapter Chapter 4, $1). §1). ItIt follows follows that that d, dm = 0 for for m 2 3, in m? in the the Atiyah-Hirzebruch Atiyah-Hirzehruch spectral spectral sequence, sequence, whence whence j GK;(RP”, *) F~ ~H’(WP”; Ki”(*)), GK~(lRpk,*) LHi(IRpk;Ki (*)), j20 j?O

(8.13) (8.13)

where + s,s, h(k) where lc k == 8q 8q + h(k) === 4q 4q ++ X,, À .. , Xl = 1, x2 = 2, x3 = 2, x4 = 3, x,5 = 3, xfj = 3, A, = 3. The The following following relations relations hold: hold: -u2 _u2 = 2u, 2u, 2h(“)U 2 h(kl u

= 0, 0,

+

7&u) 'ljJ1(u) = (1(1 + ?A)1 u)l -

1, 1,

where where til 'ljJ1 is is the the “Adams "Adams operation”, operation", shortly shortly to to be he defined. defined. is ‘u = U, For -4 = @, the generator of Kg(lRP”) For A = C, the generator of Kg (IRpk) is v = cu, cu, the the complexification complexification of of u, so that again d, = 0 for m > 3 in the Atiyah-Hirzebruch spectral sequence, so that again dm = 0 for m ? 3 in the Atiyah-Hirzebruch spectral sequence, yielding yielding Kg(Ilw”)

” &4)

-II2 = 2v,

&J)

=

(1+

?I)1 -

1.

0

(8.14) (8.14)

By one can define on on the the By means means of of real real and and complex complex representations, representations, one can define Grothendieck group Ki, n = R, @, various operations commuting with conGrothendieck group K~, A = IR, C, various operations commuting with continuous These are the K-theoretic of the ordinary cohoK-theoretic analogues analogues of the ordinary cohotinuous mappings. mappings. These are the mology operations. We now define the Adams operations of complex mology operations. Wc now define the Adams operations of complex K-theory. K -theory. Given a complex bundle q over K/L, we may form the exterior power Given a complex bundle TJ over K / L, wc may form the exterior power bundle Aj(v), j > 0. The formal power series in t: bundle Aj (TJ), j ? O. The formaI power series in t:

L

At(v) = c llq?$j j = 1 + lll(q)t + A2(q)t2 2 + ” , At(TJ) = Aj(TJ)t = 1 + Al(TJ)t + A2(TJ)t +"', j?J j?O

defines an operator defines an operator 4 : K’(K,

L) -

K”(K

L) [Ml,

whose coefficients Aj, however, are not additive since (see Chapter 4, 51) whose coefficients Aj, however, are not additive sinee (see Chapter 4, §1) (8.15) A(% @ 772) = ~t(vd~t(772). (8.15) To obtain additive operations consider the coefficients of the series Ta obtain additive operations consider the coefficients of the series - log(1 - tnl +. .) = -log n-, = c -log(l- tA l + ... ) = -logA_t = k>l

L

~k t k . $tk.

k?l

Here qk(nl,. l . , nk) is a polynomial over Z Here 'ljJkqk(A in, ... , Ak) is a polynomial over Z express terms of symmetric polynomials. express 'ljJk in terms of symmetric polynomials.

in the Aj. It is convenient to in the Aj. .It. beis the convenient Let si, ~2, elementaryto Let 81, 82, ... be the elementary

$8. §8. Definition Definition and and properties properties of of K-theory K-theory

117 117

symmetric . .;; then symmetric polynomials polynomials in ‘1~1, Ul, ‘112, U2, .... then the the symmetric symmetric polynomial polynomial ut u~ +. + .... .++ U: may be expressed as a polynomial in ~1,. . . , Sk: u~ may expressed as polynomial in S1,·.·, Sk: uf + . . .U; = pk(sl,s’J,. ... . .,Sk), U~+··'U~=Pk(Sl,S2, ,Sk),

kk = 1,2,. . 1,2, ....

The The following following expression (the (the k-th k-th Adams Adams operation) operation) is is very very convenient convenient for particular calculations: particular calculations: $k(v)

=

pk(r],

A$,.

. . A”+

ItIt follows follows that that $k(%

a3 72)

=

$J”(rll)

@ $k(rl2),

Q1

=

A1

=

1.

IfIf qTJ is is aa line Hne bundle, bundle, then then

(8.16) (8.16) (8.17) (8.17)

Qk(rl) = rlk.

For consequently For line line bundles bundles (considered (considered as as representing representing elements elements of of KE) K~) and and consequently their direct sums, it is easy to establish the following properties: their direct sums, it is easy to establish the foUowing properties: '1j!k('f/l'f/2)

'1j!k('f/l)'1j!k('f/2),

ch”(Gk(4) ch n ('1j!k('f/)) = = kn(chnrl), kn(chn'f/),

(8.18) (8.18)

where ch is the Chern character. These properties then follow without where ch is the Chern character. These properties then follow without culty for all vector bundles (or rather elements of Kg). cult y for aU vector bundles (or rather elements of Kg). Denoting by Q the generator of Kg(S2n, *) g Z, for which chq = Denoting by fJ the generator of Kg (s2n, *) ~ Z, for which ch'f/ = 2n. Z ), the dual of the fundamental class, we have H2”(S H 2n (s2n;3Z), the dual of the fundamental class, we have gkr/ = k”q.

diffidiffip E Il E

(8.19) (8.19)

The formula (8.19) implies the following rule for commuting the operator Gk The formula (8.19) implies the foUowing rule for commuting the operator '1j!k and the Bott periodicity operator w (see above): and the Bott periodicity operator w (see above): $,“(wz) = kw?lk(z). '1j!k(WZ) = kw'1j!k(z).

This This qkw '1j!kw

motivates the extension of $J~ to all KG’(K, L) with j > 0 by setting motivates the extension of '1j!k to aU Kc j (K, L) with j > 0 by setting = kw, and exploiting multiplicativeness: = kw, and exploiting multiplicativeness: $k(4

= $JkkNk(9

Extension of the operators $” to the Ki with j > 0 is possible only after Extension of the operators '1j!k to the K~ with j > 0 is possible only after tensoring with Z [i] , i.e. going over to the groups K& @Z [i] . tensoring Z [iJ,theLe.Adams going operations over to the$J” groups K~ ® Zin a similar fashion. In real with K-theory are defined In real K-theory the Adams operations '1j!k are defined in a similar fashion. These operations commute with the complexification homomorphism: These operations commute with the complexification hornomorphism: c: K;(K,L) --+ K;(K,L), $“.c=cqb’“.

[JJ

118 118

Chapter Chapter 3. 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-cornplexes. Homology Homology

The 1960s have turned The operations operations $I”, 'lj;k, introduced introduced by by Adams Adams in in the the early early 1960s, turned out to be extremely useful in the solution of several problems. In out to be extremely in the solution of In order order to to formulate formulate these problems, we we first first need need some sorne further further definitions. definitions. Let S(7]) denote the the spherical fibration fibration associated associated with with a vector vector bundle bundle q, 7], Let S(q) i.e. S(q) has the same base as Q, but each fiber lP of the vector bundle S(7]) the same base as 77, but fiber ]Rn of vector bundle vri is replaced by 772over the same by the the sphere sphere S”-l sn-1 cC R”. ]Rn. Two Two vector vector bundles bundles 771, 7]1, 7]2 the same base K with fiber lP or @” are said to be fiber-wise homotopy equivalent to fiber-wise homotopy equivalent ifif base K with fiber ]Rn there is a map there is a map

en

S(771) -~ S(r/2) S(772)

S(Q)

between fibrations, inducing inducing the the between the the total total spaces spaces of of the the associated associated spherical spherical fibrations, identity to the the identity map map on on the the base, base, and and of of degree degree +l +1 on on the the fibers. fibers. This This leads leads to concept concept of of stable stable fiber-wise fiber-wise homotopy homotopy equivalence equivalence of of real real or or complex complex vector vector 2 bundles 81,l, f.:N &Nz, 7]1 -rv 772 rl2 if if there there exist exist trivial trivial bundles bundles f.:N such that that bundles (namely (namely 71 , such are fiber-wise homotopy equivalent as above). The set 71 7]1 e3 œEN1 f.:N l and and r]2 772 $ œ~~2 f.:N 2 are fiber-wise homotopy equivalent as above). The set of Dold in the of classes classes determined determined by by this this stable stable equivalence equivalence (first (first considered considered by by Dold in the mid-1950s) are denoted by JR(K) (or Jc(K)). There are obvious epimorphisms mid-1950s) are denoted by JR(K) (or Jc(K)). There are obvious epimorphisms K;(K) K~(K) -~ JR(K), JIR(K),

K;(K) Kg(K) -~ Jc(W. Jc(K).

Let sphere Sq (i.e. of of homohomoLet G, G q be be the the group group of of homotopy homotopy equivalences equivalences of of the the sphere sq (i.e. topy classes of maps 5’4 -+ 5’4 of degree l), and let G = lim G,, the Hm G q, the direct direct topy classes of maps sq ~ sq of degree 1), and let G q-00 q ...... oo limit of the groups G, under the natural inclusions G, c G,+r. The natural limit of the groups G q under the natural inclusions G q C G q+ 1 . The natural embedding induces a BO ~ --f BG, BG, whence embedding 0 0 cC G G (see (see Chapter Chapter 4, 4, $4) §4) induces a map map jj :: BO whence Kg(K) = [K, BO] , K~(K) ~ [K, BO],

JR(K) = j*(K~(K)) ~*(K~(K)) Cc [K, [K, BG], =I, JIR(K) where (as before) [X, Y] denotes the set of homotopy classesof maps X + Y. where (as before) [X, Y] denotes the set ofhomotopy classes ofmaps X ~ Y. For smooth manifolds M” the group &(AP) has several important appliFor smooth manifolds Mn the group JIR (Mn) has several important applications: cations: 1. To obtaining lower bounds for the orders of homotopy groups of spheres 1. To obtaining lower bounds for the orders of homotopy groups of spheres (M” = P). (Mn = sn). 2. To the study of smooth structures on spheres (AP = P). 2. To the study of smooth structures on spheres (Mn = sn). 3. To estimating the number of independent vector fields on spheres S2+l 3. To estimating the number of independent vector fields on spheres S2n-1 (Mn = RP).

(Mn

= ]Rpn).

4. To classifying the stable normal bundles of smooth closed manifolds (see 4. To classifying the stable normal bundles of smooth closed manifolds (see Chapter 4, §3,4). Chapter 4, §3,4). It is appropriate to mention here also the following fact, a corollary of a It isofappropriate to mention here also the following fact, a corollary of a result Atiyah: result of Atiyah: Let qN be a vector bundle over Mn. The class [q] E JL$(M~) contains the Let rlN bundle be a vector Thehomotopy ela,58 [rll equivalent E J'JI. (Mn) contains the normal over bundle M (i.e.over' q isMn. stably to the normal normal bundle over M (i. e. 7] is stably homotopy equivalent to the normal bundle) if and only if the cycle 4( [MI) E HN+,(TQ; Z) is spherical. bundle) if and only if the cyele 4>( [M]) E H N +n (T77; Z) is spherical.

58. Definition Definition and properties properties of K-theory K-theory §8. and of

119 119

(Here T1] TV is the the Thom Thorn space, space, and cp4 is the the Thom Thorn isomorphism isomorphism cp: 4 : Hn(Mn) H,(Mn) % (Here HN+~(T~); a cycle IL: E H,(X; Z) is spherical, if it belongs to the image of H N +n (T1]); cycle x E H * (X; 7l.) spherical, it belongs to image of the Hurewicz homomorphism 7r* (X) + H, (X; Z).) By the Browder-Novikov the Hurewicz homomorphism 11'. (X) ----> H.(X; 7l.).) By Browder-Novikov theorem (see (see Chapter Chapter 4, 4, §4), §4), for for aa simply-connected simply-connected manifold manifold Mn Mn of of dimendimentheorem sion n # 41c + 2, n 2 5, the above condition that +([Mn]) be spherical, sion n f. 4k + 2, n 2: 5, the above condition that cp([Mn]) be spherical, together with with the the Hirzebruch Hirzebruch formula formula for for the the signature, signature, suffice suffice for for the the vector vector together bundle 77 to be realizable as a normal bundle of a possibly different manifold bundle 1] to be realizable as a normal bundle of a possibly different manifold homotopy equivalent equivalent to to Mn. M”. LL homotopy Consider the the subgroup subgroup H H of of K~ K:(K) generated by by aIl all elements elements of of the the form form Consider (K) generated (8.20) (8.20)

PN”(lClk - l)%

where 1]n is is any element of of K~ K:(K), ranges over over aIl all primes, primes, and and Np iVp over over ail all where any element (K), pp ranges sufficiently large integers. According to a result of Adams the quotient group sufficiently large integers. According to a result of Adams the quotient group JR(K)

”

K:(K)/(pNp(+lc

-

1)~)

=

K;(K)/H

provides for lITt(K) JR(K) since since the the kernel kernel of of the the epimorphism epimorphism provides aa lower lower estimate estimate for is contained in Adams’ subgroup H. In the case K = sn, S”, K;(K) JR(K) . K~(K) lJP1.(K) is contained in Adams' subgroup H. In the case K Adams was able to establish the reverse estimate (to within the factor 2 in Adams was able to establish the reverse estimate (to within the factor 2 in certain dimensions), thereby obtaining a measure (to within the indicated accertain dimensions), thereby obtaining a measure (to within the indicated accuracy) of the orders orders of of the the groups Jn(Sln) cC 11'ln-l 7&-i = 11'4n_1+N(SN). 7r~~-r+~(S~). This This groups lITt(s4n) curacy) of the estimate of Adams coincides (up to the factor 2 for even n) with a lower estiestimate of Adams coincides (up to the factor 2 for even n) with a lower estimate obtained by Kervaire-Milnor (see Chapter 4, $3). For K = RPzn Adams’ mate obtained by Kervaire-Milnor (see Chapter 4, §3). For K = lRp2n Adams' lower estimate yields lower estimate yields K;(W’2n) ” Jw(Rl=2n), K~ (lRp2n) ~ lITt (lRp2n) , from which there follows, by means of certain reductions due to Toda, an from which there follows, by means of certain reductions due to Toda, an exact value for the largest number of linearly independent vector fields on exact value for the largest number of linearly independent vector fields on an odd-dimensional sphere S2qp1. If 2q = 2m(21 + l), m = c + 4d, where an odd-dimensional sphere S2q-l. If 2q = 2m (2l + 1), m = c + 4d, where 0 < c < 3, this number is 2’c + 8d - 1 (Adams, early 1960s); and this number o s:: c s:: 3, this number is 2 + 8d - 1 (Adams, early 1960s); and this number of linearly independent vector fields may be constructed on the sphere by of linearly independent vector fields may be constructed on the sphere by using infinitesimal rotations. using infinitesimal rotations. Adams’s conjecture that all elements of K:(K) of the form ICN($” - 1)~ Adams's conjecture that aIl elements of K~(K) of the form kN('ljJk --- 1)1] are, for sufficiently large N, J-trivial (i.e. lie in the kernel of the epimorphism are, for sufficiently large N, l-trivial (i.e. lie in the kernel of the epimorphism K:(K) JR(W), was proved in the late 1960s by Sullivan and Quillen K~(K) ----> JJP1.(K)), was proved in the late 1960s by Sullivan and Quillen by means of elegant general categorical ideas involving augmentation of the by means of elegant general categorical ideas involving augmentation of the homotopy category in such a way that multiplication by k becomes invertible. homotopy eategory such geometric a way thatinterpretation multiplicationofbyhomological k becomes invertible. Since there is no in simple K-theory, is no simple geometric interpretation of homological -theory, Sinee there the appropriate K-theoretic analogue of Poincare duality for closed K manifolds the appropriate K-theoretic analogue of Poincaré duality for closed manifolds requires special treatment. In particular the following questions arise: What requires special treatment. particularWhen the foIlowingit questions What is a “fundamental class” inInKn(Mn)? exist? An arise: essentially n )? When does is a "fundamental class" in Kn(M does it exist? An essentially equivalent, but more accessible,question is the following one: For which vector but K, more accessible, following one: For which vector equivalent, bundles 77over with fiber Rn,question is there isathe “Thorn isomorphism” bundles 17 over K, with fiber lR n , is there a "Thom isomorphism" qSK: K;(K) + K;+j (Trl), (8.21) OK = .+k(l)> n = R@. ? (8.21 )

120 120

Chapter 3. 3. Simplicial Simplicial Complexes Complexes and CW-complexes. CW-complexes. Homology Homology Chapter

For For aa complex complex vector vector bundle bundle qT/ with with fiber fiber @“, cm, nn = 2m, the the value value of of 4~(1)
= /l.=(q)

L

lY”=y~) Aeven(T/) = = c n2qr& A2j(T/), QO j~O

- Aodd

E K;(T&

L

@d(q) Aodd(T/) = = c @+1(q). A2j+l(T/). j>O j~O

The The following following formula formula is is valid: valid:

&rl(cW~(l)))
ii:: K -~ Trl, TT/,

where Thorn isomorphism where T(q) T(T/) is is the the “Todd "Todd genus” gerlUs" of of 77, T/, and and 4~
= f! : K”(M”)

+

(i.e. having (i.e. having there is an an there is

their stable their stable analogue of analogue of

K’(N”),

satisfying satisfying

(8.22) (8.22) (This is a generalization of the Riemann-Roth theorem.) For embeddings is a generalization of the Riemann-Roch theorem.) For embeddings (This AP Nk the operator J, f! coincides coincides with with the the Thom Thorn isomorphism isomorphism for for the the Mn cc Nk the operator normal bundle VMVLCNL of the embedding. normal bundle vMnCNk of the embedding. In real K-theory (A = Iw) the construction of the Thorn isomorphism reIn real K-theory (A = !R) the construction of the Thom isomorphism requires a spinor structure to exist on the bundle q (dim7 = n), which is the n), which is the quires a spinor structure to exist on the bundle T/ (dim T/ case if the Stiefel-Whitney classesWI(V), ~~(77) = 0. One then defines case if the Stiefel-Whitney classes Wl (T/), W2 (T/) O. One then defines cw!(d)uNk)

= f!(CWVPY~“)).

4~(1) = A+(v) - A-(v)

E KidW>

(8.23) (8.23)

where A* are the “semi-spinor” representations of the group Spin(n) on the where LJ± are the "semi-spinor" representations of the group Spin(n) on the vector space n (EP), and A*(q) are the vector bundles associated with these vector space A (!Rn), and LJ±("/) are the vector bundles associated with these representations. The Thorn isomorphism 4~ in real K-theory is used to define representations. The Thom isomorphism
§8. Definition Definition and properties of K-theory 58. properties of K-theory

121

As a result investigations by by several several people people throughout the 1960s As result of of investigations throughout the 1960s and and early the ring ring K;(X) has been been calculated calculated for for most most of early 1970s 1970s, the KÂ (X) has of the the important important homogeneous spaces X. Of Of particular result that compact, homogeneous spaces X. particular interest interest is the the result that for for a compact, and simply-connected Lie group C, the ring Kê(C) is generated by connected connected and simply-connected Lie group G, the ring K;(G) generated by the "fundarnental representations" Pl, ... , Pm E KI (C) (m = rank C) as the the “fundamental representations” pi,. . . pm K’(G) (m rank G) the . exterior exterior algebra algebra in in these generators over the ring these generators over the ring Z[w,wZ[w, w-l]: I ]:

K:(G) g 4~1,.. . , pm), n = z[w,d], where w the Bott periodicity operator operator (Hodgkin’s (Hodgkin's theorem, where w is the Bott periodicity theorem, proved proved in in the the ear ly 1960s). 1960s). For For C (n) the representations Pl, just the early G = U U(n) the representations pr, ... ,,Pn pn are are just the exterior exterior powers of of the canonical bundle 1] over over BU(n): BU(n): pj Pj ::= For the the classifying classifying powers the canonical bundle 77 = Aj(1]). nj(v). For space BG BC of of a compact (or finite) group, we space compact (or finite) Lie Lie group, we have: have: K;‘(BG)

= 0,

K:(BG)

” &

= Z + I/I2

+ 12/13 + . . . ,

the completion the complex complex representation representation ring with respect respect to the completion of of the ring Re RG with to powers powers of a prime ideal Il with with RG/I Rc/ l ~ in the of prime ideal E Z Z (Atiyah, (Atiyah, in the mid mid - 1960s). 1960s). In computing the K-theory K-theory of CW-complexes addiIn computing the of infinite-dimensional infinite-dimensional CW-complexes X additional difficulties arise; in elements of tional difficulties arise; in particular, particular, elements of infinite infinite filtration filtration may may occur, occur, any n-skeleton Le. nontrivial nontrivial elements of K;(X), KÂ(X), w whose restriction to to any n-skeleton X” is i.e. elements of h ose restriction the late appearance of of zero (Mishchenko, (Mishchenko, Buchstaber, zero Buchstaber, Hodgkin, Hodgkin, in in the late 1960s). 1960s). The The appearance elements filtration is a cornmon common feature feature of generalized cohomology of infinite infinite filtration of generalized cohomology elements of theory, not not occuring occuring in in the theory. theory, the ordinary ordinary theory. One of the the most most elegant elegant applications K-theory occurs One of applications of of K-theory occurs in connexion connexion with with the operators on closed closed manifolds. the index index formula formula for for elliptic elliptic pseudo-differential pseudo-differential operators manifolds. First we give the the requisite definitions. Let Let 71, 1]1, 1]2 be vector over a First we give requisite definitions. ~2 be vector bundles bundles over e inear space of smooth cross-sections13 closed manifold Mn, and r (1]i) the linear space of smooth cross-sections 13 of of closed manifold M”, and r(~i) th 1 the bundle 1]i, i = 1,2. A general differential operator L of order m is defined the bundle vi, 1,2. A general differential operator of order defined on the the space space r(1]r) its values space r(rJ2): on r(ni) taking taking its values in in the the space r(r/z):

xn

L : T(rll)

-

F(q2).

In the form form In local local coordinates coordinates L has has the

(x) is a matrix-valued operator of of order order ail.im (x) matrix-valued function function and and L1 LI is an operator where ai,...i, <<m. m.

When L isis elliptic, elliptic, both bath kernel and cokernel finite-dimensianal, and When kernel and cokernel are are finite-dimensional, and the index Ind Ind LL = the difference difference of of these these dimensions dimensions is by by definition definition the the index dim L, of of the the operator The "index “index problem” to give give dim Ker Ker LL - dim dim Coker Coker L, operator L. L. The problem" is to a description description of the integer Ind L in terms of topological data implicit in the of the integer Ind in terms of topological data irnplicit in the 13Here the vector spaces are assumed be endowed endowed with with a proper norm determining the 13Here the vector spaces are assumed to to be proper norm determining the structure of a Sobolev Sobolev space on r(vi). structure of space on r(1)i)'

Chapter Chapter 3. 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Homology

122 122

elliptic elliptic operator operator L. L. The The following following interpretation interpretation of of the the highest-order highest-order terms terms in in the the operator operator L, L, provides provides the the key key to to the the solution solution of of the the “index "index problem”. problem". Let Let 7r 7r :: T*(M) r*(M) +--+ A4 M be be the the projection projection of of the the cotangent cotangent bundle bundle over over AP, Mn, and and let let 7; (M) == r*(M) To(M) T*(M) \\ A4 M be be the the subset subset of of nonzero nonzero vectors vectors inin r*(M). T*(M). The The highest highest terms r*rlz, terms of of LL then then determine determine aa homomorphism homomorphism ~TL (J'L :: 7r*qi 7r*7)1 + 7r*7)2, which which in in local co-ordinates is given by: local co-ordinates is given by:

(8.25) where where pP == (pi,...,p,) (Pl, ... ,Pn) EE r*(M) T*(M) isis aa covector. covector. The The homomorphism homomorphism PL (J'L isis called the symbol of the operator L. The operator L is elliptic if and only called the symbol of the operator L. The operator L is elliptic if and only ifif co-ordinates, its its symbol symbol a~ (J'L isis an an isomorphism isomorphism on on r;(M), TÔ (M), or, or, inin terms terms of of local local co-ordinates, ifif and only if each map a~(z,p) is invertible, i.e. det a~(z,p) # and only if each map (J'L(X,p) is invertible, Le. det (J'r-{x,p) i= 00 ifif pP #i= 0. O. The or class [gL] The isomorphism isomorphism (TL (J' L -or rather rather its its stable stable homotopy homotopy class [(J'L] - .- isis to to be be thought operator L, thought of of as as aa topological topological “twist” "twist" of of the the elliptic elliptic operator L, and and determines determines the operator up to the addition of a compact operator. the operator up to the addition of a compact operator. For operators we m = = 0. In the the case = 51 S1 and and we have have m O. In case A4 M = For singular singular integral integral operators any elliptic operator L of order m = 0, the symbol (TL reduces to a pair of any elliptic operator L of order m 0, the symbol (J'L reduces to a pair of matrix-valued functions g:(z) = QL(Z, l), c;(z) = ~L(z, -1) on the circle matrix-valued functions (J't(z) (J'dz,l), (J'L(z) (J'L(Z, -1) on the circle S’ @ 1 1 ]z] l}. The The Noether-Muskhelishvili formula (1920-1930) 51 = {z{z EE C Izi = 1}. Noether-Muskhelishvili formula (1920-1930) for for the in this is as the index index of of LL in this case case is as follows: follows:

f

2-/rind

f

L = det (J'L aL (z)dz. 27rlnd L = f log log det det crl (J't (t)dz (z)dz -- f log log det (z)dz.

The general construction early 1960s) is as folThe general construction of of Atiyah-Singer Atiyah-Singer (from (from the the early 1960s) is as follows: given our two bundles I*, r*(rjz) over the total space r*(M) an lows: given our two bundles 7r*(7)t), 7r*(7)2) over the total space T*(M) and and an isomorphism go : rr*(~) --+ n*(rjz) between their restrictions to the submanisomorphism (J'L : 7r*(7)1)-+ 7r*(7)2) between their restrictions to the submanifold r;(M), there is defined a difference element

ifold TÔ (M), there is defined a difference element

Let D(M) and S(M) stand for the unit disc and unit sphere bundles obtained Let D(M) and SeM) stand for the unit disc and unit sphere bundles obtained from r*(M) endowed with some Riemannian structure. Going over to the from T* (M) endowed with some Riemannian structure. Going over to the Thorn complex Tr* = D(M)/S(M), which is homotopy equivalent to the Thom complex TT* = D(M)jS(M), which is homotopy equivalent to the complex r*(A4)/70*(M), we obtain a well-defined element

complex T*(M)jTô(M), we obtain a well-defined element d(r*(m), E K’(TT*). d(7r*(7)I), r*(m),oL) 7r*(7)2), (J'L) E KO(TT*). The

Atiyah-Singer

index

theorem

consists

in the formula:

The Atiyah-Singer index theorem consists in the formula: Ind L = (4-‘(ch where

T(M)

is the Todd

d) .T(Mn),

[M”]),

genus of the complexification

of the tangent

bundle

where T(M) is the Todd gemls of the complexification of the tangent bundle r over M. T over M. One of th_e most beautiful applications of this formula yields the interpretaOne of the most beautiful applications of this formula yields the=interpretation of the A-genus of a Spin-manifold M4”4k (i.e. for which WI(M) w?(M) = tion of the Â-genus of a Spin-manifold M (Le. for which wl(M) = w2(M) = 0) as the index of the Dirac operator, yielding the integrality of the A-genus. 0) as the index of the Dirac operator, yielding the integrality of the Â-genus.

$8. Definition Definition and properties properties of of K-theory K-theory §8.

123 123

At the the present time several several different different proofs proofs of of this this theorem theorem are known, known, At present time well as various various generalizations generalizations of of it. it. Atiyah-Singer's Atiyah-Singer’s first proof proof was, was, like like as weB first that the Riemann-Roch-Hirzebruch Riemann-Roth-Hirzebruch theorem, based based on cobordism cobordism theory. theory. that of of the theorem, Both of these these results results extend extend to to manifolds-with-boundary, manifolds-with-boundary, provided that that the the Both of provided restriction of the operator L to the boundary is Noetherian. restriction of the operator boundary is Noetherian. Thus we we see see that that the link link with with the theory theory of linear operators has has been been of great benefit for topology, leading as it has to the discovery of several deep great benefit for topology, leading as it has to the discovery of several deep algebro-topological connexions. algebro-topological connexions. By way of of an an example example we we shaH shall now now give give the the Atiyah-Bott Atiyah-Bott formula formula for for the the By way number of isolated fixed points of a holomorphic self-map number of isolated fixed points of a holomorphie self-map

of aa compact compact Kahler Kahler manifold manifold Mn. M”. Let Let Pl, pi,. ... . . ,, Pk pk be the fixed fixed points points of of such such of be the a map f, all assumed to be isolated and non-degenerate. Let T(f) : T -+ a map f, aH assumed to be isolated and non-degenerate. Let T(f) : r r7 be the derivative of the map f, inducing bundle maps be the derivative of the map f, inducing bundle maps T(f) rll T(f) :: 7]1 T(f)

-

: 772 -

71,

71

=

772,

Q?=

P-7,

LPddT.

Xj, # 1, denote the eigenvalues of the induced self-map Let Let Xlq,...,Lq, À lq , ... , À nq , Àjq f. 1, denote the eigenvalues of the induced self-map

T(f) of space 7(pq) to Mn T(f) of the the tangent tangent space r(pq) to Mn at at Bott formula (an analogue of the Lefschetz Bott formula (an analogue of the Lefschetz points, is then as follows: points, is then as follows:

each fixed fixed point each point formula) the formula) for for the

pp. The The Pq. number number

AtiyahAtiyahof fixed of fixed

x(f) = & j=l fi --!--= & (n-T(f)l,,(P,,) - TrPI,&)) 7 1 h, ( ) q=l

q=l

where Tr denotes the trace. (The Lefschetz formula for the number of fixed where Tr denotes the trace. (The Lefschetz formula for the number of fixed points is:

points is:

x(f) = c (-l)“n:

~(f)*IHoP(M)r

m

m

where HOP(M) denotes the vector space of holomorphic m-forms on M”.) where Holm(M) denotes the vector space of holomorphie rn-forms on Mn.) The above formula has several analogues, including a real analogue where The above formula has several analogues, including a real analogue where M” is endowed with a Riemannian metric and the map f is an isometry, or, Mn is endowed with a Riemannian metric and the map f is an isometry, or, more generally, one has a f-invariant elliptic complex (or f-invariant elliptic) more generally, one +has T(r/2) a f-invariant elliptic complex (or f-invariant elliptic) operator L : F(Q) over M”. In the latter case the spaces Ker L, L ; r(7]l) ---> r('f]2) over Mn. In the latter case the spaces Ker L, operator Coker L and the fibers of r]l, ~2 above the fixed points p,, become representaCoker L andforthe of 7]1,T(f), 7]2 ab ove the fixed points Pq, become representation spaces thefibers operator and the following formula (of Atiyah-Bott) tion spaces for the operator T(f), and the following formula (of Atiyah-Bott) holds:

holds:

?1-T(f)ker L -n Tr T(f)IKer

L

T(f)koker L = c cn- T(f)i,,(pq) -n T(fh&,)). L = I: Q (Tr T(f)1'11(p'l) - Tr T(f)1'12(pq») .

Tr T(f)lcoker

q

124 124

Chapter Chapter 3. 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Homology

In In the the early early 1970s 1970s, in in the the course course of of investigating investigating the the “Euclidean” "Euclide an" 44dimensional dimension al theory theory of of Yang-Mills Yang-Mills fields (connexions (connexions on vector vector bundles), “in"instantons” stantons" were were discovered. These are special special solutions in in W4 JR4 of of the the Yang-Mills Yang-Mills equations, Ixl -+ -) 00, 00, so that, that, in in particular, particular, they they may may be equations, decreasing decreasing to to zero zero as as 1x1 extended extended to to the the sphere S4 54 = IX4 JR4 UU 00 00 (Belavin, (Belavin, Polyakov, Polyakov, Schwarz, Schwarz, Tyupkin). Tyupkin). Of Of particular particular interest interest is the the self-duality self-duality equation equation of of Belavin-Polyakov: Belavin-Polyakov: *F = fF,

F = Fabdxa A dxb,

(8.26) Fab

=

adb

-

ab&

i-

[&,

Ad,

for which in fact satisfor the the l-form 1-form of of aa connexion connexion on on aa vector vector bundle bundle over over S4, 54, w hi ch is is in fact satisfied full second-order fied by by instantons. instantons. (Of (Of course course instantons instantons satisfy satisfy the the full second-or der MaxwellMaxwellYang-Mills is first-order.) first-order.) ItIt turns turns Yang-Mills system system of of equations, equations, but but the the equation equation (8.26) (8.26) is out out that that all all functions functions globally globally minimizing minimizing the the functional functional -s= -5

=

!

!

S* 54

54 54

Tr (P (FA*F), 1"ab ) &d4x Tr (&,Fab) Jg d4x == s Tr À *1"), s Tr (1"ab

satisfy the self-duality equation, for given first first Pontryagin class satisfy the self-duality equation, for a a given Pontryagin class Tr (FA*F), PI = =const. Pl const.! s Tr (1" À *F),

G= SU(2). G = 5U(2).

54 54

By exploiting of the the manifold of self-dual self-dual connexions connexions on on aa given given By exploiting properties properties of manifold of manifold M4, Donaldson discovered, in the early 1980s deep topological prop4 manifold M , Donaldson discovered, in the early 1980s, deep topological properties of 4-dimensional closed manifolds. He was able to show, for instance, erties of 4--dimensional closed manifolds. He was able to show, for instance, that for a simply-connected manifold M44 the quadratic form defined by the that for a simply-connected manifold M the quadratic form defined by the intersection index on H2(M4; Q) can be positive definite only if it is reducible intersection index on H2(M4; Q) can be positive definite only if it is reducible over Z to a sum of squares. The remarkable equation (8.26) seemsto be speover Z to a sum of squares. The remarkable equation (8.26) seems to be specific, in a certain sense, to the geometry of 4-dimensional manifolds. Thus cifie, in a certain sense, to the geometry of 4-dimensional manifolds. Thus the theory of non-linear elliptic equations also turns out to be very useful for the theory of non-linear elliptic equations also turns out to be very useful for topology. topology. It is appropriate to mention also the analogues of K-theory for the category It is appropriate to mention also the analogues of K -theory for the category of G-spaces-the Kc-theory developed by Atiyah and Segal in the mid-1960s. of G-spaces - the Ke-theory developed by Atiyah and Segal in the mid-1960s. Of particular interest is the special case of the category of spaces with an Of particular interest is the special case of the category of spaces with an involution (G = Z/2), w here the group Z/2 acts on the fibers of vector bundles involution (G Z/2), where the group Z/2 acts on the fibers of vector bundles via complex “anticonjugation”. ( “KR-theory” was developed by Atiyah, and via complex "anticonjugation". ("KJR-theory" developed by Atiyah, and by W&'l Anderson.) These K-theories the important special case “KSC-theory”, the important special case "K 5C-theory", by Anderson.) These K-theories allow one to obtain various relations between invariants associated with fixed allow one to obtain various relations between invariants associated with fixed points and global invariants of a manifold, and to clarify the algebraic nature points and global invariants of aK-theory. manifold, and to clarify the algebraic nature of the g-fold periodicity of real of the 8-fold periodicity of real K-theory.

19. Bordism and cobordism cobordism theory theory as generalized generalized homology homology and and cohomology cohomology 125 §9. Bordism and 125

59. Bordism Bordism and and cobordism cobordism theory theory as generalized generalized homology homology §9. and cohomology. Cohomology operations in cobordism. and cohomology. Cohomology operations in cobordism. The Adams-Novikov spectral sequence. FormaI Formal groups. groups. The Adams-Novikov spectral Actions of cyclic groups and the circle on manifolds Actions of cyclic groups and the circle manifolds point of bordism theory the most most natural natural hoFrom a geometrical point of view, view, bordism theor'y is the homology theory. We start start with the cla..'lsical classical situation situation where where the the cycles cycles are are taken taken mology theory. We with the to In unoriented unoriented bordism bordism theory theory .rW fIp(.)(-) = = 5)1* n,( (.), .), aa to be be any any smooth smooth manifolds. manifolds. In singular of aa space space X X is is aa pair pair (Mn, (M”, 1), f), where M” is is aa closed closed manifold manifold singular n-cycle n-cycle of where Mn and ff is of the the manifold X: and is aa map map of manifold to to X: (Mn,f),

f : Mn --+ X.

Two singular n-cycles n-cycles (Mj,fI), (MT-, fl) , (M!i,h) (M; , fi) are are equivalent equivalent (or (or cobordant): cobordant): Two singular (Mr, fi) (MF, f2), if there exists a manifold Lnfl with boundary dL”+l == (Mj,fI) '" (M!i,h), ifthere exists a manifold Ln+l with boundary âLn+l Mp disjoint union) union) and and aa map map 9g :: Ln+’ + X such that that Mf UU M; M!i (the (the disjoint Ln+l ~ X such glhy

=

fl,

91&f;”

=

f2.

Here all manifolds involved into ]RN RN for for N N > >>> n. Here aIl manifolds involved are are assumed assumed to to be be embedded embedded into n. We now define the n-dimensional bordism group by We now define the n-dimensional bordism group by 0,0(X) = {n-cycles}/ N , il;; (X) = {n-cycles} / '" , where group structure is defined defined in the disjoint disjoint union union where the the (abelian) (abelian) group structure is in terms terms of of the of n-cycles: of n-cycles: [(MT-, f2)1 = [(Mf u 12)]. fi)] . [(Mf, fdl fd] + + [CM;, [(M!i, 12)] [(Mf u U M;, M!j, flfI U For a pair of spacesY c X the relative bordism groups fi,“(X, Y) are defined For a pair of spaces Y C X the relative bordism groups (X, Y) are defined analogously, using maps of manifolds (Mn, f) now possibly with boundary analogously, using maps of manifolds (Mn, 1) now possibly with boundary dM” # 0, where the boundary 8M is mapped into Y, and defining appropriâMn -:f. 0, where the boundary âM is mapped into Y, and defining appropriately the equivalence -. ately the equivalence "'. In the general case, there is a bordism theory @(X,Y) associated with In the general case, there is a bordism theory (X, Y) associated with any stable sequence of Lie groups G = (G,), G, c 0,. The basic examples any stable sequence of Lie groups C = (C n ), C n C On. The basic examples are are G = 0, SO, U, SU, Sp, e, C 0, SO, U, SU, Sp, e, where G, = 0,) G, = SO,, Gan = U,, G2n = SW,, Gdn = Spn, G, = 1. where Cthese, C nexamples SOn, may C Zn be = Un, C 2n bySUn, C4n Spn, C n = 1. n On, Besides more produced considering representations Besides these, more examples may be produced by considering representations

il;;

il:

P : G ---+

0,

P = (P,),

in

: Gn

-

On,

p: C ~ 0, P (Pn), Pn: C n On, pn : Spin, --+ SO, (a double for instance G = Spin = (G,) = (Spin.,), for instance C = Spin = (C = (Spin n ), Pn: Spin ~ SOn (a double ) n n cover). coyer). A manifold M is called a G-manifold when its stable normal bundle is is called aC-manifold stableis normal is A manifold endowed with aMG-structure. The bordism when theory itsa$(.) defined bundle as above,

endowed with a C-structure. The bordism theory

,a;? (-)

is defined as above,

126

Chapter 3. Simplicial Complexes and Homology Chapter Simplicial Complexes and CW-complexes. CW-complexes. Homology

assuming that all into ]RN, IWN, N > >>> n, n, and assuming that all manifolds manifolds involved involved are are embedded embedded into and that their normal bundles are furnished with a G-structure. The bordism that their normal bundles are furnished with G-structure. The bordism groups Y) have all the the basic basic properties properties of of a generalized homology thethen[?(X, Y) have aIl generalized homology groups O,“(X, ory: homotopy invariance, functoriality, fulfilment of the Excision Axiom for ory: homotopy invariance, functoriality, fulfilment of the Excision Axiom for pairs Y c X of CW-complexes: L’z(X,Y) = 02(X/Y,*), and existence of n[?(XjY, *), and existence of pairs Y c X of CW-complexes: n[?(X, Y) an exact sequence of a pair (X, Y): an exact sequence of a pair (X, Y): ” +- + ...

@(Y) + + n;;(X) f&y(X) -++ n;;(X, L?,G(X,Y)Y) -+ + n;;_l(Y) L?:-,(Y) --t+ .. .. .. .. n;;(Y) -

The bordism groups groups n[? O,“(*)(*) of the one-point one-point space space * * coincide coincide with with those those of of The bordism of the spheres; from the spheres; this this follows follows from the suspension suspension isomorphism isomorphism

c : fq(X, Y)

3

q!f+,(ZX,

ZY),

in view of in view of which which

.n,“(*) ” f2,“(* u *‘, *‘) 3 fq+&Y’“, *“). The bordism groups of aa point point are are called called the cobordism groups. groups. Their Their direct direct The bordism groups of the cobordism sum L?,G = @, a:(*), f urnished with the product operation induced by the sum n;, EBn n[? (*), furnished with the product operation induced by the direct product of manifolds, is the cobordism ring. In Chapter 4, $3, the reladirect product of manifolds, is the cobordism ring. In Chapter 4, §3, the relationship of the cobordism groups Og(*) to the spaces” MG (MG,) ) n;;;'(*) to the “Thorn "Thom spaces" MG = (MG tionship of the cobordism groups n of universal vector G,-bundles with base BG, is indicated. “Thorn’s theorem” of univers al vector Gn-bundles with base BG n is indicated. "Thom's theorem" yields yields .n,(*) 2 r&(MG) = rr,+k(A4Gk), > m m + + l. 1. n~(*) ~ 7r:n(MG) = 7r +k(MG ), kk > m

k

(Note that space MG, (n-1)-connected.) In particular particular for for G G == e, e, (Note that the the Thorn Thom space MG n is is (n-1)-connected.) In the groups 0, coincide with the stable homotopy groups of spheres, which the groups n;;;' coincide with the stable homotopy groups of spheres, which were end of of §7 57 above. above. Results concerning the the rings for at the the end Results concerning rings 0,” for were considered considered at other groups G will be given in Chapter 4, 53. other groups G will be given in Chapter 4, §3. Dual to the bordism theory @(.) is the cobordism theory JIG(.), the corDual to the bordism theory is the cobordism theory G(-), the corresponding cobordism groups being defined by responding cobordism groups being defined by

n;,

n;, (.)

fl,-(X, *) = Q$-m-l(SN

n

\ X, *‘),

N --+ co,

where the finite CW-complex X is embedded in a sphere of sufficiently large where the finite CW -complex X is embedded in a sphere of sufficiently large dimension. dimension. By proceeding according to the scheme laid out by Thorn (see Chapter 4, By proceeding according to the scheme laid out by Thom (see Chapter 4, $3) one obtains the following natural isomorphism: §3) one obtains the following natural isomorphism: f2Gm(X, *) E ,llm [Cn+mX, MG,] , nëm(x,*) ~ lim [En+mX,MG n ], n-.oo

(9.1) (9.1)

where [ , ] denotes the set of homotopy classes of maps. The direct sum where [ , 1 denotes the set can of homotopy of maps. The direct be given a classes multiplicative structure turningsum it f4gx, Y) = CB, .n,m(X, Y) n(;(X, Y) EBm n(j(X, Y) can be given a multiplicative structure turning into a skew-symmetric ring. This multiplication is defined as follows. For eachit For each intothea skew-symmetric ring. G, Thisare multiplication of above G, the groups “closed” with is defined respect as to follows. the operation of of the above G, the groups G are "closed" with respect to the operation of taking the direct sum of vectorn bundles in the sense that this operation gives taking the direct SUffi of vector bundles in the sense that this operation gives

39. Bordism cobordism theory homology and cohomology cohomology 127 §9. Bordism and cobordism theory as generalized generalized homology rise the corresponding classifying spaces spaces and of vector vector bundles bundles over over rise to to maps maps of of the corresponding classifying and of them: them:

(9.2) BGxBG+BG. BG x BG

---;. BG.

The universal The resulting resulting map of of univers al vector vector bundles

induces induces aa “multiplication” "multiplication" between between the the corresponding corresponding Thorn Thom spaces: spaces:

where VA1\ W W = V W/VIV V W is is the where V V xx W VW the “tensor” "tensor" spaces V, W. Applying the map to M Pn1 ,n2 to spaces V, W. Applying the map MQ,,,,, f:C

m1+n1x -

or smashproduct of the pointed or smash product of the pointed the product of maps f, g: the product of maps f, g:

g : Cm*+nnX ---+ M(Gn2),

M(G,,),

representing, elements of groups, we we obrepresenting, via via (9.1), (9.1), elements of the the appropriate appropriate cobordism cobordism groups, obtain the map tain the map

M@n,,&xg)

: ‘r ,,,1+,,iz+“1+“2X~M(G?11)~M(~,,)~~(~,,+,2),

yielding desired biliniar, associative multiplication on the direct sum sum yielding the the desired biliniar, associative multiplication on the direct L?;(X) of the corresponding cobordism groups: nê(X) of the corresponding cobordism groups:

L?,m’(X)~ f&y(X)(X) cc fl(yf”‘(X), n;.;n 1(X) . œ;'/l ne 1+m2 (X), Zl . z2 = (-lpm2Z2 A closed manifold IP with a normal A closed manifold Mn with a normal class [API E L$(AP) defined dament al class [Mn 1 E nr~ (Mn) defined

[M”] = [(iv,

There is a cap product operation which There is a cap product operation which and bordism classes, analogous to that and bordism classes, analogous to that cohomology (see Chapter 2, $3): cohomology (see Chapter 2, §3): O,-(X)

fJë!(X)

with the property with the property

n O/f(X)

n nf(x)

111.

can be defined between cobordism can be defined between cobordism defined in ordinary homology and defined in ordinary homology and

--) Lp(X),

nk-m(x) G ,

f*(a)flb=anf,(b)

f*(a) n ban f*(b)

for maps f of CW-complexes. In the case X = M” the map for maps f of CW-complexes. In the case X = Mn the map aHan[M”]

(9.3) (9.3)

G-structure possessesa natural funG-structure possesses a natural funby the identity map by the identity map

damental

1 : M” --+ M”;

..zl.

128 128

Chapter Simplicial Complexes Complexes and and CW-complexes. Homology Chapter 3. Simplicial CW-complexes. Homology

defines duality isomorphism defines the the Poincar&Atiyah Poincaré-Atiyah duality

This induces the operation operation of intersection intersection of bordism bordism classes classes This isomorphism induces dual to the classesdefined above: the product product of cobordism classes aob=D-‘(Da.Db). a 0 b D-1(Da· Db). As any generalized generalized cohomology cohomology theory, is aa spectral spectral sequence sequence (of (of As for for any theory, there there is rings) in cobordism theory (the Atiyah-Hirzebruch spectral sequence): rings) in cobordism theory (the Atiyah-Hirzebruch spectral sequence): d, : EP$ +

(Ekq, dm) i

Eg+W-mfl,

dk =O,

(9.4)

E;lq = HP(X; a;(*)),

L. Ekq E~q = G&$(X). GDê:(X).

with ring c with adjoined adjoined ring

p,q

Examples. G= = 0, 0, by theorem of Thorn from from the the mid-1950s, mid-1950s Examples. 1. 1. In In t;l the case case G by aa theorem of Thom the groups L’;(X) are isomorphic to direct sums of ordinary Z/2-cohomology the groups Dë(X) are isomorphic to direct sums of ordinary Zj2-cohomology groups dimensions. The The ring structure of of L’;(X) is given given by groups of of X X of of various various dimensions. ring structure Dâ(X) is by a canonical isomorphism a canonical isomorphism O;(X)

E H*(X;

a(;(*)),

where polynomial algebra algebra is the the polynomial where L?;(*) Dô (*) is

Dô(*)

= Zj2[V2,V4,V5,V6,VS, . .. ]

with generators vj of dimensions j = 2,4,5,. . ., j # 2”k with generators Vj of dimensions j 2,4,5, ... , j :f. 2 Hirzebruch sequence collapses (d, = 0 for m 2 2, Ehq = Hirzebruch sequence collapses (dm 0 for m 2': 2, E~q adjoined ring Go;(X) = c Egq is isomorphic to L’;(X). adjoined ring GDô(X) E~q is isomorphic to Dô(X).

L.

p,q 2. In the caseG = SO, EG structure of the ring Q&,(X)

1. The Atiyah-

1. The AtiyahEi”), and and the the EP,q) 2 ,

@Z(z), where 2~2)

2. In the case G = SO, the structure of the ring Dsa (X) ® Z(2), where Z(2) is the ring of rationals with odd denominators (i.e. Z localized at the prime is the ring of rationals with odd denominators (Le. Z localized at the prime 2) also reduces to that of the ordinary cohomology ring with coefficients from 2) also reduces to that of the ordinary cohomology ring with coefficients from Z(z) (by results of Novikov, Rohlin and Wall from around 1960). On the Z(2) (by results of Novikov, Rohlin and Wall from around 1960). On the other hand, the theory LZ&,(.) @Z@) does not reduce to ordinary cohomology other hand, the theory D'Sa (.) ® Z(p) does not reduce to ordinary cohomology for any prime p # 2. In this case the ring L’&, @izc,) is a polynomial ring for any prime p :f. 2. In this case the ring D'Sa ® Z(p) is a polynomial ring z(,) [XI, ~2, . .I with generators in each negative dimension of the form 4k Z(p) [Xl, X2, ... ] with generators in each negative dimension of the form 4k (Milnor, Novikov; around 1960). (Milnor, Novikov; around 1960). 3. The case G = U is the most interesting one, and provides a basis for 3. The case G =algebraic U is the most interesting one, and provides a basis applications of the techniques of cobordism theory. The ring 06 for is applications of the algebraic techniques of cobordism theory. The ring nû a polynomial ring with one generator of each even negative dimension (Milnor,is aNovikov; polynomial with 4, one53): generator of each even negative dimension (Milnor, see ring Chapter Novikov; see Chapter 4, §3): n* "'"

'7J[U l,U2,,···,1

Jt[J=IL.J

n- 2j =J&2j' nU

UjEHu

$9. homology and 129 §9. Bordism Bordism and and cobordism cobordism theory theory as as generalized generalized homology and cohomology cohomology 129

The Atiyah-Hirzebruch The Atiyah-Hirzebruch sequence sequence collapses collapses only only for for complexes complexes X X with with torsiontorsionfree such X free integral integral cohomology cohomology groups; groups; however however even even for for such X the the adjoined adjoined ring ring Go;(X) need L?;(X). Gnû(X) need not not be be isomorphic isomorphic to to nû(X)' 4. The the ring @ Zt,), > 2, of The structure structure of of the ring Q& ® Z(p), for for pp > 2, as well weIl as some some of 2-primary torsion of fig,, were described by Novikov in the early 1960s. The 2-primary torsion of were described by Novikov in the early 1960s. The structure (2) was completely elucidated structure of of fig, 8® ZZ(2) was subsequently subsequently completely elucidated by by Conner Conner and Floyd (in the mid-1960s). and Floyd (in the mid-1960s). 5. The The case turns out 5. case of of the the symplectic symplectic cobordism cobardism ring ring 0:” n:p turns out to ta be be much much more complicated than those of the above cobordism rings. It is known that more camplicated than those of the above cobordism rings. It is known that p is 2-primary all torsion of the ring Q,Sp (Novikov, 1960), and that the Ray aIl torsion of the ring is 2-primary (Novikov, 1960), and that the Ray elements ifJj $j E E L’fJT-s (indecomposables 2) play exclusively imporimpor(indecomposables of of order order 2) play an an exclusively elements tant role in all computations of the ring R, sp. The additive structure of tant role in all computations of the ring n:p. The additive structure of L’s” n: p has computed up 120 by Adams spectral has been been computed up to to dimension dimension 120 by means means of of the the Adams spectral sesequence (Kochman, mid 1980), and the ring structure in the same dimensions quence (Kochman, mid 1980), and the ring structure in the same dimensions has essentially been determined by by Vershinin the Adams-Novikov Adams-Novikov has now now essentially been determined Vershinin via via the spectral sequence shows that ring of spectral sequence (1990). (1990). Vershinin Vershinin also also shows that the the cobordism cobordism ring of symplectic manifolds manifolds with symplectic with singularities singularities (Ray (Ray elements) elements) reduces, reduces, in in fact, fact, to to the the cobordism ring Botvinnik has described the cobordism ring 0,“. n,;'. Using Using this this Botvinnik has described the Adams-Novikov Adams-Novikov spectral sequence for symplectic cobordism in terms of cobordism singuspectral sequence for symplectic cobordism in terms of cobordism with with singularities; further computations have led to the detection of nontrivial elements larities; further computations have led to the detection of nontrivial elements cl of 0:” p of order 2” Kochman, 1992). of of order 2k for for any any Ic k (Botvinnik, (Botvinnik, Kochman, 1992). 0

nsu,

n'Su

n;!_5

nsu

n:

n:

The Conner and and Floyd Floyd focussed the fact The work work of of Conner focussed attention attention on on the fact that that there there exist Chern characteristic classes in complex cobordism theory ofi exist Chern characteristic classes in complex cobordism theory nû (.) (and, (and, analogously, there there exist theanalogously, exist Stiefel-Whitney Stiefel-Whitney classes classes in in unoriented unoriented cobordism cobordism theory L?G(.)). These characteristic classes with values in cobordism groups ory nô (.)). These characteristic classes with values in cobordism groups are are direct analogues of the Chern and Stiefel-Whitney classes in ordinary cohodirect analogues of the Chern and Stiefel-Whitney classes in ordinary cohomology. mology. In the case of a manifold M” and a vector bundle r] over M”, this implies In the case of a manifold Mn and a vector bundle T] over Mn, this implies that the cycles in M dual to the characteristic classes cj(q), wi(n) may always that the cycles in M dual to the characteristic classes Cj(T]), Wi(T]) may always be realized as images of manifolds. The first indication of this property of the be realized as images of manifolds. The first indication of this property of the characteristic cycles was given by Gamkrelidze (in the early 1950s) when he characteristic cycles was given by Gamkrelidze (in the early 1950s) when he proved that the characteristic cycles of algebraic varieties in CP may be proved that the characteristic cycles of algebraic varieties in Cpn may be realized by algebraic subvarieties. realized by algebraic subvarieties. As will be indicated in Chapter 4, $3 (see also $6 above) there exists a close As will be indicated in Chapter 4, §3 (see also §6 ab ove) there exists a close link between the Stiefel-Whitney characteristic classes in Z/Zcohomology and link between the Stiefel-Whitney characteristic classes in Zj2-cohomology and the Steenrod squares, given by the formula of Thorn; in fact one has a reprethe Steenrod squares, given by the formula of Thom; in fact one has a representation of all Steenrod operations a E AZ in the cohomology of the product sentation Steenrod operations a E A 2 in the cohomology of the product lrw,” x . .of . xall llw,-:

IRPi x ... x IRP~:

al--> a(u),

U

Ul ...

Un

E

Hn(IRPi

X ... X IRP~).

By means of this representation one can define appropiate analogues of the By meansoperations of this representation one can defineQ(;(.) appropiate analogues of the Steenrod for the cobordism theories and L$(.). Steenrod operations for the cobordism theories nô(-) and nûC)' In what follows we confine ourselves to complex cobordism theory L?; (.). In theory what follows weanalogous, confine ourselves to complextrivialization cobordism theory (The a;(.) is with considerable occuring.)nûo. In

(The theory

nô (-)

is analogous, with considerable trivialization occuring.) In

130 130

Chapter Simplicial Complexes Complexes and CW-complexes. Homology Chapter 3. Simplicial and CW-complexes. Homology

the cobordism L’:(X) of any CW-complex CW-complex X, X, there is aa subset subset W(X) W(X) cobordism group Db(X) of cobordism” classes, classes,defined as as follows. Since Since the Thom Thorn space space of ‘Lgeometric "geometric cobordism" M(U2) (CP” = K&2) (see Chapter 4, §3), 53), there M(U2) may be identified identified with with Cpoo = K(Z,2) (see Chapter 4, corresponds to class z E E H2(X; homotopy class class to each cohomology class H 2 (X; Z) aa unique homotopy of maps maps fifz :: X x -+ --$ M(U2) M(U2) = = Cpoo, (CP”, such that f,‘(u) = z, where u u E H2((CP”;Z)Z) is generator; the the map map fifz such that J;(u) = z, where E H2(Cpoo; is aa generator; thus determines a cobordism class (geometric cobordism class) [z] E L@(X). thus determines a co bordism class (geometric cobordism class) [z J E D3 (X). The set W(X) of all geometric cobordism classes corresponds one-to-one with The set W(X) of aIl geometric cobordism classes corresponds one-to-one with 2 H2(X; additive. H (X; Z); Z); however however this this correspondence correspondence is is not not additive. 2n )) set For U-manifolds M2n one obtains a dual (to W(M2n)) set of of divisors divisors For U-manifolds M2n one obtains a dual (to W(M

also also not not additive additive in in Q~~,_2(A42n). Dfn_2(M 2n ). For each complex vector bundle over X X we we define its first Chern characcharacFor each complex vector bundle 1Jq over define its first Chern teristic class al(n) E L’s(X) to be the geometric cobordism class [cl(n)] corteristic class (Jl(1J) E D3(X) to be the geometric cobordism class [Cl(1J)] cor,in ordinary cohomology. responding, as above, to the first Chern class ci(q responding, as above, to the first Chern class Cl (1JJ2,in ordinary cohomology. The further Chern classes(Jj(1J) aj(q) E E Ou3(X), 1< < jj S 5 dimcr], are The further Chern characteristic characteristic classes DJ(X), 1 dimC1J, are then determined by general functorial properties and the Whitney formula: then determined by general functorial properties and the Whitney formula: 4%

@ 772)

=

o(q) = 1 + 01(n) +

4771)4772)1

az(T7)

+

. ‘.

(9.5)

The images of the ordinary again the the aj(~) (Jj(1J) in in the ordinary cohomology cohomology groups groups are are again the The images of the ordinary Chern characteristic classes. The Chern characteristic classes are ordinary Chern characteristic classes. The Chern characteristic classes are closely related to cohomology operations in complex cobordism theory O;(.); closely related to cohomology operations in complex cobordism theory Dû (.); the latter are determined by the following conditions: the latter are determined by the foIlowing conditions: 1. For each finite sequence w = (wi, . . . , wk) of integers there is a stable, 1. For each finite sequence W = (Wl,'" ,Wk) of integers there is a stable, additive operation

additive operation

s, : O&(X) + 0,“2’“‘(X), 2lwl 8 w : Dt(X) -+ Db+ (X),

degs, = 21~1,

deg 8 w

= 21wl,

k

Iwi = I: Wi· i=l Jw[

=

$wi.

(9.6) (9.6)

i=l

2. The following product formula holds:

2. The following product formula holds:

I:

s,(ab) = (ab) =

c sc./(ah/~ (b). 8 ,(a)sw,,(b). (w’,d’)=w w

8w

(w',w")=w

3. The operation SOis the identity operator: se = 1; if a E Q;(X) is a 3. The operation 8Q is the identity operator: 8Q 1; if a E Db(X) is a geometric cobordism class, then

geometric cobordism class, then

aW1+’ for s,a = i

0

w = (WI),

= (wt),

for

W

for

w=(wi

,..., wk),

k>l.

4. If a product s,~ o S,~Jof total degree < 2n, vanishes on the element 8 w ' 08 w " of total degree < 2n, vanishes on the element

4. If a product

$9. Bordism and cobordism cobordism theory theory as generalized generalized homology homology and and cohomology cohomology 131 §9. Bordism and 131 Ul . . u, E f2$yCP,oo

x . . . x CP,-),

where the Uj uj EE n'b(Cpr) fl~(CPJ~) are are the standard generators, then the product product is is the null operation: sWl o sw” = 0. From this condition the composition formula the null operation: Sw' 0 SW" O. condition follows: follows: SW’ SW”(IL) = = cLÀw(w',w")sw, XJW’, W”)& sw' 00 sw,,(u) w w

where the Àw &,,(w’, are integers. integers. It It follows follows also that the the element element sw S,(U), where the (w', w”) w") are also that (u), 11 = Ul”‘U,, form of of a monomial "symmetrized “symmetrized with with respect respect to to U U1 ... Un, has has the the form a monomial WI,. , wn’): s,(u) Sw(U)

(L U~l . . . . . thy; u~~,)> Ul . .

= (-&z =

u,. U1 ...... Un·

.....

The s, are are called called Landweber-Novikov Landweber-Novikov operations. operations. These These afford afford The operations operations Sw a Landweber-Novikov (LN) (LN) algebra algebra SS over over :E, Z, which which is is aa Hopf Hopf a basis basis for for the the Landweber-Novikov algebra the obvious obvious (diagonal) comultiplication, defined defined according according to to the the algebra with with the (diagonal) comultiplication, same scheme as Milnor uses for defining the classical Steenrod algebra: the same scheme as Milnor uses for defining the classical Steenrod algebra: the didiagonal formula” for the action action of of the the operations operations agonal is is determined determined by by the the “Leibniz "Leibniz formula" for the on the (see above). above). Such of Hopf algebras on on the product product of of two two elements elements (see Such actions actions of Hopf algebras on modules equipped with a multiplication, are therefore called Milnor modules. modules equipped with a multiplication, are therefore caUed Milnor modules. ItIt is is not difficult to to show show that that the the algebra algebra AU of aU all stable stable operations in not difficult AU of operations in complex cobordism theory consists precisely of the linear combinations complex cobordism theory consists precisely of the linear combinations

where the Xj E A = ~7; have negative dimensions (degrees):

nû

where the Àj E A

xj

have negative dimensions (degrees):

E QG2'j

=

/ie2qj,

degXj

=

-zqj,

and the elements s, have positive degrees:

and the elements

Sw

have positive degrees:

k

degs, = 21~1= 6w.j.

degs w

21wl

LWj.

j=l j=1

The algebra AU has the grading

The algebra AU has the grading

where Ayn consists of the operations of degree 2n, i.e. of the linear combinaconsists of the operations of degree 2n, Le. of the linear combinawhere tions tions 9 deg Xj + deg s,(j) = 2n. cL XjS&) ÀjSw(j), deg Àj + deg Sw(j) 2n.

Arn

j

Here infinite formal series are allowed, with deg s,(j) -+ co as j + 00. Here infinite formaI series are aUowed, with deg Sw(j) ~ 00 as j ~ 00.

132 132

Chapter Chapter 3.3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Homology

The The action action of of the the operations operations s, Sw on on the the cobordism cobordism ring ring J2’; nô = AA can can be be described may be described explicitly explicitly using using particular particular manifolds. manifolds. Each Each element element of of nAmay be represented classes the cobordism cobordism classes represented as as aa polynomial polynomial with with rational rational coefficients coefficients in in the of the manifolds (cP1, . , cCP, . . The tangent bundle r((CP) decomposes of the manifolds Cp!, ... cpn, .... The tangent bundle r(cpn) decomposes as foUows (see (see Chapter Chapter 4, 51): §1): as follows T(cPn) l)rj, r(cpn) ++ E$ e:~ == (7% (n ++ 1)1'/, where have where q1'/ isis the the canonical canonical line Hne bundle bundle over over U’“, cpn, for for which which we we have q(r)) D(u)

= u E n$(@Pn),

= p-l]

E L?&-2(@P”).

ItIt follows identified with follows that that the the stable stable normal normal bundle bundle v(@P) v(cpn) may may be be identified with -(n -(n ++ 1)~. 1)1'/. For For the the element element -77 -1'/ EE KO((CP”) KO(cpn) the the Chern Chern characteristic characteristic classes classes in follows: in cobordism cobordism are are as as follows: q(q)

= (-1)&j,

Duj = [CP”-j]

We define the class uw, w = (WI, . ,,Wk), wk), as as follows. follows. ConConWe now now define the characteristic characteristic class (1w, W (W1, ... sider the symmetric polynomial sider the symmetric polynomial QW(ulr...,uk)=

xu;la(l)

. . ..$+).

where all elements elements (Ia of on kk letwhere summation summation is is over over aU of the the symmetric symmetric group group on letters. The polynomial Q(ui, . . . , Uk) is uniquely expressible as a polynomial ters. The polynomial Q(Ul>"" Uk) is uniquely expressible as a polynomial P”(ol, 02, ....)) in in the generators (1 aj,j, where where uj is the jth elementary symmetric pw ((11, (12, the generators (1 j is the jth elementary symmetric polynomial in ui, . . , uk. Let Ow, w = (WI,. . , wk), denote the characteristic polynomial in Ul, ... , Uk. Let (1w, W (W1, ... , Wk), denote the characteristic class class u w- - P”(fq, c72,. . .). (1w PW((11,(12, .. ')· We now apply the class ow to the bundle -(n + 1)~ (stably equivalent to the We now apply the class (1w to the bundle -(n + 1)17 (stably equivalent to the normal bundle v(@.P”)), w h ose Chern class a(-(n + 1)~) is the following: normal bundle v(cpn)), whose Chern class 0'( -(n + 1)1'/) is the following: o(-(n

0'( -(n

+ l)T/)

=

1 + dl

+ a2 + ‘. . =

1

(1 ++ &+I. + 1)1'/) = 1 + 0'1 + 0'2 + ... = (1 u)n+1 .

A simple computation shows that A simple computation shows that O,$” ((CP) is the cobordism class n~lwl(Cpn) is the cobordism class determines the cobordism operation

the dual of the class a,(-(n

+ 1)~)

E

the dual of the class (1w( -(n + 1)1'/) E [UY”-l”1] times an integer X(w). This [cpn-1w l] times an integer ,x(w). This s, as follows: determines the cobordism operation Sw as follows: &JCP”) = X(w) [@P”+q ) sw(cpn) = 'x(w) [cpn-1wl],

k IWI = 5wj

Iwi = 2:Wj. j=l

(9.7)

(9.7)

j=1

A complete description of the action of the cobordism operations on the coborA complete actionfrom of the(9.7) cobordism operations obtained using the properties:on the cobordism ring 0;description = n is nowof the

dism ring

nô = A is now obtained from

(9.7) using the properties:

$9. cobordism theory cohomology 133 §9. Bordism Bordism and cobordism theory as generalized generalized homology homology and cohomology

L

8 w' (p,)

. 8 w" (p,),

P, E

A,

(w',w")=w

(9.8)

L

8W

'

(p,)

0

8 w ",

P, E

A,

(w',w")=w

together with SO == 11 (for (for the the empty u). (This (This was was established established by by Novikov together with 80 empty w). Novikov in the mid-1960s.) This representation of the algebra AU of operations of in the mid-1960s.) This representation of the algebra AU of operations of complex cobordism theory Q;(.) by means of its action on the coefficient complex cobordism theory Dû (.) by means of its action on the coefficient ring Q; is faithful, i.e. if if aa E AU acts acts trivially trivially on then aa == O. 0. This This Dû == A, A, is faithful, i.e. E AU on AA then ring property represents a cardinal distinction between complex cobordism theory property represents a cardinal distinction between complex cobordism theory and cohomology. and ordinary ordinary cohomology. It is to note note that the algebra as the the complet completion It is of of interest interest to that the algebra AU AU may may be be given given as ion of the product of two algebras with respect to the topology of formal series: of the product of two algebras with respect to the topology of formaI series: AU = (A ~8 S)T. This is isomorphic as vector vector space space to to the tensor product of these these algealgeisomorphic as the tensor product of This product product is bras, but the multiplication is not the usual one. They form non-commuting bras, but the multiplication is not the usual one. They form non-commuting subalgebras; formula for for their given above above (see (see 9.8) 9.8) subalgebras; the the Novikov Novikov formula their commutator commutator given depends on the action of the Landweber-Novikov algebra S on the complex depends on the action of the Landweber-Novikov algebra S on the complex cobordism described above. above. What significant here this action action cobordism ring ring described What is is significant here is is that that this determines a “Milnor module” over the Hopf algebra S, i.e. that it satisfies determines a "Milnor module" over the Hopf algebra S, Le. that it satisfies the Leibniz product formula with respect to the comultiplication. The algebra the Leibniz product formula with respect to the comultiplication. The algebra AU is realized as the algebra of operators on the (left) Milnor module A over AU is realized as the algebra of operators on the (left) Milnor module A over the Hopf algebra S, realized in turn as the algebra of differential operators the Hopf algebra S, realized in turn as the algebra of differential operators with constant coefficients, acting on the functions “belonging to the ring A”. with constant coefficients, acting on the functions "belonging to the ring A" . This is a special case of a rather general construction of “operator algebras” This is a special case of a rather general construction of "operator algebras" by Novikov (in the early 1990s) in the context of the theory of Hopf algebras by Novikov (in the early 1990s) in the context of the theory of Hopf algebras and Milnor modules over them. and Milnor modules over them. In 1978 Buchstaber and Shokurov observed that the natural dual Hopf In 1978 Buchstaber and Shokurov observed that the natural dual Hopf algebra to the Landweber-Novikov algebra S, contains the ring .4 of “complex algebra to the Landweber-Novikov algebra S, contains the ring A of "complex cobordisms for the one-point space”:

cobordisms for the one-point space":

A@Q=S@Q. ncx*, Ac X*, A ® Q = S* ® Q. (Via the Chern numbers (giving the integral structure of A as known from (Via the Chern the integral of A asby known from earlier results of numbers Milnor and(giving Novikov) the ring structure A was analysed Buchstaber earlier results of Milnor and Novikov) the ring A was analysed by Buchstaber from this new point of view in the late 1970s; its structure turned out to be from new pointWeof may viewtherefore in the late 1970s; structure turned out tocase be highly this non-trivial.) regard this itsconstruction as a special highly non-trivial.) We may therefore regard this construction as a special case of the very general construction of the “operator double” of the Hopf algebra of the very general construction the "operator Hopf algebra S and its dual S*, in the sense ofof Novikov (1992).double" There of arethe natural actions Sand its dual S*, in the sense of Novikov (1992). There are natural actions R;t, Lb* of an arbitrary algebra S on its dual space S*, which are adjoint to R~, Lb of an arbitrary algebra S on its dual space S*, which are adjoint to right and left multiplication respectively:

right and left multiplication respectively: %(a)

Rb(a)

= L,(b)

= ah (Rj(z),z)

La(b) = ab, (RZ(z),x)

= (z, R&))

(z,Ry(x))

= (z,zy).

(z,xy).

134 134

Chapter Homology Chapter 3. 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology

In situation of Hopf algebras one has i.e. S* is a In the situation Hopf algebras has the Leibniz Leibniz property, property, Le. left (or right) Milnor module over S in the natural way; in fact S* can can be natural left right) Milnor made into a left Milnor module over the larger Hopf algebra S @I St using the into left Milnor Hopf S@ representation p* satisfying representation P± satisfying

st

p±,a®b(X) "" R~(L;±lb(x)), Hopf algebra, and St denotes the algealgewhere 8s indicates indicates the “antipodes” "antipodes" of aa Hopf st denotes bra S with consequently with the opposite opposite bra S with “transposed” "transposed" comultiplication comultiplication and and consequently with the antipodes. of the action of of p*i to the diagonal Ll(S) A(S) C c S S@ the action P±l to the diagonal @ St st antipodes. Restriction Restriction of yields an “ad-module”. diagonal restriction restriction of of the the operator operator algebra algebra actactyields an "ad-module". The The diagonal ing S* (as Milnor module) leads to to Drinfeld's Drinfeld’s "quantum “quantum double" double” of of the the ing on on S* (as Milnor module) leads Hopf again a with diagonal diagonal as in S* S* and and st. St. Hopf algebra algebra S, S, again a Hopf Hopf algebra, algebra, with as in Buchstaber and Shokurov have identified identified the algebra AU A’ with the algeBuchstaber and Shokurov have the algebra with the algebra of differential on the the group group of of formaI formal diffeomorphisms diffeomorphisms of of the the bra of differential operators operators on real line fixing with first first derivative derivative at at 00 equal equal to to 1. 1. It It follows follows (as (as was was real line fixing 00 and and with pointed 1992) that the algebra AU is is the the operator double that the algebra AU operator double pointed out out by by Novikov Novikov in in 1992) of the S corresponding corresponding to to the the Milnor Milnor module S* of the Landweber-Novikov Landweber-Novikov algebra algebra S module S* equipped with the action Rz; this affords a basis for the general notion of the equipped with the action R~; this affords a basis for the general notion of the “quantum analogue of rings rings of differential operators" operators” according according to to Novikov's Novikov’s "quantum analogue of of differential scheme (1992), where such an analogue was considered in the case of Fourier scheme (1992), where such an analogue was considered in the case of Fourier transforms. Various “almost-Hopf” properties of such operator doubles transforms. Various "almost-Hopf" properties of such operator doubles were were investigated 1992) and Buchstaber (1994). (1994). A special case case of of investigated by by Novikov Novikov (in (in 1992) and Buchstaber A special this construction of operator doubles was investigated independently in 1992 this construction of operator doubles was investigated independently in 1992 by Faddeev and who used used instead instead the the term by Semenov-Tyanshanskii, Semenov-Tyanshanskii, Faddeev and Alexeev, Alexeev, who term “Heisenberg double”. "Heisenberg double" . It turns out that complex cobordism theory 0;(.) may be used very efIt turns out that complex cobordism theory Dû (.) may be used very effectively for computing stable homotopy classesof maps of finite complexes fectively for computing stable homotopy classes of maps of finite complexes by means of the Adams-Novikov spectral sequence(an analogue of the Adams by means of the Adams-Novikov spectral sequence (an analogue of the Adams spectral sequence described in §6,7 above). The group of stable homotopy spectral sequence described in §6,7 above). The group of stable homotopy classesof maps K --+ L is defined by

classes of maps K ---- L is defined by

[K, L]’ = /imJCjK,

ZjL].

[K,LJs = lim [EjK,EiLJ. J-OO

Set

Set

[K, L]; = [CqK,L]‘,

[K,L]~

[PK,L]S,

[K, L]; = @[K,

[K,L]!

L];.

EB[K,L]~. 4

q

Thus from the stable homotopy classesof maps K -+ L, Thus from the stable homotopy of maps K ---- L, abelian group [K, L]S,. For K =classes L = So, in particular, abelian group [K,LJ:. For K = L = SO, in particular, sum of stable homotopy groups of spheres:

we obtain the graded we we obtain obtain the thegraded direct

we obtain the direct

sum of stable homotopy groups of spheres:

[SO,so]: = @r; = @rN+q(SN), [SO, SOl! = EB7r~ EB7rN+q(SN), q20 n20

q~O

N > q + 1. N > q + 1.

q~O

How does one compute [K, L]“,? Let M, N denote the graded Au-modules How one compute [K,L]!? Letexists M, Na denote graded (the AU-modules Q;(K),doesO;(L) respectively. There spectralthe sequence AdamsDû (K), Dû (L) respectively. There exists a spectral sequence (the AdamsNovikov spectral sequence)

Novikov spectral sequence)

19. §9. Bordism Bordism and and cobordism cobordism theory theory as as generalized generalized homology homology and and cohomology cohomology 135 135

Ekq,

°

EP,q --+ ~ Eg-mVq+m-l, EP-m,q+m-l d, dm·:. Ekq m m'

2 = 0, dm d; ='

E;lI

= H(E;*,d,),

(9.9)

with with second second term term E;,q E~,q = Ext;: Ext~3 (N, (N, M) M),, and and adjoined adjoined group group G[K,L]; G[K,Ll~ =

L

c Egq E~? p-q=n p-q=n

Note Note that that Ext:$ Ext~3 (N, (N, M) M) coincides coincides with with the the set set of of Au-module AU -module homomorphisms homomorphisms N = R;(K)

+

f2;(cqL)

= CQM,

of q: of degree degree q: Ext>z Ext~3 (N, (N, M) M) a! ~ Horn:, Hom~u (N, (N, M). M). Here CQM coincides with M, but the grading is shifted Here Eq M coincid es with M, but the grading is shifted by by q. q. The definition of the functor Ext>* ( , ) (basic to homological The definition of the functor Ext~* ( , ) (basic to homological algebra), algebra) , where (in the case when when A Given any where A A isis aa ring, ring, is is as as follows follows (in the case A == A’). AU). Given any AUAU_ module N, we can form its free acyclic resolution (respecting the grading module N, we can form its free acyclic resolution (respecting the grading on on N); N); this this is is aa complex complex C C of of Au-modules AU -modules of of the the form form . . . +cc,~c,-$+c,-z d ... ~ Cn ~ Cn -

d l ~

Cn -2

-%...~C1%C,,$V--,O, d d d ~ ... ~

Cl

~

Co

d ~

N

~

0,

where Cj are free Au-modules (whose free generators can be chowhere all aIl of of the the C j are free AU -modules (whose free generators can be chosen compatibly with the grading), the operators the grading, sen compatibly with the grading), the operators dd preserve preserve the grading, and, and, finally, Ker d = Im d everywhere. We therefore have N g Co/Im d. We now finaIly, Ker d lm d everywhere. We therefore have N ~ Co/lm d. We now form the dual complex C* = HomAu(C, M): form the dual complex C* = HomAu (C, M):

...

8" 8" 8* ~ C~ ~ C~-l ~

...

8* ~

Ci

8" ~ C~ <-

0,

where

where c; = @qq 9

= @HomP,U(C,, 4

q

q

M) = @Homiu(&, Q

CqM).

q

The homology groups of the complex C* are then the objects Ext;::

The homology groups of the complex C* are then the objects Ext~~: @ ExtpAz (N, M) = @ Ker a*/Im d*. Ext~3 (N, M) = Q Ker â* /Im â*. 4

EB

EB

q

q

It can be shown that these groups do not depend on the choice of the free It can be shown that these do not depend on the choice of the free Au-module resolution C withgroups the prescribed properties.

A U -module resolution C with the prescribed properties.

Remark. The objects Ext>* for an arbitrary ring A, were first introduced for an arbitrary in ring were first The introduced The objects intoRemark. homological algebra Ext~* by Eilenberg-MacLane theA,late 1940s. followinto homological algebra by Eilenberg-MacLane in the late 1940s. The following theorem is due to them: ing to them: If Atheorem = Z[T], isthdue e integral group ring of a group x, and Z is given the trivial If A Z[1I'], the integral a group 11', for andany Z isA-module given theM:trivial A-module structure, then group there isring an of isomorphism

A-module structure, then there is an isomorphism for any A-module M:

136 136

Chapter 3. Simplicial Simplicial Complexes and CW-complexes. CW-complexes. Homology Chapter 3. Complexes and Homology

Ext>(Z,

M) ” Hq(K(n,

1); M).

0 o

In case K spectral sequence sequence converges converges to to In the the case K = So, SO, the the Adams-Novikov Adams-Novikov spectral the groups of complex L: L: the stable stable homotopy homotopy groups of the the complex A4 = n .n;,ù, M == nA =

N Q;(L), N == nù (L),

EZpyq= Ext”A;(N, E~,q Ext~~(N, /l). A). The existence existence of of the the Adams-Novikov Adams-Novikov spectral sequence sequence (9.9) (9.9) with with second second The spectral term Ext>: represents aa quite general categorical categorical property of aa generalized generalized represents quite general property of term Ext~~ cohomology theory theory hh*(.) on the category of of stable homotopy cohomology * (.) defined defined on the stable stable category stable homotopy types and stable stable homotopy homotopy classes classes of (Note that the stable stable category category is is types and of maps. maps. (Note that the not in the the sense although stable stable homotopy homotopy classes classes of of not abelian abelian in sense of of Grothendieck, Grothendieck, although maps always come natural structure structure of abelian group. group. Note maps always come with with the the natural of aa graded graded abelian Note also K-theory there is is no spectral sequence sequence since since that that theory theory also that that in in K -theory there no such such spectral is defined of non-stable objects.) The usefulness of of aa generalized generalized is defined in in terms terms of non-stable objects.) The usefulness cohomology theory h*(.) f or solving particular problems depends of course course cohomology theory h*(-) for solving particular problems depends of on the specific properties of that theory: how efficiently the algebra Ah of of on the specifie properties of that theory: how efficiently the algebra Ah stable operations can be computed; to what extent the structure of the Ahstable operations can be computed; to what extent the structure of the Ah_ module be worked out for for particular particular spaces spaces X, X, and, finally, how how module h*(X) h*(X) can can be worked out and, finally, many non-trivial differentials d, there are in the Adams-Novikov spectral many non-trivial difIerentials dm there are in the Adams-Novikov spectral sequence corresponding to the the theory theory h*(-) h*(.) (the fewer, the the better). better). In In the the sequence corresponding to (the fewer, latter respect complex cobordism theory Q,$(.) has a significient advantage latter respect complex cobordism theory nùO has a significient advantage over ordinary Z Z/pcohomology theory (especially (especially for for primes primes pp > > 2), 2), and and is is over ordinary / p-cohomology theory essentially no worse as far as computations of the appropriate algebraic objects essentially no worse as far as computations of the appropriate algebraic objects are concerned. concerned. For example for the stable stable homotopy groups of of spheres spheres one one has has are For example for the homotopy groups E;l* = Ext=;;(n,(A /l), E 2*,* -- EX t*,* AU , A) , and the groups ExtiT, contain the lower estimate of Kervaire-Milnor for the and the groups Ext~~ contain the lower estimate of Kervaire-Milnor for the groups Jr,- 1(SO), which is not the case for the ordinary Adams spectral groups J7r q _1(SO), which is not the case for the ordinary Adams spectral sequence. (For the definition of the J-homomorphism, see $8 above, and Chapsequence. (For the definition of the J-homomorphism, see §8 above, and Chapter 4, $3.) The groups Ext”*AU yield much new information, as do the higher ter 4, §3.) The groups Ext~~ yield much new information, as do the higher terms. In the 1970s several authors carried out far-reaching calculations of terms. In the 1970s several authors carried out far-reaching calculations of the Adams-Novikov spectral sequence for spheres (Miller, Ravenel, Wilson, the Adams-Novikov spectral sequence for spheres (Miller, Ravenel, Wilson, Buchstaber and others) .14

Buchstaber and others) .14

The first Chern class 01 in complex cobordism theory has certain remarkThe first Chern class al in complex cobordism theory has certain remarkable properties. It was mentioned above that for a complex line bundle q, the able properties. It was mentioned above that for a complex line bundle "7, the class a(q) is a geometric cobordism class. How may one compute the class class 0'(1]) is a geometric cobordism class. How may one compute the class al(ql@ r/z) for line bundles ~1, r]z? 0'1(1]101]2) for Une Write (T~(T]I) = U,bundles gl(r/z) =1]1,V.1]2? It can be shown that the class gl(ql 8~~) is Write 0'1(1]1) U,0'1(1]2) be shown in that class 0'1(1]10"72) is representable as a power series v.inItu,can 2, (Novikov, thethe mid-1960s):

representable as a power series in u, v (Novikov, in the mid-1960s):

l4 7Vunslator’s note: For a general account of the results on computations in the Adamsnote: For aforgeneral the results on computations in the AdamsNovikovTranslator's spectral sequence spheres account see the ofbook D. Ravenel, Complex Cobordism and Novikov spectral sequence spheres see the book Stable Homotopy of Spheres, forAcademic Press, 1986. D. Ravenel, Complex Cobordism and

Stable Homotopy of Spheres, Academie Press, 1986.

$9. §9. Bordism Bordism and and cobordism cobordism theory theory as as generalized generalized homology homology and and cohomology cohomology 137 137

al C'7l 0 T/2)

ÀijUiV j

u+v + L

f(u, v)

Àij E nû

,

A,

i,j~l

for for which which the the defining defining conditions conditions of of aa commutative commutative formal formaI group group hold: hold:

f(u, f(v, w))

f(f(u, v), w), (9.10)

f(u,v)

=

f(v,u).

Let line bundle. Let 7-l T/-1 be be aa complex complex line line bundle bundle such such that that 77 T/ @ 0 q-l T/-l isis the the trivial trivialline hundle. The The Chern Chern class class ai(v-‘) al (T/-l) == G ü say, say, isis aa power power series series in in uu == ~1 (JI (r]) (1J) of of the the following following form: form: ii=-u+ PiUi, Pi ü -u + cLJ.LiUi, J.Li E E n7 A, i>2

i~2

with with

(9.11) (9.11)

f(u,G) f(u, ü) == 0. o.

The The resulting resulting formal formaI group group explicitly calculated explicitly calculated in in term term

f(u, classes can can he be f( u, w) v) of of geometric geometric cobordism cobordism classes of the series of the series

un+l "'cpn_ _ L....t n+1'

g(u)

n~O

namely as (Mishchenko, late 1960s): 1960s): namelyas (Mishchenko, in in the the late (9.12) (9.12)

f(‘lL, + g(w)). f(u, w) v) = L7-%7(4 g-l(g(u) + g(v)).

By means of the formal group f(u, w) one may define (and calculate) anaBy means of the formaI group f (u, v) one may define (and calculate) analogues in fiG(.)-theory of the Adams operations constructed in K-theory using logues in nûO-theory of the Adams operations constructed in K-theory using representations of the groups U,. We define representations of the groups Un. We define !Pk(u) = ;g-ykg(u)),

u E L$(w=),

where u is as before the geometric cobordism

class cl(q).

We require that the

where u is as before the geometric cobordism class al (1J). We require that the operations P” have the following properties, analogous to those of the Adams operations 1ji'k have the following properties, analogous to those of the Adams operations in K-theory: operations in K -theory:

!Pk(z+ y) = !Pk(x)+ + ljik(y) *k(y),,

(9.13)

Ijik(X + y) = ljik(x) !P(a:

. y)

=

!P(x)

(9.13)

S"(y).

ljik(x. y) = ljik(x) ljik(y). The operation

Pk is then well-defined

in the theory

Q;(.)

@ Z [i]

[iJ

by (9.13)

is then by (9.13) The operation (Novikov, in the1ji'klate 1960s).well-defined in the theory nù(-) 0 lE (Novikov, in the late 1960s). Let k be any algebra over the rationals. In many situations it is useful any algehraofover In many Qsituations to Let havek abeclassification the the ring rationals. homomorphisms : 0; + it k.is Ituseful was to have a classification of the ring homomorphisms Q : Dû ---+ k. It was

138 138

Chapter Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Chapter 3. Simplicial Homology

shown by such a homomorphism Q is is determined by aa series series by Hirzebruch Hirzebruch that that such Q(Z) + qlz + q2z2 q2z2 + + ... , (see Chapter 4, 4, §1-3). $1-3). Writing Writing Q(CpN) Q(@pN) for the Q(z) = = 1+ qlz + (see Chapter the value of one has (Novikov, late 60s) 60s) of Q at CP”, cpn, one has the formula formula (Novikov,

SQ(u) =c QW$

= Q(du)>,

7X10

whence

,

z

Q(z) +-- )' Q(z) = = ---=-r-( gQ Q

z

Thus determining the above formaI formal group, makes makes Thus we see see that that the series series g(u) determining an appearance also in this context. an appearance also in this context. A natural analogue of character (the Chern-Dold of the Chern character Chern-Dold character, A natural whose existence was first established by Dold) can be effectively constructed who se existence was first established by Dold) can be effectively constructed by means of the series g(u). This character is given by homomorby means of the series g(u). This character is given by aa A-algebra A-algebra homomorphism ChU ---+ H*(K;n@3Q), n == .n;, (9.14) (9.14) chu:: L?;(K) Dû(K) -+ H*(K; A 0 Q), A Dû, determined that for K == CP”O same generator by the requirement requirement that Cpoo and u the same determined by as above, as above, chU(g(u)) = t E H2((CP2;Z), (9.15) (9.15) where canonical generator. generator. (The of the the character chu and and where tt is is the the canonical (The theory theory of character chu various applications of it were introduced by Buchstaber in the late 1960s.) various applications of it were introduced by Buchstaber in the late 1960s.) At the 1960s 1960s Quillen Quillen showed that the the ab above formal group group of of At the the end end of of the showed that ove formaI geometric cobordism classes is in fact a geometric realization of the universal geometric cobordism classes is in fact a geometric realization of the universal Lazard the theory theory of of one-dimensional He was was Lazard formal formaI group group of of the one-dimensional formal formaI groups. groups. He able to calculate effectively important projective operators in the algebra able to calculate effectively important projective operators in the algebra is the at the prime p), thereby comcomAU 0 8 Z(,) iz@) is Z(p) (where Z(p) the localization localization of of Z Z at the prime p), thereby pleting the computation of the “reduced” complex cobordism theories (now pleting the computation of the "reduced" complex cobordism theories (now known as “Brown-Peterson” theories BP*(.)) into which the theory 0fi(.)@U&, known as "Brown-Peterson" theories BP*(.)) into which the theory DûO®Qp decomposes sum. (The (The existence the theories theories B BP*(.) esdecomposes as as their their direct direct sumo existence of ofthe P* (.) was was established (non-effectively) by Brown and Peterson in the mid 1960s. Such tablished (non-effectively) by Brown and Peterson in the mid 1960s. Such projectors had earlier, not sufficiently constructed projectors had earlier, though though not sufficiently effectively, effectively, been been constructed by Novikov.) This approach has proved effective for computing the groups by Novikov.) This approach has proved effective for computing the groups Ext;: @ Zc,) figuring in stable homotopy theory (see above).15 Ext~~ 0 Z(p) figuring in stable homotopy theory (see above).15 Later the early early 1970s) 1970s) two-valued two-valued analogues analogues of formal groups groups were were Later (in (in the of the the formaI introduced into cobordism theory by Buchstaber and Novikov. Through the introduced into cobordism theory by Buchstaber and Novikov. Through the 1970s the associated algebraic theory was developed by Buchstaber, who also 1970s the associated algebraic theory was developed by Buchstaber, who also l5 ‘D-anslator’s note: Formal group theory has been applied to great effect in computions 15 Translator's note: Formai group theory has been applied to great etfect in computions of the spectral sequence Wilson, and of the Adams-Novikov Adams-Novikov spectral sequence for for spheres spheres (Miller, (Miller, Morava, Morava, Ravenel, Ravenel, Wilson, and others). are other theories generalizing both K-theory and complex complex others). There There are other cohomology cohomology theories generalizing both K-theory and cobordism “Morava K-theories” that have have been been etfectively effectively cobordism theory, theory, namely namely the the "Morava K-theories" K(n)*(.), K(n)*O, that formulated of formal theory, and have have proved proved very in computations computations formulated in in terms terrns of formai group group theory, and very useful useful in with the Adams-Novikov spectral sequence for investigations investigations of the general general proppropwith the Adams-Novikov spectral sequence as as well well as as for of the erties of the stable homotopy category. (For details, Ravenel, Complex Cobordism erties of the stable homotopy category. (For details, see see D. D, Ravenel, Complex Cobordism and of Spheres, Academic Press, and Stable Stable Homotopy Homotopy of Spheres, Academie Press, 1986.) 1986.)

$9. §9. Bordism Bordism and and cobordism cobordism theory theory as as generalized generalized homology homology and and cohomology cohomology 139 gave gave several several applications applications of of this this theory theory to to symplectic symplectic and and self-dual self-dual coborcob ordism theories. Algebraic topology from the point of view of cobordism dism theories. Algebraic topology from the point of view of cobordism theory theory has has been been elaborated elaborated by by several several authors authors (Dieck, (Dieck, Gottlieb, Gottlieb, Becker, Becker, and and others); others); however however we we shall shaH not not pursue pursue these these further further developments developments here.16 here. 16 In In the the early early 1960s 1960s itit was was observed observed by by Conner Conner and and Floyd Floyd that that bordism bordism thetheory ory can can be be used used to to obtain obtain various various results results concerning concerning smooth smooth actions actions of of finite finite groups groups and and compact compact Lie Lie groups groups on on closed closed manifolds. manifolds. We We shall shan now now consider consider some sorne of of these these results. results. Suppose Suppose for for instance instance that that we we are are given given a diffeomordiffeomorphism phism of of order order pP (prime) (prime) TT :: M2” M 2n +---> M2n M 2n , Tf’ TP = = 11, where where M2” M 2n isis aa closed closed U-manifold, .‘. ,p,. U-manifold, with with isolated isolated fixed fixed points points pl, Pl, .... ,Pm. The The restriction restriction of of the the difdiffeomorphism to aa sufficiently sllfficiently small small sphere sphere in in M2” M2n centered centered at at pj Pi has has no no feomorphism TT to fixed fixed points, points, and and determines determines aa singular singular complex complex bordism, bordism, namely namely aa map map of of the the appropriate appropriate lens lens space space (as (as orbit orbit space) space) to to the the Eilenberg-MacLane Eilenberg-MacLane space space KWP, K(7l/p, 1): 1): fjfJ :: s,2”-l/z/p S;n-1/7l/p ----> Sjoo/U~ Sr /7l/p == KK (UP, (Z/p, 1)) 1),

wJ

w;

= (Sj2n-1/Z/p,fj) (S;n-l/7l/p,!j),)

(9.16) (9.16)

wJ

WjT E LIZ”,-1 (sj”/z/p) E nfn-1 (Sr/ 7l /p)·.

depends only on T(M),~ ---+ wJ i.e. depends only on the the differential differential dT dT :: T(M)pj on the eigenvalues Xi, , X, of the complex

The of wT The bordism bordism class class of at the point +-f)P, T(M)Pi at the point pj, Pj, matrix matrix A.j Aj = dTpl dT ::

--->

Le. on the eigenvalues

)'1, ... ,.À n

of the complex

pi

WT = wjT(Xl,.

, L),

A,(&)

= exp(2~q,/p),

qq E (‘UP)*,

where are non-zero residues modulo p. By removing open neighbourwhere the the xjq Xjq are non-zero residues modulo p. By removing open neighbourfrom the manifold M2n, one obtains a hoods bounded by the spheres S;n-l S7-l hoods bounded by the spheres from the manifold M2n, one obtains a manifold on which Z/p acts without fixed points. In particular, this yields the the manifold on which 7l/p acts without fixed points. In particular, this yields relation

c wjT=o. LwJ =0.

relation

j

in the group 0&-i (S$F/Z/p). in the group (Sr /7l/p). In the case p = 2 we have xjq = 1, and from knowledge of the structure of In the case P 2 we have Xjq = 1, and from knowledge of the structure of the group the group J-g-1 fsj”/z/2) = g-1 W”)

nfn-l

we infer (see below)

we infer (see below) wj T = a,,

2+lan

# 0,

2na, = 0,

so that the number of fixed points must be divisible by 2” (Conner

and Floyd,

so that the number of fixed points must be divisible by 2n (Conner and Floyd, in the mid-1960s). in the mid-1960s). 16The theory of two-valued formal groups (developed in the late 1970s - mid 1980s theory of two-valued formaItoolgroups (developed inin the 1970s - midspectral 1980s by Buchstaber) provides a powerful for computations the late Adams-Novikov by Buchstaber) computations in the Adams-Novikov spectral sequence for the provides simplectic a powerful cobordism tool ring for(Buchstaber, Ivanovskii, Nadiradze, Vershinin).

sequence for the simplectic cobordism ring (Buchstaber, IvanovskiI, Nadiradze, Vershinin).

140 140

Chapter Chapter 3. Simplicial Simplicial Complexes Complexes and and CW-complexes. CW-complexes. Homology Homology

wJ

The case complicated since The case pp > > 2 is more complicated since the the bordism classes classes wT (our invariants invariants of of the fixed fixed points) points) depend on the residues residues xj,r Xjq E E (z/p)* (71jp)* for which which of classeswjT of course course there are are more more choices choices than than before. before. ItIt turns turns out out that that the the classes can (Novikov, Mishchenko, can be be effectively effectively calculated calculated as as functions functions of of the the residues residues (Novikov, Mishchenko, Kasparov, 1960s): writing cr(xji, ... . ,xjn), the late late 1960s): writing wT = = Œ(Xjl, ,Xjn), we we have have Kasparov, in in the

wJ

wJ

n

II

cy(Xjl,... = q=l fi g-l(x;qg(u)) Œ(Xjl,"" ,Xjn) Xjn) ::;; -1( U ( q=l 9

xJqg u

))

n aŒ(l, ‘.....), 1). r-l).

(9.17) (9.17)

2n - 1 j71jp) and The (S 2”-‘/Z/p) The groups groups 06 nû (8 and 0: n~ (S’,-‘/,/p) (8 2n - 1 j71jp) for for the the lens lens spaces spaces S2n-1/Z/p escribed explicitly using formal follows. 8 2n - 1 j71jp may may bbee d described explicitly using formaI group group theory theory as as follows. (We note (We note that that the the first first partial partial results results about about these these cobordism cobordism and and bordism bordism groups were obtained by Floyd in 1960s.) Consider Consider groups were obtained by Conner Conner and and Floyd in the the early early 1960s.) the as of the the spaces spaces S,““+‘/Z/p 8;n+l j71jp cc ST/Z/pj71jp as representatives representatives of the bordism bordism classes cu(xji, . . xjn) E .nz”,+r (S”/Z/p). The geometric cobordism 00 classes Œ(Xjl,'" xin) E n:fn+1 (8 j71jp). The geometric cobordism class class u uE E 2n 1 2n 0; (S2+‘/Z/p) generates the ring 0; (S2nf1/Z/p) (as J$-module) with n'b (8 - j71jp) generates the ring nû (8 +1 j71jp) (as nû-module) with relations: relations:

8r

un+2 = Un+2 = 0, 0,

g-l(pg(u)) 0, g-l(pg(u)) = = 0,

Du” . ) 1 ) = o!,-k E nijn-k)+1 @-k)+1 (P+‘/Z/p) Du k = a( Œ( \1,. 1, ... Œn-k E (8 2n + 1 j71jp) ), ” ,1) , '-v---' n-k times

(9.18) (9.18)

n-k times Œn-k

The sets {xji,

0

Œn-l

Œn-k-l·

. , xjn} of residues (j = 1, . . . , m) actually realizable as corre-

The sets {Xjl,"" Xjn} of residues (j ::;; 1, ... , m) actually realizable as corresponding to the set of fixed points pl, . ,p, E M2” of some Z/p-action on sponding to the set of fixed points Pl, .. . ,Pm E M2n of sorne 7ljp-action on M2n, must satisfy M 2n , must satisfy m m

C”(Xjl:.

L

. . ,XjJ = 0.

Œ(Xjl,"" xin) =

O.

(9.19)

(9.19)

j=1 . . . Brmgmg m the formula (9 .:;)r >we obtain from (9.19) a system of equations in Bringing in the formula (9.17), we obtain from (9.19) a system of equations in the mn variables x:jq ( f 0 mod p) determining the set of admissible residues the mn variables Xjq ( mod p) determining the set of admissible residues “CP?. Xjq. We remark that the bordism class [M”“]2n in the group L?zn 8 Z/p can also We remark that the bordism class [M ] in the group n'bn 0 '1ljp can also be computed via the residues xjq. be The computed the residues above via discussion carriesXiq' over to oriented cobordism theory, with some The ab ove discussion to oriented cobordism sorne simplification, by meanscarries of the over homomorphism 0; + L’&.theory, Here,with however, simplification, by means of the homomorphism nû ---t n'So. Here, however, the casep = 2 is special and requires different techniques. The formula (9.17) the case P =to2the is special andwhere requires techniques. formula (9.17) generalizes situation the different fixed point set of theThe transformation is to the situation where the fixed point set of the transformation generalizes a submanifold of M with a nontrivial normal bundle. The situation where isa a submanifold M with order a nontrivial bundle. where a cyclic group of of composite acts onnormal a manifold hasThe also situation been investigated cyclic group of composite order acts on a manifold has also been investigated (by Mishchenko, Gusein-Zade, Krichever, in the early 1970s). (byEspecially Mishchenko, Gusein-Zade, in the early 1970s). interesting resultsKrichever, have been obtained in the situation where the Especially results have been in the the circle S’ acts interesting smoothly on an oriented or obtained U-manifold. Forsituation instance,where Gusein-

°

circle 8 1 acts smoothly on an oriented or U-manifold. For instance, Gusein-

$9. §9. Bordism Bordism and and cobordism cobordism theory theory as as generalized generalized homology homology and and cohomology cohomology 141 141

Zade Zade (ca. (ca. 1970s) 1970s) has has completely completely described described the the bordism bordism groups groups arising arising in in connexion on manifolds nexion with with actions of of S’ SIon manifolds such such that that there there are no no points points fixed fixed

by by the the whole whole group group S1. SI. ItIt turns turns out out that that these bordism bordism groups contain contain in in a natural natural way way invariants invariants of of (and (and relations relations between) the the fixed fixed points points for for any any smooth smooth action action of of the the circle circle on on aa manifold. manifold. Several elegant elegant results of of this this type type for on U-manifolds, for smooth smooth actions actions of of S1 SIon U-manifolds, using using complex complex cobordism cobordism theory theory and formal formaI group group theory, theory, were obtained obtained by by Krichever Krichever in in the the first first half half of of the the 1970s. We shall now describe two such results. 1970s. We shaH two 1. 1. Let Let A(k) A(k) be the the characteristic characteristic class class determined determined by by the the series series kue” A(w = &-y_l’ (see characteristic class (see Chapter Chapter 4, 4, $3); §3); then then for for each each U-manifold U-manifold M2n M 2n this this characteristic class 2n If the circle S’ acts smoothly on determines the rational number Ack,(M2”). determines the rational number A(k)(M ). If the circle SI acts smoothly on 2 aa U-manifold U- manifold M2” M 2n for for which which kk divides divides c~(M~~), CI (M n), then then Ack)(M2”) A( k) (M2n) == 0. O. In In the the 2n 2 case k = 2, the number Ap,(b~f~~) coincides with the signature of M2”. case k = 2, the number A(2)(M ) coincides with the signature of M n. (The (The latter formulae latter result result was was proved proved by by Atiyah-Hirzebruch Atiyah-Hirzebruch via via the the Atiyah-Bott Atiyah-Bott formulae for Spin-manifolds.) for Spin-manifolds.) of this action 2. U-manifold M2n, 2. IfIf S1 Si acts acts on on aaU-manifold M 2n , then then the the linearization linearization of this action on the normal bundle to the submanifold Nj” c M2” (j = 1,. . . , m) of on the normal bundle to the submanifold N} C M2n (j 1, ... , m) of fixed fixed points, has eigenvalues eigenvalues points, has ~lj~~~~~~n-k,j~

Xqj = exp(2+n,j),

where (the weights) are integers. Let lj denote the number of negative where the the n,j nqj (the weights) are integers. Let lj denote the number of negative weights n,j < 0, and and let let TTzI be the characteristic classwith values polynomials weights nqj < 0, y be the characteristic class with values polynomials in y, determined by the Hirzebruch series

in y, determined by the Hirzebruch series

(Y + + lb (y l)u Ty(4 = 1 -ex"":"::p' - exp(-u(y + 1)) - y”’ T y ( u) = -:--1 -:-(--u-':(-y-+-:-17-:-)) yu.

Then the following formula holds:

Then the following formula holds: m

2 Y’JT,(N;) = T?I(M2n). LyljTy(N}) Ty (M2n). j=l j=l

(9.20)

(9.20)

For y = 1, the value of Tl on manifolds coincides with their signature, and

For y I , the value of Tl on manifolds coincides with their signature, and this case of the above result was obtained earlier by Atiyah-Hirzebruch using this case of the ab ove result was obtained earlier by Atiyah-Hirzebruch using the theory of elliptic operators. Putting y = -1 yields the Euler-Poincare the theory of elliptic operators. Putting y = -1 yields the Euler-Poincaré characteristic: T-l(M) = x(M), and (9.20) reduces to the classical formula characteristic: T_ 1 (M) X(M), and (9.20) reduces to the classical formula for x(M).17 for X(M).17 “Around 1990 further important results were obtained concerning actions of the circle on 17 AroundCertain 1990 further resultsanalysis were obtained circle manifolds. of the important ideas of “global on loop concerning spaces” firstactions appearedof the here (Wit-on of the ideas spaces" first appeared and herecertain (Witmanifolds. ten, Taubes), Certain the elegant notion of "global of “ellipticanalysis genera”on loop (Oshanine, Landweber), ten, Taubes), the elegant "elliptic genera"functions (Oshanine, Landweber), and certain generalizations of these suchnotion as theof Baker-Akhieser (Krichever).

generalizations of these such as the Baker-Akhieser functions (Krichever).

Smooth Manifolds Manifolds Chapter 4. Smooth

142 142

Chapter Chapter 44 Smooth Smooth Manifolds Manifolds $1. §1. Basic Basic concepts. concepts. Smooth Smooth fiber fiber bundles. bundles. Connexions. Connexions. Characteristic Characteristic classes The The topology topology and geometry geometry of of smooth smooth (differentiable) (differentiable) manifolds is is the most important important area of of study study in topology, topology, the the source of of and most fruitful fruit fui field for for applications with applications of of the the whole complex complex of of topological topological methods, closely linked linked with analysis and, and, particularly particularly in in recent times, with with contemporary contemporary mathematical mathematical physics. physics. The The elementary elementary theory theory of of smooth smooth manifolds was was created by Whitney Whitney in the the mid-1930s. mid-1930s. The The internal internaI definition definition of of a differentiable differentiable manifold manifold is as as follows: itit is in the first (seethe conclusion of 51 of first place an (n-dimensional) (n-dimensional) topological manifold manifold (see of §1 Chapter Chapter 2), i.e. Le. a Hausdorff Hausdorff topological topological space space X for which there is aa covering collection V, = collection of of open sets sets V,, Va, lJ, Ua Va = X, X, each each homeomorphic to (an open region of) ]Rn. Such homeomorphisms homeomorphisms of) W.

then . ,x2) set (or (or chart then determine determine local local co-ordinates co-ordinates (xk, (x~, ... ,x~) on on each each open open set chart or or local neighbourhood) Va. V,. The said to be smooth local co-ordinate co-ordinate neighbourhood) The manifold manifold X X is is said to be smooth of V, n V,a VP the transition functions of class class Ck C k ifif on on each each region region of of intersection intersection Va the transition functions expressing of class C”,k , expressing one one set set of of co-ordinates co-ordinates in in terms ter ms of of the the other other are are smooth smooth of class C i.e. the and 404;~ appropriate regions of Euclidean Euclidean i.e. the maps maps 4,$,’ <Pa O. 0. (If = CXJ oo the the manifold manifold is is said said to to n-space are are functions of class C k , Ic k > (If kk = be infinitely diflerentiable, or just smooth. A real-analytic manifold is one for be infinitely differentiable, or just smooth. A real-analytic manifold is one for which are real-analytic.) real-analytic.) which the the transition transition functions functions are Manifolds will be denoted by IP, N”,k , etc. Note that that the the Jacobian etc. Note Jacobian Manifolds will be denoted by Mn, N det(G’xA/dxi) of the transition function function xa(x,a) xa(xa) is is non-zero on Va V, n n V,a, Vo, since since of the transition non-zero everywhere everywhere on ifif itit were zero at any point, then the inverse function xp(xa) would not were zero at any point, then the inverse function x,a(x a ) would not be be smooth. smooth. The development, at at the the elementary elementary level, level, of of techniques for studying studying The development, techniques for smooth manifolds, requires the existence of partitions of unity, i.e. the exisexissmooth manifolds, requires the existence of partitions of unit y, Le. the tence, for each open cover {I&} of a manifold M”, of open sets IV, c IV; C V, tence, for each open cover {Va} of a manifold Mn, of open sets W a C W~ C Va such that U, W, = M”, and of real-valued functions qcl 2 0 of class C” desuch that Ua W a = Mn, and of real-valued functions 'ljJa ~ 0 of class COQ defined on on Mn, with the the following following properties (seeFigure 4.1): fined Mn, with properties (see Figure 4.1): $J~== 3 11 on Wa:; +,, z== 00 outside outside W~; WA; 'ljJa on W a ; 'ljJex 00 > C,$)a(x)

> 0 for each x E M”;

C&

= 1, where & = $)a/x+or.

$1. Basic Connexions §1. Basic concepts. concepts. Smooth Smooth fiber fiber bundles. bundles. Connexions

143 143

'l'(x)

Disk D ({ii)

Fig. Fig. 4.1 4.1

&le. Example. For For compact compact manifolds manifolds M”, Mn, partitions partitions of of unity unity exist exist for for all aU fifinite covers {Va} of M” nite covers {Va,} of Mn by by local local co-ordinate co-ordinate neighbourhoods. neighbourhoods. IfIf we we multimultiply the local co-ordinates V, by (see x~ on on each each region region Va by the the function function Q, '!/Ja (see ply the local co-ordinates xj, above) of a corresponding partition of unity, then the resulting functions ab ove ) of a corresponding partition of unit y, then the resulting functions q!~~xj, = y: say, are defined and smooth on the whole manifold Mn. If the cover '!/JQx~ y~ say, are defined and smooth on the whole manifold Mn. If the coyer VN}, then the collection of functions {yi} ,j = 1,. . . , n, WY> = vl,... {Va} {VI," .,, V 1, ... , n, N }, then the collection of functions {vÜ ,j (Y = 1, . , N, defines a C”-embedding Cl: = 1, ... , N, de fines a Coo-embedding {yi}

: M” +

RWnN,

since $,cy on W, W, are are distinguished distinguished by since '!/Ja = 1Ion WQ and and any any two two points points of of W by their their local local Q co-ordinates x”,. (For non-compact manifolds the analogous argument yields co-ordinates x~. (For non-compact manifolds the analogous argument yields an embedding into Hilbert space.) Using the idea of “general position” (see an embedding into Hilbert space.) Using the idea of "general position" (see below) and projections of the above embedded manifold onto k-dimensional below) and projections of the above embedded manifold onto k-dimensional hyperplanes, it can be shown that for k 2 2n+l the set of smooth embeddings hyperplanes, it can be shown that for k ;::: 2n+ 1 the set of smooth embeddings Mn -+ lRk is everywhere densein the space of smooth (even just continuous) Mn ___ lR k is everywhere dense in the space of smooth (even just continuous) maps Mn -+ iRk, and the set of smooth immersions likewise for k > 2n. In maps Mn ---t lR k , and the set of smooth immersions likewise for k ;::: 2n. In the case n = 1 this is visually obvious - see Figure 4.2. 0 the case n 1 this is visually obvious see Figure 4.2. 0 A smooth map f : Mn --+ Nk between two smooth manifolds is a A smooth map f : Mn ___ Nk between two smooth manifolds is a each of whose expressions in terms of local co-ordinate systems map map each of whose expressions in terms of local co-ordinate systems {xja Ij=l,... , n} on the charts V, of Mn and {y+ ) s = 1,. . , k} on the {x~ 1 j l, ... , n} on the ch arts Va of Mn and {v~ 1 s I , ... , k} on the charts U, of N”k is made up of smooth functions y$(zi, . ,x”,) (in the usual charts U"( of Neach is domain made upofofdefinition smooth functions y~(x~, ...has ,x~)the(inform the usual sense). (Here of these functions sense). (Here each do main of definition of these functions has the form

At each point x of Mn the rank (rank f(x)) of the smooth map f is defined to At the eachrank pointofxthe of Mn the (ayt/axj,) rank (rank fevaluated (x)) of theatsmooth defined to be matrix x. The map rank fatis each point be the rank of the matrix (ay~/ax~) evaluated at x. The rank at each point is an invariant of f (in the sense that it is independent of the particular local

is an invariant f (in the sense that it is independent of the particular local co-ordinates in of use). co-ordinates in use).

144 144

Chapter 4. Smooth Smooth Manifolds Chapter 4. Manifolds

S’ --> --+ ]R3. lR3. (i) SI In general general position In position this is is an an embedding embedding this

(ii) SI S’ --> lR*. (ii) ]R2. The projection to ]R2 R2 The projection to has self-intersections self-intersections not has not removable by means of of smal! small removable by means perturbations: the tangent tangent perturbations: the vector field (in (in general general vector field position) is non-degenerate non-degenerate position) is

Fig. 4.2 Fig. 4.2

An immersion N” is a smooth map with the property that An immersion ff :: M” Mn -+ --> Nk is a smooth map with the property that at every point x E Mn the rank of ff is (so that we must must have have nn :::; 5 k). A at every point x E Mn the rank of is nn (so that we k). A submersion is defined by the requirement rank f E k (so that n 2 k). An emsubmersion is defined by the requirement rank f k (so that n ;::: k). An embedding f is a one-to-one immersion: f(x) # f(y) if x # y. A diffeomorphism bedding fis a one-to-one immersion: f(x) =1= f(y) if x =1= y. A diffeomorphism MT- * MT between two smooth manifolds is a smooth homeomorphism ff :: Mï --> Mll between two smooth manifolds is a smooth homeomorphism with smooth inverse.

with smooth inverse.

The smoothness class C”k of a manifold is, beyond the basic assumption that The smoothness class C of a manifold is, beyond the basic assumption that k > 1, not of great significance for topology, in view of a theorem of Whitney k ;::: 1, not of great significance for topology, in view of a theorem of Whitney to the effect that any C”-manifold is diffeomorphic to a unique C”-manifold, to the effect that any Ck-manifold is diffeomorphic to a unique eco-manifold, in fact to a (unique) real-analytic manifold. (It was further shown by Morrey in fact to a (unique) real-analytic manifold. (It was further shown by Morrey (in the late 1950s) that any real-analytic manifold can be embedded real(in the late 1950s) that any real-analytic manifold can be embedded realanalytically in some P; this is a difficult result requiring more recent methods analytically in sorne ]Rn; this is a difficult result requiring more recent methods of the theory of complex manifolds.) In the sequel we shall therefore frequently of the theory of complex manifolds.) In the sequel we shaIl therefore frequently assume, without explicit mention, that our smooth manifolds and maps have assume, without explicit mention, that our smooth manifolds and maps have smoothness class C”,OO and shall not distinguish diffeomorphic manifolds.

smoothness class C

,

and shaH not distinguish diffeomorphic manifolds.

Beginning with Whitney, lemmas (due also to Brown, Sard, and others Beginning with Whitney, lemmas (due also to Brown, Sard, and others in the 1930s) concerning “general position” have come to play an important in the 1930s) concerning "general position" have come to play an important role in connexion with the techniques employed in studying the topology of role in connexion with the techniques employed in studying the topology of smooth manifolds. The simplest such result states that a real-valued map The Ck simplest such result large, states always that a has real-valued map fsmooth : Mn manifolds. + Iw of class for k sufficiently the property fthat: Mn ]R of class C k for k sufficiently large, always has the property the image f (2) of the set 2 of critical points (i.e. points x E 111” for that the the set Z of critical (Le. points x Eset Mnf (2)for which gradimage f(x) f(Z) = 0), of has measure zero in R. points (For compact Mn the which grad f(x) = 0), has mea..'lure zero in lR. (For compact Mn the set is closed, and its complement has positive measure.) More generally, for f(Z) any is closed, and -+ its complement measure.) generally, for any map f : Mn Nk with k Nk with k :::; n, of sufficiently high smoothness class, the

§1. $1.

Basic concepts. concepts. Smooth Smooth fiber fiber bundles. bundles. Connexions Connexions Basic

145 145

set f(2) f(Z) has has measure measure zero zero inin Nk, Nk, where where ZZ consists consists ofof the the points points xx EE M” Mn for for set which rank rank f(x) f(x) << k.k. Note Note also also that that for for any any Cl-map C1-map Mn Mn --f __ Nk N k with with kk >> n,n, which the the image image f(Mn) f (Mn) has has measure measure zero zero in in NNIc. k. In In all aB of ofthese these situations situations the the points points inin the codomain Nk outside the set f(2) (f(M”) in the latter situation the codomain Nk outside the set f(Z) (J(Mn) in the latter situation where where kk >> n) n) are are said said toto be be in in general general position position with with respect respect to to f.f. We We shall shall describe describe the the various various essential essential lemmas lemmas about about bringing bringing maps maps into into general general position, position, as as the need for them arises. the need for them arises. Recall that aa tangent tangent vector vector to to aa (smooth) (smooth) manifold manifold M” Mn at at aa point point x0 Xo of of Recall that M” is the velocity vector (dxj,/dt),, at x0 to a parametrized curve x’,(t) Mn is the velocity vector (dx~/dt)xo at Xo to a parametrized curve x~(t) (in (in local local co-ordinates co-ordinates on on a chart chart V, Vo: containing containing ~0) xo) passing passing through through x0. xo. The The vector vector space IF& of all tangent vectors to Mn at x0 is called the tangent space lR~() of aB tangent vectors to Mn at Xo is called the tangent space space to to M” Mn atat x0. xo. Let . . ,qE) Let (r&, (71~, .... , 71~) be be any any tangent tangent vector vector to to aa point point xx EE M”, Mn, in in terms terms of of local co-ordinates XL,. . . , x”, on a local co-ordinate neighbourhood containing local co-ordinates x;, ... , x~ on a local co-ordinate neighbourhood containing x. . . ,xz x. Under Under aa change change to to co-ordinates co-ordinates xb, x1, .... , x~ (with (with Jacobian Jacobian matrix matrix lap Ia{3 = (G’x$/axi)), (8x~ / 8x~)), the the components components of of the the tangent tangent vector vector transform transform according according to to the the rule rule

ax;

.8x~ (with summation over j). $71(3k == qig (1.1) 71~ J::l j (with summation over j). (1.1) UX aa On gradient, and, l-forms given On the the other other hand hand the the gradient, and, more more generally, generally, differential differentiall-forms given in local co-ordinates by C Q&X; = Q&X:, transform according to the the rule rule in local co-ordinates by L 71io:dx~ 71io:dx~, transform according to 71i{3 = 71ko:

8x k J::l ~. uX{3

(1.2) (1.2)

More general tensors tensors of of type (m, k), type (m, k), with with components components More general ail,,·i rn

Th .. .jk transform under co-ordinate changes relative to local co-ordinates xi, . . , x:, transform relative to local co-ordinates x;, .... ,x~, under co-ordinate changes LY -+ ,D in the appropriate way (obtained by “extrapolating” from (1.1) and a -- f3 in the appropriate way (obtained by "extrapolating" from (1.1) and (1.2)). A little more detail is in order. In terms of local co-ordinates xk, . . , x”, (1.2)). A litt le more detail is in order. In terms of local co-ordinates x~, ... , x~ in some neighbourhood of the point x of interest, there are the usual standard in some neighbourhood of the point x of interest, there are the usual standard basis vectors el,, . . . , ena for the tangent space at x, and the correspondbasis vectors elo:, ... , eno: for the tangent space at x, and the corresponding dual basis of covectors ek, . . . , en, for the dual space, with scalar product ing dual basis of covectors ez., ... , e~ for the dual space, with scalar product eja) = Sj, given by evalulating each ej, at the eja. Thus each vector 77in (eh, ejo:) (e~, oj, given by evalulating each e~ at the eja. Thus each vector 71 in rWz has the unique form r]&eja E rW2, and each covector q* the unique form lR~ has the unique form 71Lejo: E lR~, and each covector 71* the unique form QaeL E IQ”. At each point x E M” we then have the space of tensors of type 71iae~ E lR~*. At each point x E Mn we then have the space of tensors of type h k) (m,k) I%; 63 ’ . ’ @RpR~*@-.@R~*, lR~ 0 ... 0 lR~ 0 lR~ * 0 ... 0 lR~* , -' - v -m- - - ' \" vk .J m k with standard basis (relative to the prevailing local co-ordinates) consisting with standard basis (relative to the prevailing local co-ordinates) consisting of the expressions of the expressions

146 146

Chapter Smooth Manifolds Manifolds Chapter 4. Smooth

An arbitrary (m, Ic)) the point An arbitrary tensor tensor (of (of type type (m, k)) at at the point zx has has the the form form co-ordinates ICY, . . , x~) z”,) co-ordinates x~, . ..

(in the the local local (in

where summation convention suppressed). where the the summation convention is assumed assumed (and (and the the index index cr 0: suppressed). The eja correspond correspond naturally to the operators a/ax~, a/ax&, and and the The basic basic vectors vectors ejQ naturally to the operators the dual-basis covectors e~ ej, to the differentials dxj,: d ual- basis covectors to the differentials dx~: eia H

c?/~x$;

H dxj,. dx jQ. ee:jQ f-'>

A type (m, (m, k) is then aa smooth function function assigning assigning to each each point A tensor tensor field field of type k) is point 5 of M” a tensor of type (m, Ic). x Mn a type k). The usual tensor (field) permutation of of The (field) operations (contraction, (contraction, product, product, permutation indices, etc.) and the Einstein summation convention (whereby any index indices, etc.) and the Einstein summation convention (whereby any index as both expression is is autoappearing as both superscript superscript and subscript subscript in a single expression automatically summed be used used without without comment in what matically summed over) will will be what follows. Skew-symmetric tensors (more precisely tensor indices fields) with with lower lower indices Skew-symmetric tensors tensor fields) are of particular importance; they have have the the alteralteronly, only, i.e. of of type type (O,Ic), (0, k), are of particular importance; they native name differential forms are usually usually written as follows follows (in (in local local native name differential forms and and are written as co-ordinates): : co-ordinates) fik = iTi,,,,ikdxil

A’.’

Adxi”,

(1.3)

in view of in integration integration theory. theory. The The exterior exterior product of in view of their their use use in product fik nk A A 01 ni of differential is defined defined algebraically tensor product product and and differential forms forms is algebraically by by means means of of the the tensor the operation making making the product skew-symmetric: skew-symmetric: the “alternation” "alternation" operation the exterior exterior product

In (1.3) for exterior product product In terms terms of of the the local local notation notation (1.3) for differential differential forms, forms, the the exterior is by the conditions of of associativity, commutativity with with is determined determined uniquely uniquely by the conditions associativity, commutativity scalars, scalars, and and dxi A dxi = -dxi A dxi. (1.4) The differential on forms defined by the following The differential operator operator dd on forms is is defined by the following conditions: conditions: functions f; (i) (i) df df = = -$dx$ af dx~ (locally) for for real-valued real-valued functions Q ax~

(ii) d( d(dxil A ... . . . AA dX dx2k)ik ) == 0; 0; (ii) dX i1 A

(111) .nlc) = df A A& + ffdnk, d&, ff any (iii) d(f d(fnk) = df nk + any scalar scalar function. function. From infer From these these we we infer dod=O,

d(.&Ai2~)=df&A~~+(-l)“f&Adf&

Finally the important concept of or pseudoimportant concept of aa Riemannian Riemannian or pseudoFinally we we mention mention the Riemannian metric on a manifold IMn; this is a non-degenerate symmetric Riemannian metric on a manifold Mn; this is a non-degenerate symmetric

Il. Basic concepts. Connexions §1. Basic concepts. Smooth Smooth fiber fiber bundles. bundles. Connexions

147 147

tensor (field) (0,2) smoothly on the point (field) of type type (0, 2) depending smoothly point 5x of Mn. Mn. In local co-ordinates (XL), it will have the form (gij), with gij = gji and det (.9ij) (gij) # # (gij) , with gij = gji co-ordinates (x~), it will 0. Such a tensor determines a symmetric scalar product of pairs of tangent O. symmetric product of tangent vectors tensors of vectors (at (at each point point x), x), pairs of covectors (and in general pairs of tensors any way: any type) type) in in the the natural natural way: =

(771,7?2)

rlggij,

As tool for the topology topology of of manifolds, manifolds, usually usually Riemannian Riemannian metmetAs aa tool for investigating investigating the rics are used, i.e. those for which det (gij) > 0. However pseudo-Riemannian rics are used, i.e. those for which det (gij) > O. However pseudo-Riemannian metrics they are important not in the the theory of metrics occur occur not not infrequently; infrequently; they are important not only only in theory of relativity, but also for instance in studying the geometry of semisimple Lie relativity, but also for instance in studying the geometry of semisimple Lie groups. groups. A smooth manifold manifold IP equipped with smooth binary binary A Lie Lie group group is is aa smooth Mn equipped with a a smooth operation operation P:M”xMn*Mn;

@(X,Y) =x.y,

X,Y E M”,

under which Mn (x. y) (y . z); identity element element Mn is is aa group: group: (x· y) .. zz = z. x· (y. z); there there is is an an identity under which 1; each element x of M” has an inverse element 5-l 1 (xx-l l = x-1x = l), and 1; each element x of Mn has an inverse element x- (xx- = X-lX = 1), and this inverse 2. this inverse should should vary vary smoothly smoothly with with x. Let x1, . . . , xn be local co-ordinates of Lie group group Mn M” in neighbourhood n Let Xl, ... ,x be local co-ordinates of aa Lie in a a neighbourhood of xc = 1. In terms of such co-ordinates the group operation z = !P(z, y) has of Xo 1. In terms of such co-ordinates the group operation z !li (x, y) has the form the form zi= !P(xl,... Jn,YIY.,Yn), j ,T,j ( 1 n l n) , Z = Y! X , ... ,X ,y , ... ,y whence by Taylor’s theorem

whence by Taylor's theorem

zi = xi + yj (~-l)j

=

-,j

+ bj,,x”y” + 0(/x1 . lyl), (1.5) (1.5)

+ o(lxl).

The Lie algebra of the Lie group M” is the tangent space to M” at x0 = The Lie algebra of the Lie group Mn is the tangent space ta Mn at Xo equipped with the bracket operation defined (in terms of local coordinates equipped with the bracket operation defined (in terms of local coordinates a neighbourhood of 1) by

1, 1, in

in

a neighbourhood of 1) by

[C,Vlk = CbSs -

btj)Ji77S1

(1.6) (1.6)

where < = (cl,. . . ,<“), n = (T$, . . , qn). It is immediate that this operation where en), "1 ([E, ("Il,q]...= , -[v, rr)·
e (e ,... ,

KI’ll7Cl+ h~17Jl+ K751~Vl = 0. o.

[[e,1J],(] + [[7], (], e] + [[(,e],1J] =

(1.7)

(1.7)

The most important Lie groups are groups of matrices over various fields, Theparticular most important groups are groups matrices overThe various in Iw, CcandLie also the skew-field Ml of of quaternions. groupfields, O,,, in particular IR, C and also the skew-field !HI of quaternions. The group consists of the linear transformations of the vector space Rn, n = p Op,q + q, consists of the linear transformations of the vector space Rn, n = p + q,

148

Chapter Chapter 4. 1. Smooth Smooth Manifolds Manifolds

preserving preserving a symmetric symmetric scalar scalar product product of of signature signature p, p, q. The The orthogonal orthogonal group group 0, On (= (= On,,) On,O) isis of of course course of of particular particular importance. importance. Analogously, Analogously, the the group group UP,, Up,q consists consists of of the the linear linear transformations transformations of of @“, en, nn == pp ++ q, preserving preserving an an Hermitian Hermitian scalar scalar product product of of signature signature (p, (p, q), q), and and lJ, Un = U,,e Un,O isis the the unitary unitary group. group. The The respective respective subgroups subgroups SO,,,, SOp,q, SO,, SOn, SUP,,, SUp,q, SU,, SUn, and and SL,(R) SLn(JH!) cC w h ere SG G&(R), GLn(lH!), s-L(@) SLn(C) cC GL(@) GLn(C) ((where SG denotes denotes the the subgroup subgroup of of matrices matrices of of determinant G) are Finally, Sp, determinant 1 of of the the matrix matrix group group G) are also also important. important. Finally, SPn denotes denotes the the subgroup subgroup of of nn xx n orthogonal orthogonal quaternionic quaternionic matrices. matrices. The The Lie Lie groups groups O,, On, U,, SO,, SU,, Sp, are all compact, and by the theorem of Cartan-Killing Un, SOn, SUn, SPn are aIl compact, and by the theorem of Cartan-Killing there there are are only only finitely finitely many many (namely (namely 6) 6) compact, compact, simply-connected, simply-connected, simple simple Lie groups not locally isomorphic to a group in this list of matrix groups. Lie groups not locally isomorphic to a group in this list of matrix groups. In In fact of the the form form fact every every compact compact Lie Lie group group G G isis of (T” x G1 x G2 x . . x Gk) /D, where Tn isis the the n-torus, n-torus, the the Gi G i are are compact, compact, simply simply connected, connected, simple simple Lie Lie where Tn groups (and so figuring in the Cartan-Killing list), and D is a finite subgroup groups (and so figuring in the Cartan-Killing list), and D is a finite subgroup contained the center. center. contained in in the A smooth fiber defined as as in case (see (see Chapter 3, §6) $6) A smooth fiber bundle bundle is is defined in the the general general case Chapter 3, except that now the spaces E, B, F are all required to be smooth manifolds, except that now the spaces E, B, Fare all required to be smooth manifolds, the the transition transition functions functions are likewise smooth, smooth, the projection projection pP is is aa smooth smooth map, map, the are likewise and the structure group G c Diff F, i.e. is a subgroup of the group and the structure group G C Diff F, Le. is a subgroup of the group of of difdiffeomorphisms of the fiber F. (In the most important situations G is Lie feomorphisms of the fiber F. (In the most important situations G is aa Lie grow) group.) AA (smooth) the fiber fiber FF is is (smooth) vector vector bundle bundle is is aa smooth smooth fiber fiber bundle bundle where where the aa vector space on which the structure group G acts as a group of linear vector space on which the structure group G acts as a group of linear transformations. The two most important types are real vector bundles, where transformations. The two most important types are real vector bundles, where and complex vector bundles, where F g C”, G c FF ”9:! IR!n, R”, G G C c GLn(lR!) GL,(iR), , and complex vector bundles, where F 9:! en, G c G-L(Q. GLn(C). An important example is the tangent bundle r = r(M”) of a smooth manAn important example is the tangent bundle T T(Mn) of a smooth manifold M”, with F g Iw”, G = GL,(IR) and base B = Mn; the total space is ifold Mn, with F 9:! IR!n, G = GLn(lH!) and base B = Mn; the total space is denoted by T(M”). Note that a cross-section of the tangent bundle is just a denoted by T(Mn). Note that a cross-section of the tangent bundle is just a (tangent) vector field on Mn. If a Riemannian metric is given on M, then the (tangent) vector field on Mn. If a Riemannian metric is given on M, then the structure group reduces to 0,. structure group reduces to On. From the tangent bundle over Mn one can construct many important assoFrom the tangent bundle over Mn one can construct many important associated vector bundles over Mn, with the group GL,(lR) acting on the various ciated vector bundles over Mn, with the group G Ln (IR!) acting on the various fiber-manifolds. For example if we take as fiber the dual space lQ* at each fiber-manifolds. For example if we take as fiber the dual space IR!~* at each point z E M” (i.e. the space of tensors of type (0,l)) we obtain the cotanx E Mn (Le. the space of tensors of type (0,1)) we obtain the cotanpoint gent vector bundle T* with total space denote by T*(M”). More generally, gent vector bundle T* with totalonspace denote by T*(Mn). More generally, the vector space of tensors of type (m, k) the action of the group GL,(IK) the action of the group GLn(lR!) on the vector space of tensors of type (m, k) yields an associated vector bundle over Mn with fiber this space of tensors; yields an associated vector bundle over Mn with fiber this space of tensors; this bundle is denoted by this bundle is denoted by T

------

® ... ® T ® T* ® ... ® T*

~ m times m times

k times k times

.

51. §l.

Basic Basic concepts. concepts. Smooth Smooth fiber fiber bundles. bundles. Connexions Connexions

149

Similarly, Similarly, one one can can form form the the exterior exterior powers powers Lo” Ak(T*) of of the the cotangent cotangent space space r*. Differential forms on Mn are then cross-sections of such vector T*. DifferentiaI forms on Mn are then cross-sections of such vector bundles. bundies. AA Riemannian Riemannian metric metric on on M” Mn may may analogously analogously be be regarded regarded as a cross-section cross-section of of the symmetric square S2r of the tangent bundle, obtained as the symmetric square S2 T of the tangent bundle, obtained as the the symmetric symmetric part part of of 7T 8~. ® T. AA further further example example of of a vector vector bundle bundle associated associated with with the the tangent tangent space, space, is that where the fiber above each point consists of the ordered is that where the fiber ab ove each point consists of the ordered I+tuples k-tuples of of linearly linearly independent independent tangent tangent vectors vectors (k-frames). (k-frames). In the presence presence of of aa Riemannian Riemannian metric metric itit isis appropriate appropriate to to use use representarepresentaIn the tions of the group 0, rather than GL,(IW) (by means of linear transformations tions of the group On rather than G Ln (]R) (by means of linear transformations of of the the fiber). fiber). Important Important homogeneous homogeneous spaces spaces for for the the orthogonal orthogonal and and unitary unitary groups are Stiefel manifolds and Grassmannian manifolds. The points the groups are Stiefel manifolds and Grassmannian manifolds. The points of of the real (or complex) Stiefel manifold Vzk (or V,“,) are the orthogonal k-frames real (or complex) Stiefel manifold V:'k (or Vn~k) are the orthogonal k-frames or in easy p;,,;;;; (Vl, ... , Vk) inin R]Rn71 ((or in @” en endowed endowed with with the the Hermitian Hermitian metric). metric). ItIt isis easy

to see that

v;k

2 son/so,-k

g on/on-k,

(1.8) (1.8) vzk g sun/sun-, V~k 2:! SUn/SUn-k 2 &l&-k. Un/Un-k. The GE,kk (or (or G~ Gz,,)k) The points points of of the the real real (or (or complex) complex) Grassmannian Grassmannian manifold manifold G~ are the k-dimensional subspaces of Iw” (or U?). Since the usual action of 0, are the k-dimensional subspaces of]Rn (or en). Since the usual ~ction of On on lP induces a transitive action of that group on the set of all k-dimensional on ]Rn induces a transitive action of that group on the set of an k-dimensional subspaces is not that subspaces (and (and analogously analogously for for U, Un on on P), en), itit is not difficult difficult to to deduce de duce that G;,k g so,/(so,-k on/(&-k G~,k 2:! SOn/(SOn-k xx sok) SOk) ”2:! On/(On-k GE k 2 su,l(su,-,, x suk) C~,k ~ SUn/(SUn-k x SUk)

x ok),

Ok),

X

(1.9) (1.9) z

i&/(&-k

Un/(Un-k

The orientable Grassmannian manifold GE k has as The orientable Grassmannian manifold ê~ k has as dimensional subspaces of Iw”. This is a double cover dimensional subspaces of ]Rn. This is a double cover corresponding Grassmannian manifold Gt,k:

x

X

uk).

Uk).

points the oriented kpoints the oriented k(with fiber Z/2) of the

(with fiber '1./2) of the

corresponding Grassmannian manifold C~,k:

There are natural

embeddings

There are natural embeddings

induced

by the standard

embeddings

IW* --+ lP+‘,

Cc” +

(Cnfl.

We denote

induced by the standard embeddings ]Rn ]Rn+1, en e n +1, We denote by Gf , Gf, Gz the direct limits of the respective sequences of spaces and by C~, ê~, Cf the direct limits of the respective sequences of spaces and embeddings: embeddings: ---)0

GF = lim GF,k, C~ = 12’00 Hm C~ k' n-+oo

It is easy to see directly

'

G^F = lim c:,,, n+m

---)0

GE = lim Gz,k. 71’00 lim C~ k' n-+oo

'

that GT is contractible, Gy = RP”, and Gy = CP”.oo It is easy to see directly that ê~ is contractible, C~ ]Rpoo, and = ep .

cf

150

Chapter Chapter 4. 4. Smooth Smooth Manifolds Manifolds

From From the the isomorphisms isomorphisms (1.13) (1.13) below, below, itit follows follows that that 7rj(Vt,) 7rj(V,~\) == 00 for for jj << nn - k, k, rj(V$) 7rj(Vn~k) = 00 for for jj << 2(n 2(n - k). k).

(1.10) (1.10)

The The principal principal bundles bundles E,” GF,,+, II!. = Vck v:IR + En n,k ----> GII!.n,k'

FF = 01, G, 0 k = G,

@ = vTk --+ @,,

F = SOI, = G,

E,” = Vzk +

FF = Uk uk = G, G,

Gz,k, G~,k'

(1.11) (1.11)

(where the obserobser(where the the projections projections are are the the obvious obvious ones), ones), taken taken together together with with the vation that the spaces vation that the spaces E& = lim EE, n-CC

@$ = lim ,!?,“, n-+m

Ez = lim E,@ lim E~ n-+00 n--+oo

are conclusion that: that: are contractible, contractible, lead lead to to the the conclusion The go in 00 to the universal universal GGThe principal principal bundles bund les (1.11) (1.11) go in the the limit limit as as nn --+ 00 ta the ’. bundles for the Lie Groups G = oki SOI,, uk. Hence the classifying spaces for bundles Jar the Lie Groups G = Ok, SOk, Uk. Hence the classifying spaces for these groups these groups are are BOI,

= GE,

Bsok

= C?;,

Buk

= G;.

(1.12) (1.12)

Recall that any any map -+ BG BG determines determines up to equivalence (principal) Recall that map M” Mn ----> up to equivalence aa (principal) G-bundle over M” as the induced bundle (and this determines a one-to-one G-bundle over Mn as the induced bundle (and this determines a one-to-one “classifying” correspondence between the equivalence classesof (princi"classifying" correspondence between the equivalence classes of (principal) G-bundles over M” and the set [M”, BG] of homotopy classesof homotopy classes of pal) G-bundles over Mn and the set [Mn, BG] of maps I@’ + BG). In the case of the tangent bundle of a manifold M” the maps Mn ----> BG). In the case of the tangent bundle of a manifold Mn the corresponding map Ivan -+ BOk (or M” --+ BSOk) arises naturally as a corresponding map Mn BOk (or Mn BSOk) arises naturally as a generalization of the spherical Gauss map of a surface. One first embeds Mn generalization of the spherical Gauss map of a surface. One first embeds Mn in IWN for sufficie nt 1y large N (this is possible by Whitney’s theorem), and in ]RN for sufliciently large N (this is possible by Whitney's theorem), and then associateswith each point x of M” the tangent plane to Mn at x (with then associates with each point x of Mn the tangent plane to Mn at x (with an orientation indicated if M” is orientable) parallel-translated to the origin an orientation indicated if Mn is orientable) parallel-translated to the origin in KP. This defines an appropriate classifying map in ]Rn. This defines an appropriate classifying map f : Mn + G Gt,n, (or f : M” + g”,,,, if Mn is orientable), IR (or J: Mn â~ ,n' if Mn is orientable), f .. Mn N,n' which is covered in the natural way by the induced map from the tangent which is covered in the natural way by the induced map from the tangent bundle of M” to the universal bundle. bundle of Mn to the universal bundle. The generalization of this construction to any smooth vector bundle with The generalization of this construction to any smooth vector bundle with base B a manifold presents no difficulty. Note that the total space E of such a B a manifold presents no difficulty. Note that the total space E of such a base bundle is contractible to B (identified with the zero-th cross-section). Embed bundle is for contractible to B N, (identified with the zero-th cross-section). E in IWN suitably large and consider at each point of B c E theEmbed plane E in ]RN for suitably large N, and consider at each point of BeE the plane ---j.

---j.

---j.

---j.

$1. Basic concepts. Smooth Smooth fiber fiber bundles. bundles. Connexions Connexions §l. Basic concepts.

151 151

IWN tangent tangent at at xz to to the the fiber fiber Fx F, c c IRN IWN through through x. z. On parallel-translating parallel-translating in IRN these planes to to the the origin origin in in RN, we obtain obtain the desired desired map map B --+ GE,nn (or these planes IR N , we t GIJt, -w if B is orientable). Since for any finite CW-complex K there is a B t êIJt"n if Bis orientable). Since finite CW-complex the;e is a *GN?l homotopy equivalent equivalent smooth smooth manifold manifold U U (obtained (obtained by by embedding embedding K K in in IRN lKN for for homotopy some N, N, and taking as as U U aa small small neighbourhood neighbourhood of of K K having having Kas K asdeformation deformation sorne and taking retract), above construction of of the the smooth smooth Gauss Gauss map determined by by aa retract), the the ab ove construction map determined vector bundle is, in essence, no less general than the construction of a universal vector bundle is, in essence, no less general than the construction of a universal G-bundle over finite CW-complex. CW-complex. G-bundle over aa finite For general smooth fiber bundles with G c Diff Diff F, F, the universal G-bundle G-bundle For general smooth fiber bundles with GC the universal may be constructed as follows. Denote by EN(F) the space of embeddings may be constructed as follows. Denote by EN(F) the space of embeddings F + t IWN. group Diff Diff F F then then acts acts in in the the obvious obvious way way on on EN(F), EN(F), and and F IR N . The The group the direct limit E,(F) = Jirrr EN(F) is contractible. The orbit space under the direct limit Eoo(F) lim EN(F) is contractible. The orbit space under N-too the action of G on E,(F) then gives gives the the desired desired base BG of of the the univeruniverthe action of G on Eoo(F) then base BG sal G-bundle. In the literature an alternative construction (due to Milnor in saI G-bundle. In the literature an alternative construction (due to Milnor in the late 1950s) is generally employed, based on a different idea, allowing the the late 1950s) is generally employed, based on a different idea, allowing the construction of wide class G. construction of BG BG for for aa wide class of of groups groups G. Returning to vector bundles, note that the exact sequences sequencesof of Returning to vector bundles, note that the homotopy homotopy exact the following principal bundles: the following principal bundles: so,-1 --+ SOn so, --+t so,/so,-1 SOn-l SOn/SOn-l ”~ s-l, sn-l, Un- 1 Un - t Un/Un- l ~ s2n-l (where SO,-1 4 SO,, U,-i -+ U, are embeddings) taken together with (where SOn-l - t SOn, Un- 1 - t Un are embeddings) taken together with the fact that 7rj(SkV1) is trivial for j 2 k - 2, yield isomorphisms: the fact that 1f'j(Sk-l) is trivial for j ::; k 2, yield isomorphisms:

nk(SOn) E 7rk(SOn-l), k < n - 1, 1f'k(SOn) ~ 1f'k(SOn-t}, k < n - 1,

(1.13) (1.13) TQIJ~) CL!T,#,-I), k < 2(n - 1). 1f'k(Un) ~ 1f'k(Un-t), k < 2(n - 1). Hence the natural embeddings son-k + SO,, un-k -+ U, (and Rence the natural embeddings SOn-k - t SOn, Un-k - t Un (and Sun-k --+ Sun) induce isomorphisms between the corresponding homotopy SUn-k - t SUn) induce isomorphisms between the corresponding homotopy groups of dimensions < n - k - 1, 2(n - k) respectively. From this (1.10) above groups of dimensions < n - k -1, 2(n - k) respectively. From this (1.10) above follows. (For Sp, one has, analogously, Tk(Spn) 2 ?Tk(Sp+l) for k < 4n - 2.) follows. (For SPn one has, 1f'k(SPn-l) for 2.) The homotopy groups rk(C,)analogously, g xk(SCn)1f'k(SPn) for k <~n-l, .TTk(Un) fork k<<4n2n-2, The homotopy groups 1f'k(On) ~ 1f'k(SOn) for k < n-1, 1f'k(Un) for k < 2n 2, and flk(S&) for k < 4n - 2, are called stable, since they are independent of and 1f'k(SPn) for k < 4n - 2, are called stable, since they are independent of n in these ranges. They were first computed by Bott (in the late 1950s) using n in these ranges. They were first computed by Bott (in the late 1950s) using the global calculus of variations (see $2 below). theThe global calculus geometry of variations (see §2 below). differential of smooth G-bundles with structure group G a The differential geometry of smooth G-bundles with structure group G a Lie group, depends crucially on the concept of a “connexion”. A differentialLie group, depends crucially on the concept of a "connexion". A differentialgeometric G-connexion on a principal G-bundle over a manifold B of dimengeometric on a principal G-bundle over manifold sion n, is aG-connexion smooth G-invariant field associating with aeach point Bof of thedimentotal sion n, is a smooth G-invariant field associating with each point of the total space E an n-dimensional subspace of the tangent space to E, transverse to space E through an n-dimensional of the E, transverse to the fiber the point. subspace (Recall that thetangent group Gspace acts to freely on E (on the the fiber through the point. (Recall that the group G acts freely on E (on the left say) with the orbits of the action as the fibers F g G.) The n-dimensional left say) with the or bits of the action thesaid fibers G.) The n-dimensional subspaces attached to each point of Easare to Fbe~horizontal. subspaces attached to each point of E are said to be horizontal.

152

Chapter Chapter 4. Smooth Smooth Manifolds Manifolds

On group G G there On the the Lie Lie group there is a standard standard right-invariant right-invariant l-form I-form zua Wo with with values values (at identified with (at tangent tangent vectors vectors to to G) G) in in the the Lie Lie algebra algebra 6\!) of of G, G, where where 6\!) is is identified with the at the algebra algebra of of right-invariant right-invariant vector vector fields fields on on G, G, namely namely the the l-form I-form which which at each tangent G, takes each tangent vector vector 1777to to G G at at x E G, takes as value value the the unique unique right-invariant right-invariant vector The l-form satisfying E(x) ç(x) == v. 17. The I-form then then satisfies satisfies vector field field eç satisfying

1 -- ;[wo;Zuol (Ad s)wo, 2 [wo, wo] = dwo, dwo, g*wo g*wo = = (Ad g)wo,

(1.14) (1.14)

where left multiplication h H gh, and where gg acts acts by by left multiplication h I---t gh, and [[ ,, ]1 denotes denotes the the exterior exterior product G with \!). (Recall (RecaIl that that forms forms and and product operation operation on on forms forms on on G with values values in in 6. cochains any ring the cochains with with values values in in any ring may may be be multiplied, multiplied, although although in in general general the multiplication will not be Lie multiplication will not be skew-commutative. skew-commutative. Note Note also also that that for for aa matrix matrix Lie group G of the the linear linear transformation transformation Ad Ad gg of the (matrix) (matrix) Lie Lie algebra algebra 6 \!) is is group G the effected by A effected by conjugation conjugation of of the the matrices matrices uu E E 8 \!) by by g: g: Ad Ad g(u) g(u) = gug-I.) gug- 1 .) A connexion on B, G, G,p) is (E,B,G,G,p) is then then determined determined by by any any connexion on a a principal principal bundle bundle (E, l-form algebra 6, satisfying I-form w w on on EE with with values values in in the the Lie Lie algebra \!), satisfying g*w g*w = = (Ad (Ad g)w, g)w,

wIF = wo, WIF = WO,

(1.15) (1.15)

where is the where FF (E G) G) is the fiber fiber over over any any point; point; the the horizontal horizontal n-dimensional n-dimensional tantangent subspaces at each point y of E are defined by the equation gent subspaces at each point y of E are defined by the equation w w= = 0 0 at at y. y. ItIt can any connexion can be be shown shown that, that, conversely, conversely, any connexion on on the the G-bundle G-bundle determines determines such 8uch aa l-form I-form w. w. The on EE is The curvature curvature form form Q fl on is the the 2-form 2-form defined defined by by 1 n=dw+;[w,w]. dw + 2 [w,w].

(1.16)

(1.16)

It follows that g*Q = (Ad g)Q for all g E G. Furthermore the form R is, as It follows that g* fl = (Ad g)fl for aIl g E G. Furthermore the form fl is, as they say, “purely horizontal”: it vanishes on any pair of tangent vectors of they say, "purely horizontal": it vanishes on any pair of tangent vectors of which at least one is vertical, i.e. tangent to the fiber. Thus 0 has non-zero which at least one is vertical, Le. tangent to the fiber. Thus fl has non-zero values essentially only on pairs of “horizontal” tangent vectors. values essentially only on pairs of "horizontal" tangent vectors. Let x1,. , xn be co-ordinates on any (distinguished) chart U c B satisfyLet xl, ... , Xn be co-ordinates on any (distinguished) chart U C B satisfying p-‘(V) Z U x G (via &J say). These then furnish coordinates on U x {l}, ing p -1 (U) '2:! U x G (via 1>u say). These then furnish coordinates on U x {1}, and in terms of such local co-ordinates the form w of the connexion, restricted and in terms of such local co-ordinates the form w of the connexion, restricted to U x {l}, is given by to U x {1}, is given by

(where the A, are (where the Aa are form 0 has locally

form fl has locally

wq,u = WIUX{l} = Aa dxa , B-valued functions of the point (xa)), and the curvature \!)-valued functions of the point (x a )), and the curvature the form the form f&,, = L’luxfll

= Fabdx” A dxb, flu fllux{l} Fabdxa 1\ dx b, where the Fab are B-valued functions of (xa). From (1.16) it follows that where the Fab are \!)-valued functions of (x a ). From (1.16) it follows that Fab = aaAb - ab& + [A,, Ab] , a, = a/ax? (1.17) (1.17)

$1. Smooth fiber Connexions §l. Basic Basic concepts. concepts. Smooth fiber bundles. bundles. Connexions

153 153

In view of w and and n R to to U xx {1} { 1) In view of of (1.15), (1.15), these these local local formulae formulae for for the the restrictions restrictions of determine completely on on the p-i(V) E U x G. G. determine w w and and 0n completely the whole whole region region p-l(U) 8:' How expressions for for Wu wou and and nu $,, change change under under changes changes How do do the the above above local local expressions of bundie bundle coordinates, the diffeomorphism Let 'l/Ju $)rr be of coordinates, i.e. Le. changes changes of of the diffeomorphism &? t/Ju? Let another G, and and consider another diffeomorphism diffeomorphism p-‘(V) p-l(U) ---+ U U x G, consider the the diffeomorphism diffeomorphism &,$qjl G ---> + U U xx G, t/Ju'l/J"i/ :: U U xx G G,

(1.18) (1.18)

h g) 9) 1-+ +-+ (x, (x7 JG)g), (x, À(x)g),

where appropriate transition transition function, G. In particular, de fines the the appropriate function, X(z) À(x) EE G. In particular, where XÀ defines the cross-section U x 1, where 1 E G is the identity element, goes to different the cross-section U xI, where 1 E Gis the identity element, go es to aa different one: one: qh/!+ :: (x, (x, 1) 1) 1-+ H (x,X(x)), G. (1.19) t/J'l/J-l (x, À(x)), X(x) À(x) EE G. (1.19) (A transformation of the the type (1.19) is called aa gauge gauge transtrans(A transformation of type (1.19) is in in this this context context called formation.) Under this transformation the connexion and curvature forms formation.) Under this transformation the connexion and curvature forms transform as follows (assuming that G and @3 are comprised of matrices): transform as follows (assuming that Gand Q; are comprised of matrices):

(9

Wd+-+w*, A,dx”

w6 = A,dxa,

H Ahdx”,

wq,j = Ahdx”,

A:, = XAJ-l

+ (&X(x))

X-‘(x); (1.20) (1.20)

Fab H +)Fabx-1(X)

= Fib.

Example 1. Let G be an abelian Lie group (for example G = S1 g Example 1. Let G be an abelian Lie group (for example G SI 8:' VI). Then in view of (1.16) th e curvature form is in essence the closed U ). Then in view of (1.16) the curvature form is in essence the cIosed on1 the base defined in terms of the connexion form w by

SO2 g S02 8:' 2-form

2-form

on the base defined in terms of the connexion form w by p*R = dw. p* dw.

n

The cohomology class [0] E H’(B;lR) turns out to be independent of the The cohomology cIass [n] E H 2 (B; IR) turns out to be independent of the choice of connexion, and to be “integral” in the sense that the integrals of the choice of connexion, and to be "integral" in the sense that the integrals of the form R over 2-cycles are integers.

form nover 2-cycIes are integers.

Example 2. Let G = U,. In this case the form Tr Q (= Tr F,bdx” A dxbb in Example 2. Let Gof =any Un.curvature In this case the form Tr n (= Tr FabdXa A dx in local co-ordinates) form 0, is again a closed 2-form on the local co-ordinates) of any curvature form n, is again a cIosed 2-form on the base with cohomology class

base with cohomology cIass

[Tr n] = Cl E IP(B; independent

of the connexion,

and integral.

R) This cohomology

class is called

independent cohomology cIasspowers is called the first Chernof the classconnexion, of B. The and formsintegral. Tr(Qm)mThis obtained by taking of the first Chern class of B. The forms Tr( n obtained by taking powers of ) the matrix (Fab), represent integral cohomology classes & E H2m(B; R) (or 2 the , represent integral cohomology classesfor Cmthe E H m(B; IR) (or over matrix Cc), and (Fab) are also called Chern classes of B. However integral Chern over q, and are also called Chern classes of B. However for the integral Chern classes (see below for their precise definition) there is a more appropriate basis classes (see below for their precise definition) there is a more appropriate basis

154 154

Chapter Chapter 4. 4. Smooth Smooth Manifolds Manifolds

in terms of of which which the the classes classes & Ci have integer integer polynomial polynomial expressions expressions (but (but not not vice viee versa) zi =Pi(cl,...,ci). Ci = Pi (Cl, ... , Ci)' ClrC2..., Cl, C2 ... ,

The ci , .... . ,,Ci) ci) are are in the Newton expressThe polynomials polynomials Pi( Pi (Cl, in fact fact the Newton polynomials polynomials expressn

of the symmetric L ~1 uk inin terms terms of the elementary elementary symmetric

ing ing the the symmetric symmetrie polynomials polynomials 2

k=l k=l

polynomials polynomials cj Cj =

L

c uil’...‘uij: Uil·· ... Uij: i,<...
E1=cl, Cl Cl,

ci - 2C2,

~2=+2c2, C2

..... ..

This This representation representation of of the the classes classes &, Ci, ci Ci as as symmetric symmetric polynomials polynomials has has topotopological significance, and turns out to be very useful (see below). 0 logical significance, and turns out to be very useful (see below). D Example SO,, the curvature form any connexion) Example 3. 3. In In the the case case G G ==: SOn, the curvature form R [] (of (of any connexion) 4 yields Pontryagin classes pi = [Tr n2i] E H4i(B; IR) of B, also yields the the Pontryagin classes Pi = [Tr []2i] E H i(B; IR) of B, also integral. integral. As As in caseof shall choose in the the case of the the Chern Chern classes, classes, we we shaH choose below below a a different difIerent (“indivisible” ("indivisible" over of which are once Pi for for the the Pontryagin Pontryagin classes, classes, in in terms terms of which the the pi Pi are once over Z) Z) basis basis pi again expressed as Newton polynomials. cl D again expressed as Newton polynomials. The field of subspaceson space The field of horizontal horizontal n-dimensional n-dimensional tangent tangent subspaces on the the total total space of a principal G-bundle, determining, in accordance with the definition, of a principal G-bundle, determining, in accordance with the definition, aa connexion likewise transverse connexion on on the the bundle, bundle, induces induces such such a a field, field, likewise transverse to to the the fibres, fibres, on the total space E’ of any associated bundle (E’, B, F, G, p’). Via on the total space E' of any associated bundle (E', B, P, G, p'). Via this this field field of horizontal tangent subspacesevery smooth or piecewise smooth path y(t), of horizontal tangent subspaces every smooth or piecewise smooth path "((t), a 5 t 5 b, in the base B, is covered in the total spaceE’ by a unique horizontal a 5 t 5 b, in the base B, is covered in the total space E' by a unique horizontal once the initial point T(a) is prescribed (see Figure path path T(t), ;Y(t), p’?(t) p';Y(t) == -dt), "((t), once the initial point ;Y(a) is prescribed (see Figure 4.3). 4.3).

Fig. 4.3 Fig. 4.3

~(tJ

The covering motion of the point y(t) ab ove y(t) is termed parallel transThe This covering motion of represents the point ;Y(t) is termed parallel translation. construction the above smooth"((t) version of the “homotopy lation. This introduced construction represents smooth of the“covering "homotopy connexion” in Chapter 2, the $3, with the version concomitant ho-

connexion" introduced 2, §3, with the concomitant "covering homotopy property”, and in theChapter homotopic consequences flowing thence. motopy property", and the homotopie consequences flowing thence.

$1. §l. Basic Basic concepts. concepts. Smooth Smooth fiber fiber bundles. bundles. Connexions Connexions

155

AA connexion connexion on on a principal principal G-bundle G-bundle determines determines an an operation operation of of covariant covariant differentiation the direction direction of of any any tangent tangent vector vector to to the the base base B, B, on on the the difJerentiation in in the cross-sections cross-sections of of any any associated associated G-bundle. G-bundle. As As noted noted above, above, in in terms terms of of local local co-ordinates of the the base base BB the the connexion connexion l-form I-form is is given given co-ordinates on on suitable suit able regions regions UU of by by w4” Wq,u == A,(z)dz”. Aa(x)dxa. Since Since A, Aa isis aa matrix-valued matrix-valued function, function, we we may may write write A, Aa = (A,): (Aa)~ == (A$), (A~j), == 1, q. AA cross-section 1, ... ,,q. cross-section of of an an associated associated G-bundle G-bundle can can locally locally be be reprepresented (1Ji(X)) with with values values in in Iw”. ]Rn. The The covariant covariant resented as as aa vector-valued vector-valued function function (@(z)) derivatives then the the operators operators D, Da defined defined by by derivatives are are then i,i, jj

D,=a,+A,, Da = 8a + Aa,

D&(x) Dar/(x) = a,$ 8a'r/i ++ A;$. A;/I{

(1.21) (1.21 )

The The commutators commutators [Da, [Da, Db] Db] yield yield the the curvature curvature components components (1.17): (1.17): [Da,

Db]

=

adb

-

ab&

-t

[Aa,

hi

=

&b.

(1.22) (1.22)

In the In the the case case of of the the tangent tangent bundle bundle 7(Mn) r(Mn) the the above above discussion discussion yields yields the classical concepts of Riemannian geometry. It appears that the idea of a conclassical concepts of Riemannian geometry. It appears that the idea of a connexion was first introduced in 1930s by Weyl in the 1930s by Weyl in aa special special nexion on on aa fibre fibre bundle bundle was first introduced in the case, and in general form by Cartan. Physicists came to the concept case, and in general form by Cartan. Physicists came to the concept somewhat somewhat later the mid-1950s), mid-1950s), although although doubtless doubtless independently independently later (Yang (Yang and and Mills Mills in in the - at least of Cartan. In physical terminology connexions called "gauge “gauge at least of Cartan. In physical terminology connexions are are called fields”; they are of fundamental importance in the physics of elementary fields"; they are of fundamental importance in the physics of elementary partiparticles. Weyl) is that of The simplest simplest example example of of such such aa field field (due (due to to Weyl) is that of aa connexion connexion cles. The on Ui Z~ SO2 on aa G-bundle G-bundle with with G G= U S02 2~ Si, Sl, having having the the physical physical interpretation interpretation 1 as an electromagnetic field with field strength the curvature tensor Fab. As as an electromagnetic field with field strength the curvature tensor Fab. As noted above (see (1.20) the gauge transformations are just those transformanoted above (see (1.20) the gauge transformations are just those transformations of the local formula for the connexion, arising from changes from the tions of the local formula for the connexion, arising from changes from the identity cross-section to another: (z, 1) H (IC, X(X)). In Einstein’s general theidentity cross-section to another: (x, 1) ~ (x, À(x)). In Einstein's general theory of relativity one considers connexions on the tangent bundle arising from ory of relativity one considers connexions on the tangent bundle arising from the gravitational field. the gravitational field. Returning to the main line of development of our exposition, we now inReturning to the main line of development of our exposition, we now introduce the concept of a “characteristic class” of a smooth G-bundle. troduce the concept of a "characteristic class" of a smooth G-bundle. Definition 1.1 A characteristic class 0 with respect to the category of GDefinition 1.1 A characteristic class () with respect to the èategory of Gbundles (E, B, F, G, p) is a function associating with each G-bundle an element (E, B, F, G, p) is a function associating with each G- bundle an element bundles B of H’(B) (over some coefficient ring) in such a way that under bundle maps ()@of: E H*(B) (over some coefficient ring) in such a way that under bundle maps ---+ E’, 4 : B + B’, the characteristic class behaves naturally (more rI> : E --+ E', rj; : B --+ B', the characteristic class behaves naturally (more precisely, as a contravariant functor): precisely, as a contravariant functor): +*(O’) = 0, c) : B --f B’. rj;*(()') (), rj;: B --+ B'. It is easy to seethat the characteristic classesof G-bundles correspond oneIt is to easy see that of thethe characteristic oneto-one thetoelements cohomology classes ring of of theG-bundles base BG correspond of the universal to-one to the elements of the cohomology ring of the base BG of the universal G-bundle: For, as noted earlier, every G-bundle with base B is induced by a G-bundle: For, as noted earlier, every G-bundle with base B is induced by a

156

Chapter Chapter 4. Smooth Smooth Manifolds Manifolds

map (BG) is map 4rp :: BB --+ ~ BG BG unique unique up up to to homotopy, homotopy, so so that that ifif 0’ e' E H* H*(BG) is any any given element, then we obtain a characteristic class 0 of the G-bundle over given element, then we obtain characteristic class of the G-bundle over BB induced induced by by 4, rp, as the the pullback pullback of of 8’: e': 0e== $*(Q’). rp*(e /). For For compact compact abelian abelian groups groups G G the the spaces spaces BG BG are are known: known:

e

G=S1, G = SI,

BG BG == @Pm; Cp=;

G = Tn,n , G=T

BG CP,-; BG == @Plm Cpre xx ... ... xx CP,:;';

G = (Z/2)n,

BG = iRP,oo x . . x IWPF;

G G = Z/m, 7/.,/m,

BG lim S~/Z/m, BG = 7X-m Hm S2n-1/Z/m s2n-I/7/.,/m = SOO /7/.,/m, n--+oo

(where, lens space). space). The (where, in in the the last last case, case, P/Z/m SOO /7/.,/m is is the the infinite-dimensional infinite-dirnensionallens The cohomology rings of these BG are also known; of particular importance cohomology rings of these BG are also known; of particular importance are are the the following following ones: ones: H*(@P,“x...~@p,“;Z)“Z[u~,...,u,l, H* (CPf' X ... x CP,:;'; 7/.,) ~ 7/., [U1,'''' un] , (the E H22 (“Pj”; Z)); (the polynomial polynomial ring ring over over ZZ in in uj Uj E H (CPt'; 7/.,) ); H’ (RP,oo x .

x RP,“;

Z/2) ” Z/2 [q, . . . , v,] ,

uj (“Pj” ; 7/.,/2) vq ;. v EE H1 Hl (lRpoo. J

J"

H* UP) = 44~) Z/P 1~1, H* (WP; (B7/.,/p; 7/.,/p) Ap(v) 0@ 7/.,/p ru],

(1.23) ( 1.23) u = ,Bu E H22 (BZ/p; Z/p) , ‘u E H1 (B~/P; UP), v E Hl (BZ/p; Z/p) , U = {3v E H (BZ/p; 7/.,/p) , where homomorphism (see Chapter 3, §5), §5), Ap(v) A,(w) is is the the where ,Ll {3 is is the the Bockstein Bockstein homomorphism (see Chapter 3, exterior algebra over Z/p generated by w (v” = 0), and the tensor product has 2 exterior algebra over 7/.,/p generated by v (v 0), and the tensor product has imposed on it skew-commutativity. imposed on it skew-commutativity. The most important facts about the cohomology rings H* (BG) for G = O,, The most important facts about the cohomology rings H* (BG) for G On, SO,, U,, SU, with coefficients from Z, Z/2 and IR are as follows: SOn, Un, SUn with coefficients from 7/.,,7/.,/2 and IR are as follows: In the group U, we have the subgroup Tn,n a maximal torus, comprised In the group Un we have the subgroup T , a maximal torus, comprised of the diagonal matrices g, and the Weyl group S,, of permutation matrices of the diagonal matrices g, and the Weyl group Sn of permutation matrices g normalizing Tn:n gTng-’ 1 = Tn.n The restriction of the cohomology ring of 9 normalizing T : gTng- = T . The restriction of the cohomology ring of BU, to BTn has zero kernel, i.e. the homomorphism BUn to BTn has zero kernel, Le. the homomorphism i* : H*(BU,;Z)

-

H*(BT”;Z)

=Z[u~,...,u,],

invariant under induced by the inclusion i : BTn ----+ BU,, is a monomorphism induced by the inclusion i : BTn ~ BUn, is a monomorphisrn invariant under the action of the Weyl group. It follows that the cohomology ring of the space the action of the Weyl group. It follows that the cohomology ring of the space BU, is given via the homomorphism i’ by that of BT”, and that the image unBUn is given via the homomorphism i* by that of BT n , and that the image under i* consists of all symmetric polynomials in ~1, . . , u,, thus yielding a comder consists of aIl symmetric in U1, ... , Un, thus yielding a completei* description of the ring H* polynomials (Bun; Z). The Chern class ci E H2i2i (Bun; Z) plete description of the ring H* (BUn; Z). The Chern Ci E H (BUn; 7/.,) is defined as the element mapped by i* to the i-th class elementary symmetric is defined as the element mapped by i* to the i-th elementary symmetric polynomial in the “Wu variables”, i.e. the elements uj E H22 (@Pjoo; Z): polynomial in the "Wu variables", Le. the elements Uj E H (CPt; 7/.,):

Basic concepts. concepts. Smooth Smooth fiber fiber bundles. bundles. Connexions Connexions §51. 1. Basic Ci f---t H Ci

157 157

n71

L

uj, . . ... . . . Uji' uj; ) Ujl··

c

L uj.

Ei f---t H Ci c

jl<...<ji h<···<ji

(1.24) (1.24)

U;.

j=l j=l

Thus the the elements elements Ci ci form form a basis basis for for H* H’ (BUn; (Bun; Z). Z). (The (The elements elements Ci Ci were were Thus defined earlier earlier in in terms terms of of the the curvature curvature form.) form.) defined For the the groups SOzn, S02n+l SOzn+i the the maximal maximal torus torus Tn T” is is as as for for Un, U,, but but the the groups S02n, For Weyl groups are are larger. larger. The The kernel kernel of of the the restriction restriction homomorphism homomorphism Weyl groups H*(BSOk;Z) Z) + t H*(BTn;Z), H*(BSOk; H*(BT n ; Z),

kk=2n,2n+l, = 2n, 2n

+ 1,

analogous to i’ above, above, contains contains only only the the 2-torsion 2-torsion elements. elements. The The Weyl Weyl group group of of analogous toi* SOzn, modelled via via the the action action on on H* (BTn;n ; Z), Z), is is generated generated by by the the symmetric symmetric S02n, modelled H*(BT group all permutations permutations of U1, ui, ... . ,, Un, u,, together together with with the the transformations transformations group S, Sn of of all of aij defined defined by (J'ij by 1 Ui H

Uij

-Ut,

UjH-Uj,

ukwuk,

k#i,j.

(1.25) (1.25)

For needs instead instead of of these the transformations transformations For SOzn+i S02n+l one one needs these (as (as generators) generators) the given by given by Us 1 Ui H -Ui, Uj H Uj, j # i. (1.26) (1.26) Thus the set invariant polynomials polynomials for for SOzn+i than for for S02n. SOZ,. Thus the set of of invariant S02n+l is is smaller smaUer than The invariance of the homomorphism H*(BSOZ~; Z) * H*(BTn; Z) The invariance of the homomorphism H*(BS0 2n ; Z) -> H*(BTn; Z) under under the of the the Weyl Weyl group yields as as image image the linear span the elemenelementhe action action of group yields the linear span of of the tary symmetric polynomials in the squares u;, representing the Pontryagin tary symmetric polynomials in the squares u;, representing the Pontryagin classes pi, supplemented by the the polynomial ~2~ == Ul ui ... Un, u,, representing representing the the classes Pi, supplemented by polynomial X2n Euler characteristic class. The image under the restriction homomorphism Euler characteristic class. The image under the restriction homomorphism Z) on the other hand, is spanned only by the H*(BS02,+1; 4, -+ H*(BTn; H*(BS02n+1iZ),--+ H*(BTn;Z) on the other hand, is spanned only by the images of the Pontryagin classes. In summary (for both SOa,, SOzn+i):

images of the Pontryagin classes. In summary (for both S02n, S02n+1): Pi H

Pi

f---t

L j,<...<ji

2 uj;...

. uji E H4i(BTn; Z),

UJl" ... UJi E H 4i (BT n ; Z),

c

11<· .. <ji

X2n

X2n

H

Ul

f---t

Ul

(1.27) (1.27)

E H2”(BTn; 2n (BT n ; Z), ... Un E H Z),

.

u,

pi E ff4i(BSok;Z), k 2 2n, ~2~ E H2”(BS02,;Z). Pi E H 4i (BSOk; Z), k:2: 2n, X2n E H 2n (BS0 2n ; Z). Over the field IR of reals (or over Q) the situation is the same for all comOver the field lR of reals (or over Q) the situation is the same for aU compact Lie groups G (with maximal torus Tn): restriction homomorphism n ): the pact Lie groups G (with maximal torus T the restriction homomorphism H* (BG; IR) 4 H*(BTn; JR) has trivial kernel and image consisting all polyH*(BG;lR) H*(BTn;lR) has trivial kernel and image consisting polynomials in 211,. . . , u,, invariant under the induced action of the WeylaUgroup. 11,1,"" Un, invariant under the induced action of the Weyl group. nomials in In order to describe the mod-2 cohomology of BG, with G = O,, SO,, with the G =subgroup On, SOn, order describe the mod-2 cohomology of BG, oneInuses thetoobvious discrete analogue of the torus, namely of one uses the obvious discrete analogue of the torus, namely the subgroup of diagonal matrices (Z/2)n c 0,. Here the restriction homomorphism

diagonal matrices (Z/2)n COn. Here the restriction homomorphism H’(B0,;

Z/2) +

H*(B(Z/2)n;

uj E H1(RPjoo; Z/2),

Z/2) 2 Z/2 [q, BZ/2 = RP”,

. . , un] ,

158 158

Chapter Chapter 4. Smooth Smooth Manifolds Manifolds

has has trivial trivial kernel, kernel, and and the the image image consists consists of of the the symmetric symmetric polynomials polynomials in in ~1, . , u,. elements of of H*(BO,; Z/2) sent VI,···, v n · The The elements H*(BO n ;Z/2) sent under under this this homomorphism homomorphism to to the symmetric polynomials . ,,V II,,n , are the elementary elementary symmetric polynomials in in ~1, VI, ... are called called the the StiefelStiefel-

Whitney characteristic Whitney characteTistic classes classes wj: W{ wj Wj H f--->

1 Vi,I '‘. .... . . ’ Vi; Vi,? E LVi E H’(B(Z/2)“; Hi (B(Z/2)n; Z/2). Z/2).

(1.28) (1.28)

il<...
The natural homomorphism The natural homomorphism H*(BO,;Z/2) H*(BO n ;Z/2) ---+ -> an an epimorphism) epimorphism) with with kernel kernel the the principal principal ideal ideal of of wr. of multiples multiples of Wl. Note that reduction of the Note that reduction modulo modulo 22 of the Chern Chern Whitney classes ~2%~ and of ~2~ yields Whitney classes W2i, and of X2n yields ~2~: W2n:

~2% = ci (mod 2), W2i = Ci (mod 2),

H*(BSO,;Z/2) H*(BSO n ;Z/2) isis onto onto (i.e. (i.e. generated generated by by wr, Wl, i.e. Le. the the set set classes classes ciCi yields yields the the StiefelStiefel-

(mod 2). (mod 2).

= ~2% X2n

~2~

=

W2n

Observe also that Observe also that P,(T)

(mod

2)

=

w4i(cV)

L

=

1

w2j(~)w2k(~)~ W2j(TJ)W2k(TJ),

2j+2k=2i

2j+2k=2i

(where c denotes LLcomplexification”), = c,. (where c denotes "complexification"), and and for for U,-bundles Un-bundles xsn X2n en. The characteristic classes of a manifold are defined as the characteristic The characteTistic classes of a manifold are defined as the characteristic classes of the tangent bundle of the manifold. classes of the tangent bundle of the manifold. Such are the basic facts concerning characteristic classes. We now describe Such are the basic facts concerning characteristic classes. We now describe some of their further properties. some of their further properties. Note first that the condition wr(qn) = 0 is nessesary and sufficient for Note first that the condition wdTJn) = 0 is nessesary and sufficient for orientability of the G-bundle qn, i.e. for reducibility of the bundle structure orientability of the G-bundle TJn, i.e. for reducibility of the bundle structure group G to SO,. group G to SOn' Recall that with any real (or complex) vector bundle qn with base B one Recall that with any real (or complex) vector bundle rI" with base B one may associate a fibration (over B) with fiber

may associate a fibration (over B) with fiber F = Vin-lci

(or F = V$-k),

the corresponding Stiefel manifold. The Stiefel-Whitney (or Chern, in the the corresponding Stiefel manifold. The Stiefel-Whitney (or Chern, in the complex case) classes arise as the “first obstructions” to the existence of crosscomplex case) classes arise as the "first obstructions" to the existence of crosssections (see Chapter 3, 56) of this fiber bundle:

sections (see Chapter 3, §6) of this fiber bundle:

v~,-k(rln)

vk-, -

B (or V&-k(?)

V?Y+k B

A

in the complex

case),

where rj(F)

= 0,

j < k - 1,

7Q-l(F)

@ 2/2

” Z/2,

F = Vn!jnmkr

or

or

7rj(F) = 0, j < 21c - 1,

7~~~-r(F)

g Z,

F = V&wk.

51. Basic Smooth fiber bundles. bundles. Connexions §l. Ba.'lic concepts. concepts. Smooth fiber Connexions

159 159

The group TrI ni(B)(B) acts on the group 7rk.-r(F)@Z/2 the The fundamental fUlldamental group acts trivially trivially on the group Trk~l (F)0Zj2 in in the first case (F = V:n-k), and on nzk-i(F) (F = V&k), where the structure first case (F V,iRn~k)' and on Tr2k~I(F) (F V~n~k)' where the structure groups bundle Tt vn are U, (or (or may be GLn(JR), GL,(IR), groups of of the the original original vector vector bundle are O,, On, Un' may even even be GL, (@)) respectively. G Ln (C)) respectively. The the Poincark duals DWi, Dwi, DCi Dci of The construction construction of of the Poincaré duals of the the characteristic characteristic cycles cycles w,, in the the appropriate appropriate homology groups, is is more more direct. direct. For For the the Wi, c, Ci as as cycles cycles in homology groups, classes any On-bundle rln with fiber R” and base smooth closed closed classes wi, Wi, take take any On-bundle TJn with fiber lR n and base B B aa smooth manifold, field on on B of k-tuples Ic-tuples of of vectors vi(z), . . , Vk(X), v~(z), manifold, and and consider consider aa field B of vectors Vl(X),,,,, II: B, in position; the set of aU all points points zx E E B B where vectors ;r E B, in general general position; the set of where the the vectors ‘~1 (xc), , uk(z) are are linearly dependent then cycle (mod (mod 2) 2) of of Vl(X)"",Vk(X) linearly dependent th en constitutes constitutes aa cycle dimension Dwi. In other words words ifif one one dimension nn -- k/c ++ 11 representing representing the the element element DWi. In other considers fiber bundle cross-section of the associated associated fiber bundle Vf,-,(v”) V!n~k(TJ") over over B B and and aa cross-section of considers the this then all’ of this this section section form form the the cycle cycle this bundle bundle in in general general position, position, then aU “zeros” "zeros" of representing the case of an an oriented (or SO,,-) SO,-) bundle TJ", qn, nn = = 2m 2m representing Dwi. DWi' In In the case of oriented (or bundle and is just form an an integral integral and Ic k == 1, 1, the the above ab ove field field is just aa vector vector field, field, whose whose zeros zeros form cycle representing the element element Dxzm. The duals of the the Chern Chern cycle of of dimension dimension nn representing the DX2m' The duals of classes may be obtained similarly. classes may be obtained similarly. For subgroup Gi cC G G will will frequently serve as as structure structure group; For G-bundles G-bundles aa subgroup G frequently serve group; I the operation of taking a subgroup as a structure group, where possible, is the operation of taking a subgroup as a structure group, where possible, is called reduction of the structure group G. A structure Lie group G may always called reduction of the structure group G. A structure Lie group G may always be compact subgroup in aa class class of of equivalent equivalent be reduced reduced to to aa maximal maximal compact subgroup in 0C”-00 _ bundles. (However for a class of equivalent complex-analytic G-bundles bundles. (However for a class of equivalent complex-analytic G-bundles this this is not always possible.) is not always possible.) Any complex-vector bundle q may be regarded as real, by “forgetting” Any complex-vector bundle TJ may be regarded as real, by "forgetting" the complex structure: q H rv; this operation corresponds to the embedding the complex structure: '1] ~ 7'TJ; this operation corresponds ta the embedding 2 SOzn (or GL,(@) A GLz,(JR)). In the other direction, one can un Un S02n (or GLn(C) ~ GL 2,,(lR)). In the other direction, one can apply to any real vector bundle 77 the operation of complexijication r] H CT, apply ta any real vector bundle TJ the operation of complexification TJ ~ C17, using the embedding GL,(IR) 5 G&(C). Given any representation (i.e. using the embedding GL,,(lR) ~ GL,,(C). Given any representation (Le. one obtains, in the homomorphism) p : Gi ---+ Ga of one group to another, homomorphism) p : G I --+ G2 of one group ta another, one obtains, in the obvious way, from any Gi-bundle q a Gz-bundle pq. We now list these and obvious way, from any G1-bundle TJ a G2-bundle PTJ. We now list these and other operations on vector bundles (some mentioned earlier):

other operations on vector bundles (some mentioned earlier): 1. The direct sum 71 $ r/2. 1. The direct sum TJl œTJ2. 2. The tensor product ~1 8 772(over R or C). 2. The tenso7' p7'Oduct TJl 0 TJ2 (over lR or C). 3. The exterior powers Ajq (coinciding when j = n = dimv with the deter3. The exte7'io7' powe7'S Aj TJ (coinciding when j = n = dim TJ with the dete7'minant of the bundle: det 7 = nnq). minant of the bundle: detTJ A"TJ). 4. The symmetric powers Sjq (the symmetric part of the tensor power &q). 4. The symmetTic powe7'S sj TJ (the symmetric part of the tensor power 0 j '1]). Thus Thus (1.29) A27 CB s2q = 7 @ q. (1.29) 5. The dual bundle q*; the complex conjugate bundle 7j (when Q is a complex 5. The dv,al bundle 17*; the complex conjugate bundle'fi (when 1] is asatisfying complex bundle); the operations r and complexification c noted above, bundle); the operations 7' and complexification c noted above, satisfying

7'C( TJ)

TJ

œ1]*,

cTJ

cTJ,

cr( 1])

9i. 1]

œ'fi.

(1.30)

160 160

Chapter 4. Smooth Smooth Manifolds Manifolds Chapter

It follows from their their definitions definitions that that the Pontryagin Pontryagin classes classesof aa vector vector bundie bundle It n coincide (up to sign) with the Chern classes of the complexified bundle cq: TJ sign) with classes complexified CTJ: (-l)“Pi(77)

=

Czi(Crl),

2C2i+1(crl)

=

(1.31)

0.

For direct direct sums sums of vector vector bundies bundles the following following formulae formulae are are valid: (9 (i)

Wi(Vl @rl2) Wi(1)l EB 1)2)

= =

L

Wjbl)Wk(~2)> Wj(rl1)Wk(TJ2),

L

cjh)ck(r/2), Cj(TJ1)Ck(TJ2),

c

wo Wo

= 1; 1; =

j+k=i j+k=i

(ii) (ii)

ci(rll(TJ1 @ Ci EB 72) TJ2)

=

c

CO = Co

1;, l'

j+k=i j+k=i (iii) (iii)

Pi(Q(1)1 EB @ TJ2) rl2) Pi

c L

=

(1.32) (1.32) pj(qi)pk(r]Z), to within within 2-torsion; 2-torsion; to Pj(1)1)Pk(TJ2),

j+k=i j+k=i Xzn(rl1 @ 772) = X2k(%)X2m(~2), + m = = n, 72, (iv) (iv) X2k (1) r) X2m (TJ2) , kk+m X2n(TJ1 EBTJ2) where in the case assumedthat that dim771 2k, dim dim772 = 2m. 2m. where in the case (iv) (iv) itit is is assumed dim TJ1 = = 2k, TJ2 = It follows from these formulae that except for ~2~ these classes are all all It follows from these formulae that except for X2n these classes are stable, i.e. if E; (or E,“) is the trivial bundle with fiber RN (or CN), then stable, Le. if €ft (or €[!) is the trivial bundle with fiber ]RN (or eN), then wi(rl (TJ @$) Wi(rl), Wi œ€fr() = Wi( TJ),

ci(rl @ $) Ci( TJ œ €[!)

= ci(rl), = Ci (TJ),

Pi(rl &,“) Pi (TJ @ EB éft)

=

Pi(V).

Pi (TJ)·

The behaviour of characteristic characteristic classes classesunder the tensor product operation The behaviour of under the tensor product operation (and likewise the operation of forming exterior and symmetric of powers) of (and likewise the operation of forming exterior and symmetric powers) vector bundles over B is somewhat more complicated. This behavior may vector bundles over B is somewhat more complicated. This behavior may be described in rational (or real) cohomology by means of “Wu generators” be described in rational (or real) cohomology by means of "Wu generators" of the cohomology ring H*(Tn; Q) of the maximal torus. For U,~l,...IwI U1,"" Un of the cohomology ring H*(Tn; Q) of the maximal torus. For Unbundles q the Chern character ch(v) is defined as the formal sum bundles TJ the Chern character ch(TJ) is defined as the formaI sum ch(v) = $exp(ui) ch(TJ)

&x(v)

= 2 j=lm>O

C

2

= C

$G

= C

77220

= $m = P,(cI,

chm(q), (1.33) (1.33)

7QO

. . , cm),

and for Son-bundles 7, and for SOn-bundles TJ,

Nrl) = ch(crl), ch(TJ) = ch(cTJ), where ch(q) is given by (1.33). Here we identify the elementary symmetric where ch(cTJ)inis 211, given Here we identify the(and elementary polynomials . . . ,by u, (1.33). with the Chern classesci in VA:,. .symmetric . , U: with Ul, ... ,Un with the Chern classes Ci (and in u~ with polynomials in the Pontryagin classespi). One then has the simple formula the Pontryagin classes Pi)' One then has the simple formula (1.34) (1.34) In what follows we shall often omit the symbol 8 for the tensor product, In what follows shaHproduct often omit the symbol ® for the product, since there is nowe other for bundles. Returning for tensor the moment to sinee there is no other produet for bundies. Returning for the moment to

ur, ... ,

concepts. Smooth Smooth fiber §$1. 1. Basic Basic concepts. fiber bundles. bundles, Connexions Connexions

161 161

characteristic classes of sums qi @ r/2 of of vector vector bundles over B, B, we we consider consider characteristic classes of sums 171 EB 172 bundles over the formal sums (defined for for each bundle 77): the formaI suros (defined each vector vector bundle 17):

1 EB 17 EB A2 17 EB A3 17 EB",

A(17),

1 EB @ 17 q@ s3rl 69.. S(q), EB s2v 8217 @ EB 8317 EB",. == 8(17), which are easily easily seen seen to satisfy which are to satisfy 4% @r/2) A(171 EB 172)

=

4771)4v2), A(171)A(172),

(1.35) (1.35)

S(rll r/2) == S(q)S(q2), 8(171 63EB 172) 8(171)8(172)'

Similarly, ifif we write (formally) (formally) Similarly, we write w=1+wi+w2+..., w 1 + Wl + W2

+ ' ,.,

c=1+ci+c2+..., 1 + Cl + C2

C

+ ' , ., p=l+p1+p2+..., P 1 + Pl + P2 + . , , ,

then (1.32) take take the the simple simple form: form: then the the formulae formulae (1.32) (4 (i)

4% 63 r/2) W(171 EB172)

=

W(VlMV2),

(ii) (ii)

4771

c( 171 @EB 772) 172)

=

4rll)C(r/2),

(iii)

~(771

=

p(r]i)p(r/2)

@ r/2)

(modulo 2-torsion)

appropriate elements of The generators ‘1~1, . . . , U, (representing The symbolic symbolic generators U1"", Un (representing appropriate elements of H* R)) are also convenient convenient for defining additive additive and multiplicative exH* (BF; (BT n ; IR)) are also for defining and multiplicative expressions f(v) (like w(q), c(q) and p(q)) in th e ch aracteristic classes. It turns pressions f(17) (like w(r/), C(17) and P(17)) in the characteristic classes, Tt turns out that all such additive expressions (i.e. such that f(qi@qz) = f(ni)+f(q2)) out that aH such additive expressions (Le, such that f(r/l EBr/2) = f( 171) + f( 172)) can be obtained from those in a single variable u, expressed as a power series can be obtained from those in a single variable u, expressed as a power series (for complex and real bundles respectively): (for complex and real bundles respectively): (i)

(i)

f(U)=

f(u)

ConrUm, m amu ,

L

?TL>O

n

f(~)=f(cl,...,Ck,...)=f:f(uj);

f(17)

f(C1,'''' Ck,''')

m2:0

L

j=l j=1

f(uj);

n (ii) f(u2)= Cbmu2m, f(o)=f(P1,...,Pk,---)=~f(U32). f(u 2 ) = L bmu 2=, f(17) = f(P1,'" ,pk,"') = L f(u;).

(ii)

j=l j=1

T?QO

=2:0

Hence all additive expressions f(u) are of the form (again in the cases of Hence aIland additive expressions f(u) are of the form (again in the cases of complex real bundles respectively): complex and real bundles respectively): 6)

(ii) (ii)

f(v)

= f(cl,

. . , Ck, . . .) = c

m!a,~;,

= c

77X20

m>O

=2:0

=2:0

f(q) = f(pl,. . . ,plc,. . .) = C f(17) f(pl,'" ,Pk",,)

L

m2:0

~%,&, = C m!b mC2m

L

m2:0

m!a,ch,(~); (1.36) (1.36)

m!b,&,(q). mlb mch 2m (17),

162 162

Chapter Manifolds Chapter 4.4. Smooth Smooth Manifolds

Analogously, satisfying f(~ 8~~) Analogously, all all multiplicative multiplicative characteristics characteristics f(q), f ('TI), i.e. satisfying f (7)1 181 '1]2) == f(~) determined by power series series in in one variable, of of the the following following f('TId . f(qs), f('TI2) , are are determined by power one variable, form: form: n

L

(i) (i)

m f(v) = f(cl, . . . ,ck,. .) = fi f(%); f(u) = C amu amurn, f(u) = , f(7)) = f(C1,"" Ck,"') = II f(71,j); m>o j=l j=l m~O

(ii)

f(u’)

n

= C

Luzrn,

f(7)

= f(Pi, . . . ,Pk,.

.) = fi

?TL>O m~O

f(u;).

j=l j=l

On such aa multiplicative multiplicative charOn closed closed manifolds manifolds (real (real or or complex) complex) the the form form of of such characteristic is determined acteristic ff is determined by by the the requirement requirement

f(Mî x Mf')

= f(Mî)

. f(Mf'),

yielding: yielding: (i) (i)

2n J) f(i@-“) (f(ci, . . . , cn), c,), [M [M2n]) f(M 2n ) == (f(C1,""

(the complex case); case); (the complex

(ii) (ii)

f(M4n)4n) = (f(P1,'" (f(pi,. ,p,), [M [M”“]) 4n]) f(M ,Pn),

(the real real orientable orientable case). case). (the

In particular, (i) above have for Euler characteristic characteristic X x [M [AP”]2 n] the case case (i) above we we have for the the Euler In particular, in in the that that f(u)=l+u, c~,...)=l+ci+c2+ .... f(u) = 1 +u, f(Cl,..., f(Cb'" ,Ck, ... ) = 1 +C1 +C2 + .... We shall meet important cases cases below. We shan meet with with other other important below. Ezam$e For the of n-dimensional real Example 1. 1. For the tangent tangent bundle bundle TT = TT(IRP) (!Rpn) of n-dimensional real projective space, we have the simple bundle isomorphism projective space, we have the simple bundle isomorphism T

(!Rpn)

œEJR

~ 7)

œ... œ7),

------n+l times

where ER denotes the trivial bundle over RPn with fiber IR1, and 7 is the where EJR denotes the trivial bundle over !Rpn with fiber !RI, and 17 is the unique nontrivial line bundle over IRP with structure group 01 Z Z/2. Over unique nontrivialline bundle over !Rpn with structure group 0 1 ~ 7l../2. Over lRP1 the total space of q is the usual Mobius band (see Figure 4.4) - though !RP1 the total space of 7) is the usual M6bius band (see Figure 4.4) - though with the fiber taken to be an open interval. The Stiefel-Whitney classes of q with the fiber taken to be an open interval. The Stiefel-Whitney classes of 7) are as follows: are as follows: 0 # WI(V) = ‘u E H’(WP; Z/2), Wi(V)

=

0,

i > 1.

The cycle Dv E H,-i(IRP; Z/2) is represented The E Hn_l(!Rpn; 7l../2) is represented IRP”-l cycle c liw”;Dv see Figure 4.5. Using the formula for !Rpn-l C !Rpn; see Figure 4.5. Using sum of vector bundles, we obtain thence the formula for su~ of vector bundles, we obtain thence W(T)

W(T)

= 1 + WI(T) + . . + W*(T) = (1+ = 1 W1(T) wn(r) (1

+

+ ... +

?Jy+l

by the hypersurface by the the classhypersurface of a direct the class of a direct

E H’(RP”;

Z/2).

+ vt+ 1 E H*(!Rpn; 7l../2).

$1. Basic Basic concepts. concepts. Smooth §1. Smooth fiber fiber bundles. bundles. Connexions Connexions

163 163

b

a

a b

Fig. 4.4. The Mijbius band: simplest nontrivial nontrivial bundle, bundle, Fig. 4.4. The M6bius band: the the simplest with and fiber fiber an with base base S1 Si and an interval interval

a

b

Fig. 4.5 Fig. 4.5 Example 2. The tangent bundle r = r(CPn) is a complex GL,(C)-bundle. Exa'Tnple 2. The tangent bundle T = T(cpn) is a complex GLn(C}-bundle. For reasons similar to those applying in the previous example we have For reasons similar to those applying in the previous example we have T(CP) CBE@ ” 17 T(cpn) El) EC ~ T] El) ... El) T], '--v---" n+l times

n+l

Cl

C(T)

=

1 +

Cl(T)

+

=

.

u

+

E H2(@P;

C,(T)

=

times Z),

(1

+

tqn+l

E H*(cP;Z),

where u is a generator of H2(CPn; Z) E Z. The Poincare dual cycle DU E where u is a generator of H 2(cpn;:2::) ~ :2::. The Poincaré dual cycle Du E H2n-2(CPn; Z), is represented by the cycle @P”-l c CP”. H 2n _2(cpn; :2::), is represented by the cycle cpn-l C cpn. Example 3. Let M” be a smooth submanifold of a smooth manifold N” Example 3. Let Mn be a smooth submanifold of a smooth manifold Nk (n < k). We then have, in addition to the tangent bundle r = 7(Mn), the (n < k).bundle We then the tangent bundle T = T(Mn), normal v = have, v(M”) ininaddition N”k withtorespect to some Riemannian metric the on normal bundle v v(Mn) in N with respect to sorne Riemannian metric on N”k (although in fact the metric is not significant here). One has the simple N (although in fact the metric is not significant here). One has the simple formula formula i*7(Nk)

= T(AP)

@ v(AP)

(where i’ is induced by the inclusion i : Mn -+ N”), whence (where i* is induced by the inclusion i : Mn ---+ N k ), whence

164

Chapter Chapter 4. Smooth Smooth Manifolds Manifolds

i*w(N”)

= W(T)W(V), W(T)W(V),

i*p(Nk) i*p(Nk)

=

i*c(Nk) i*c(Nk)

C(T)C(V), = C(T)C(V),

p(~)p(v) p( T )p(v)

(modulo (modulo the the 2-torsion), 2-torsion),

for for complex complex U,-manifolds Un-manifolds M” Mn cc Nk N k in in the the last last case. case. In In the the case case that that Nk N k == lKk, ]Rk, the the tangent tangent bundle bundle r(Rk) T(]Rk) is trivial, trivial, so that that i’w(R”) i*W(]Rk) = 1,

i*p(@) = 1. i*p(]Rk)

Consequently W(T)W(V) == 11 and p(r)p(v) == 11 (modulo Consequentlyw(T)w(v) andp(1')p(v) (modulo the the 2-torsion), 2-torsion), yielding yielding (see (1.32)) formulae expressing the corresponding characteristic classes of of the the (see (1.32)) formulae expressing the corresponding characteristic classes tangent and normal bundles of M” in terms of one another. tangent and normal bundles of Mn in terms of one another. Example T of orientable Example 4. 4. For For the the tangent tangent bundle bundle l' of an an even-dimensional even-dimensional orientable 2n manifold M2n the Euler characteristic class satisfies manifold lvf the Euler characteristic class satisfies

[hif2”])= X(M2”),

(X27x(7),

i.e. evalulated at i.e. yields yields the the Euler-Poincare Euler-Poincaré characteristic characteristic when when evalulated at the the fundamenfundamental homology class [Mu”] . (For Umanifolds one replaces the class xsn by by the 2n tal homology class [M J. (For U-manifolds one replaces the class X2n the Chern class c,.) Chern class en.) Example tangent bundle and aH all of of the 5. For For Lie Lie groups groups G G the the tangent bundle is is trivial trivial and the Example 5. characteristic classes are zero. Lie groups are parallelizable manifolds, i.e. characteristic classes are zero. Lie groups are parallelizable manifolds, Le. admit fields of n-frames (n = dimG) that are everywhere non-degenerate admit fields of n-frames (n = dim G) that are everywhere non-degenerate (obtained left (or right) translation translation of of any of the the tangent tangent (obtained by by means me ans of of left (or right) any basis basis of space at the identity element). space at the identity element). Example 6. Suppose a submanifold M” of RN is defined by a set of siExample 6. Suppose a submanifold Mn of ]RN is defined by a set of simultaneous equations fi = 0,. , fN-, = 0, where the gradients of the fi, 0, ... , fN-n 0, where the gradients of the fi, multaneous equations fI i = l,..., N - n, are everywhere linearly independent on the manifold of i = 1, ... , N n, are everywhere linearly independent on the manifold of common zeros: common zeros: hfn = (5 E RN 1 fi(x)

Mn = {x

E]RN

1

= 0,.

. , f&,(x)

= o} .

fI (x) = 0, ... ,fN-n(x) = O}.

The normal bundle .(Mn) (with fiber lRNpn ) is then trivial and in view of The normal bundle v(Mn) (with fiber ]RN-n) is then trivial and in view of the formula (1.32) all of the stable characteristic classes wi, pk of the manifold the formula (1.32) aIl of the stable characteristic classes Wi, Pk of the manifold Mn will be zero. Only the Euler class x2k may be non-zero, provided n = 2lc. Mn will be zero. Only the Euler class X2k may be non-zero, provided n = 2k. (For instance x(S’~~) = 2.) The tangent bundle T(M") is here stably trivial, (For instance =E;. X(S2k) = 2.) The tangent bundle T(Mn) is here stably trivial, i.e. . 7 CBE,N+N-n N 00 l.e. T

œEIR

= EIR .

At various points above we have mentioned the concept of a complex At various points above we have real mentioned concept a complex manifold: this is an even-dimensional smooth the manifold with ofcharts V, cothis is an even-dimensional real smooth manifold with charts comanifold: ordinatized by complex co-ordinates z;, . .zz, and transition functions V onQ the ordinatized by complex co-ordinates z~, . .. z~, and transition functions on the regions V, n VP of overlap complex-analytic. Here n is the complex dimension regions VQ n V/3 of overlap complex-analytic. Here n its is the complex isdimension of the complex manifold M”; as a real manifold dimension 2n. The of the complex manifold Mn; as a real manifold its dimension is 2n. The

52. §2. The The homology homology theory theory of of smooth smooth manifolds. manifolds. Complex Complex manifolds manifolds

165 165

simplest examples examples of of compact compact complex complex manifolds manifolds are the the complex complex projective projective spaces spaces @P”, epn, and the the even-dimensional even-dimensional tori tori T2” T 2n = P/r, en / where where r is a lattice lattice determined determined by by any any 2n 2n linearly linearly independent independent real vectors vectors in U? Cn g~ R2n. ]R2n. Of Of particular particular importance importance are are the the compact compact complex complex submanifolds submanifolds M” M k cC Alan, cpn, the the projective projective algebraic algebraic varieties. varieties. An An Hermitian Hermitian metric metric on on aa complex complex manifold manifold Mn Mn is is given given in in terms terms of of local local co-ordinates on each chart V, by co-ordinates on each chart Va by

r,

2

..

ds2 ds == g,idzhdZi, gijdz~az~,

r

gij 9ij == gji gji.

With With such such aa metric metric one one may may associate associate the the real real 2-form 2-form given given locally locally by by

The The requirement requirement that that parallel parallel translation translation of of tangent tangent vectors vectors along along any any path path determine a unitary transformation, leads to the condition that the 2-form determine a unitary transformation, leads to the condition that the 2-form 0 be be closed: dosed:

n

dL? = 0. dn o.

Complex associated 2Complex manifolds manifolds endowed endowed with with an an Hermitian Hermitian metric metric whose whose associated 2form R is closed, are called Kiihler manifolds. A Kahler manifold AP’ for which which form n is dosed, are called Kiihler manifolds. A Kiihler manifold Mn for the represents cohomology class, intethe form form R n2-cycles represents an an integral integral cohomology dass, i.e. i.e. is is such such that that its its integrals over in Mn are integer-valued, is called a Hedge manifold. The grals over 2-cydes in Mn are integer-valued, is called a Hodge manifold. The complex projective space CP” and its complex submanifolds (varieties) complex projective space cpn and its complex submanifolds (varieties) are are Hodge In fact, every Hodge Hodge manifold algebraic varifact, every manifold is is a a projective projective algebraic variHodge manifolds. manifolds. In ety; this was proved by Siegel for complex tori, and by Kodaira in the general ety; this was proved by Siegel for complex tori, and by Kodaira in the general case (in the late 1940s). A complex torus endowed with an Hermitian metric case (in the late 1940s). A complex torus endowed with an Hermitian metric with constant components gii is Kahler, but is a complex projective algebraic with constant components gij is Kiihler, but is a complex projective algebraic variety if and only if the defining lattice r c C=” satisfies the conditions of variety if and only if the defining lattice r c en satisfies the conditions of Riemann for a corresponding e-function to be definable. Riemann for a corresponding B-function to be definable. All complex submanifolds of Cc” without boundary and of dimension > 0, AlI complex submanifolds of en without boundary and of dimension> 0, are non-compact. Further results concerning the topology of complex maniare non-compact. Further results concerning the topology of complex manifolds will be given in the following sections. folds will he given in the following sections.

52. The homology theory of smooth manifolds. §2. The homology theory of smooth manifolds. manifolds. The classical global calculus of variations. Complex manifolds. The classical global calculus of variations. H-spaces. Multi-valued functions and functionals Complex

H-spaces. Multi-valued functions and functionals

Every differentiable manifold Mn of smoothness class Ck, k > 1, can be k , k 2': 1, can be Every differentiahle Mn of smoothness triangulated, i.e. turned manifold into a simplicial complex. A dass smoothC simplex is a pair triangulated, turned simplex into a simplicial A smooth simplex is a pair consisting of aLe. standard gkk c IWkcomplex. and a regular smooth embedding of consisting of a standard a such C ]Rkthat andoka regular smooth emhedding of some k-dimensional opensimplex region U c U c Iw”, into the manifold: sorne k-dimensional open region U such that a k eUe ]Rk, into the manifold:

166 166

Chapter 4. Smooth Smooth Manifolds Chapter Manifolds

dlor, a, ~2]=[OI,oc,] + .. . d [a, ~~ a,]=-[q,a,]+ .. .

&3

Fig. 4.6 4.6 Fig.

A is then decomposition of of Mn Mn into into A smooth smooth triangulation triangulation of of M” Mn is then aa simplicial simplicial decomposition smooth simplexes: smooth simplexes: M” = u fk,&lc), 4 making NIn M” a such that that the the simplicial complex made made up up a “PL-manifold,” "PL-manifold," i.e. i.e. such simplicial complex making of all simplexes containing any given simplex is combinatorially equivalent to of aIl simplexes containing any given simplex is combinatorially equivalent to the standard n-simplex on (see Chapter 3, $1). the standard n-simplex (7n (see Chapter 3, §l). Up to combinatorial equivalence equivalence any any smooth smooth manifold has just one Up to combinatorial manifold M” Mn has just one smooth triangulation (Cairns, Whitehead, in the 1930s). Hence the standard smooth triangulation (Cairns, Whitehead, in the 1930s). Hence the standard apparatus of simplicial homology and cohomology (H, (Mn), H’ (Mn), the apparatus of simplicial homology and cohomology (H*(Mn), H*(M n ), the multiplication on H* making it a ring, intersections of cycles in H,(Mn), multiplication on H* making it a ring, intersections of cycles in H*(Mn), Poincare duality, etc. - see Chapter 3, §1,2) goes over in its entirety, via Poincaré duality, etc. see Chapter 3, §1,2) go es over in its entirety, via smooth triangulations, to smooth manifolds M” (see Figure 4.6). smooth triangulations, to smooth manifolds Mn (see Figure 4.6). For closed submanifolds IV”, Vj c Mn, the intersection operation has a For c\osed submanifolds W k , V J C Mn, the intersection operation has a clear geometric interpretation. First one subjects W”k to an arbitrarily small clear geometric interpretation. First one subjects W to an arbitrarily small perturbation to bring it into “general position” relative to Vj, in the sense perturbation to bring it into "general position" relative to V J , in the sense that at each point x E W”k n V j, the vectors tangent to W”k or Vj should that at each point x E W n vj, the vectors tangent to W or vj should together span the full tangent space EXEof the ambient manifold Mn. Thus together span the full tangent space IR~ of the ambient manifold Mn. Thus if Ic + j < n the intersection W” n Vj will, in general position, be empty. If if k + j < n the intersection W k n vj will, in general position, be empty. If Ic + j = n, then (in general position) the intersection will consist of finitely k+j n, then (in general position) the intersection will consist of finitely many isolated points x1, . . . , x, (see Figure 4.7); in this case the number m many isolated points Xl,'" ,X m (see Figure 4.7); in this case the number m of these, reduced modulo 2, gives the intersection index modulo 2: of these, reduced modulo 2, gives the intersection index modulo 2: W”k o Vjj s m (mod 2). w 0 V m (mod 2).

(2.1) (2.1)

If the manifolds Mn, Wk, Vj come with orientations, then each point xj of If manifoldsW” Mn, Wk, vjassociated come with then(Wk each pointnamely Xj of thethe intersection n Vj has withorientations, it a sign sgn,,? n Vj), k n vj has associated with it a sign sgn . (W k n VJ), namely the intersection W the sign of the determinant of the change from a framex .Jrzj for the tangent the sign thethe change a frameofT XjM*,for tothethetangent space to of Mntheat determinant xj, agreeing of with givenfrom orientation frame space to Mn at Xj, agreeing with the given orientation of Mn, to the frame

52. smooth manifolds. manifolds. Complex Complex manifolds manifolds §2. The The homology homology theory theory of of smooth

167 167

rlJ) where T~. rLj is a tangent Wk at xj (T~.,7 T~) tangent frame for W cc, k k

XJ' according with with the given orientation W”, and similarly similarly for rgjJ with with respect to Vi. VT. We We then have orientation of of W J

J

J

T:;

,

w”ovj

=

L: j sgn.+ sgn Xj (W Ci (Wkk n vj) Vj) ,,

Wowk

=

(_l)k (-l)“j j (W (Wkk 00 Vj). vi)

(2.2) (2.2)

For aa smooth manifold Mn, Mn, one one may choose smooth orientable orientable manifold choose a pair pair of bases bases {q},

{qJ>,

CYj= 1,“. ,bj,

for the the groups (Mn; Z), Z), jj = 0,1, mutually Poincaré-dual in pairs: for groups Hj Hj(Mn; 0, 1, ... . , n, n, mutually Poincark-dual in pairs:

(2.3) zo = 20 = l,z,

= & = [Aq.

For n = particular two bases bases{z$m}, (52}, am ay, = 1, 1,.... ,., b b,, For n = 2m, one has has in particular {z;:::''''}, {Z~n}, m , for the group group H”(M2m; Hm(M 2 m; Z), Z), satisfying satisfying

non-orientable manifolds there exist analogous analogous and so so Poincare-dual. Poincaré-dual. (For non-orientable mutually basesmodulo 2.) mutually dual bases Example. Let closed, orientable orientable manifold, and f : Mn Mn ---Jo -+ M” Let M” Mn be a closed, manifold, and Mn any regular cycles (the regular smooth self-map. Consider the the following following cycles (the diagonal and the graph of 1) Mn xX Mn: Mn: the f) in M” A .1

=

.1 41 =

{(x, E Hn(M Mn;Q); {(x,x)}x)} E H,(Mn n Xx M”;Q);

{(x, f(x))} EE Hn(Mn Mn; Q). {(x,f(x))) &(Mn xX MYQ).

position the intersection intersection .1 As noted above, in general position Ann .1 A,1 consists consists of isolated points xj: x j: ff(xj) (x j) == xj, x j, which, in view view of the the regularity regularity of f and the assumption that the cycles .1, .1 1 are in general position relative that A, A, general position relative to one one another, another, are are nonneighbourhood of the degenerate, i.e. in terms of local co-ordinates x'lx x9, in aa neighbourhood point xj, point Xj,

Fig. 4.7 4.7 Fig.

168 168

Chapter 4. 4. Smooth Smooth Manifolds Manifolds Chapter

det

(Opq ~~~)

X=Xj

~ o.

intersection index is is then then the sum sum of the the attached attached signs signsof the fixed fixed points The intersection don,=

c

(2.4) (2.4)

x,EAnAf

From bases bases {z;j}, {z? }, {z;j} { 27 } satisfying satisfying (2.3), (2.3), one one obtains obtains the the basis basis z;' ~7 Ci?> @z~:'? Zzy;J From for H,(AP x Mn; Q), using which one can establish (via (2.4)) the Lefschetz for Hn(Mn x Mn; Q), using which one can establish (via (2.4)) the Lefschetz formula (ca. (ea. 1920) 1920) formula

2".)

A o.dt 0 A, == ~W)“W~IH~~M~;Q~. .d -l)kTrf*IHkCM";\Q).

(2.5)

k/O k2: 0

It follows follows that that if if ff is is null-homotopie null-homotopic then then .d A o.dt o Af = = 1. 1. Applying Applying this this to to It M” = S”, and considering the disc D” as a hemisphere of S”, Dn c S”, one Mn sn, and considering the disc Dn as a hemisphere of sn, Dn c sn, one can deduce deduce Brouwer's Brouwer’s theorem (ca. 1910) 1910) on on the the existence existence of of aa fixed fixed point point of of can theorem (ca. any map Dn ---+ D”. any map Dn Dn. 00

t=1 t=1

WXI

t=o t=O

Fig. 4.8. The intersection index is not changedby a deformation of cycles Fig. 4.8. The intersection index is not changed by a deformation of cycles

Returning to our closed submanifolds Wk,k l/j j c M”, we now consider the Returning to our closed submanifolds W , V c Mn, we now consider the case lc + j > n. Here the intersection Wkk n Vj (if non-empty and assuming case k + j > n. Here the intersection W n vj (if non-empty and assuming general position) is always a submanifold (with an orientation imposed if the general position) is always a submanifold (with an orientation imposed if the manifolds are oriented) of dimension j + k - n: manifolds are oriented) of dimension j + k - n: v.i o wk = Nj+k-n (2.6) (2.6) Thus in this casethe intersection of cycles has immediate geometric meaning. Thus in this the intersection of cycles immediate geometric meaning. However it iscase far from being the case that allhas cycles in a manifold (representing However it is far from being the case that all cycles in a manifold (representing elements of the integral or Z/2-homology groups) are representable as subelements of integral or Ilj2-homology groups)In are as submanifolds or the linear combinations of submanifolds. fact representable in integral homology manifolds or linear combinations of submanifolds. In fact in integral homology not all cycles need even be representable as images of closed manifolds not aU cycles need even be representable as images of closed manifolds f+[Nk]k = z E &(Mn;Z), f : Nk + Mn. f*[N ] = z E Hk(M n ; Il), f: N k ---+ Mn.

52. homology theory theory of of smooth smooth manifolds. manifolds §2. The The homology manifolds. Complex Complex manifolds

169 169

The The problem problem of of determining determining when cycles are so so representable is called caIled Y%eenrod’s problem”. (Of course the fundamental is trivially so "Steenrod's problem". (Of course fundamental cycle [M”] [.!VIn] is triviaIly so representable.) In Z/2-homology Steenrod's Steenrod’s problem has has aa positive positive solution In Zj2-homology (Thorn, 1950s). We shan shall discuss Thorn’s theory theory in connection (Thom, in in the the early early 1950s). discuss Thom's connection with this sort of of problem below (see $3). with this (see §3). As noted earlier, orientability is determined by the orientability of a manifold manifold M” Mn is vanishing first Stiefel-Whitney class wi(M”), Poincare dual vanishing of of the first Stiefel-Whitney class wl(Mn), whose whose Poincaré Dwi Z/2) is “zeroes DWl E E H,-i(M”; Hn-l (Mn; Zj2) is represented by the cycle consisting of the "zero es of of the the determinant” determinant",, i.e. the points where aa field of tangential tangential n-frames r”(z) rn(x) in general position becomes degenerate (Le. (i.e. linearly linearly dependent). It It may also also position becomes be obtained image under the Bockstein obtained as as the the image Bockstein homomorphism p* : H,(M”;

Z) --+ H,-1(M”;

Z/2)

of class [Mn]: fundamental class [Mn]: of the fundamental DWl = /3*[M”].

(2.7)

(Recall from Chapter 3, end end of 54, that Bockstein homomorphism is is deChapter 3, of §4, that the Bockstein defined (on integral chains) by integral by P*(Z) = = 8zj2, az/2, (3*(i)

zz == z 5 mod mod 2.) 2.)

From infer the invariance of sequel we we shaIl shall From (2.7) (2.7) we we infer the homotopy homotopy invariance of WI; Wl; in in the the sequel discuss the homotopy homotopy invariance invariance of of all of the the StiefelStiefeldiscuss Thorn’s Thom's theorem theorem on on the an of Whitney wi, and the explicit explicit formula formula of of Wu Wu for calculating them Whitney classes classes Wi, and also also the for calculating them (dating from the early 1950s). (See also the preceding section.) (dating from the early 1950s). (See also the preceding section.) For an orientation orientation epimorphism epimorphism (j CT:: For a a non-orientable non-orientable manifold manifold M” Mn there there is is an rl(M”) Z/2 = = {-l, {-l,l}), 1}), defined follows: we set (j( o(a) -1 'll'l (Mn) -+ -+ Z/2 Zj2 (where (where Zj2 defined as as follows: we set a) == -1 ifif parallel tangent n-frame loop y'Y in in the the homotopy parallel transport transport of of a a tangent n-frame around around a a loop homotopy class changes the orientation of of the the n-frame; otherwise set o(a) = = 1. 1. (Thus class a a changes the orientation n-frame; otherwise set (j(a) (Thus in sufficiently small neighborhood of of aa non-self-intersecting non-self-intersecting loop loop in any any surface surface a a sufficiently small neighborhood y'Y parallel around which changes the of aa tangent nparaIlel transport transport around which changes the orientation orientation of tangent nframe, band (see (see Figure Figure 4.6).) is therefore an action action frame, will will be be aa Mobius Mübius band 4.6).) There There is therefore an o of rl(Mn) on Z which simply multiplies Z by &l; the homology groups (j of 'll'l(Mn) on Z which simply multiplies Z by ±1; the homology groups Hi(Mn; cr) with from the corresponding representation are the the Hi (Mn; (j) with coefficients coefficients from the corresponding representation are Poincare of the integral cohomology groups H”-j(Mn; Z) (see (see Poincaré duals duaIs of the usual usual integral cohomology groups Hn-j (Mn; 71.) Chapter 3, $5) (and analogously for Q, Q, Iw, Chapter 3, §5) (and analogously for IR, @). C). Cellular and consequently consequently cellular homology, Cellular decompositions decompositions of of manifolds, manifolds, and cellular homology, arise in connection with the following situation (considered by Morse the arise in connection with the following situation (considered by Morse in in the late 1920s): Suppose we are given a real-valued smooth function f on a closed late 1920s): Suppose we are given a real-valued smooth function f on a closed or with the the following properties: open manifold manifold M”, Mn, with following properties: or open 1. “Compactness”: every closed region Mn determined by an an inequality 1. "Compactness": every closed region of of Mn determined by inequality of the form f(z) < c should be compact. of the form f(x) ::::; c should be compact. 2. (or stationary) (points where grad f vanishes) 2. All AIl critical critical (or stationary) points points xj Xj (points where grad f vanishes) should be non-degenerate, i.e. the quadratic form d2f tangent space space should be non-degenerate, i.e. the quadratic form d 2 f on on the the tangent

170 170

Chapter Chapter 4. Smooth Smooth Manifolds Manifolds

Tw:, determinant; in ter terms is ]R~. should have non-zero determinant; ms of local coordinates this is .1 the condition condition that that at Xj xj det det (

éPf

ôx~ôx~

)

of O.

0 0 c;

Such functions functions are called Morse functions. “height functions" functions” on closed closed Such functions. The "height orientable surfaces embedded in ]R3, IK3, furnish examples of orientable surfaces suitably suitably embedded of Morse functions-see squares in the functions-see Figure Figure 4.9. 4.9. The number of of negative squares the diagonaldiagonal2 ized quadratic form d2f the point point xj, is called called the Morse index of the the ized quadratic form d f at at the Xj, is the Morse index of critical critical point. point. j=o 9=0

0

g> 1 9”

9=1

4 2 7..

Cl

@

Z 7..

\

Z Z

0

Fig. Fig. 4.9 4.9

Note function ff on on aa manifold always be chosen Note that that aa Morse Morse function manifold M” Mn may may always be chosen in such a way, that for any critical value c of f there is an E > 0 such that in such a way, that for any critical value c of f there is an ê > 0 such that f-l [c - E,c + E] contains only one critical point ~0, f (x0) = c. If x0 is such a f-l[C - ê, C + ê] contains only one critical point xo, f(xo) = c. If Xo is such a critical point of index A, then the manifolds critical point of index .\, then the manifolds M:+,

(f(x)

I c+ E) and

M,“_,

(f(x)

I c - E),

and their boundaries and their boundaries iVz;$ (f(x)

= c + E) and

iv,“--,’

(f(x) (f(x)

= c - E),

(f(x)

C

are related in the following way: Consider the “n-dimensional handle of index are related in the following way: Consider the "n-dimensional handle of index A” X' H,” = D”-A x DA, which has boundary which has boundary tlH,n = (S”-‘-l

ôHf

x DA) u (D,-’ n

(sn-..\-1 x

D..\) U (D

x Sx-‘)

.

-..\ x S..\-1).

One identifies the portion D”-A x Sx-l of the boundary 8Hr with a suitable One identifies the portion Dn-"\ S..\-1 of the boundary ôH): with a suitable small tubular neighborhood of anxembedded (A - 1)-sphere in iV,“--,’ = aA&, sm aU tubular neighborhood of an embedded (.\ -l)-sphere in N::::/ ôM;)"_t:' via a diffeomorphism via a diffeomorphism $ : Dn-’ x S’-’ ----f N,“--,‘. n- 1 of • • D n -..\ S..\-1 ---; N C 'f/ • X -é'

$2. The The homology homology theory theory of of smooth smooth manifolds. manifolds. Complex Complex manifolds manifolds §2.

171 171

One then then attaches attaches the the handle handle H~ H,” ta to Mi-ê MF-, via via the diffeomorphism diffeomorphism 'Ij;, 111, smoothOne smoothing out out "corners" “corners” to to obtain obtain aa smooth smooth manifold-with-boundary. manifold-with-boundary. It can can he be ing It shown that that this this can can he be effected effected in in such such aa way way as as to to obtain obtain aa manifold manifold diffeodiffeoshawn morphic ta to M:;+ë' IkfF+,, Le. i.e. one one obtains obtains morphic M”C+&= - M:-,

u+ (ITA

x DA).

This procedure procedure induces induces aa "Morse “Morse surgery" surgery” This N,“--,’ H N;JE1, where where

N;cE1 ~ E (N:;~/ N,“--,’ \\ î5 En-’ IJ+ S-‘-l n À 1 x Ox). À 1 n- À x N;!;/ x 8S’-’ -x DÀ ) . - ) Ut,b (8 > (

(Here î5 En-’ denotes the interior of of the the disc disc Dn-À.) PWx.) n - À denotes (Here the interior

"-

M(-f)

,,-

-a.iH

"'

,/

'\

\

J]

.l.\

])n.-À.

\---1

1

'\

/

/

--

"-

1 f

1 ---1 \ '\

M (-f) -a.LH

"-

)111_ Dn-R=6n-2

Fig. 4.10 Fig. 4.10

Thus a Morse function on M” provides a “handle-decomposition” of the Thus aMn, Morse Mn provides aof"handle-decomposition" of the manifold i.e. function leads to on a decomposition the manifold into a “sum” of manifold Mn, Le. leads ta a decomposition of the manifold into a "sum" of handles. (Figure 4.11 shows the two possibilities for attaching a handle Hf to handles. (Figure 4.11 shows the two possibilities for attaching a handle Hf ta a disc Hz, and then in Figure 4.12 a “Smale function” (i.e. a Morse function a disccriticaland thenordered in Figure 4.12 a to "Smale (Le. atoMorse with values according their function" index) is used effect function a Morse with criticai values ordered according to their index) is used ta effect ahandles Morse surgery on the torus by starting with Hi and successively attaching surgery on the torus by starting with and successively attaching handles as one passesthrough the critical points.) as Morse’s one passes through the critical theorem consists in thepoints.) following lower estimate for the number Morse 's theorem consists lower estimate for themanifold numher rnA of critical points of indexinXthe of afollowing Morse function f on a closed mÀ of critical points of index À of a Morse function f on a closed manifold M”, relating that number to the integral homology of Mn: Mn, relating that number ta the integral homology of Mn:

HJ,

Hg

eH12 e-

172 172

Chapter 4.4. Smooth Smooth Manifolds Manifolds Chapter

Xl

==

H,2

X0

DH~ CI)

D<J (JI)

Fig. 4.11 4.11 Fig.

mxm2 b, mx(f)2 by,

(2.8)

(2.8)

where bx is is the the Àth Xth Betti Betti number number (= (= rank rank Hx(iP; Z)), and and b?) the Àth Xth mod-p mod-p where b).. H)..(Mn; Z)), b~) the Betti number (see Chapter 3, $2). Betti number (see Chapter 3, §2). Writing for the least number of generators of the torsion subgroup Writing qx q).. for the least number of generators of the torsion subgroup Tor IJ,(hl”;Z) between the Tor H)..(Mn;z) of of Hx(AP;Z), H)..(Mn;z), we we infer infer from from the the relationships relationships between the homology groups over different coefficient groups (see Chapter 3, homology groups over different coefficient groups (see Chapter 3, 32) §2) that that

(2.9)

(2.9)

These These are are the the Morse Morse inequalities. inequalities.

-w-e%fy+y-

-

-

00 00 Fig. Fig. 4.12 4.12

x=2 h=? h=? h=O

§2. The The homology homology theory theory ofof smooth smooth manifolds. manifolds. Complex Complex manifolds manifolds 52.

173 173

For simply-connected simply-connected manifolds manifolds of of dimension dimension >::::: 66 these these inequalities inequalities are are For strict in the sense that there exists a Morse function f on Mn for which strict in the sense that there exists a Morse function f on M” for which equality obtains obtains in in all an of of the the equations equations (2.9) (2.9) (Smale, (Smale, in in the the early early 1960s). 1960s). equality AA Morse Morse function fun ct ion ff on on aa closed closed manifold manifold M” Mn can can be be used used to to construct construct aa CW-complex the same same homotopy homotopy type type as as the the manifold, manifold, with with the the same same CW-complex of of the number ceUs of of each each dimension dimension XÀas as the the Morse Morse function fun ct ion has has critical critical points points of of number of of cells index index X. À. The The basic basic idea idea of of the the construction construction is is as as follows: Endow Endow the the manifold manifold Mn Mn with with aa Riemannian Riemannian metric metric gij gii and and consider consider the the vector vector field field -grad -grad f,f, which which in in local local coordinates coordinates {XL} {x~} defines defines the the dynamical dynamical system system given given locally locally by by dx; dx~ _ ij af âf - -Py& dt -g âx~' dt

(2.10) (2.10)

where (x0) = cc gij gjq = IS:. 8~. Let Let x0 Xo be be aa critical critical point point of of index index X À with with ff(xo) where gijgj, say, and, as before, denote by M:wE, MF+, the manifolds-with-boundary say, and, as before, denote by M;)'_e' M;)'+e the manifolds-with-boundary determined 5 cc -- E, f(x) 5 cC ++ E, determined respectively respectively by by the the inequalities inequalities f(x) f(x)::; IE:, f(x)::; E, where E is sufficiently small for there to be no critical points other than where lE: is sufficiently small for there to be no critical points other than xc Xo in in f-l xj} in U of f-1 [c [e - E, lE: , c C+ +s]. 4 Choose Choose a a coordinate eoordinate system system {{xi} in aa neighborhood neighborhood U of x0 Xo so that the identity so that the identity f(x)

= c - (x1)2

- . . . - (XX)”

+ (xA+y

+ .

+ (xc”)2

holds U for Morse funetion function f.f. Now an embedding holds throughout throughout U for the the Morse Now consider eonsider an embedding 4 : DA U of the X-disc, determined by the formula À + cp : D U of the À-dise, determined by the formula (x1)” + + .... + + (x’)~ E and (x À )2 5 ::; lE: and

(X 1 )2

xx+l = 0, 0,..... ,,xn 0. XÀ+l xl> = O.

...

//"

/

1

1 1

1

1

1

1

\

\

\

/

:-1 \

H-e \

\ \.

/

'

....

_-_ ...

", ....... ----t ... ...Fig. 4.13 Fig. 4.13

Note that: Note that:

(i)

(i)

4(O) =

x0;

cp(O) = Xo;

(ii) the function cj*f on the disc DÀA has the center of the disc as its only (ii) the funetion cp* f on the dise the center of the dise as its only (non-degenerate) critical pointD (ahas maximum); (non-degenerate) eritical point (a maximum);

174 174

Chapter Chapter 4. 4. Smooth Smooth Manifolds Manifolds

(iii) (iii) the the image image 4(a@) cjJ(aD À) of the boundary boundary 8Dx aDÀ == SxP1 8 À- 1 is is contained contained in in the the of the manifold manifold E)MCn_. aM:;_é == {x {x E M” Mn 11 f(s) f(x) == c -- E}. The The manifold manifold Mc+E M;}+é is is homotopy homotopy equivalent equivalent to to the the space space M:-, M;}_é U$ Uq, DA. DÀ. An An appropriate appropriate homotopy homotopy (Pi
k

j=o j=O

j=o j=O

2:) -l)jmj(j) :::: 2) -l)jb

j ,

(2.11) (2.11)

&l)jm3(f) 'L) -l)imj(j) = = x(Mn). X(M n ).

(2.12) (2.12)

so that that in particular taking taking Ic has so in particular k = n, one one has n

j=o j=O

The Morse Morse inequalities inequalities do do not not apply apply if if the the critical critical points points of of aa real-valued real-valued The smooth function function on on aa closed closed manifold manifold Mn are isolated isolated but but are are allowed allowed to to smooth Mn are be degenerate. There is is however however another another useful useful homotopy homotopy invariant invariant -- the the be degenerate. There Lyusternilc-Shnirelman category category of of the the manifold manifold (introduced (introduced in in the the late late 1920s) 1920s) Lyusternik-Shnirelman provides aa lower lower bound bound for for the the number number of of critical critical points points of of smooth smooth -- which which provides functions on on Mn M* regardless regardless of of non-degeneracy. non-degeneracy. For For an an arbitrary arbitrary topological topological functions space X, X, this this invariant, invariant, denoted denoted by by cat( cat(X), is defined defined as as the the least least number number klc space X), is of closed closed sets sets Xl, Xi,. ... ,Xk, , Xk, such such that that X X == U Ui Xi, Xi, and and each each Xi Xi is is contractible contractible of i in X X (to (to aa point). point). One One then then has has the the result result that that for for aa closed closed manifold manifold Mn Mn the the in number m(f) of critical points of an arbitrary smooth, real-valued function number m(j) of critical points of an arbitrary smooth, real-valued function ff on Mn M” satisfies satisfies on n ). m(f) :::: > cat(M cat(Mn). m(j) The invariant invariant cat(Mn) cat(Mn) may may in in turn turn be be estimated estimated from from below below using using the the cohocohoThe mology ring ring of of Mn Mn (with (with appropriate appropriate coefficients): coefficients): Defining Defining the the cohomological cohomological mology

$2. §2. The The homology homology theory theory ofof smooth smooth manifolds. manifolds. Complex Complex manifolds manifolds

175 175

length length I(Mn) l(M n ) of of M” Mn toto be be the the largest largest number number 1l for for which which there there exist exist nonnonzero . . ,,0:1 cyl of products LYE 0:1 .... ~1 0:1 of of elements elements al, 0:1, .... of positive positive degree degree in in H*(M”; H*(Mn; Z) Z) zero products (H* (iW; Z/2) ifif Mn (H*(Mn;Zj2) Mn isis non-orientable), non-orientable), we we have have cat(Mn) cat(M n ) >> l(M”). l(M n ). One One always always has has cat(Mn) cat(Mn) 5~ nn ++ 1. ItIt isis not not difficult difficult to to see that that cut(Sn) cat(sn) = 2, 2, and and that that cat(M’) cat (M 2 ) == 3 for for every every closed closed surface surface M2 M 2 other other than than S2, S2, and and also also that that cat(RP) cat(IRpn) == nn ++ 1, cat(@P) cat(cpn) = nn ++ 1. In In the the sequel se quel we we shall shaH consider consider inequalities and Lyusternik-Shnirelman Lyusternik-Shnirelman type type in in the the context context of of certain certain inequalities of of Morse Morse and path path spaces spaces arising arising in in the the calculus calculus of of variations. variations. Finally Finally we we consider consider smooth smooth functions functions ff on on a manifold manifold Mn Mn whose whose critical critical in the points points constitute constitute aa submanifold submanifold of of Mn; Mn; such such functions functions arise arise inevitably inevitably in the presence presence of of aa Lie Lie group group of of symmetries symmetries of of M”, Mn, where where itit is is required required that that the the level level sets sets of of ff be be preserved preserved by by the the Lie Lie action action (see (see Figure Figure 4.14). 4.14).

Line of degenerate singularities

Fig.

Fig. 4.14 4.14

Let ff be be aa smooth smooth function function on on Mn. Mn. A A submanifold submanifold W W” k is is called called critical critical if if Let grad flWk E 0. Note that the Hessian d2f a quadratic form on a 2 determines gr ad flwk O. Note that the Hessian d f determines a quadratic form on a subspace Vpwk c TM: orthogonal to W” with respect to some Riemannian subspaee Vxn - k C TM:; orthogonal to W k with respect to some Riemannian metric on Mn. A connected critical submanifold W” k is called nondegenerate metric on Mn. A connected critical submanifold W is called nondegenerate of index X if for any x E W” the quadratic form d2flvz-b is nongenerate and of index À if for any xE W k the quadratic form d2 flvn-k is nongenerate and has index X. (The index is independent of the point x since W” k is connected.) has index À. (The index is independent of the point x :ince W is connected.) A particular case of a critical manifold was considered by Pontryagin in the A particular case of a critical manifold was considered by Pontryagin in the late 1930s in connection with a calculation of the mod-2 Betti numbers of Lie late 1930s in connection with a calculation of the mod-2 Betti numbers of Lie groups; the general theory together with its most important applications are groups; the general theory together with its most important applications are due to Bott in the 1950s. due to Bott in the 1950s. Over a critical submanifold W,” c M” there is a naturally defined vector Over a critical submanifold W~ C Mn there is a naturally defined vector bundle yxu --f Wk with fibers lIX$- given by the directions of “steepest debundle v Àa W~ with fibers IR~" given by the directions of "steepest descent” (i.e. decrease in f) at each point x E W,“, i.e. those determined by the seent" (i.e. decrease in f) at each point x E W~, Le. those determined by the negative squares in the Hessian d2f restricted to the normal vector bundle of negative squares in the Hessian d2 f restricted to the normal vector bundle of W,.k WQ'Suppose now that the solution set of the equation gradf = 0 constitutes a Supposeofnow that critical the solution set of theW,$ equation 0 constitutes collection smooth submanifolds of M”, gradf with indices X,. If fora collection of smooth critical submanifolds W~ of Mn, with indices Àc.. for every Wk the vector bundle uxU ---+ W,“, just defined, is orientable (i.e.If one every W~ the vector bundle lI Àa -+ W~, just defined, is orientable (i.e. one

176

Chapter 4. Smooth Smooth Manifolds Manifolds Chapter

À" to may reduce stucture group SOJ,, ), then then one has has the the may reduce the the stucture group of of vuxa to the the group group SOÀ,,)' ineaualities inequalities (2.13) (2.13) Q

for the the ranks groups (and (and also also over over 7ljp). Z/p). for ranks bj bj of of the the jth jth rational rational homology homology groups IfIf the Y’U -+ IV! are not not aIl all orientable then the the inequalities inequalities the vector-bundles vector-bundles vÀc. w~ are orientable then (2.13) hold only over 7lj2. Z/2. These These inequalities may be explained in in the the following following inequalities may be explained (2.13) hold only over way: Each s-dimensional cycle in a critical submanifold IV,” corresponds to way: Each s-dimensional cycle in a critical submanifold w~ corresponds to aa relative cycle of dimension s + A, in each of the groups Hs+x, (A4=, M,-,) relative cycle of dimension s + À in each of the groups HS+À
Example 1. 1. If If M” which there there exists exists aa Morse Morse funcExample Mn is is aa closed closed manifold manifold on on which function with just 2 critical points, then it is not difficult to see that Mn must be tion with just 2 critical points, then it is not difficult to see that Mn must be homeomorphic (even piecewise piecewise linearly) linearly) to to S”. By By the Smale-Wallace theorem theorem homeomorphic (even the Smale-Wallace (of the for nn 2: 2 55 there there exists exists on manifold Mn M” homotopihomotopi(of the early early 1196Os), 960s), for on every every manifold a Morse function with exactly critical tally equivalent to the sphere S* a Morse function with exactly 22 critical caBy equivalent to the sphere points. The idea for such a characterization of spheres is due to Reeb (in points. The idea for such a characterization of spheres is due to Reeb (in the early 1950s); the first deep application of it was made by Milnor in the the early 1950s); the first deep application of it was made by Milnor in the late 1950s 1950s in with his his discovery of exotic exotic smooth smooth structures structures on late in connection connection with discovery of on the sphere S7, in the sense that the manifolds with these smooth structures the sphere S7, in the sense that the manifolds with these smooth structures are to the standard smooth smooth 7-sphere, 7-sphere, though they are are PLPLthe standard though they are not not diffeomorphic diffeomorphic to homeomorphic to it. In 54 below we shall discuss the more elaborate Milnorhomeomorphic to it. In §4 below we shan discuss the more elaborate MilnorKervaire homotopically equivalent to to sn S” for for nn 2: 2 5. 5. Kervaire theory theory of of manifolds manifolds M” Mn homotopically equivalent

sn.

sn

o Example 2. Let W n+l be a manifold-with-boundary, with boundary a disExample 2. Let Wn+l be a manifold-with-boundary, with boundary a disjoint union: IYW”+’ and suppose that Wn+’ is contractible onto 1 = VT U VP, n n n joint union: ôW + Vr u V2 , and suppose that W + 1 is contractible onto each of these portions of the boundary: each of these portions of the boundary: wn+l - vr - v;, nj (wn+lJy) = 0, j 2 0. W n+ 1 rv Vr rv V n , 1f'j (W n+ 1 , Vin) 0, j 2: O. 2

In this situation we say that the manifold Wn+’ is an h-cobordism between In this situation we say that the manifold Wn+l is an h-cobordism between Vr and VT, and that the latter manifolds are h-cobordant. The “h-cobordism Vr and V2n , and that the latter manifolds are h-cobordant. The "h-cobordism theorem” of Smale states that if n 2 5 and VT, V2 are simply-connected then then theorem" of Smale states that if n 2: 5 and Vr, V n are simply-connected there exists a smooth, real-valued function f on 2 Wn+‘, constant on Vln and n +1 , constant on Vr and there exists a smooth, real-valued function f on W VT, without any critical points (and with gradient nowhere tangential to VP without any critical points gradient nowhere or2n,Vi%). It follows that under the(and samewith hypotheses, IV+’ is tangential diffeomorphicto Vr to n ). It follows that under the same hypotheses, W n + 1 is diffeomorphic to or V 2 0 the cylinder VT x I, and VP is diffeomorphic to Vzn. the cylinder Vr x l, and Vr is diffeomorphic to V2n . 0 Example 3. The closed surfaces M22 may be readily classified by considering ExampleMorse 3. Thefunctions closed surfaces M The maygeneralization be readily classified considering particular on them. of this byapproach, on particular Morse functions on them. The generalization of this approach, on

v

$2. §2.

The The homology homology theory theory of of smooth smooth manifolds. manifolds. Complex Complex manifolds manifolds

177 177

the the basis basis of of the the h-cobordism h-cobordism theorem theorem above, above, leads leads to to the the classification classification of of cercertain tain classes classes of of simply-connected simply-connected manifolds, manifolds, for for instance instance those those having having non-zero non-zero homology ;]. The The method method of of Novikov Novikov homology groups groups only only in in the the middle middle dimension dimension [[~]. (of the early 1960s) for classifying arbitrary simply-connected manifolds (of the early 1960s) for classifying arbitrary simply-connected manifolds uses uses aa different difIerent approach approach although although itit does does invoke invoke the the h-cobordism h-cobordism theorem theorem (see (see $4 §4 below). below). On On any any connected, connected, closed closed S-dimensional 3-dimensional manifold manifold M3 M 3 there there exists exists aa Morse Morse function function ff with with one one maximum maximum point, point, one one minimum minimum point, point, a collection collection of critical points of index 1 all on one level surface f = cl, and of critical points of index 1 aU on one level surface f Cl, and a collection collection of of critical critieal points points of of index index 22 all aU on on another another level level surface surface ff == cz. C2. (The (The number number of of critical critical points points of of index index 11 can can then then be be shown shown to to be be equal equal to to the the number number of of index index 2; 2; we we denote denote this this number number by by g.) g.) IfIf the the manifold manifold isis cut cut along along an an intermediate intermediate level surface f = c, cl < c < cz, one obtains the two submanifolds level surface f C, Cl < C < C2, one obtains the two submanifolds My {x EE M M 11 f(z) f(x) I~ c} c},, M~ = {cx

M;

= {x c} {x EE M M )1 f(x) f(x) 2~ c}

M; asas common common boundary: boundary:

with with the the surface surface Mi

M; == {x M 1 f(z)(x) = = c} M; {x EE Mlf c} . ItIt is see that is not not difficult difficult to to see that ifif M3 M3 is is orientable orientable then then the the manifolds manifolds M-, M_, M+ M+ are each homeomorphic to the solid handlebody obtained as the region of ]R3 Iw3 are each homeomorphie to the solid handlebody obtained as the region of filling the sphere-with-g-handles Mi embedded in standard fashion in Iw3. It filling the sphere-with-g-handles Mi embedded in standard fashion in ]R3. It follows that the structure of the S-manifold M3 wholly by the 3 is determined follows that the structure of the 3-manifold M is determined wholly by the way (identical) solid handlebodies MA and M+ M+ are way in in which whieh these these two two (identieal) solid handlebodies M_ and are glued glued together along their boundaries, i.e. by a diffeomorphism h : Mi -+ Mi. (Such (Such together along their boundaries, Le. by a difIeomorphism aa decomposition 3-manifold is is called called aa Heegaard Although the decomposition of of aa 3-manifold Heegaard splitting.) splitting.) Although the homotopy and topological classes of diffeomorphisms Mi + Mi have been homotopy and topological classes of difIeomorphisms Mi ---. Mi have been classified (by Nielsen in the 193Os), this does not yield a classification of the classified (by Nielsen in the 1930s), this does not yield a classification of the (closed, connected) S-manifolds since since difIerent different difIeomorphisms diffeomorphisms Mi ---. --f M; Mi (closed, connected) 3-manifolds Mi may yield homeomorphic manifolds M3. In the case g = 1 one obtains, as may yield homeomorphic manifolds M3. In the case 9 = 1 one obtains, as the only 3-manifolds with Heegaard splittings of genus one, S3, S2 x S1, and the only 3-manifolds with Heegaard splittings of genus one, S3, S2 X SI, and Cl certain lens spaces. 0 certain lens spaces.

h:M; M;.

Example 4. If g(z) is a holomorphic function in general position on a comExample 4. If g(z) is a holomorphie function in general position on a complex manifold M” of complex dimension n, then the real part f(z) = Reg(z) Reg(z) plex manifold Mn of complex dimension n, then the real part f(z) of g(z) is a Morse function all of whose critical points have index n. This is of g(z) is a Morse function aU of whose critieal points have index n. This is true in particular for the manifold obtained from a projective algebraic variety true in particular for the manifold obtained from a projective algebraie variety M” c (CP” by deleting the hyperplane section @PNP1 n M” = W”-ln l say: Mn c cpn by deleting the hyperplane section CpN -1 n Mn W - say: on the remaining afline portion Mn \ W”-l there exist many everywhereon the remaining affine portion Mn \ W n- 1 there exist many everywhereholomorphic functions g in general position, and their real parts will then be holomorphie functions 9 in general position, and their real parts will then be Morse functions with all critical points of index n. From this one can conclude Morse functions with aIl critical points of index n. From this one can conclude that: Every map K -P Mn, where K is a CW-complex of dimension 5 n - 1, that: Every map K Mn, where K is a CW-complex of dimension ~ n 1, is contractible onto the hyperplane section. This has the consequence for the is contractible onto the hyperplane section. This has the consequence for the homology groups that the inclusion homomorphism homology groups that the inclusion homomorphism Hj (W”-‘) + Hj (Mn) Hj (W n - l ) ---. Hj (Mn) is an isomorphism for j < n - 1 and an epimorphism for j = n - 1. is an isomorphism for j < n - 1 and an epimorphism for j = n - 1.

178 178

Chapter Chapter 4.4. Smooth Smooth Manifolds Manifolds

The of projective The homology homology theory theory of projective algebraic algebraic varieties, varieties, developed developed by by LefLefschetz the 1920s 1920s, including including intersections intersections of of cycles, cycles, the the theory theory of of algebraic algebraic schetz in in the cycles the hyperplane cycles and and the the special special role role of of the hyperplane section, section, will will be be examined examined in in detail detail in volume surveying We shall shall confine confine in a cognate cognate volume surveying complex complex algebraic algebraic geometry. geometry. We ourselves of particular to the the consideration consideration of particular properties properties of of meromorphic meromorphic ourselves here here to maps maps f:M”-CP1rS2 J : Mn ---..
The the The induced induced self-map self-map Aj Aj :: H,(F,) Hj(Fe) + Hj Hj (FE) (Fe) of of the the homology homology groups groups has has the form form Aj=l Aj 1 for for jfn-1, j i- n I , A,ml(z) = z zt (z. z)zo, z, A n - 1(z) = z ± (zo . z)zo, z, zo Zo E E fLl(F,). Hn-l (Fe)· (Note that the fiber F, is a complex manifold of complex dimension 7~- 1.) (Note that the fiber Fe is a complex manifold of complex dimension n - 1.) The monodromy transformations associated with degenerate critical points The monodromy transformations associated with degenerate critical points of algebraic functions have been the object of deep investigations in algebraic of algebraic functions have been the object of deep investigations in algebraic geometry over a long period (including the last decade), by Deligne and many geometry over a long period (including the last decade), by Deligne and many others. One of the most interesting situations is that where there arise such others. One of the most interesting situations is that where there arise such “fiber bundles with singularities” as a result of sectioning a variety Mn c "fiber bundles with singularities" as a result of sectioning a variety Mn c CPN by a one-parameter family of hyperplanes, yielding fibers F which are all
52. The The homology theory of of smooth manifolds. Complex Complex manifolds §2. homology theory smooth manifolds. manifolds

179 179

with deeper topological topological and algebro-topological algebro-topological properties, properties, inwith some sorne of of their their deeper cluding been considered considered earlier by Poincaré Poincark in connection connection cluding applications, applications, had been with theory of his investigations investigations of Hamiltonian Hamiltonian with the theory of automorphic automorphic forms and his systems; however the full due to full systematic systematic theory theory is due to Cartan. Cartan. n Consider the algebraic complex A* (Mn), defined to have as as its elements elements the complex A*(M ), the smooth globally-defined differential forms & on M”, given in terms of of the smooth globally-defined differential for ms k on Mn, given in terms local co-ordinates on each chart V, c Mn by local co-ordinates on each chart Va C Mn by

n

f& =

c j,<...<j,

jk r\dxj,“. '·Jl"'Jk. dx ha A.. /1. .... /1. dx a' 1Tj,mjkdxj,

Thus form, and and the k-forms flk together comcomThus here here kk is is the the degree degree of of the the form, the k-forms [2k together prise differential operator operator d: d : Ak(Mn) is defined defined prise Ak(Mn). Ak(Mn), The The differential Ak(Mn) -+- t Akfl(Mn) Ak+1(Mn) is locally locally by by

DTJ' J' dxJ" A Jk' c %,,,; dxj,’ . ,j/1. dx dxe, J A /1. dx /1. ... df& = L dfh= !ClJ a a a' UX acx j,<...<j, j,<"'<jk 1:' k

!

(2.14) (2.14)

dxj a A dxSa = -dxS a A dxj a dx~ /1. dx~ = -dx~ /1. dx~.

The . ,, n, n, together together with The real real vector vector spaces spaces Ak(Mn), Ak(M n ), kk == 0,. 0, ... with the the differential differential dd the algebraic complex A* ( Mn) : determine the algebraic complex A*(Mn):

determine

0-

llO(M”)

5

A’(M”)

+ .” 3

fl”(M”)

---f 0.

(2.15)

(2.15)

may be defined as the sheaf of germs of k-forms (of class Note Note that that Ak(Mn) Ak (Mn) may be defined as the sheaf of germs of k- forms (of class C”), and the sequence (2.15) gives rise to an exact sequence of sheaves (see COO), and the sequence (2.15) gives rise to an exact sequence of sheaves (see below). The cohomology groups of the manifold Mn are now defined via the below). The cohomology groups of the manifold Mn are now defined via the homology of the algebraic complex (2.15): homology of the algebraic complex (2.15): H$(M”;IW)

= Ker d/Im d,

Im d

c

Ker d c Ak(Mn),

and their direct sum forms a graded ring and their direct sum forms a graded ring

IiJr,(M”) = c H;(M”;Et), HiJM n ) = LH~(Mn;IR), k>O

k?::O

where the multiplicative operation is the exterior product of forms. where the lemma” multiplicative operation the exterior ofproduct of of forms. “Poincark’s states that in someisneighbourhood every point Mn states that in sorne neighbourhood of every point of "Poincaré's lemma" every closed form 0 E Ak(Mn), n k > 0, (i.e. da = 0) is exact (i.e. 0 = dfl’).Mn every closed form [2 E Ak(M ), k > 0, (Le. d[2 = 0) is exact (i.e. [2 dr2'). For compact, simply-connected Lie groups and certain homogeneous For compact, simply-connectedsymmetric Lie groups andof certain homogeneous spaces, namely simply-connected spaces covariantly constant spaces, namely simply-connected symmetric spaces of covariantly constant Riemann curvature (i.e. where the covariant derivative of the Riemann curvaRiemann curvature (Le. where the covariant derivative of the Riemann curvature is identically zero), the cohomology ring Hs(Mn) is readily calculated. ture is identically zero), the cohomology ring HÂ(M n ) is readily calculated.

180 180

Chapter 4. 4. Smooth Smooth Manifolds Manifolds Chapter

Each Each class class zz EE Hi HÂ (M”) (Mn) contains contains aa unique unique two-sided two-sided invariant invariant k-form, k-form, (and (and conversely conversely each each such such k-form k-form belongs belongs to to some sorne cohomology cohomology class) class) on on M” Mn = G G (in (in the the Lie Lie group group case), case), or or a unique unique G’-invariant G'-invariant k-form k-form on on Mn Mn == G’/Gb G' /G'o in in the the case case of of aa symmetric symmetric space. space. Note Note that that in in the the latter latter case case the the Lie Lie algebra algebra 6’ (5' of of the the Lie Lie group group G’ G' always always admits admits a Z/2-grading: 1Z/2-grading: 6’ = 6; + q, (2.16) (2.16) [@3b,q

c ($7

p&q

c @%,

p5;,cq

c (5;.

AA Lie Lie group group G G endowed endowed with with aa Riemannian Riemannian metric metric may may be be regarded regarded as as aa symmetric G, with symmetric space space by by taking taking the the group group G’ G' == G G xx G G as as acting acting on on G, with the the action action given given by by (917 92) :: 9g +-+ ~ 9ld. glgg:;l. (gl,gZ) This Mn == G G ”2=! G’IG, G' / G, and and This yields yields M* cY=@;+s;, where where

t$, {(z, z)} (5~ = {(x, x)}),

(2.17) (2.17)

S: { (5, -x)} (5~ = {(x, -x)}..

In any homogeneous homogeneous space of aa compact compact Lie G the the computation In fact fact for for any space of Lie group group G computation of the cohomology ring reduces to the computation of the complex of the cohomology ring reduces to the computation of the complex -4zn,(hiln) Ainv(Mn) consisting restriction of of the the operator operator dd consisting of of the the G-invariant G-invariant forms, forms, although although the the restriction to to &, ATnv may may act act like like the the null null operator. operator. In the 1930s de Rham showed that these cohomology (i.e. defined defined In the 1930s de Rham showed that these cohomology groups groups (i.e. in terms of differential forms) are naturally isomorphic to the usual simplicial in terms of differential forms) are naturally isomorphic to the usual simplicial cohomology groups with with coefficients coefficients in in 1Ft Iw. We We now now sketch sketch the the modern modern proof proof cohomology groups of this result using sheaf theory. Consider the following sheaves on a manifold of this result using sheaf theory. Consider the following sheaves on a manifold IP: Mn: A” = the sheaf of germs of /c-forms (of class C”); Ak the sheaf of germs of k-forms (of class COO); 2” = the sheaf of germs of closed k-forms. Zk = the sheaf of germs of closed k-forms. (Thus 2’ is the sheaf of constants.) One has the following exact sequence of (Thus ZO is the sheaf of constants.) One has the following exact sequence of sheaves sheaves 0 --f z” 2 Ak L z”+l ---+ 0 (2.18) (2.18) (where i is the inclusion), which , as noted in Chapter 3, $4, gives rise to the (where i is the inclusion), which , as noted in Chapter 3, §4, gives rise to the exact cohomology sequence exact cohomology sequence . . + H.?(z”>

.!.z+ @(A”>

.&+ Hj(zk+l)

2

@+l(zk)

+ ...,

(2.19) (2.19) (where Hj(F) denotes the jth cohomology group with coefficients from the (where Hj (F) denotes the jth cohomology group with coefficients from the sheaf .F, i.e. the group Hj(M”; F)). Using the fact that sections of n”(h 2 0) sheaf F, i.e. the group Hj(MnjF)). and Using fact that of Ak(h 2': 0) not the analytic, it issections not difficult to show are required only to be of class C”OO are required only to be of class C and not analytic, it is not difficult to show that that (2.20) Hj(Ak)=O forj>O. (2.20)

$2. theory of of smooth smooth manifolds. §2. The The homology homology theory manifolds. Complex Complex manifolds manifolds

181 181

Note also that (M”), since since by definition H’(A”) coincides with Note also that H”(Ak) HO(Ak) = .4k Ak(Mn), by definition HO(Ak) coincides with the all k-forms k-forms of of class class C” on the whole of Prom (2.20) (2.20) the group group of of all Coo defined defined on the whole of Mn. Mn. From and exactness of the sequence it follows for j > > 0 the the homohomothe exactness of the sequence (2.19), (2.19), it follows that that for and the morphisms morphisms d* : Hi(Zkfl) + H i+l(Zk) are isomorphisms. we have have a sequence sequence of of isomorphisms isomorphisms are isomorphisms. Thus Thus we

H1(Zk) 5 H2(Zk-l) -‘* ... ---i ‘* Hk+l(Zo),

(2.21) ) (2.21

where where H”+l(Z’)

” Hk+‘(Mn;

R),

(2.22) (2.22)

the cohomology group, group, since Z” is is the the sheaf of constants. constants. We We the ordinary ordinary real real cohomology since ZO sheaf of have also from (2.19) and (2.20) the exact sequence have also from (2.19) and (2.20) the exact sequence H”(Ak)

5

H”(Zkfl)

-% H1(Zk)

+ 0,

where all k-forms class C”) globally defined defined on on where H’(A”) HO(Ak) isis the the group group of of all k-forms (of (of class COO) globally M” (so that d, = d), and H”(Zkfl) the group of closed (Ic + 1)-forms on Mn. Mn (so that d* = d), and HO(ZkH) the group of closed (k+ l)-forms on Mn. Prom (2.20), (2.21) have immediately immediately From this this exact exact sequence, sequence, together together with with (2.20), (2.21) we we have Hk+l(Mn;IR) E Hl(Zk) H1(Zk) 9:! “KerKer d/lm d/Im d d Z9:! Hi+l(Mn), HkH(Mn; IR) 9:! H~+l(Mn), Ker d c Ak(Mn), Ker de Ak(Mn), which is de Rham’s theorem. (Note that the Poincare lemma is used implicitly which is de Rham's theorem. (Note that the Poincaré lemma is used implicitly in this argument to establish local exactness, i.e. to obtain the exactness of in this argument to establish local exactness, i.e. to obtain the exactness of the sequence of sheaves (2.18).) the sequence of sheaves (2.18).) We next glance at a different kind of invariance of the cohomology groups We next glance at a different kind of invariance of the cohomology groups Hi (M”) , namely their invariance under smooth homotopies of manifolds H~ (Mn),

namely their invariance under smooth homotopies of manifolds F : M; x I + M2m, F: Mf x 1 ---+ M!!",

I = [a,b].

1

[a,b].

Observe first that for any form w E Ak(MF) the pullback has the form Observe first that for any form w E Ak(Mr) the pullback has the form Fqw

F*w

= I$!‘) dxi’ A.. . A dxik--l A & + Qj!‘) jl...jkdxii A . . A dxj”. 21”‘2k-l pCO) , dXil /\ ... /\ dxik-l /\ dt + pCl) , dx j1 /\ ... /\ dxjk. "1''''k-l

Jl"'Jk

If we define an operator D on forms w E Ak(Mp) by If we define an operator D on forms w E Ak (Mr) by Dw = s,” F*w, ab F* w,

Dw

J.

Dw E A,-,(MF),

where the integral is with respect to t, then the contribution to Dw from the where the withexpression respect to for t, then contribution Dw from second termintegral in the isabove F*w the is zero, since thattoterm does the not second term in the above expression for F*w is zero, since that term do es not involve dt. Using this it is straightforward to verify that involve dt. Using this it is straightforward to verify that DdfdD= f, - f;, Dd ± dD = lb ,

f:

182

Chapter Smooth Manifolds Manifolds Chapter 4. Smooth

where It follows follows (much (much as in in Chapter Chapter 3, §2, $2, the the where fa fa = F( F(,,a), a), fb = F( F(,,b). b). It analogous conclusion from equation equation (2.17) (2.17) there) there) that that for for any any cocycle cocycle analogous conclusion followed followed from zz (dz (dz == 0) do(z), coboundary. Hence 0) the the difference difference f,*(z)-f;(z) f; (z) - lb (z) == dD (z), and and so is a co boundary. Hence the induced homomorphisms the induced homomorphisms

coincide aIl j). j). coincide (for (for all As far as homotopy theory is is concerned, category of of CW CW-complexes As far as homotopy theory concerned, the the category -complexes and cellular maps may be replaced by the category of manifolds and smooth smooth and cellular maps may be replaced by the category of manifolds and maps, since any CW-complex K, after being embedded in IWN for some N, maps, since any CW-complex K, after being embedded in ]RN for sorne N, may be replaced by a neighborhood of it in IWN contractible to K, and then may be replaced by a neighborhood of it in ]RN contractible to K, and then cellular maps arbitrarily closely by by smooth smooth ones, ones, or or cellular maps may may be be approximated approximated arbitrarily closely replaced by smooth maps deformable to them by means of arbitrarily small replaced by smooth maps deformable to them by means of arbitrarily smaU homotopies. and the the homotopy invariance of of the the cocohomotopy invariance homotopies. From From this this consideration consideration and homology groups Hi in outline outline (together with the the fact fact that that homology groups H~ just just established established in (together with they satisfy cohomology axioms, axioms, given given in in Chapter Chapter 3, 3, §3), §3), it it is is clear they satisfy the the other other cohomology cIear that the coincidence of the Hi with the usual cohomology groups follows. In that the coincidence of the H~ with the usual cohomology groups follows. In fact there is no difficulty in taking this route to establishing the equivalence of fact there is no difficulty in taking this route to establishing the equivalence of cohomology theory defined via forms on manifolds, with the usual cohomology cohomology theory defined via forms on manifolds, with the usual cohomology theory; however rigorous exposition of this line of of argument argument theory; however aa detailed detailed and and rigorous exposition of this line would not seem to be worthwhile since it would not yield a proof shorter than would not seem to be worthwhile since it would not yield a proof shorter than other known proofs. other known proofs. There operator on forms on on an orientable manifold manifold AP, There is is aa natural natural operator on forms an orientable Mn, dedetermined by a Riemannian metric gij on M”, called the duality operator termined by a Riemannian metric 9ij on Mn, called the duality operator * : nyiq n - An-k (Mn). It is given locally by * : Ak(M ) An-k(Mn). It is given locally by dxil A

*w = (*T)il...i7L-k

w = Tj,...j,dxil

A

A &+-k

1

. A dxik.

(The operator * is well-defined for oriented manifolds M” since for (The operator * is well-defined for oriented manifolds Mn since for local expression local expression da = det(gij) dxi A . . A dx”, da = -\i Vdet(9ij) dx~ fi··· fi dx~ determines a well-defined n-form.) The duality operator * has the determines a well-defined n-form.) The duality operator * has the properties: properties: 1) (*)2 = (-l)“(“-k) on k-forms; 1) (*)2 = (_l)k(n-k) on k-forms; 2) the operator 6 = *-‘de : nk+‘(iVP) + nk(AP) n is formally the operator = *-ld* : Ak+l(Mn) with2) respect to the (jscalar product given by -+ Ak(M ) is formally with respect to the scalar product given by (w,w’)

(w,w')

= /

w A *w’,

{Wfl*W', M”

lMn

these the these the

following following

dual to d dual to d

(2.23)

§2. The of smooth Complex manifolds The homology homology theory theory of smooth manifolds. manifolds. Complex manifolds

183 183

that that is, is, f(dw,w’) (w,Sw’). ±(dw, w') = (w,ow'). 3) The Laplace + od 6d = = (d (d + + S)2 commutes with with both and 3) The Laplace operator operatoT A Ll == d6 do + 0)2 commutes both d and S (and (and is is self-dual). self-dual).

o

For manifold Mn one one has has the following orthogonal “Hodge decomdecomFor any any closed closed manifold Mn the following orthogonal "Hodge n position” of fl”(jV) considered as a Hilbert space with respect to the above position" of Ak(M ) considered as a Hilbert space with respect to the above scalar product (2.23): scalar product (2.23): Ak(AP)

= Im d @Im S @ Ker A,

(2.24) (2.24)

where the harmonic on Mn. One has also L\ consists consists of of the harmordc forms forms on Mn. One has also where Ker Ker A Kerd=Imd@Ker L\, Ker d = lm d EB Ker A, Kerd=Imh@Ker A, Ker 0 = lm 0 EB Ker L\,

Ker = Ker n Ker 6, Ker A L\ = Ker d dn Ker 0,

(2.25) (2.25)

whence follows that dimension of space of forms in in Ak n”(iU”) whence itit follows that the the dimension of the the space of harmonic harmonie forms (Mn) is equal to the lath Betti number bk(kP). It can be shown that the Poincari, is equal to the kth Betti number bk(Mn). Tt can be shown that the Poincaré duality homomorphism action of of the the duality * the action duality operator operator * duality homomorphism isis determined determined by by the on the harmonic forms. on the harmonie forms. The Laplace operator A can can be defined also for simplicial complexes, in The Laplace operator L\ be defined also for simplicial complexes, in fact for CW-complexes more generally, by taking d* and d in place of d fact for CW -complexes more generally, by taking 8* and 8 in place of d and S respectively, that the Laplace operator in this context is defined as and 0 respectively, so so that the Laplace operator in this context is defined as A = da* + d*a. With this understanding, the formulae (2.24) and (2.25) for Ll 88* + 8*8. With this understanding, the formulae (2.24) and (2.25) for the Hodge decomposition hold for any CW-complex K, with respect to the the Hodge decomposition hold for any CW -complex K, with respect to the scalar product determined by taking the cells to form an orthogonal basis. scalar product determined by taking the cells to form an orthogonal basis. If the CW-complex K is not simply connected, then as described in ChapIf the CW-complex K is not simply connected, then as described in Chapter 3, $5, any (unitary) representation p : rl(K) -+ U, determines a chain ter 3, §5, any (unitary) representation p : 7rl(K) ....... Un determines a chain complex C, (K; p) with the structure of a Z[Ti]-module. In the case where complex C*(K; p) with the structure of a Z[7rl]-module. In the case where all homology groups of positive dimension of this chain complex are zero, ail homology groups of positive dimension of this chain complex are zero, the following formula for the Reidemeister torsion holds (Singer, in the late the following formula for the Reidemeister torsion holds (Singer, in the late 1960s):

1960s):

~WNK,P)I = 2 log IR(K, p)1 =

L9 q( -1)q log det L\IC'I(K;p). Cq(-l)qlogdetAl~,(~;p).

(2.26) (2.26)

q

By extending this formula to the complex of differential forms on a manifold, By extending thisdefined formula to the complex of differential forms on a manifold, using a suitably relative determinant of two different representations, using a suitably defined relative determinant of two different representations, Ray and Singer (in the early 1970s ) were able to give an analytic treatment Ray and Singer (in the early 1970sin ) particular were able that to give an analytic treatment of Reidemeister torsion, showing their analytic version of of in particular their analytic version the Reidemeister concept was torsion, independentshowing of the metric gij. that A rigorous verification thatof the concept was independent of the metrie g'ij. A rigorous verification that the Ray-Singer torsion actually does coincide with the usual combinatorially the Ray-Singer torsion actually do es coincide with the usual combinatorially defined Reidemeister torsion was given in general form somewhat later (by defined Reidemeister torsion was given in general form somewhat later (by Miiller and Cheeger in the late 1970s). Müller Cheeger It is and of interest to in notetheinlate this 1970s). connection A. S. Schwarz’ curious quantumIt is of interest to note in connection Schwarz' curious integral quantumtheoretical interpretation of this Ray-Singer torsionA. S.using a functional of

theoretieal interpretation of Ray-Singer torsion using a functional integral of

Chapter Chapter 4. Smooth Smooth Manifolds Manifolds

184 184

a simple simple gauge-invariant gauge-invariant expression expression in in differential differential forms forms where where the the torsion torsion appears hypoappears in in the the process pro cess of of constructing, constructing, in in the the standard standard way, way, a theory theory hypothetically thetically violating violating gauge-invariance.’ gauge-invariance. 1 Differential DifferentiaI forms forms are are especially especially important important in in the the homology homology theory theory of of Kahler manifolds AI” of n complex dimensions, with Kahler metric 9kj. ReReKahler manifolds Mn of n complex dimensions, with Kahler metric &j. call (from end of of the section) that definition this caU (from the the end the preceding preceding section) that by by definition this metric metric is is Hermitian, given locally (on a chart V, co-ordinatized by .zA , . . . , z,“) Hermitian, given locally (on a ch art Va co-ordinatized by z~ , ... , Z~) by by ds2 = gkjdzkdZj a ar with real 2-form with the the property property that that the the associated associated real 2-form L? = ;gkidZg i dZak A1\ d.Zi d-Zaj 29kj is (and the sheaf Ak is closed: closed: dQ df? == 0. O. The The space space Ak(Mn) Ak(Mn) of of k-forms k-forms (and the associated associated sheaf Ak on AP) decomposes naturally as a direct sum of forms of type (p, q): on Mn) de composes naturally as a direct sum of forms of type (p, q): A”(W) Ak(Mn ) ==

L

c LPq, Ap,q, p+q=n p+q=n

(2.27) (2.27)

where defined (locally) as where aa form form of of type type (p, (p, q) q) isis defined (locaUy) as c

fil..&jl...j<,

dz~r\...r\dz~r\dz~r\...ndz~,

(2.28) (2.28)

Differentiation Differentiation of of forms forms on on the the Kahler Kahler manifold manifold Mn Mn is is defined defined on on O-forms O-forms (functions) by (functions) by d=d,+&,

d,f = gdz,

and extended essentially as for real forms to and extended essentially as for real forms to Kahler metric is used to define the formally Kahler metric is used to define the formally 6 = *-‘d*, 6, = *-‘d,*, and the corresponding Laplace operator and the corresponding Laplace operator A, = &d, + d,&, de = 8e de + de 8e ,

&f

= $dz,

(2.29)

forms on M” of degree > 0. The forms on Mn of degree > O. The dual operators dual operators s, = *-l&*, (2.30) (2.30)

z\, = A,, lie = de,

(2.31) (2.31 ) A = dS + 6d. d d8 + 8d. By “Hodge’s lemma,” provided the metric is Kahler the real and complex By "Hodge's lemma," provided the metric is Kahler the real and complex Laplace operators differ only by a constant factor: Laplace operators differ only by a constant factor: lAn up-to-date survey was written in 1983-1985 and published in Russian in 1986. The was written published in 1986. The work l An of up-to-date A. S. Schwarz survey mentioned above in was1983-1985 completed and around 1980 inandRussian was well-known at worktime of A.toS.certain Schwarz mentioned above was completed 1980 and well-known that mathematical physicists (specialists around in quantum field was theory). Starting at that to certain mathematical from time the late 1980s these ideas werephysicists developed (specialists by Witten in quantum and others. field Thistheory). theory Starting is now from 1980s these ideas were by and others. This theory now andWitten has become an extremely active is area known the as late “Topological Quantum Field developed Theory”, known Quantum Field Theory", and has become an extremely active area of modernas "Topological topology.

of modern topology.

$2. homology theory theory of of smooth smooth manifolds. manifolds. Complex manifolds §2. The The homology Complex manifolds

185

A const. xX .de. A,. .d = const. If follows If follows hence hence the the

that M” a closed manifold the space of forms and and that for for Mn closed manifold the space of harmonic harmonie forms cohomology groups decompose as direct sums: cohomology groups decompose as direct sums:

L

H”(M;@) Hk(M;q == c

Hp~Q(Mn;@), Hp,q(Mn;q,

(2.32) (2.32)

p+q=k p+q=k

in accordance accordance with the decomposition (2.27). This decomposition depends depends of of in with the decomposition (2.27). This decomposition course on the metric. Note that the form L? determined by the metric (see course on the metrie. Note that the form n determined by the metrie (see above) (1,l): above) has has type type (1,1): [L?] E C), [n] E HlJ(M”; H 1 ,1(Mn ; q, and that since the the integral integral and that [P] [nn 1 #i- 00 since

I 0” M”

is aa non-zero the volume volume of of AP. is non-zero multiple multiple of of the Mn. The operation of forming the exterior of any any form w on Mn M” with with The operation of forming the exterior product product of form won the form R is usually denoted by L: the form n is usually denoted by L: L(w) = nnw, L(w) = !lw,

n

and the by A. w are are defined to be be those for and the dual dual operator operator by A. Primitive Primitive forms for-ms w defined to those for which Aw = 0. Every form may be obtained from the primitive ones by means which Aw = O. Every form may be obtained from the primitive ones by means of applications of operators given by expressions in L and A. Note the result of applications of operators given by expressions in Land A. Note the result of Dolbeault: of Dolbeault: Hp,q(Mn; Hp,q(M n ;C) q ”~ Hp(M”; HP(M n ; nq), Aq), where here LI’J denotes the sheaf of germs of holomorphic q-forms on Mn. If where here Aq denotes the sheaf of germs of holomorphie q-forms on Mn. If the form 0 represents an integral cohomology class, then by Kodaira’s theothe form n represents an integral cohomology class, then by Kodaira's theorem the manifold Mn can be algebraically embedded in @PN in such a way rem the manifold Mn can be algebraieally embedded in CpN in such a way that the cohomology class of 0 is dual to the hyperplane section (mentioned that the cohomology class of n is dual to the hyperplane section (mentioned earlier in connection with its special role in the homology theory of algebraic earlier in connection with its special role in the homology theory of algebraic varieties). Note also the result of Moisheson (mid-1960s) to the effect that varieties). Note also the result of Moisheson (mid-1960s) to the effect that a complex manifold endowed with a (not necessarily Hodge) Kahler metric aadmits complex manifold endowed with a (not necessarily Hodge) Kahler metrie an algebraic embedding in @PN if the space of meromorphic functions admits an algebraie meromorphic functions on the manifold has embedding transcendencein CpN degreeif the (i.e. space largestof number of algebraically on the manifold has transcendence degree (Le. largest number of algebraically independent functions) equal to its dimension. independent functions) equal to its dimension. It is a useful observation that algebraic cycles, i.e. subvarieties (possibly It is a useful algebraic cycles,variety Le. subvarieties (possibly with singularities) observation iVnek of athat projective algebraic Mn, determine hon k of a projective with singularities) mology classes dual Nto - elements of H”>“: algebraic variety Mn, determine homology classes dual to elements of Hk,k: D [IV”-“] E Hk>k(Mn; C) n H2k(Mn;Z). (2.33) (2.33) According to sharpened version of a conjecture of Hodge every element of According to sharpened version ofside a conjecture Hodge every element of the intersection on the right-hand of (2.33) isof realizable in this way by the intersection on the right-hand side of (2.33) is realizable in this way by

186

Chapter Manifolds Chapter 4. Smooth Smooth Manifolds

means linear combination combination of of algebraic algebraic cycles. cycles. The The case kIc = = 1 of of means of of a rational rational linear this conjecture settled earlier earlier by in st stronger form: every every class class in in ronger form: this conjecture was was settled by Lefschetz Lefschetz in II’)’ (as in in (2.33)) by means means of the fundamental fundamental HI,1 n n N2(Mn; H2(Mn; Z) 7l,) can can be realized realized (as (2.33)) by of the cycle of of an an algebraic n AP (a so-called so-called "divisor"). “divisor”). For k > > l, 1, algebraic subvariety subvariety N”-l Nn-l n Mn (a For cycle Thorn’s theory theory (see below) shows shows that arise even even to to the the realizarealizaThom's (see below) that obstructions obstructions arise tion of (AP; 7l,) ;Z) as images images in Mn of non-singular complex complex H 2 (n-k) (Mn; in Mn of non-singular tion of elements elements of of H2cn--k) (or even quasi-complex) varieties Nnek (under maps Nn-” ---+ AP). We note (or even quasi-complex) varieties Nn-k (under maps Nn-k -+ Mn). We note here also the result of Hironaka of the mid-1960s according to which every vahere also the result of Hironaka of the mid-1960s according to which every variety with singularities may be obtained via contraction from a non-singular riety with singularities may be obtained via contraction from a non-singular variety is therefore continuous image image of of it). variety (and (and is therefore aa continuous it). We shall not discuss in the present survey the original original circle circle of ideas, initiinitiWe shaH not discuss in the present survey the of ideas, ated by Grothendieck in the late 1950s for formulating a homology theory of ated by Grothendieck in the late 1950s, for formulating a homology theory of projective algebraic varieties over (algebraically closed) fields of characteristic projective algebraic varieties over (algebraicaHy closed) fields of characteristic pp >> O. 0. These depend on elegant use use of of the the category category of of finite-sheeted, finite-sheeted, These ideas ideas depend on elegant unbranched coverings of open regions of @PN (with respect to the Zariski Zariski unbranched coverings of open regions of CpN (with respect to the topology, where the closed sets are just the algebraic subvarieties) involving topology, where the closed sets are just the aigebraic subvarieties) involving the “&ale topology” of Grothendieck. the Grothendieck Grothendieck theory, theory, the "étale topology" of Grothendieck. Analogues Analogues in in the of the Lefschetz formula for the number of fixed points of a smooth self-map of the Lefschetz formula for the number of fixed points of a smooth self-map M to the the Galois Galois group group of of the closure of of aa finite finite M, applied applied to the algebraic algebraic closure M + M, field, yield, by means of homological algebra along the lines of the Weil-Tate field, yield, by me ans of homological algebra along the Hnes of the Weil-Tate program, asymptotics for of solutions solutions of of diofor the the numbers numbers of dioprogram, number-theoretic number-theoretic asymptotics phantine equations modulo p, and the rationality of the mod-p analogue of phantine equations modulo p, and the rationality of the mod-p analogue of the Riemann <-function (which is considerably simpler than the ordinary cthe Riemann (-function (which is considerably simpler than the ordinary (function). This and related questions, questions, including the relevant history and and later later function). This and related including the relevant history developments, will doubtless be dealt with in those volumes of the present sedevelopments, will doubtless be dealt with in those volumes of the present series devoted to algebraic geometry. What is important for us here is that the ries devoted to algebraic geometry. What is important for us here is that the Grothendieck homology theory of real varieties defined by polynomials over Grothendieck homology theory of real varieties defined by polynomials over the integers yields the same information as the ordinary homology theory of the integers yields the same information as the ordinary homology theory of the complex variety defined by the same set of polynomials; the situation is the complex variety defined by the same set of polynomials; the situation is more complicated only in mod-p homology where p > 0 is the characteristic more complicated only in mod-p homology where p > 0 is the characteristic of the field of interest. of the field of interest. We now return to the theory of ordinary real smooth manifolds. One of the We now return to the theory of ordinary real smooth manifolds. One of the most important areas of application of homology and homotopy theory is the most important areas of application of homology and homotopy theory is the study of global properties of extremals of functionals of paths on M” with study of global properties of extremals of functionals of paths on Mn with one or the other of two types of boundary conditions: one or the other of two types of boundary conditions:

1) s {y} = s” L(s,k)&, y = z(t), x(to) = 20, S{--y}= t1L(X,±)dt, "Y = x(t), x(to)=xo, to

1)

lto

x:(t1)

= Xl,

X(tt}=XI,

(2.34)

(2.34)

the extremal problem where the end-points of the paths y are fixed at ~0, ~1; the extremal problem where the end-points of the paths "Y are fixed at xo, Xl; 2)

2)

S(Y) =

i

-qz, kc)&

Y(t + T) = 7(t),

Sb) = f Y L(x, ±)dt, "Y(t + T) = "Y(t),

the periodic variational problem. the periodic variational problem.

(2.35)

$2. The The homology theory of smooth smooth manifolds. manifolds. Complex Complex manifolds manifolds §2. homology theory

187 187

The quantity L, the the Lagrangian Lagrangian of of the the variational variational problem, problem, is a real-valued real-valued The quantity L, smooth function function L(x, L(x, v) U) defined defined on the the points points (x, (x, v) v) of of the the total total space space T(Mn) T(Mn) smooth of the the tangent tangent bundle bundle of of Mn, M”, and and is always always required required to to satisfy satisfy the the positivity positivity of condition condition 2

&

vi+

> 0

for all non-zero vectors

r].

(2.36) (2.36)

The standard of Lagrangians Lagrangians arising arising in in geometry geometry or or in in the the classical classical The standard examples examples of mechanics of non-charged particles, are as follows: mechanics of non-charged particles, are as follows: 2 , the a) lv12, the Lagrangian Lagrangian appropriate appropriate to to the the motion motion of of aa ~ gijwiwj gi.iVivj = ~+ Iv1 a) LL = Jj particle in aa Riemannian manifold Mn; Mn; provided provided the the parameter is natural, natural, parameter tt is particle in Riemannian manifold i.e. proportional to arc arc length, length, the the extremals extremals are are the the parametrized geodesics i.e. proportion al to parametrized geodesics of M”; of Mn; b) U(Z), appropriate appropriate to to the the motion of aa particle particle in in the the b) LL = ;~ gijvivj %ViV j - U(x), motion of Riemannian manifold Mn in the presence presence of of aa force force field field with potential U U(Z); Riemannian manifold Mn in the with potential (x); c) L(x, v) == Ivl [VI = = yi ,,/‘s,gijViV j , yields yields the the length length functional, with the the geodesics geodesics c) L(x, v) functional, with as extremals; as extremals; d) c), functionals functionals of of Finsler length type type where where the the d) more more generally generally than than c), Finsler length Lagrangian satisfies Lagrangian satisfies

L(x, XL(x, v), > 0; (2.37) (2.37) L(x, Xv) ÀV) == ÀL(x, v), xÀ> 0; these on submanifolds Banach spaces, spaces, with they these occur occur on submanifolds of of Banach with the the induced induced metric; metric; they are usually required to satisfy the positivity condition are usually required to satisfy the positivity condition

i

L(x,f)dt > 0. l(Y) = lb) s Y L(x, x)dt > o.

(2.38) (2.38)

e variational problem is said to be reversible; this is If L(x, If L(x, -v) -v) == L(x, L(x, w), v), th the variational problem is said to be reversible; this is the case for instance for the length functional c) with respect to a Riemannian the case for instance for the length functional c) with respect to a Riemannian metric. metric. According to the principle of Maupertuis (or Fermat-Jacobi-Maupertuis) According to the principle of Maupertuis (or Fermat-J acobi-Maupertuis) the trajectories extremizing functionals of Lagrange type (see b) above) are the trajectories extremizing functionals of Lagrange type (see b) ab ove ) are just the geodesics with respect to the new metric defined in terms of the total just the geodesics with respect to the new metric defined in terms of the total energy E = a lv122 + U(Z) of the particle by energy E = ~ Ivl + U(x) of the particle by lj: = 2(E - U)gij(x), lE($

=

I

Y

,/vdt,

dxj k:j = dt ’

(2.39) (2.39)

Provided E > rnaxZEMIL U(x), this clearly defines an ordinary Remannian Provided > maxxEMn U(x),by this clearly“principle defines an ordinary metric. TheE extremals yielded Fermat’s of least time” Remannian are essenmetric. The extremals yielded by Fermat's "principle of least time" are essentially the geodesics with respect to a metric of the form tially the geodesics with respect to a metric of the form gij = C-l(X)Sij1 in Iw3, gij c- (x)8 ij in ]R3, where c(x) is a smooth scalar function (originally the speed of light). (As is where c(x) isFermat a smooth scalar function (originally the rays speed of light). well known, introduced the principle that light travel along (As pathsis well known, Fermat introduced the principle that light rays travel along paths

188

Chapter 4. Smooth Chapter Smooth Manifolds Manifolds

of explain the the observed of light light rays passing of least least time time in in order order to to explain observed bending bending of rays in passing from one isotropic medium to another, linking this to the differing speeds of from one isotropie medium to another, linking this to the differing speeds of light, then unknown, in the different media.) light, then unknown, in the different media.) In general one one considers considers functionals functionals defined on the the path path space {}(xo, fl(zc,zi) Xl) In general defined on consisting of all all piecewise piecewise smooth parametrized paths M” joining joining Xo ~0 to to ~1, consisting of smooth parametrized paths in Mn Xl, or (in the the space of of of aU all piecewise piecewise smooth, smooth, or (in the periodic periodic problem) problem) on the space fl+(AP) n+(M n ) of directed, maps 8 S11 -; -+ M”, + 27r) = f(v). The directed, parametrized parametrized loops, loops, i.e. maps Mn, f(cp f(

$2. The The homology homology theory theory of of smooth smooth manifolds. manifolds. Complex Complex manifolds manifolds §2.

189 189

context also also the the Morse Morse inequalities inequalities context w(S) L bi(il(xo, b(fqso, a)), mi(8) ::::: x!)),

(2.40) (2.40) m(S) ::::: 2 bi(il+), h(fl+), mi(8) where mi mi is is the number of of extremals extremals of of index index i.i. where the number In certain circumstances circumstances the the first of these inequalities may may be be used used to to infer infer In certain first of these inequalities the existence existence of of distinct distinct geodesics geodesics from from Xo zc to to Xl. zi. However in the the periodic periodic the However in problem closed geodesics geodesics may may be be illusory illusory for for instance instance in in the the sense sense problem aa plurality plurality of of closed that they all be multiples of of aa simple simple closed closed geodesic geodesic (i.e. (i.e. obtained obtained by by that they might might all be multiples going round geodesic aa varying varying number number of of times). times). In In the case M” that geodesic the case Mn == 8S”n going round that with nn ::::: > 3, 3, itit is is known known that that there there are are at at least least two two geometrically geometrically distinct distinct with extremals in the periodic problem (A. I. Fet, in the mid-1960s). For any closed extremals in the periodic problem (A. 1. Pet, in the mid-1960s). For any closed manifold Mn with the property that the Betti numbers bi(R(xc, zi)) increase manifold Mn with the property that the Betti numbers bi (il (xo, Xl)) increase at least least linearly as ii +-+ 00, oo, itit has shown (by (by Gromoll Gromoll and and Meyer Meyer in in the the at linearly as has been been shown late 1960s) that there exist infinitely many geometrically distinct, periodic late 1960s) that there exist infinitely many geometrically distinct, periodic extremals. extremals. Working context of concerning three non-selfthe context of Poincare’s Poincaré's problem problem concerning three non-selfWorking in in the intersecting geodesics on the 2-sphere S 2‘, Lyusternik and Shnirelman exintersecting geodesics on the 2-sphere 8 , Lyusternik and Shnirelman extended their inequality (see above) to show that m(S) > cot(X/Mn), for tended their inequality (see ab ove) to show that m(8) ::::: cat(X/Mn) , for arbitrary reversible functionals S on the quotient space X/Mn of the space arbitrary reversible functionals 8 on the quotient space X/Mn of the space of non-self-intersecting paths on Mn (or more precisely the completion of of non-self-intersecting paths on Mn (or more precisely the completion of this space) modulo the space Mn c X of constant paths on Mn. (Here nonthis space) modulo the space Mn c X of constant paths on Mn. (Here nondegeneracy of the critical points is again not required.) By transferring to the degeneracy of the critical points is again not required.) By transferring to the space of non-directed paths and showing that X/S2 2 (taking M” = S2) 2 has space of non-directed paths and showing that X/8 (taking Mn = 8 ) has the homotopy type of IRP2, they were able to show, around 1920, that for any the homotopy type of lRP2, they were able ta show, around 1920, that for any reversible functional S on X/S2, 2 the number of critical points, regardless of reversible functional 8 on X /8 , the number of critical points, regardless of degeneracy, satisfies degeneracy, satisfies m(s) 2 3,

m(8) ::::: 3, thereby solving Poincare’s problem.2 2 In conclusion we note again that this thereby solving Poincaré's problem. In conclusion we note again that this theory applies only to reversible variational problems. theory applies only to reversible variational problems. An important chapter of the global calculus of variations is devoted to An important chapter of the global calcul us of variations is devoted to the study of the length functional and geodesics determined by a metric of the study of the length functional and geodesics determined by a metric of Cartan-Killing type on a Lie group or symmetric space (mainly due to Bott in Cartan-Killing type on a Lie group or symmetric space (mainly due to Bott in the late 1950s). On the group SU zn, for instance, equipped with the invariant the late 1950s). On the group 8U , for instance, equipped with the invariant metric determined by the Killing 2n form (X, Y) = Re Tr(XY*) on the Lie metric determined by the Killing form (X, Y) = Re Tr(XyT) on the Lie algebra, the minimal geodesics from zc = 1 to ~1 = -I (I the identity matrix) the minimal geodesics from to Xo = [to Xl (I the identitymanifold matrix) algebra, form a submanifold diffeomorphic the complex - [ Grassmannian to the complex Grassmannian manifold form a2 submanifold diffeomorphic Uzn/(Un x U,), and the minimal geodesics, as critical paths of the Gn,n G~n n ~functional U2n /(Un a), X Un), and the minimal geodesics, as critical paths of the action have index 0, while the non-minimal geodesics (from acti~n functional a), while that the non-minimal (from TCOto ~1) have index have > 2n index + 2. It 0,follows the inclusion geodesics of the space of Xo to Xl) have index::::: 2n + 2. It follows that the inclusion of the space minimal geodesics (” G!&, ) in n(zc,zi) induces an isomorphism betweenof minimal geodesics (~ G~n,n) in il(xo, Xl) induces an isomorphism between 2A complete and rigorous exposition of the Lyusternik-Shnirelman theory has been pubcomplete see andtherigorous of the Lyusternik-Shnirelman lished2 A recently; survey exposition by I. A. Taimanov in Russian Surveys, theory 1992, v.has2. been pub-

lished recentlYi see the survey by 1. A. Taimanov in Russian Surveys, 1992, v. 2.

190 190

Chapter Chapter 4.4. Smooth Smooth Manifolds Manifolds

the the corresponding corresponding homotopy homotopy groups groups of of dimensions dimensions up up to to and and including including 2n. 2n. From From the the homotopy homotopy exact exact sequence sequence of of the the principal principal fiber fiber bundle bundle V&,

+

G!&,

with

fiber

U,,

(see (see 51 §l (l.ll)), (1.11)), we we infer infer that that T.~(G:~,,) 1Tj(G~n,n) 2s::: T?-l(U,), 1Tj-l(Un),

11 <~ jj <~ 2n 2n - 2. 2.

(2.41) (2.41 )

Putting Putting these these various various facts facts together together we we obtain obtain (after (after some some work) work) the the result result known known as as Bott Bott periodicity periodicity for for the the stable stable homotopy homotopy groups groups of of the the unitary unitary group: group: 7rj(Un)

2 7rj-z(Un),

2 5 j 5 2n - 1.

(2.42) (2.42)

For For the the orthogonal orthogonal and and symplectic symplectic groups groups the the argument argument is is somewhat somewhat more more complicated. For the orthogonal groups one obtains 8-fold periodicity: complicated. For the orthogonal groups one obtains 8-fold periodicity: ~~(5’0,) E 7rj_8(SOn),

8 < j < n - 2.

(2.43) (2.43)

The first groups of of On 0, and given in in the the The first seven seven stable stable homotopy homotopy groups and U, Un are are given following table: following table:

jj

0 0

11

22

33

44

55

6 6

7 7

88

1Tj(On)

z/2 7/.,/2

z/2 7/.,/2

0 0

z 7/.,

0

0

0 0

0 0

z 7/.,

z/2 ·‘.’. 7/.,/2

7rj(Un)) 1Tj(U n

0 0

z7/.,

0 0

z 7/.,

0 0

z 7/.,

0 0

z 7/.,

Tj(Od

0 0

·....

·“..

One has the following isomorphisms for the symplectic group Spn: One has the following isomorphisms for the symplectic group SPn:

nj (04

=

~j-4(%J,

n’j (SPn)

=

7r.j-4(0,),

j 5nj 5 n-2.

2,

(2.44) (2.44)

In fact the Bott periodicity operator is defined for any CW-complex K in the In fact the Bott periodicity operator is defined for any CW -complex K in the sensethat sense that [C8K,0,] = [K,O,] if dim K < n - 10, [17 8 K, On] = [K, On] if dim K < n 10, (2.45) (2.45) [C’K, Un] = [K, Un] if dim K < 2n 4, [17 2 K, Un] [K, Un] if dim K < 2n 4, where C denotes the suspension and [X,Y] the set of homotopy classesof where denotes the the suspension [X, Y] inthe set of classes of maps X17 ---f Y. (Often index n and is omitted (2.45) withhomotopy the understanding maps X -Jo Y. (Often the index n is omitted in (2.45) with the understanding that it is sufficiently large.) that it is sufficiently large.)

§2. The The homology homology theory theory of of smooth smooth manifolds. manifolds. Complex Complex manifolds manifolds

191

Taking the the direct direct limits limits Taking 00= = lim lim O,, On, n-+oo n-+03

U U == lim lim U,, Un, n-+oo 12’00

Sp Sp == lim lim Sp,, SPn, n-+oo n-+cc

we infer infer from from Bott Bott periodicity periodicity that that we f140 n4 0 = sp, Sp,

n4sp n4 Sp = 0,0,

f12u n2 U = u,U,

where loops on at ~0. where 0X nx = 0(X, n(X, zc) xo) is the the space space of ofloops on XX based based at xo. (The (The base base point point isis often often left left implicit.) implicit.) Remark. Without Without entering entering into into detail detail we we mention mention the the fact fact (established by Remark. (established by sever al authors) that the nonparallelizability of the spheres sn for n =f=. 1, 3, several authors) that the non-parallelizability of the spheres S” for # 1,3,7 7 ean be dedueed from Bott periodicity independently of the results of Adams can be deduced from Bott periodicity independently of the results of Adams originally used used for for this this purpose. purpose. Thus Thus for for all aIl nn #=f=. 1,3,7, the tangent tangent bundle originally 1,3,7, the bundle 7(sn) isis non-trivial, non-trivial, and and moreover moreover for for odd odd nn represents represents (via (via the the classification classification T(,!P) of $6) an of Son-bundles SOn-bundles - see Chapter Chapter 3, §6) an element element of of r,-r(S0,) 11"n-1(SOn) of of order order 2. Note Note that that for for the the unitary unitary group group U, Un the the first first non-stable non-stable homotopy homotopy group group is is

11"2n(Un ) = cl!. Zn!. mn(Un) On of elements elements On the the other other hand hand Adams’ Adams' result result on on the the non-existence non-existence of of 7r2n-1(Sn), 11"2n-1 (sn), n >::::: 8, of of Hopf does not not seem to follow from Bott of Hopf invariant invariant 1 does seem to follow from Bott 0 periodicity $6). periodicity in in this this way way (see (see Chapter Chapter 3, §6). 0 We of Lie We now now turn turn to to the the cohomology cohomology algebras algebras of Lie groups groups and, and, more more generally, generally, “H-spaces.” with a continuous continuous "H-spaees." An An H-space H -space is a topological topological space space X X equipped equipped with multiplication y) E X multiplication zx·. y = $(z:, 1jJ(x,y) X (i.e. (i.e. $1jJ : X X xx X X +---+ X) X) with with respect respect to to which “homotopic identity such which there there is a "homotopie identity element,” element," i.e. i.e. an an element element 1 E X X sueh that the the maps maps that 7/Q,%) --) x, $(Z, 1) : X x -----f 1jJ(1, x) : xX ---+ X, 1jJ(x, ---+ x X defining multiplication by 1, are the identity identity map map lx. lx. are both both homotopic homotopie to to the defining multiplication by Note that H-spaces may be more more or or less less like for instance Note that H-spaces may be like groups, groups, for instance in the the sense sense that “homotopically associative,” the case of the loop loop spaces spaces that they they are are "homotopically associative," as in in the of the ~(X,Q). with Lie Lie groups, furnish the the most most important important f?(X,xo). The The latter, latter, together together with groups, furnish examples of instance it it follows follows from from Bott Bott periodicity that the the examples of H-spaces. H-spaces. For For instance periodicity that space homotopy type of the the loop space Q(U, space BU BU = J@m lim BU,, BUn has has the the homotopy type of loop space fl(U, 1) : n->oo

BU rv fl(U, 1), similarly for for BO BO and BSp, so that that these regarded BU N f2(U, l), and and similarly and BSp, these may may be regarded as H-spaces H -spaces as far far as their their homology or homotopy is eoncerned. Thus homology or homotopy concerned. Thus these, these, also 0, 0, U, Sp, Sp, are homotopieally associative as also are homotopically associative and and commutative, commutative, although although for for finite for the the Lie Lie groups U,, Spn, Sp,, this this is not not the the ease. case. finite n, n, i.e. for groups O,, On, Un, We now now consider consider the the cohomology cohomology algebra -space X, We algebra of of an arbitrary arbitrary H H-space X, with with coefficients natural k-algebra k-algebra isomorphism coefficients from from any any field field k. Ic. From From the the natural isomorphism Hm(X X; k) k) ”~ Hm(X x X;

c L

Hj(X; k) C.9Hq(X; k), Hj(X; k) ® Hq(X; k),

j+q=m j+q=m and the multiplication 1c,on X, X, one obtains obtains a k-algebra k-algebra homo homomorand the H-space H-space multiplication 1jJ morphism phism

192 192

Chapter Chapter 4. 4. Smooth Smooth Manifolds Manifolds ?y 'IjJ* :: H*(X; H*(X; k)k) 4~ H’(X; H*(X; k)k) @ ® H*(X; H*(X; k). k).

(2.46) (2.46)

From From the the existence existence ofof aa homotopic homotopic identity identity element element itit follows foUows that that for for any any zz EE H*(X; Ic) we may write G*(z) in the form H*(X; k) we may write 'IjJ*(z) in the form T),+(z) 'IjJ*(z) == 163 1 ® zz ++ zz @ ® 11 ++ cL z; zj @ ® z:‘, zj',

(2.47) (2.47)

j

where where dimz;, dim zj, dimzj’ dim zj >> 0. O. The The conditions conditions (2.46), (2.46), (2.47) (2.47) determine determine the the strucstructure e existence ture of of aa Hopf Hopf algebra algebra on on H*(X; H*(X; k). k). Th The existence of of an an algebra algebra homomorhomomorphism phism of of the the form form (2.47) (2.47) turns turns out out to to be be aa strong strong restriction restriction on on the the structure structure of of the algebra H*(X; k); according to Hopf’s theorem, a graded, skew-symmetric, the algebra H* (X; k); according to Hopf's theorem, a graded, skew-symmetric, associative associative algebra algebra HH == CH”, H” k -C LH k , dim dimH < co, 00, k>O k;:::O

over over aa field field of of characteristic characteristic 00 (a, (Q, R, lR, @ C for for instance) instance) for for which which there there exists exists aa homomorphism H + H @ H of the form (2.47), is a free skew-commutative homomorphism H -----> H ® H of the form (2.47), is a free skew-commutative graded = graded algebra, algebra, i.e. i.e. satisfies satisfies no no relations relations other other than than skew-commutativity: skew-commutativity: zw zw = (-l)ab~~, where a, b are the dimensions of z, w. Hence the algebra H*(X; k) (_1)ab wz , where a, b are the dimensions of z, w. Hence the algebra H*(X; k) has and aa set set of has aa set set of of exterior exterior generators generators ej,, ejo E E H’j+‘(X; H 2 j+l(X; Ic), k), and of polynomial polynomial 2 generators uj, E H2j(X; Ic), together generating H*(X; Ic) without further generators Uj{3 E H j(X; k), together generating H*(X; k) without further relations. Since the free polynomial generators uj, satisfy uyO # 0 for all it relations. Since the free polynomial generators Uj{3 satisfy uJ{3 i= 0 for aU n, n, it follows finite-dimensional H-spaces X (in (in particular for follows immediately immediately that that for for finite-dimensional H -spaces X particular for Lie groups), the algebra H*(X; k) is a finitely generated free exterior algebra. Lie groups), the algebra H*(X; k) is a finitely generated free exterior algebra. Over of characteristic > 0, can be shown that that there there is is aa set set of of characteristic pp > 0, itit can be shown Over fields fields kk of multiplicative generators of H* (X; k) in terms of which there are only relations multiplicative generators of H* (X; k) in ter ms of which there are only relations of the form up’” = = 0. Furthermore in several several cases cases it it is is known known that that although although of the form uP" O. Furthermore in +J’” = 0, there is in the “dual algebra” a non-zero element of that dimension. uP" 0, there is in the "dual algebra" a non-zero element of that dimension. Here the the dual dual (Pontryagin (Pontryagin dual) algebra to the Hopf algebra H*(X; k) of an dual) algebm to the Hopf algebra H*(X; k) of an Here H-space X, is obtained from the direct direct sum sum H*(X; H,(X; k) k) of of the the homology homology groups groups H-space X, is obtained from the (vector spaces) by defining a multiplication on it via the homomorphism (vector spaces) by defining a multiplication on it via the homomorphism $L : H,(X;

k) @ H,(X;

k) -

H,(X;

k)

These results have been obtained by means of close scrutiny of the cohomology These results have been obtained by means of close scrutiny of the cohomology of H-spaces over finite fields, initiated by Browder in the early 1960s. of H-spaces over finite fields, initiated by Browder in the early 1960s. In particular Browder proved a Poincare-duality law for the homology of In particular Browder proved a Poincaré-duality law for the homology of finite-dimensional H-spaces. However for a considerable period no nontrivial finite-dimensional H-spaces. However for a considerable period no nontrivial examples of finite-dimensional H-spaces were known, until in the early 1970s examples of finite-dimensional H-spaces were known, until in the early 1970s Mislin, Hilton and others found such examples using the “plocalization techMislin, Hilton and others found such examples using the" p-Iocalization technique for the category of homotopy types” invented by Quillen and Sullivan nique for the category of homotopy types" invented by Quillen and Sullivan around 1970 in connexion with the “Adams conjecture”. (The idea behind this around 1970 in connexion with the" Adams conjecture". (The idea behind this technique actually dates back to work of Serre of the first half of the 1950s.) technique back the to work of Serre of the first half of the 1950s.) These new actually H-spaces dates resemble homotopy-theoretical plocalizations of certhe homotopy-theoretical p-Iocalizations of cerThese new H-spaces resemble tain classical Lie groups and H-spaces for various primes p. tain classical Lie groups and H-spaces for various primes p.

$2. smooth manifolds. manifolds. Complex Complex manifolds manifolds §2. The The homology homology theory theory of of smooth

193 193

As Q(Y, Yo), yc), it it should be out that that over As regards regards the the H-spaces X = Q(Y, be pointed pointed out aa field characteristic zero, the Pontryagin Pontryagin ring H,(X; k) is is isomorphic isomorphic to to of characteristic zero, the ring H*(X; k) field kk of the algebra of of the the Lie-Whitehead Lie-Whitehead superalgebra of the the homotopy the enveloping enveloping algebra superalgebra of homotopy groups groups n*(X)@‘,® Q, 1l'i(X) %(X) =%+1(Y), 1l'*(X) 1l'i+l(Y), where the grading is as in X and the Whitehead (Lie superalgebra) superalgebra) multimultiwhere the grading is as in X and the Whitehead (Lie plication [z, y] of elements as in Y. To construct the enveloping algebra one plication [x, y] of elements as in Y. To construct the enveloping algebra one takes all elements x~j of some graded basis for the homotopy groups (tentakes aB elements Xj of some graded basis for the homotopy groups (tensored identity element element 1, 1, and the algebra algebra of of aU all sored with with Q), Q), adjoins adjoins an an identity and considers considers the noncommutative k-polynomials in the variables Z, subject to the relations noncommutative k-polynomials in the variables x, subject to the relations x.x. _ (-l)dim xi dim q,.,. = [x.21 x.~ 3’ XiXj'" Xj]. 2 3 (._I)dim Xi dim Xj XjXi 3 2 [Xi,

This of theorems Serre and This result result is is aa consequence consequence of theorems of of Serre and Cartan. Cartan. Example. loop space of the the sphere we have the Ioop space of sphere S” sn we have Example. For For the Hj(LY; Z) Hj(nsn; Z) == 0, 0,

jj f. # k(n k(n -- 1), l),

H+Y3”;Z) Z) E Z, Hj(nsn; Z,

jj = k(n l), k k ~ 2 O. 0. k(n - 1),

Denoting a (suitable) Denoting by by VI, Vk a (suitable) the following multiplicative the following multiplicative 1)

1)

n = 2k:

generator of of the the group 09; Z), one has generator group H”(+‘)( Hk(n-l)(nsn;z), one has relations in H*( QSn; Z): relations in H*(nsn; Z):

VT = 0,

VI2-0 ,

n = 2k :

(2.48) (2.48)

V2p V 2q

=

(p ~ q)

V2(p+q);

(2.49)

(2.49) 2)

2)

n=2k+l: n 2k + 1:

upup= vpvq

(p ~ q)

YP+q)'

v(p+q)'

The isomorphisms (2.48) and the relations (2.49) h ave been shown (by Serre) The isomorphisms (2.48) and the relations (2.49) have been shown (by Serre) to follow automatically from the Leray cohomology spectral sequence of to follow automatically from the Leray cohomology spectral sequence of the appropriate fibrations; however they were first obtained by Morse and the appropriate fibrations; however they were first obtained by Morse and Lyusternik around 1930 by considering geodesicson the sphere Sn. 0 Lyusternik around 1930 by considering geodesics on the sphere sn. 0 Anymap$:SnxSn--tSn, can be extended naturally (using a construcAny map 'ljJ : sn X sn --'> sn, can be extended naturally (using a construction of Hopf) to a map tion of Hopf) to a map f$ : s2n+l ---i sn+l, f 1/1 .• s'2n+l --'> sn+l , where we consider S” c Snfl and S” x S” c SZnfl by means of the following where we consider sn C sn+! and sn X sn C s2n+1 by means of the following canonical embeddings: S” is identified with the equator of Snfl: canonical embeddings: sn is identified with the equator of sn+1: Sn+l = D;+’ U D,nfl, where Sn = D;“+l n or+l; and S” x S” with the following subspace of S2n+1: and sn X sn with the following subspace of S2n+1: S”

sn where

x

X

Sn

=(D;+’

sn = (D?+1 S2n+1 = (0:”

x

X

Sn)n(Sn

sn) n (sn

x

D;+l),

X D~+1),

x Sn) U (5’” x DC+‘).

194 194

Chapter Manifolds Chapter 4.4. Smooth Smooth Manifolds

The “Hopf $) of of the the map map f$ to coincide with the the The "Hopf invariant” invariant" h(f, h(J,7jJ) il/; turns turns out out to coincide with product degrees of the maps product mlrn2 mlm2 of the degrees the maps

7jJ(s, so) : sn deg7jJ(s, so) = ml, ml, +(S,SCI) 5’” xX {so} {SO) ----> sn, S”, degti(s,so) $(so,s) : {so} {SO} xX Sn 9’ --+ -----fS”, m2. 7jJ(so,s): sn, deg$(sc,s) deg7jJ(so,s) = m2.

sn

If S” is an H-space with 1c,then clearly degreesml, ml, m2 m2 == 1, 1, If with multiplication multiplication 7jJ clearly the the degrees so that f$ : s2n+l S2n+1 ---t invariant 1. 1. Hopf Hopf showed showed so that there is aa map il/; --+ Sn+’ sn+! of Hopf Hopf invariant q - 1, and they do that if n has has the form n = 224 that such su ch maps maps can exist only only if they do exist - 1, 1,3,7. The with Hopf invariant 11 The theorem that that maps maps S2nf1 s2n+! --+ --+ Sn+’ sn+! with Hopf invariant for n == 1,3,7. do not exist for n other than 1, 3, 7, was proved by Adams in the late 1950s. 1950s. exist n other than 1, 7, was Theorems of this type on nonparallelizability of spheres or the above this type nonparallelizability of spheres result of Adams (which is much stronger) yield as a corollary the nonresult of (which is stronger) yield as corollary existence of real finite-dimensional division algebras of dimensions other than existence of real finite-dimensional division algebras of dimensions other than 2, 4 and 8. (As already already noted, Bott periodicity, periodicity, noted, this follows alternatively alternatively from Bott as was shown by Milnor, Kervaire and Atiyah.) as was Milnor, Kervaire Atiyah.) In the early 1980s in the course of investigations investigations of classicalIn the early 1980s in the course of of certain certain classicalmechanical problems such as that of the motion of a rigid body about mechanical problems such as that of the motion of a rigid body about aa fixed fixed point gravitational field, field, or or of moving freely freely in in an an ideal ideal fluid, fluid, point in in a a gravitational of aa rigid rigid body body moving and of contemporary contemporary mathematical physics (in (in particular particular and also also in in problems problems of mathematical physics that of giving a detailed topological analysis of Dirac monopoles), there arose that of giving a detailed topological analysis of Dirac monopoles), there arose a class of multi-valued functionals (for instance as generalized analogues a class of multi-valued functionals (for instance as generalized analogues of of Dirac defined on space of fields with with aa Dirac monopoles) monopoles) defined on one one or or another another space of paths paths or or fields large number variables (Novikov). (Novikov). The The topology of multi-valued functions large number of of variables topology of multi-valued functions and functionals, estimates of the number of their critical points, and the study and functionals, estimates of the number of their critical points, and the study of the structure of their level sets (in the finite-dimensional case), which we of the structure of their level sets (in the finite-dimensional case), which we shall now describe, constitute a non-classical analogue of the classical global shall now describe, constitute a non-classical analogue of the classical global calculus of of variations. variations. calculus A functional with single-valued gradient gradient on finite- or or A multi-valued multi-valued functional with single-valued on aa finiteinfinite-dimensional manifold closed l-form w (dw (dw == 0) 0) on on M, is is given given by by a a closed I-form w infinite-dimensional manifold M, M. multi-valued function function S itself is is then by the the integral integral M. The The multi-valued S itself then given given by z S(x) = fixed point. S(x) = s w, w, where where xc Xo is is aa fixed point. x0 Xo

lX

The value S(x) depends, in general, on the path 'Y y connecting connecting xx with with the point The value S(x) depends, in general, on the path the point ~0. The multi-valued function S becomes a single-valued one when lifted xo. The multi-valued function S becomes a single-valued one when lifted to to some space sorne regular regular infinite-sheeted infinite-sheeted covering covering space p :: M M --+ 4 M, p M,

where p*w = dS. dS. where p*w =

(2.50) (2.50)

Assuming H,(M; Z) # 0, 0, this this covering group that Hl(M; Z) 1= covering space space has has monodromy monodromy group Assuming that isomorphic to Z x Z x. . . x Z (k factors), where (k 1) is called the “irrationality isomorphic to Z x Z x ... x Z (k factors), where (k -1) is called the "irrationality degree” of the form w. w. An of generators Tl, ... . . . ,,Tk Tk of monodromy degree" of the form An action action of generators Tl, of the the monodromy group Z X Z X . . x Z on M is determined by numbers ~1, . . . , Kk, given by the the group Z x Z x ... x Z on M is determined by numbers KI, ... , Kk, given by

§2. The of smooth smooth manifolds. manifolds. Complex manifolds The homology homology theory theory of Complex manifolds

195

“periods” of the follows. First, First, one basis "(1, yl, ... . . ,"(m , -ym E E "periods" of the form form w w as follows. one chooses chooses some sorne basis HI (M; Z), and then ~j are given by the integrals (periods) of of the HdM; Z), and then the the numbers numbers /'ij are given by the integrals (periods) the form w 7.j: form w around around the the "ij:

1 I

w == Kj, w /'ij, s 'Yj 7.7

j 2$ k, k (2.51) (2.51)

w = 0, jj > > k. wO, k. I 'Yj ri (The numbers /'il, ~1,. ... . . ,, /'ik & are are linearly over Z.) The action action of of the (The numbers linearly independent independent over Z.) The the above monodromy group then is given by above monodromy group then is given by S(T,x) == S(x) /cir S(Tix) S(x) ++ K,i,

1,..... ,, k, k, ii == 1,

(2.52) (2.52)

If {Va} {Va} is cover of of the the manifold manifold M each member of which which w w If is an an open open coyer M on on each member of is exact: w Iv,, = dSca) f or some single-valued function s(a) (x) defined on V,, is exact: wlv = ds(a) for sorne single-valued function s(a) (x) defined on Va, then overlap Va V, nn Vp VP the the difference Sea) - Sep) S(p) will will be be locally then on on each each region region of of overlap difference s(a) locally constant, i.e. constant on on each connected component V, n n Vp. VP. This This affords affords aa constant, Le. constant each connected component of of Va canonical method constructing aa multi-valued multi-valued function function S on aa manifold manifold M canonical method of of constructing S on M (or any topological space) with single-valued gradient, via an open cover {Va} {Va} (or any topological space) with single-valued gradient, via an open coyer of M functions {Sea)}, S’ca) defined defined of M and and aa corresponding corresponding collection collection of of functions {s(a)}, each each s(a) and single-valued on V,, with the property that Sea) S’(fl) is locally constant and single-valued on Va, with the property that s(a) - Sep) is locally constant on V, that if if there function S on M M such such that that on Va I? n Vo. Vp. Note Note that there is is aa single-valued single-valued function S on S So) is constant on each V,, then the collection {S(o)} essentially defines S s(a) is constant on each Va, then the collection {s(a)} essentially defines that single-valued function. that single-valued function. If S then the the If S is is aa multi-valued multi-valued function function on on M M with with single-valued single-valued gradient, gradient, then level sets S = const. are well-defined and determine a “foliation” (possibly level sets S const. are well-defined and determine a "foliation" (possibly with singularities) of M, with “leaves” which may be, in general, non-compact, with singularities) of M, with "leaves" which may be, in general, non-compact, even if M is compact. The study of such foliations, which are of considerable even if M is compact. The study of such foliations, which are of considerable topological interest, has begun only recently. topological interest, has begun only recently. As for single-valued smooth functions on a manifold M, so also for multiAs for single-valued smooth functions on a manifold M, so also for multivalued functions S (as above) is one interested in the critical points XC~ : valued functions S (as above) is one interested in the critical points Xj : = 0. We shall consider, in the infinite-dimensional as well as the finitea3 dSlxj = O. We shall consider, in the infinite-dimension al as well as the finitedimensional case, only l-forms w for which the critical points are all isolated dimension al case, only I-forms w for which the critical points are aIl isolated and non-degenerate and have finite Morse index (essentially the number of and non-degenerate and have finite Morse index (essentially the number of negative squares in the quadratic form d2Slzj on the tangent space at xj, negative squares in the quadratic form d2 SIxj on the tangent space at Xj, with respect to appropriate local coordinates). If the monodromy group is Z with respect to appropriate local coordinates). If the monodromy group is Z with generator yi (or in another words the l-form w has zero irrationality with generator "(1 (or in another words the I-form w has zero irrationality degree) and its only period c degree) and its only period ü

/'il

11

W

is an integer (see above), then w represents an integral cohomology class is an integer (see above), then w represents an integral cohomology class (2.53) [w] E Hl(M;Z).

(2.53)

This determines a single-valued map This determines a single-valued map

196 196

Chapter 4. Smooth Smooth Manifolds Chapter Manifolds

{2&S} : A4 S1. 'I)Ij; = exp {21TiS} M -t ---+ SI.

(2.54)

In this situation is said said to be quantized. In situation the the form w is 1. Let M = Mn be aa finite-dimensional, finite-dimensional, closed closed manifold, manifold, and w Mn be Examples. 1. a closed l-form on M. How may one estimate the number of critical points of of a closed 1-form on M. How may one estimate the number of critical points w? The above-mentioned classical situation of a quantized form (see (2.53), w? The above-mentioned classical situation of a quantized form (see (2.53), (2.54)) is case M* the homotopy homotopy type type of of (2.54)) is especially especially simple: simple: in in this this case Mn must must have have the a fibration over S1. There are various topological criteria for the existence of a fibration over SI. There are various topological criteria for the existence of maps 11, : M --+ S1 without critical points, obtained in the 1960s by Browder, maps'lj; : M ---+ SI without critical points, obtained in the 1960s by Browder, Levine, Hsiang and 55 below). below). If If M” is not not of of this this type, type, there there will Levine, Hsiang and Farrell Farrell (see (see §5 Mn is will be critical points, estimates for which are given below. be critical points, estimates for which are given below. 2. Let M O(Nk,za,si) consisting of of the the paths paths in in N N”k 2. Let M be be the the path path space space n (N k , XO, Xl) consisting k of directed loops (any closed manifold) from 50 to xi, or the space R+(N”) (any closed manifold) from Xo to Xl, or the space n+ (N ) of directed loops y : S’ Nk.k Note that the space O+(Nk)k may be regarded as a fiber bundle 'Y: S1 -+ ----7 N . Note that the space n+(N ) may be regarded as a fiber bundle over N” with map over Nk with projection projection map p : L?+(Nk) +

N”,

P(Y) = Y(O)?

whose E N” is the the loop loop space R(N”,x) consisting of k , x) consisting whose fiber fiber over over aa point point xX E Nk is space n(N of the loops beginning and ending at x. There is also the obvious cross-section the loops beginning and ending at x. There is also the obvious cross-section cp: N” + Qn+(Nk), 1, where p(x) is the constant at x. x. If If Nk Nk is is tp: Nk ---+ n+(Nk), pep ptp = 1, where tp(x) is the constant loop loop at endowed with a Riemannian metric then any closed 2-form R on Nk k (which endowed with a Riemannian metric then any closed 2-form n on N (which may be called a magnetic field" field” for for kk 2 3, or an “electromagnetic may be called a “generalized "generalized magnetic '$ 3, or an "electromagnetic field” for k = 4) gives rise to the functional field" for k = 4) gives rise to the functional (2.55) (2.55) (the “action functional”), and the functional (with the same extremals) (the "action functional"), and the functional (with the same extremals) (2.56) (2.56) (the “Maupertuis functional”). Here E is the “energy,” e the “charge,” and y (the "Maupertuis functional"). Here E is the "energy," e the "charge," and 'Y is from Q(Nk,xs,xi) or O+(Nk). Provided Nk is simply-connected and the is from n(Nk,xo,xt} or n+(N k ). Provided N k is simply-connected and the cohomology class of the form R is non-zero: cohomology class of the form is non-zero:

n

0 # [fl] E H2(Nk;R), the formulae (2.55), (2.56) defi ne multi-valued functionals on the space M = the (2.56) define multi-valued functionals on the space M = fi(N”,formulae ~1) (2.55), (or @(N”)). k , ~0, k (If the manifold N”k is not simply-connected, then n(N Xo, Xl) (or n+(N )). (If the manifold N is not simply-connected, then it can be embedded in the (suitably realized) Eilenberg-MacLane complex it can1)be embedded in the in (suitably realized) K(T, where 7r = ri(Nk), such a way that Eilenberg-MacLane the cells of K(x, 1) \complex Nk all K(1T,l) where 1T = 1Tl(Nk), in such a way that the have dimensions 2 3. The induced homomorphism cells of K(1T, 1) \ Nk aIl have dimensions ~ 3. The induced homomorphism H2(7rl(Nk); lit) = H2(K(n, 1); R) + H2(Nk; R)

52. §2. The The homology homology theory theory of ofsmooth smooth manifolds. manifolds. Complex Complex manifolds manifolds

197 197

isis then then aa monomorphism, monomorphism, from from which which itit follows follows that that on on the the subspace subspace @(IV”) ilt(Nk) of of null-homotopic null-homotopic free free loops loops the the functionals functionals (2.55) (2.55) and and (2.56) (2.56) are are singlesinglevalued.) valued.) 3. 3. Consider Consider aa smooth smooth fiber fiber bundle bundle p:E--+B, p:E---+B, with with base base BB aa smooth smooth closed closed manifold manifold or or manifold-with-boundary manifold-with-boundary Wq. W q • SupSuppose pose we we are are given given aa closed closed (q (q ++ 1)-form l)-form Q il (dR (dil == 0) 0) on on the the total total space space E, E, and and some sorne single-valued single-valued functional functional 5’0 So {$} {'IjJ} defined defined on on the the cross-sections cross-sections $J 'IjJ :: BB -+ EE (p$ (p'IjJ = 1) 1) of of the the fiber fiber bundle. bundle. The The formula formula

1

sS == so d-‘(f-9, So {$I+ {'IjJ} + d- 1 (st), .ICBS++) (B,'M

(2.57)

of of the the general general type type of of (2.55) (2.55) and and (2.56), (2.56), defines defines aa functional, functional, in in general general multimultivalued, valued, on on the the space space CC of of cross-sections cross-sections of of the the bundle, bundle, with with prescribed prescribed values values on on the the boundary boundary dWq, âW q , ifif there there is is any any boundary. boundary. The The most most interesting interesting case case is the is that that where where EE = 5’4 xx G, G, G G aa Lie Lie group. group. Here Here the the cross-sections cross-sections are are the so-called 1x1 ---+ so-called “chiral "chiral fields” fields" g(z), g(x), zx EE IRq, IRq, where where g(z) g(x) -+ ---+ go go as as Ixl ---+ 00. 00. Note Note that simple Lie group G G there defined aa nonthat on on every every compact compact simple Lie group there can can always always be be defined nontrivial 3-form or 5-form (for if G G = SUn, SU,, n n >;:::: 3), 3), trivial two-sided two-sided invariant invariant 3-form or 5-form (for instance instance if 0 appropriate appropriate to to the the cases cases qq == 22 or or 44 respectively. respectively. 0

sq

We now formula (2.57) (2.57) (and (and so so also also (2.55) (2.55) We now explain explain in in more more detail detail how how the the formula and (2.56)) d e fi ne multi-valued functionals. Let 0 be a closed form (as above) and (2.56)) define multi-valued functionals. Let st be a closed form (as above) on space of E ---+ + B, and {Va} {Va} be be an an open open coyer cover on E, E, the the total total space of the the fibration fibration pp :: E B, and of E, with the following two properties: of E, with the following two properties: a) form 0st is is exact exact on on each V,: a) the the form each Va:

b) for any smooth cross-section 1c,of the fibration p : E -+ B there should b) for any smooth cross-section 'IjJ of the fibration p : E ---+ B there should exist an index Q such that q!~(B) c V,. exist an index 0: such that 'IjJ(B) C Va. Let C be the space of cross-sections of the fibration p : E --+ B. The forms Let C be the space of cross-sections of the fibration p : E ---+ B. The forms Qa determine “local functionals” Pa determine "local functionals" +-%a

= iB

~) @a

on the regions W, c C consisting of those cross-sections contained in V,. On on the regions W a C C consisting of those cross-sections contained in Va. On the regions of overlap we have the regions of overlap we have

1

Sy$,) - Sqb) = J-Bp (Pa -- P(3). @P).

S(al('IjJ) - S(f3)('IjJ) =

(B,1/1)

Since d( Qa - Gp) = 0 on V, n Vo, the difference S CL21 - SC@)is locally constant Since d( Pa P(3) = 0 on Va n Vf3, the difference s(a) S(f3) is locally constant on W, II Wp by Stokes’ Theorem, so that the collection { Sccl)} does determine W a n Wf3 by l-form Stokes'wTheorem, so that the collection {s(a)} does determine on a well-defined on C. a well-defined I-form won C.

198

Chapter Chapter 4. Smooth Smooth Manifolds Manifolds

If, the form If, in in this this context, context, the form w w is quantized, quantized, i.e. Le. [w] E E H1(C; H1(C; Z), Z), then then the the value the resulting (single-valued) map value at at '+1jJ of of the resulting (single-valued) map exp exp (27riS) {271'iS} : C C -+ --; S1, SI, is the cross-section The condition is called called the the Feynman Feynman amplitude amplitude of of the cross-section +. '1jJ. The condition that that the such fields the form form w w be be quantized quantized is is essential essential in in the the quantum quantum theory theory of of such fields with as with multi-valued multi-valued action action functionals. functionals. This This represents represents the the procedure procedure known known as “topological "topological quantization quantization of of coupling coupling constants” constants" in in Quantum Quantum Field Field Theory, Theory, as formulated by Deser-Jackiv-Templeton in the special special case case as formulated by Novikov Novikov (1981), (1981), Deser-Jackiv-Templeton in the of Chern-Simons (1982), and Witten (1983). of Chern-Simons (1982), and Witten (1983). In the case F +----+ SQ, the the form form fl R is In the case of of aa trivial trivial fibration fibration EE == Sq xx F is actually actually defined on the fiber F, the space C of cross-sections reduces to the space of defined on the fiber F, the space C of cross-sections reduees to the spaee of maps of the form w domain F, and and the the integrals integrals of the form w along along paths paths in in the the domain maps 5’4 +----+ F, function space C C coincide function space coincide with with integrals integrals of of the the form form fifl on on FF over over appropriate appropriate images under the Hurewicz homomorphism rq+l(F) + H,+l(F). images under the Hurewicz homomorphism 71'q+l(F) ----+ Hq+1(F). For general bundles E of R For general bundles E -+ B B (as (as above) above) the the association association of of integrals integrals of fl over cycles in E with contour integrals of w along paths in C, requires individover cycles in E with contour integrals of w along paths in C, requires individual topological analysis (not depending on the particparticual topological analysis (not usually usually very very difficult) difficult) depending on the ular bundle in question and the boundary conditions. We note in conclusion ular bundle in question and the boundary conditions. We note in conclusion that if the (2.55) or that if the functional functional (2.57) (2.57) (or (or (2.55) or (2.56)) (2.56)) is is actually actually multi-valued multi-valued (i.e. (i.e. not single-valued) then on the covering space C (cf. (2.50)) = SS not single-valued) then on the covering space C (cf. (2.50)) where where p;w PÎw = oS (pi C), the on all (Pl:: 2; C +----+ C), the functional functional S S takes takes on al! values values (-oo, (-00, co). (0). We now turn to the known topological estimates of the the number number of of critical critical We now turn to the known topological estimates of points of multi-valued functions and functionals. We begin with the points of multi-valued functions and functionals. We begin with the simplest simplest finite-dimensional finite-dimensional case. case. Let Let w w be be aa closed closed l-form I-form on on M” Mn of of irrationality irrationality degree degree zero. We then have a covering space A?l of M” with monodromy group Z: zero. We then have a covering spaee M of Mn with monodromy group Z:

sq

sq,

sq

p:M+Mn, P : M --; Mn,

p*w=dS, p*w dS,

where the generating monodromy transformation T : A? + A? satisfies where the generating monodromy transformation T : M ----+ M satisfies S(Tx) = S(z) + n, S(Tx) S(x) + 1\:,

n # 0. 1\:

i= o.

By suitably decomposing Mn and fi as CW-complexes, we can endow the By suitably decomposing Mn and M as CW-complexes, we can enclow the cell-chain complex of&l with the structure of a Z[T, T-‘l-module (see Chapter cell-chain complex of M with the structure of a Z[T, T- 1 ]-module (see Chapter 3, 55). Denote by K+ the completion of the ring Z[T, T-l] 1 in the direction of 3, §5). Denote by K+ the completion of the ring Z[T, T- ] in the direction of positive powers of T only, i.e. the ring of formal Laurent series of the form positive powers of T only, i.e. the ring of formai Laurent series of the form q=

q=

c

L

ajTj,

ajTj,

ajEZ. aj E

Z.

j~-N(q)

Consider the homology groups of the chain complex of 2 with coefficients in Consider the homology groups of the(inclusion) chain complex of M with---+ coefficients the ring Kf via the representation p : Z[T,T-l] Kf; thesein the ring K+ via the representation (inclusion) p : Z[T, T-1] these homology groups Hj (M; K+) may then be regarded as K+-modules.K+; Since homology groups Hj(M; K+) may then be regarded as K+ -modules. Sinee K+ is a principal ideal domain, the groups Hj(h;r; K+) (which we may write K+ is a principal ideal domain, the groups Hj(M; K+) (which we may write

$2. §2. The The homology homology theory theory of of smooth smooth manifolds. manifolds. Complex Complex manifolds manifolds

199

appropriately appropriately as as Hj(h;l; Hj(M; [w]) [w]) ttoo indicate indicate their their dependence dependence on on [w]) [w]) have have wellwelldefined the Betti Betti numbers) numbers) and and torsion torsion defined torsion-free torsion-free ranks ranks (the (the analogues analogues of of the ranks K+, which which we we shall shaH denote denote respectively respectively by by ranks over over K+, bJM”,

[w]) = rk Hi@&

K+),

e rank and qj(M n , [u]) [w]) (for (for th the rank of of the the torsion torsion submodule). submodule). Thus Thus bj bj ++ qj qj is is the the and qj(Mn, least number of generators of the K+-module Hj(i@; Kf). Provided the critleast number of generators of the K+ -module Hj(M; K+). Provided the critical are isolated the multi-valued multi-valued function function SSare isolated and and all aH non-degenerate, non-degenerate, ical points points of of the the following lower bounds for the number mi(5’) (or m,(w)) the following lower bounds for the number mi (S) (or mi (w)) of of critical critical points points of index i can be established (Novikov’s inequalities, from the early of index i can be established (Novikov's inequalities, from the early 1980s): 1980s): mi(w)

> bi(M”,

[w]) + qi(M”,

[w]) + qi-l(Mn,

[WI).

(2.58) (2.58)

More the mid mid 1980s) 1980s) Farber Farber has has shown shown by by constructing constructing aa suitable suitable More recently recently (in (in the map M” --+ S1, where rl(Mn) E Z, n 2 6, that these map Mn -----> SI, where ?Tl (Mn) Ë:' Z, n 2:: 6, that these inequalities inequalities are are strict strict in in general. general. IfIf the the irrationality irrationality degree degree kk - 11 22:: 1, 1, so so that that the the covering covering space space &? M has has monodromy group Z x . x Z with more than one factor, then the situation monodromy group Z x ... x Z with more than one factor, then the situation is The corresponding on the the set set of corresponding ring ring KL, K;};, depending depending on of is more more complicated. complicated. The ~2 : . : Q), is defined as the enlargement of periods (2.51): K = (~1 : (KI: K2 : ... : Kk), is defined as the enlargement of the the periods (2.51): K ring ring Z[Tl, . .T Tk, , T,-’ Z[T , ... T - 1 .... T[l] Tk-1l 1

k

1

of polynomials obtained obtained by Laurent series series qq in of Laurent Laurent polynomials by including including all all formal formaI Laurent in which the coefficient am,mz...mk of Tm1Tm2 Tmk is zero unless C Kjrnj > > which the coefficient amlm2 ... mk of Tr"lT:f'2 ... T;:k is zero unless L Kjmj -N(q) for some positive integer N(qf, an: furthe:, are such that for each pair -N(q) for some positive integer N(q), and further, are such that for each pair of numbers A < B there exist only finitely many non-zero coefficients aml...mk of numbers A < B there exist only finitely many non-zero coefficients a m1 ... mk satisfying A < C mjlcj < B. The homology groups Hj (G; K,$) should again satisfying A < L mjKj < B. The homology groups Hj(M; K;};) should again yield somehow somehow aa lower lower bound for the the number number of critical points points of of each each index. index. yield bound for of critical A variant of this type of completion is the ring (K,f)Stab’e where in addition it A variant of this type of completion is the ring (K;};) Stable where in addition it is required that the condition C Kjrnj > -N(q) hold not just for a particular is required that the condition L Kjmj > -N(q) hold not just for a particular point K. = (K~ : ~2 : . . : Q) but for some small neighbourhood of it. However point K = (KI: K2 : ... : /'i,k) but for sorne small neighbourhood of it. Howcver since the ring K,’ is no longer a principal ideal domain, the exact picture since the ring K;}; is no longer a principal ideal domain, the exact picture remains unclear.3 We conclude by emphasizing the crucial fact underlying remains unclear. 3 We conclude by emphasizing the crucial fact underlying the inequalities (2.58) and their possible generalizations to the cases k > 2, the inequalities (2.58) and their possible generalizations to the cases k 2:: 2, namely that the l-form w gives rise not just to a chain complex but a complex namely that the I-form w gives rise not just to a chain complex but a complex of K,f-modules with homotopy-invariant homology groups. of K;}; -modules with homotopy-invariant homology groups. We now turn to the analogous problem for functionals of types (2.55) or We now turn to the analogous problem for functionals of types (2.55) or (2.56) on the function spaces Q(Nn,~O, ~1) and fin+(Nn).n As we have seen spaces f?(Nn,XO,Xl) wc have (2.56) on the function these functionals determine closed l-forms w =and 6s. f?+(N However). As it turns out seen that w 8S. However it turns out that these functionals determine closed I-forms for the space Q(i’V, zo,~l) the homology groups analogous to those used to for the space f?(Nn, XO, Xl) the homology groups analogous to those us cd to 3Sirokav has proved that the ring (K, + ) Stable for a generic point n is an integral domain, that the hold ring (K;:t)Stable a generic K, is an integral domain, so 3Sirokav that the has Morseproved inequalities in this more for general case point as well. Recently Pazhitnov the Morse inequalities in this general RecentlyanPazhitnov so and that Ranicki have proved further hold results alongmore these lines case and as haveweil. developed analogue and Ranicki further results along theory”). these lines and have developed an analogue of Morse theoryhavein proved this case (“Morse-Novikov

of Morse theory in this case ("Morse-Novikov theory").

200 200

Chapter Chapter 4. 4. Smooth Smooth Manifolds Manifolds

obtain obtain the the inequalities inequalities (2.58) (2.58) are are all aU degenerate, degenerate, so so that that for for the the “problem "problem with with fixed fixed end-points,” end-points," i.e. Le. for for O(Nn, n(Nn, xc, Xo, xl), Xl), there there would would appear appear to to be be no no analogue analogue of of the the Morse Morse inequalities. inequalities. On On the the other other hand, hand, for for the the “periodic "periodie problem,” problem," i.e. Le. for for functionals functionals on on L?+(Nn), n+(Nn), the the situation situation isis different. different. On On a complete complete RiemanRiemannian like, for Nn with with Riemannian Riemannian metric metric gij, gij, a functional functionallike, for instance, instance, nian manifold manifold N” (2.39) (2.39) (but (but allowed allowed to to be be multi-valued) multi-valued) always always has has the the property property that that the the oneonepoint point loops loops $(N”) 'IjJ(N n ) C c Q+(Nn) n+(Nn) constitute constitute aa submanifold submanifold of of local local minimum minimum points the functional. functional. In In the the multi-valued multi-valued case case the the complete complete inverse inverse image image points of of the

in in the the appropriate appropriate covering covering space space &l M with with projection projection pp :: Ak M +-> x,M, always always consists of local minimum points of the single-valued function consists of local minimum points of the single-valued function fE lE on on M M defined defined by the pullback pullback p*w p*w = SLE. 8l E . The The inclusions inclusions by the k

N” N n +-> Nj”, N Jn ' iE(N,n) [E(Non ) == 0, 0'i”N,JJ tE N n == &"L.....t 6&n, . J "'mJm, m=l

(2.59) (2.59)

m=l

are all homotopic > 0, consider a are aB homotopie to to one one another another as as maps maps to to h;r. M. Assuming Assuming ~1 "'1 > 0, consider a map (homotopy) map (homotopy) @:NnxI--tF, F, I=[a,b], 1 [a, bJ,

where

where Jk @(Nn 12. @(N” rI>(N n xx {al) {a}) = Ni”o,...,O) N&, ... ,O) ccM, rI>(N n xx VI) {b}) == N;,,,...,,) Nn,o, .. ,0) ccM. One One has has 0 < rn& Essentially

mgxLP(@(N”

by deforming

x I))

1

< iE(N;l,,..,,o,)

the map @ on subsets

= 0.

(2.60)

(2.60)

of the form z x I where

z

Essentially by deforming the map rI> on subsets of the form z x 1 where z ranges over the cycles in H, (Nn), in the direction opposite to the gradient of ranges over the cycles in H*(N n ), in the direction opposite to the gradient of the functional iE, one obtains the following inequalities for the number of nonthe functionalZE , one obtains the following inequalities for the number of nondegenerate critical points in the region lE > 0 (assuming general position): degenerate critical points in the region lE > 0 (assuming general position): m.+l(lE) z

2 maxb!p)(Nn), P

where p ranges over all primes (Novikov).

z

These inequalities

(2.61)

(2.61 )

thus represent

where p ranges over aU primes (Novikov). These inequalities thus represent the analogues in the “periodic problem” of the Morse inequalities (2.8). It is of the analogues in thethat "periodie problem" (2.61) of the do Morse (2.8). Tt is of interest to observe the inequalities not inequalities involve the homology of interest to observe that the inequalities (2.61) do not involve the homology the function space L’+(Nn) itself. However because of certain analytic difficul-of the function space n+ (Nn) itself. However because of certain analytic difficulties these inequalities have not been established in more general situations4 ties these inequalities have not been established in more general situations. 4 4Novikov and Grinevich (1994) established these inequalities for nonzero magnetic fields and Grinevich (1994) established inequalities for nonzero magnetic on 4Novikov the flat a-torus and on surfaces with negative these curvature. A different technique was fields used on fiat 2-torus and onin surfaces curvature. A different by the Ginzburg, beginning the late with 1980snegative for studying periodic orbits technique in a magneticwas used field by Ginzburg, the late 1980s, for studying periodic in a magnetic with fixed periodbeginning (but not innecessarily fixed energy as in the former orbits approach). In particular,field with fixed toperiod not necessarily energy as in the and former approach). particular, he helped correct (butcertain mistakes infixed papers of Novikov Taimanov of theIn mid-1980s.

he helped to correct certain mistakes in papers of Novikov and Taimanov of the mid-198Gs.

$2. §2. The The homology homology theory theory ofof smooth smooth manifolds. manifolds. Complex Complex manifolds manifolds

201 201

The The inequality inequality (2.61) (2.61) applies applies also also to to functionals functionals of of type type (2.38) (2.38) when when 1l isis aa single-valued single-valued functional functional on on Qn+(Nn), [}+(Nn), but but “Arzela’s "Arzelà's principle” principle" isis violated, violated, in in the the sense sense that that there there exist exist loops loops y'Y E @(Nn) [}+(Nn) for for which which l(y) lh) << 0 (so (so that that the the metric metric constitutes constitutes aa non-positive non-positive analogue analogue of of a Finsler-type Finsler-type metric; metric; such such metrics metrics do do in in fact fact arise arise in in mathematical mathematical physics). physics). In In this this situation situation one one needs needs to to utilize utilize inin addition addition (analogously (analogously to to above) ab ove) segments segments of of the the form form

°

CD P:: [a, b] ---f n+(N”) such such that that Q(u) p(a) isis aa one-point one-point (i.e. (Le. constant) constant) loop, loop, and and G(b) p(b) = y'Y with with l(y) lh) << 0. O. One One has has the the following following recent recent theorem theorem of of Taimanov: Taimanov: IfIf for for aa single-valued single-valued functional functional of of Finsler Finsler type type y'Y EE KP(Nn) for which l(y) < 0, and [}+(Nn) for which lh) < 0, and 1l == 0 on on exists exists aa cross-section cross-section

°

$Q : N” --a R+(N”),

(see (see (L.3’7)) (2.37)) all aU constant constant

there there loops, loops,

exist exist then then

loops loops there there

plc, = 1,

such su ch that that a) a) l(h(Nn)) l( 1./Jl (Nn)) << 0, 0, and and b) the cross-section of constant b) the cross-section $1 1./Jl is is homotopic homotopie to to the the cross-section cross-section 1c, 1./J of constant loops lOOP8 via a homotopy Sp : N” x I ---f L’+(Nn). via a homotopy P: Nn xl -4 [}+(Nn). This thus establishes might be be called called the the "principle “principle of of transfer transfer This result result thus establishes what what might of the negative region of of values values of of the the functional." functional.” The The inequality inequality of cycles cycles to to the negative region (2.61) can be inferred from this result for such single-valued functionals 1, i.e. (2.61) can be inferred from this result for such single-valued functionals l, i.e. taking as positive New results of the Morse-Novikov taking negative negative as as well well as positive values. values. New results of the Morse-Novikov theory on extreme points of locally-regular multi-valued functionals on loop theory on extreme points of locally-regular multi-valued functionals on loop spaces (or on spacesof nonselfintersecting curves) may be found in the survey spaces (or on spaces of nonselfintersecting curves) may be found in the survey by I. A. Taimanov in: Russian Surveys, 1992, v. 2. 5 by 1. A. Taimanov in: Russian 8urveys, 1992, v. 2. 5 An interesting example (due to Polyakov and Vigman) of a multi-valued An interesting example (due ta Polyakov and Vigman) of a multi-valued functional arises in connection with the “Whitehead formula” for the Hopf functional arises in connection with the "Whitehead formula" for the Hopf invariants of the elements of 7rs(S2); we recall the construction of that formula: invariants of the elements of 1T3 (8 2 ); we recall the construction of that formula: Let f : S3 2 be a smooth map, and w a 2-form on S2 2 satisfying 3 --+ S2

Let

f :8

-->

8 be a smooth map, and w a 2-form on 8 satisfying

h2

w = 1. s s2W 1.

Let w be a l-form on S3 satisfying Let v be a I-form on 8 3 satisfying formula for the Hopf invariant of f formula for the Hopf invariant of f

(2.62)

(2.62)

dv = f*(w) = W say. Then the Whitehead dv = f*(w) = w say. Then the Whitehead is given by

is given by

‘By the end of the 1980s specialists in symplectic geometry (Floer, Hoffer, A. Salomon, end others) of the 1980s specialists in symplectic Hoffer, A. D. 5By McDuffthe and had created a Morse theory for geometry multi-valued (Floer, functionals on Salomon, free loop others) haddocreated a Morse theory for functionals D. McDuff spaces. Theseandfunctionals not have, in general, the multi-valued property of local regularity,on free i.e. loop the spaces. These of functionals do not in general, the property local regularity, Le. the Morse indices critical points mayhave, be infinite. (These ideas first ofappeared in Rabinovich’s Morse points may be infinite. ideas first in Rabinovich's work onindices periodic of critical orbits of Hamiltonian systems (These of the early 1980s.)appeared There are applications on periodic orbits Hamiltonian of the on early 19808.) There are applications work of Floer’s theory to the of moduli spaces ofsystems connections 3-dimensional spheres, where the of Floer's theory to the coincides moduli spaces of connections on 3-dimensional spheres, The wherelatest the multi-valued functional with the well-known Chern-Simon functional. multi-valued functional coincides with thethewell-known functional. developments in this theory also involve rings Kf Chern-Simon of Novikov, defined above. The latest

developments in this theory also involve the rings K+ of Novikov, defined above.

202

Chapter Chapter 4. Smooth Smooth Manifolds Manifolds

H{J,w}

r vl\w.

JS

(2.63) (2.63)

3

and and has has the the properties properties of: of: a) homotopy and homotopy invariance: invariance: SH/Sf 8H/8f == 0; and b) rigidity: SH/Sw = 0 (provided (2.62) b) rigidity: 8H/8w (provided (2.62) holds). holds). We now define a multi-valued functional (determined, We now define a multi-valued functional (determined, as as earlier, earlier, by by aa closed closed 2 + S2, l-form) on the space F of null-homotopic smooth maps f : S2 by 1-form) on the space F of null-homotopie smooth maps f : 8 - t 8 2 , by

8{J}=

r

JD3+

vl\w,

(2.64)

aD!

D!.

where extended from = 8Dt to the map map ff has has been been extended from S2 82 = to the the whole who le disc disc 0:. where the The furnished by The corresponding corresponding l-form 1-form on on FF is is then then furnished by 65’. 88. Beginning of Chen in the Beginning with with the the work work of Chen and and Sullivan Sullivan in the late late 1970s 1970s, various various hohomotopically invariant integrals generalizing (2.63) have been constructed. The motopically invariant integrals generalizing (2.63) have been constructed. The most of such most transparent transparent and and general general version version of such integrals integrals (defined (defined by by Novikov Novikov in the early 198Os), also involving the concept of “rigidity,” is important for in the early 1980s), also involving the concept of "rigidity," is important for investigating the associated multi-valued functionals and their quantization. investigating the associated multi-valued functionals and their quantization. We We shall shall now now describe describe this this integral. integral. Let A = Cz>0 Ai be skew-commutative differential differential algebra algebra over over aa field field lc k Let A = Li>O Ai be aa skew-commutative of characteristic 0, with A0 = k, and coboundary operator d : Ai + Aif’, of characteristi~ 0, with AO = k, and coboundary operator d : Ai - t Ai+!) satsatisfying or each each positive construct aa free HI(A) == 0. O. FFor positive integer integer qq we we construct free differential differential isfying H1(A) extension C,(A) of A, minimal with respect to the property extension Cq(A) of A, minimal with respect to the property @(C,(A))

= 0, J’ 24,

(2.65) (2.65)

as follows: Set Ci = A. Let {yja} b e a minimal set of multiplicative generators A. Let {Yja} be a minimal set of multiplicative generators as follows: Set for the cohomology ring in dimensions j < q. We now introduce corresponding for the cohomology ring in dimensions j ~ q. We now introduce corresponding free generators v~-I,~, and set free generators Vj-l,a, and set

cg

C4’ = A[. , ujpl+, . .I, dvj-l,, = yja. C~ = A[ ... )Vj-l,a, .. '], dVj-l,a Yja. Note that the inclusion A = C$’ + Ci induces the zero homomorphism of Note that the inclusion A induces the zero homomorphism of the cohomology groups in dimensions 5 q. By iterating this construction we the cohomology groups in dimensions ~ q. By iterating this construction we obtain a sequence of inclusions obtain a sequence of inclusions

CJ

cg

C~ c C~ c ... C Cq(A). We now define the (q + l)st homotopy group (relative to Ic) of the algebra A Webenow (q + l)st homotoPY group (relative to k) of the algebra A to the define vector the space Hq+‘(C’J(A)):

A

= C~

c

to be the vector space Hq+ 1 ( cq (A)):

(2.66) (2.66) If we take A = A*(AP), the algebra of Coo-forms on a simply-connected manIf we Mn, take then A Ait * can (Mn), algebra COO -forms a simply-connected ifold be the shown that of nq+l(A) @Iw on 2 rq+l(M) @IR. Hencemanthe ifold Mn, then it ofcan shown thatgroups 'Tf q+ 1 (A) ® IR ~ 'Tf q+ 1 (M) ® IR. Hence the infinite portions thebehomotopy of any simply-connected manifold infinite portions of the homotopy groups of any simply-connected manifold nq+dA)

@k = H”+l(C,(A)).

§3. Smooth Smooth manifolds manifolds and and homotopy homotopy theory theory $3.

203 203

M” Mn may may be be obtained obtained inin aa quite quite elementary elementary manner manner from from the the algebra algebra A*(Mn) A*(Mn) of forms on Mn (although the full verification of the above isomorphism of forms on Mn (although the full verification of the above isomorphism rerequires the more more elaborate elaborate Cartan-Serre Cartan-Serre apparatus apparatus of of algebraic algebraic topology). topology). This This quires the result Rham's thetheresult may may be be regarded regarded as as aa homotopy-theoretic homotopy-theoretic analogue analogue of of de de Rham’s orem orem (see (see above). ab ove ). Using Using (2.66) (2.66) we we can can now now construct construct the the desired desired homotopically homotopically invariant invariant integrals. integrals. Given Given any any smooth smooth map map FF :: Sqfl Bq+! xx IK IR -+ ~ Mn, Mn, we we obtain obtain aa natural natural homomorphism homomorphism E’ : C,(A)

--+ A*(Sq+’

x W),

A = A*(Mn),

cg

commuting and d, by by simply simply taking taking P F == F* F* on on Ci == A A == A*(Mn) A*(M n ) and commuting with with d, 1 extending using the fact that all closed forms of degree 5 q on Sq+l x Iw extending using the fact that an closed forms of degree :S q on Bq+ X IR are are exact. q+’ ( C q( A)) represented any element element of of H Hq+!(Cq(A)) represented by by aa cocycle cocycle 2 say, say, then then exact. IfIf zz isis any P;(Z) F(z) isis aa (q (q ++ l)-form l)-form on on Sqfl Bq+! xx R IR satisfying satisfying d@(Z) dF(z) = 0. o. We We now now define define our our integral by integral by

z

H(t)

H(ft, z) =

J

F(z),

(2.67)

S'I+lx{t}

where where

ft

Fls
Since the the form P(Z) is is closed closed we dH/dt = = 0, whence the homotopy Since form F(z) we have have dH/dt 0, whence the homotopy invariance of the integral (2.67) follows fairly quickly. The definition and inininvariance of the integral (2.67) follows fairly quickly. The definition and vestigation of “rigidity” in this context is somewhat more involved. vestigation of "rigidity" in this context is somewhat more involved.

$3. Smooth Smooth manifolds and homotopy theory. Framed Framed §3. manifolds and homotopy theory. manifolds. Bordisms. Thorn spaces. The Hirzebruch formulae. manifolds. Bordisms. Thorn spaces. The Hirzebruch formulae. Estimates of the the orders orders of of homotopy groups of of spheres. spheres. Estimates of homotopy groups Milnor’s example. The integral properties of cobordisms

Milnor's example. The integral properties of cobordisms

In this section we shall first consider some ideas, based on the geometry In this section we shan first consider sorne ideas, based on the geometry of manifolds, giving access by elementary means to certain information conof manifolds, giving access by elementary means to certain information concerning the homotopy classes of maps of manifolds (in cases other than those cerning the homotopy classes of maps of manifolds (in cases other than those examined in Chapter 3 where the homotopy groups were known to be trivial examined in Chapter 3 where the homotopy groups were known to be trivial for elementary reasons). In their further development these geometrical ideas for elementary reasons). In their further development these geometrical ideas become combined with the algebraic techniques described at the conclusion become combined with the algebraic techniques described at the conclusion of Chapter 3. of Chapter 3. We remind the reader that for smooth manifolds any continuous maps and We remind the reader that for smooth manifolds any continuous maps and homotopies can be approximated arbitrary closely by smooth maps coinciding homotopies cau bemap approximated smoothsomaps with the original wherever it arbitrary happenedclosely to be by smooth, that coinciding we always with the original map wherever it happened to be smooth, so that always assume all maps and homotopies of manifolds to be of smoothness we class C” to be of smoothness class C= assume al! maps and homotopies of manifolds if need be. if need be.

204 204

Xl1 0 FiJ ‘i

Chapter 4. 4. Smooth Smooth Manifolds Manifolds Chapter

"+I

X31

f-l (*)

f-y**)

f-y***)

0

+hJt&)

Fig. 4.15 4.15 The The simplest of of homotopy homotopy invariants invariants is the the degree degree of of a map between closed closed orientable orientable manifolds manifolds of of the the same same dimension: f : Mn

4

N”,

defined as as follows: Consider a generic point point y cobian JJff of f is then non-zero at each point of then each point f-‘(y) = (51,. ,xk}. We define f-l(y) = {Xl,"" xd.

(i.e. “regular”) "regular") in N”; Nn; the Jaof the complete inverse image image

k

deg = 5 sgnJf(zj). degff = LsgnJf(Xj).

(3.1)

j=l j=l

(Figure used to illustrate illustrate the verification verification that that the degree degree of f is is (Figure 4.15 can be used independent of the choice choice of the regular point point yy of Nn, assuming assuming that that Mn M” independent and Nn N” are are connected. Figures 4.16, 4.16, 4.17 4.17 exemplify exemplify the concept.) concept.) y

x desf=l ct.eg.f=1

degf=O d.eg. f =a

Fig. 4.16 4.16 Fig. The definition definition (3.1) (3.1) of of the the degree degree of of aa map map applies applies also also to to the the following following The situations: situations:

§3. Smooth Smooth manifolds manifolds and and homotopy homotopy theory theory $3.

205 205

ap = p-l), W n - 1 ,8Nn a) Maps Maps ff :: (Mn,Wn-‘) (Mn, W n - 1 ) --+ (Nn, p-l), V n - 1 ), (d&f” (8M n = w-1, V n - 1 ), _ u) (N”, n 1 n 1 where the boundary W”-l of M” is mapped to the boundary V”-l where the boundary W - of Mn to the boundary V - of of Nn. N n. n 1 is equal to its degree Here the degree of f restricted to the boundary W”-l Here the degree of f restricted to the boundary W equal to its degree on on the the interior interior of of M”. Mn. (Hence ifif ff defines defines aa diffeomorphism diffeomorphism between the the boundaries, boundaries, then then as as aa map between between the the interiors interiors we we shall shaH necessarily necessarily have have degf deg f == fl.) ±l.) 47L 411 00 - - - - - - - - - "

tk=Z) (k=Z)

--____ yo-yo+27i

Ya 0 sgn

X1 x1=+,

s~n X1

X2

+,

sgn sVn x+=-t, Xq.= +,

6671

syn

x2=

Xz

x3

XY

-,

sgn

-,

deg d.ey f=Z f=Z

2%

sgn

x3=

XJ

+

+

Fig. Fig. 4.17 4.17 b) M” -+ Nn N” of of open i.e. maps maps with with the the b) Proper Proper maps maps ff :: Mn open manifolds, manifolds, Le. property that the complete inverse image of any compact set is again compact. property that the complete inverse image of any compact set is again compact. Here is invariant F :: Mn M” X x I -+ Nn the degree degree is invariant under under homotopies homotopies F l N n that that are are Here the also proper maps. Note that if w is an n-form on N”, then one has the formula also proper maps. Note that if w is an n-form on Nn, then one has the formula /Mn = (degf) (deaf) / ( w. w. { f’w f*w = NTL

lM"

lN"

(3.2) (3.2)

c) isolated singular singular points za of q(z) on on aa manifold manifold Mn: M”: c) The The isolated points Xo of aa vector vector field field 1](x) V(Q) = 0. On a sphere SrP1 about ICYof sufficiently small radius E (defined 1](xo) = O. On a sphere Sr;-l about Xo of sufficiently smaU radius é (defined by Iz - ~01 = E) the vector field v( z will be non-vanishing and therefore the by lx - xol = é) the vector field 1](x)) will be non-vanishing and therefore the following Gauss map is well-defined: following Gauss map is weU-defined: v(x)

y-1

1](x) . Sn-1 b?(x)1 11](x)l'

-

_

p-1.

sn-1

c&

.

The degree of this map is called the index of the singular point x0. If the The degree of this map is caUed the index of the singular point xo. If the singular point is non-degenerate, i.e. singular point is non-degenerate, Le. 81]i det(~)l,, # 0,

det( -8J,)lxo::J 0, x

8 i then the index of the singular point is equal to sgn det(g)/,, . (Figure 4.18 then the index of the singular point is equal to sgn det( 81]J, ) Ixo. (Figure 4.18 x points of a vector showsthe possible types of isolated non-degenerate singular shows thea possible types of isolated non-degenerate singular points of a vector field on 2-dimensional manifold.) field on a 2-dimensional manifold.)

Chapter Chapter 4. 4. Smooth Smooth Manifolds Manifolds

206 206

For field on the sum of the For aa vector vector field on a closed closed manifold manifold M”, Mn, the sum of of the the indices indices of the singular characteristic x(Mn) singular points points isis equal equal to to the the Euler Euler characteristic X(Mn) (Poincare, (Poincaré, Hopf). Hopf).

Fig. 4.18 Fig. 4.18

For non-orientable For non~orientable manifolds manifolds the the degree degree of of aa map map is is defined defined only only modulo modulo 2. As noted above, for connected manifolds the degree is independent 2. As noted above, for connected manifolds the degree is independent of of the the choice of the point y E N”. The homotopy invariance of the degree of a choice of the point y E Nn. The homotopy invariance of the degree of a map map as follows. homotopy ff :: Mn Mn + N” Nn can can be be seen seen as follows. Let Let F F :: Mn Mn xX I1 +--;. Nn Nn be be any any homotopy and consider the complete inverse image of of of ff (F(TO) (F(x,O) = f(z)), f(x)), and consider the complete inverse image of aa point point vfi arbitrarily close to y, with the property that at every point of F-l&) the 1 arbitrarily close to y, with the property that at every point of F- (fJ) the map map F n, as as illustrated N” = S1. The F has has rank rank n, illustrated in in Figure Figure 4.19 4.19 in in the the case case M” Mn = = Nn = SI. The coincidence the degrees of the restrictions coincidence of of the degrees of the restrictions of of F F to to the the base base and and lid !id of of the the cylinder is clear in that figure, and the general case is similar. Hopf’s theorem

cylinder is clear in that figure, and the general case is similar. Hopf's theorem

Mnx?

MnXI

0 /----MnX

--.

.\

-!-

Cl

The complete inverse image F-‘(iiJ The complete inverse image p-l Ci)) is shown in bold is shown in bold

..

a /fjn=sq

63

Fig. 4.19 Fig. 4.19 (of the late 1920s) states that for a closed manifold Mn every map M” n --+ S” (of the late 1920s) states that for a closed manifold Mn every map M __, sn from Mn to the n-sphere is determined to within a homotopy by its degree, so from Mn to the n-sphere is detcrmincd to within a homotopy by its that the degree is a complete homotopy invariant of such maps. Hopf’sdegree, actualso that the degree is a complete homotopy invariant of such maps. Hopf's actual argument was along the lines of the above sketch of the proof of the homotopy argument was along the Hnes of the above sketch of the proof of the homotopy invariance of the degree of a map, except for his use of PL-manifolds, i.e. of invariance of the degreeofofthe a map, except Inforthehis more use ofgeneral PL-manifolds, suitable triangulations manifolds. situation i.e. of of a suitable triangulations of the manifolds. In the more general situation of a simplicial (i.e. PL- ) map f : Mn+k + Nn of PL-manifolds, the complete simplicial (i.e. P L- ) map f : Mn+k --;. N n of P L~manifolds, the complete

$3. Smooth Smooth manifolds manifolds and homotopy homotopy the theory §3. ory

207

inverse image image of of the the interior interior of of a simplex simplex (Jn 8’ of of Nn N” of of largest largest dimension dimension will will inverse have the the form form have f-l (Int on) = Int 6-n x W”, (3.3) (3.3) n k for some some simplex simplex ô6-” of of Mn+k Mn+lc and and submanifold W”. . Following Following Hopf's Hopf’s idea, idea, it it submanifold W for is then then natural to seek seek invariants invariants of of the map ff using using such such complete complete preimages. preimages. is natural to the map Consider spheres: Consider aa map map between between spheres: f : snfk

-

S”.

Reverting to to our our use of points points in position, we we consider consider the complete Reverting use of in general general position, the complete inverse images of two distinct such points yi, y2 E S”; these will be submaniinverse images of two distinct such points Yl, Y2 E sn; these will be submanifolds folds yi E S”, i = 1,2, w,” = f-‘(Vi),

wt, wz” wt, W~ c C sn+k. sn+k. In the case case where two su submanifolds be linked linked In the where nn is is even even and and kk = n-l, n-1, these these two bmanifolds may may be in Sn+k = s2n-1 l with “linking coefficient” {WI”-‘, W;-‘} (see Chapter in sn+k == s2n-l, with "lin king coefficient" {W~-l, Wr- } (see Chapter 11 for the of two two loops For the definition definition ‘of of the the linking lin king coefficient coefficient of loops in in aa manifold). manifold). For for odd n the linking coefficient { Wp-‘, W;-‘} is zero. For even n (the case we l odd n the linking coefficient {W~-\ Wr- } is zero. For even n (the case we are considering) the linking coefficient is independent of the pair of regular are considering) the linking coefficient is independent of the pair of regular points y2 E S”, and is invariant under homotopies of f; it is called the Yl, Y2 E sn, and is invariant under homotopies of f; it is called the points yi, Hopf invariant Hopf invariant of of f:f: h(f)

h(f)

= {W;“-‘, { W ln -

l

WF-l}, wn-l}

'2

f : !P-1

,

n = 2q, 2q, n

---t SQ.

(3.4) (3.4)

The Hopf invariant determines the Hopf homomorphism: The Hopf invariant determines the Hopf homomorphism: r4q-1(sZq)

-

z,

f +-+ h(f),

from which it follows that the groups 7rdq-i(S2’J) are infinite. It was shown from which it follows that the groups 7f4q_l(S2q) are infinite. It was shown later (in the early 1950s) by Serre that all other 7rj(Sn) are finite; as we saw later (in the early 1950s) by Serre that aU other 7fj(sn) are finite; as we saw earlier (in Chapter 3, 57) the methods used here go considerably beyond the earlier (in Chapter 3, §7) the methods used here go considerably beyond the purely geometrical. purely geometrical. The switch from P&maps to smooth ones, leads to the possibility of obThe switch frommethod P L-maps to smooth ones, leads to the possibility obtaining a standard for investigating homotopy groups of spheres of using taining a standard method for investigating homotopy groups of spheres using complete preimages of points (Pontryagin, late 1930s). In this approach, one thisa approach, one completethe preimages (Pontryagin, late =1930s). exploits fact thatof points the inverse image W” f-‘(y) In of regular point k the fact that the inverse image W 1 (y) of a regular point exploits y E S” under a map f : Snfk ---+ S”, is a framed manifold, i.e. comes with a y E sn under a map f : sn+k --lo sn, is a framed manifold, Le. comes with a non-degenerate field vn(z) of vector n-frames determined by f as follows: Let vn(x) ofvector n-frames determined by f as follows: ULet non-degenerate field ,& be local co-ordinates in some sufficiently small neighbourhood of 41,... CPl, ... ,CPn be local co-ordinates in some sufficiently small neighbourhood of y with y as origin and with linearly independent gradients (with respect to Uthe y with y as origin and with linearly independent gradients (with respect to the standard local co-ordinates on Sn); then Lhe complete preimage Wk k = f-‘(y) standard local co-ordinates o~ sn); then ~he complete preim~ge W = f-l(y) is given by the n equations 41 = 0,. . ,q5, = 0, where the & are lifts of the is given by the n equations CPl 0, ... , CPn 0, where the cp; are lifts of the

208

Chapter Manifolds Chapter 4. Smooth Smooth Manifolds

$i gradients grad linearly independent on W Wk,k , and i will will be linearly independent on so determine a field of n-frames on Wk. so determine of n-frames Wk. Similarly, given aa homotopy --+ sn, S”, the complete inverse Similarly, homotopy F : Sn+k sn+k Xx Il --> image p-l F-‘(y)(y) of a regular point y E 5’” is a manifold-with-boundary V”+l C c of regular point E sn is a manifold-~with~boundary Vk+l Sn+li (whose boundary disjoint parts, one one in the base base and and sn+k xx Il (whose boundary falls into two disjoint one in the lid of the cylinder ,YP+~ I) equipped, much same manner one in the !id of the cylinder sn+k Xx 1) mu ch in the same as before, with to sorne some Riemannian Riemannian metric) metric) field field vn(x) Y,(Z) of of as before, with aa normal normal (relative (relative to k 1 lies in Sn+k x {CL} vector n-frames in Sn+lc x I, which on the boundary of Vk+l vector n-frames in sn+k xI, which on the boundary of V + lies in sn+k X {a} or Sn+k {b}. (Note (Note that may be be assumed assumed that that Vk+ V’“+l1 approaches approaches the the lid lid or sn+k xx {b}. that itit may k 1 and base of Sn+lc x 1 transversely.) The pair (V”+l , vn) c Snfk x 1 is called and base of sn+k xl transversely.) The pair (V + , lin) C sn+k xl is called a (or framed bordism). Two framed manifolds manifolds framed manifold-with-boundary manifold-with-boundary (or framed bordism). Two framed a framed are said said to equivalent ifif there framed bordism bordism y$‘d?),lI~l»), (W~, (W,“, lI~2») dZ’ ) are (Wf, to be be equivalent there is is aa framed k 1 v,), v”+’ 1 c snfk x I, whose respective boundary components are the k k+l, (V + , Vn), V + C sn+k X I, whose respective boundary components are the given framed of such such an equivalence may may be be called called aa given framed manifolds. manifolds. A A realization realization of an equivalence framed bordism (or framed cobordism). It is quite straightforward to show that framed bordism (or framed cobordism). It is quite straightforward to show that the equivalence classes of closed framed manifolds are in natural one-to-one the equivalence classes of closed framed manifolds are in natural one-ta-one correspondence with the elements elements of group 7f r,+k(Sn). of the the group correspondence with the n+k(sn). Before proceeding, we note that the groups are called called stable stable if if Before proceeding, we note that the groups rn+k(Sn) 7fn+k(sn) are kk < < n 1, since for these k they are independent of n; this is immediate from n - 1, since for these k they are independent of n; this is immediate from the suspension isomorphism the suspension isomorphism c : ?Tn+k(Sn) ---t 7Tn+k+l(Sn+‘),

k < n - 1.

More generally, generally, for for any any CW CW-complex construction yields yields More -complex K K the the suspension suspension construction an isomorphism in homology always (see Chapter 3, §3), and an isomorphism an isomorphism in homology always (see Chapter 3, §3), and an isomorphism C: n,+k(K) r,+k+i(CK) for k < n - 1, provided the complex K is E : 7fn+k(K) 3 7f n +k+l(EK) for k < n - 1, provided the complex K is (n - 1)-connected, i.e. if xj(K) = 0 for j 5 n - 1. (This was shown in the late (n 1)-connected, Le. if 7fj(K) 0 for j sn 1. (This was shawn in the late 1930sby Freudenthal, who was also the first to compute the groups 7rn+i (P).) 1930s by Freudenthal, who was also the first ta compute the groups 7fn +l (sn).) The following facts about framed manifolds are not difficult to establish: The following facts about framed manifolds are not difficult to establish: 1. Each framed manifold (W”, vn) of dimension k > 0 is equivalent to a con1. Each framed manifold (W k, lin) of dimension k > 0 is equivalent to a connected framed manifold. (Figure 4.20 illustrates the construction of a framed nected framed manifold. (Figure 4.20 illustrates the construction of a framed bordism (Vkfl, vn) with dV kk+l = Wf U W,” U Wt, realizing an equivalence U W;: U W~, realizing an equivalence bordism (Vk+l, lin) with 8V +l between the disjoint union (Wt U Wt, ~~l~:,~;) and the connected framed and the connected framed between the disjoint union (Wf U W~, IlnlwkuWk) 1 2 manifold (W,“, vnlwb).) manifold (W:,vnlw!,).) 2. In the case k = 1, the only stable invariant of a framed manifold is the 2. In the case k 1, the only stable invariant of a framed manifold is the homotopy classof its field of frames, which is given by a map V, : Si + SO,. homotopy class of its field offrames, which is given by a map lin : SI --> SOn' Since ni(SO,) % Z/2 for n > 3, it follows that nIT,+i(Sn) Z Z/2 for n > 3. ~ 7l,/2 for n?: 3, it follows that 7f n+l(sn) ~ 7l,/2 for n?: 3. Since 7fl(SOn) 3. In the case k = 2 there is again a single stable invariant of each equiv8. Inclass the ca.."le k = 2manifolds there is again single invariant ofiseach equivalence of framed (W’,2 av,), butstable its construction somewhat (W alence class of framed manifolds , lin), but its construction is somewhat more involved. Suppose n > 4. By statement 1 above we may assumewithout n ?: c4. llP+2 By statement a..ssume without moreofinvolved. Suppose loss generality that W2 (or W2 c1 above S”+2) we is may a connected surface 2 C JRn+2 (or W 2 C sn+2) is a connected surface 10ss of generality thatW (of genus g say) with normal field vn(lc). Each class z E H1(W2; Z/2) may normal(i.e. field lIn(X). Each classC z2 ES1 H 1in(W2; may (of represented genus 9 say)bywith be a simple embedded ) circle W27l,/2) c Rn+2. be represented by a simple (Le. embedded ) circle C ~ SI in W2 C JRn+2.

Wf

$3. Smooth Smooth manifolds §3. manifolds and and homotopy homotopy theory theory

209

Writing for the normal vector W2,2 , we field of of normal normal Writing n(z) n (x) for the normal vector field field to ta C C in in W we have have a field frames (n(z), vn(z)) to C n+2 . By above there there is a lin + 1 (x) = (n(x), IIn (X)) to C in in Iw jRn+2. By statement statement 2 above frames v,+i(z) unique invariant G(z) say (Q(z) of the the framed manifold (C,lI (C, ~,+i(z)) p(z) say (p(z) EE Z/2) Z/2) of framed manifold unique invariant n +1(X)) called the Arffunction. One has following "Arf “Arf identity”: called the Arf function. One has the the following identity": @(zi + ~2) = @(zi) + @(z2) + zi 0 z2

(modulo

2),

(3.5)

where zi 0o Z2 z2 is intersection index index of and Z2' ~2. In terms of of any any is the the intersection of the the cycles cycles zi Z1 and In terms where Z1 canonical of cycles zi, . . . , Zg, zgr Vh, ‘~1, .... .. ,, wg, i.e. satisfying canonical basis basis of cycles Z1,"" w g , i.e. satisfying zi 0 zj = w, 0 wj = 0,

z,owj

=&,

the full Arf defined by the full Arf invariant invariant is is defined by 9

L

@ = -g @(Zi)@(W& P= P(Zi)P(Wi),

@EE Z/2. z/2. P

(3.6)

i=l

i=1

(It It turns turns out out that 2 4 the (It is is independent independent of of the the (canonical) (canonical) basis.) basis.) It that for for nn 2: 4 the quantity Cp is the only stable invariant of framed manifolds (W2, vn), whence quantity P is the only stable invariant of framed manifolds (W2, lin), whence itit follows follows that 4,7r,+2 (Sn) E Z/2 (Pontryagin, in in the 1940s). that for for nn > 2: 4,7T'n+2(sn) Z/2 (Pontryagin, the late late 1940s).

Fig. 4.20

Fig. 4.20

4. There is a complicated theory (devised by Rohlin in the early 1950s) 4. There is a complicated theory (devised by Rohlin in the early 1950s) leading to the determination of the groups r,+s(Sn). As the first (non-trivial) leading to the determination of the groups 7T'n+3(sn). As the first (non-trivial) step, it is shown that every non-trivial framed manifold (W3,3 w,) is bordant bordant step, it is shown that every non-trivial framed manifold (W , vn ) is (since to a framed sphere S3 c Wnf3, situated in standard fashion in IWnf3 in to a framed sphere S3 C jRn+3, situated in standard fashion in jRn+3 (since in the stable range every disposition of S3 in 1Wn+3is equivalent to the standard the stable of S3 ininjRn+3 is equivalent one). It is range shownevery next disposition that the latitude defining a frame ta onthe S3standard E TWn+3 one). It is shown next that the latitude in defining a frame on S3 E jRn+3 determines a homomorphism (in fact an epimorphism) determines a homomorphism (in fact an epimorphism) J : 7r3(SO,) + 7r,+3(Sn). (3.7) (3.7) Since 7rs(SO,) g Z for 7~> 4, it now follows that the stable group 7rIT,+s(Sn) Since 7T'3(SOn) Z for n >reveals 4, it now is cyclic. Further~ analysis that follows that the stable group 7T'n+3(sn) is cyclic. Further analysis reveals that

Chapter 4. Smooth Manifolds Chapter 4. Smooth Manifolds

210 210 nnf3(Sn)

= Z/24

for n > 4,

n7(S4) 2 z @z/12,

“s(S3)

2 z/12.

These results to Rohlin's Rohlin’s theorem to the effect that that for for a These results are are closely closely linked linked to theorem to the effect closed manifold M4 with the property that M4 \ {ze} is parallelizable, the closed manifold lvf4 with the property that lvf4 \ {xo} paraUelizable, the first Pontryagin class pi(iU4) is divisible by 48. (Recall that characteristic first Pontryagin class Pl (lvf4) is divisible by 48. (RecaU that charactcristic classes integral cohomology cohomology groups.) groups.) the integral classes are are elements elements of of the As (3.7) one that the the equivalence equivalence classes of framed framed As a generalization generalization of of (3.7) one has has that classes of k-spheres determine in in 7r 7r,+k(Sn) of the the "Whitehead “Whitehead ho(Sk, vn) lin) determine n+k (sn) the the image image of hok-spheres (S”, momorphism” (see Chapter Chapter 3, 3, §8) $8) momorphism" (see J : 7rk(SOn)

+

7r,+k(Sn).

It is this image cyclic, and and that that its its order order (an (an important important It is known known that that this image is is finite finite cyclic, quantity) is in the case Ic = 4s 1 expressible in terms of the Bernoulli quantity) is in the case k = 48 - 1 expressible in terms of the Bernoulli numbers. (Milnor and Kervaire produced a lower bound for the order in the the numbers. (Milnor and Kervaire produced a lower bound for the order in late 195Os, and Adams an upper bound in the mid-1960s.) late 1950s, and Adams an upper bound in the mid-1960s.) 00 These results limit of of what can be be discovered about the the hohoThese results represent represent the the limit what can discovered about motopy groups of spheres using geometrical methods; further progress on this motopy groups of spheres using geometrieal methodsi further progress on this problem use of of the the methods methods of homotopy theory theory (see (see Chapter Chapter problem involves involves heavy heavy use of homotopy 3, §7). §7). 3, The theory theory of of bordisms and cobordisms cobordisms may may be regarded as as developing developing The bordisms and be regarded naturally out of the above-described geometrical ideas, providing bridge naturally out of the above-described geometrical ideas, providing aa bridge to algebraic techniques for settling questions about smooth manifolds. The to algebraic techniques for settling questions about smooth manifolds. The classical bordism groups (or cobordism groups) are defined as follows: Two classieal bordism groups (or cobordism groups) are defined as follows: Two smooth closed manifolds IV;, IV! of the same dimension Ic, are said to be smooth closed manifolds wt, w~ of the same dimension k, are said to be equivalent (or cobordant) if their sum (i.e. disjoint union) is diffeomorphic to equivalent (or cobordant) if their sum (i.e. disjoint union) is diffeomorphic to the boundary of a compact smooth manifold Nk+‘: the boundary of a compact smooth manifold Nk+l:

For each Ic 2 0 the equivalence (or cobordism) classes of closed k-manifolds For eaeh k 2: 0 the equivalence (or eobordism) classes of closed k-manifolds form a group L’p under the operation defined in the obvious way in terms of form a group under the operation defined in the obvious way in terms of the sum operation on manifolds. The direct sum the sum operation on manifolds. The direct sum

nI?

(3.8) (3.8) then forms a graded ring,6 the classical bordism ring, with multiplicative opthen farms a graded ring,6 the classical bordism ring, with multiplicative operation determined by the direct product of manifolds. It is in fact an algebra eration over Z/2.determined by the direct product of manifolds. It is in fact an algebra over 71,/2. Examples. 1. L@ Z Z/2 (the scalars). Examples. 1. n(? ~ 71,/2 (the sealars). 2. np = 0 2. O.

DP

‘There

is an another

standard

notation,

namely

6There is an another standard notation, namely

‘32,, for the bordism

ring

02.

sn., for the bordism ring n?

33. Smooth Smooth manifolds theory §3. manifolds and homotopy homotopy theory

211 211

n?

(This follows the classification for closed closed sursur3. Lrf Z~ Z/2. 71.,/2. (This follows from from the classification theorem theorem for faces. IwP2 represents element, i.e. IwP2 is not the faces. Note Note that that lRP2 represents the the non-zero non-zero clement, Le. lRP2 not the boundary compact 3-manifold. Since x(IwP2) -1, this this follows follows from from boundary of of any any compact 3-manifold. Since X(lRP2) == -1, the following general general fact: fact: the following IfIf a a closed dW”+l,n+!! then then its Euler-Poincard closed manifold manifold is is aa bounday: boundary: Mn Mn = 8W its Euler-Poincaré characteristic is even. characteristic is even. This consequenceof This is is in in turn turn a a consequence of the the equality equality XP n+l LJMTL w”+l)

= 2x(Wn+l)

- x(h!P),

wn+ln+! is n+ 1 U&p” n+ 1 (obtained where W n+l the "double" “double” of of W Wn+’ (obtained by taking two UM" W is the by taking two where W n 1 and identifying their boundaries), and the fact that any oddcopies of Wn+l copies of W + and identifying their boundaries), and the fact that any odd-dimensional clo8ed closed manifold manifold has has zero zero Euler-Poincaré Euler-Poincare characteristic). dimensional characteristic).

nr

00

3. 03” == 0 (Rohlin, early 1950s). 3. 0 (Rohlin, early 1950s).

Restricting manifolds, we obtain analogously Restricting ourselves ourselves to to oriented oriented manifolds, we obtain analogously the the oriorientable bordism groups f&f” and the orientable bordism ring entable bordism groups and the orientable bordism ring

ntO

Q,“O = L x nk'io. @“. n;'iO k>O

k?;.O

Z Z (the scalars). Examples. 1. 1. 6~‘:” n~o ~ Z (the scalars). 2. L?f” = 0 = Q$” (obviously). 2. ntO = 0 = n10 (obviously). Examples.

3. 6’:” 3.

early 1950s). n;o == 00 (Rohlin, (Rohlin, early 19508). 4. n1° L?p ” Z (Rohlin and Thorn, both in the early 1950s). Note that the 4. ~ 71., (Rohlin and Thom, both in the early 1950s). Note that the 0 bordism class [CP”] 2 generates 6’2”. bordism class [CP ] generates n1°. 0 If a closed manifold M” is a boundary, i.e. represents zero in the bordism If a closed manifold Mn is a boundary, i.e. represents zero in the bordism group 6’2, then all of its “Stiefel-Whitney numbers” (see below) are zero. In then all of its "Stiefel-Whitney numbers" (see below) are zero. In group the late 1940s Pontryagin established the following basic facts: If a closed 4kthe late 1940s Pontryagin established the following basic facts: If a closed 4kdimensional manifold M4” is the boundary of an orientable manifold then in dimensional manifold M 4k is the boundary of an orientable manifold then in addition its “Pontryagin numbers” are all zero. The Stiefel-Whitney characaddition its "Pontryagin numbers" are aH zero. The Stiefel- Whitney characteristic numbers of Mn are the values taken on the fundamental class [Mn] by teristic numbers of Mn are the values taken on the fundamental class [Mn] by homogeneous polynomials over Z/2 of degree n in the Stiefel-Whitney charhomogeneous polynomials over 71.,/2 of degree n in the Stiefel-Whitney characteristic classes. (To obtain the Pontryagin characteristic numbers of M4k acteristic classes. (To obtain the Pontryagin characteristic numbers of M 4 k one evaluates homogeneous polynomials over Z of degree lc in the Pontryagin one evaluates homogeneous polynomials over Z of degree k in the Pontryagin characteristic classes.) characteristic classes.) for computing the bordism groups is based on their The main method The main method for computing the bordism groups based on their connexion with homotopy theory; we shall now describe thisis connexion. connexion with homotopy theory; we shaH now describe this connexion. Definition 3.1 The Thorn space TV of a vector bundle 7 (or any fibre Definition 3.1IlF)The Thom space space T'I) ofE/AE, a vector bundle fibre bundle with fibre is the quotient where AE 17is (or the any subspace n ) is the quotient space E / dE, where dE is the subspace bundle with fibre lR consisting of the vectors in the fibres of length > 1. (Provided the base B of v vectors in the fibres length :2: compactification 1. (Provided the of base of 17 consisting of the is a compact space one may take the ofone-point theBtotal is a compact space one may take the one-point compactification of the total

n;;,

212 212

Chapter Chapter 4. Smooth Smooth Manifolds Manifolds

space E, as space spaee E, as the the resulting resulting space spaee is is homotopically homotopically equivalent equivalent to the the Thorn Thom space TT/.) TT)

°

For each jj > For each ::: 0 there there is is aa natural natural isomorphism, isomorphism, the the Thorn Thom isomorphism isomorphism (valid in modulo (valid modulo 2-cohomology 2-cohomology ifif the bundle is not not orientable): orientable): q6 rp :: W(B) Hj(B) --f ---> Hj+“(T,), Hj+n(TT/)'

jj::: > 0, 0,

(3.9) (3.9) 4(z) %, WI rp(z) = zZ· Un, Un = 4(l), rp(l),

where classof fibres, the where 4(l) rp(l) is is the the cohomology cohomology class of T,, TT/ taking taking the the value value 11 on on the the fibres, the Thorn class. One has the following formula of Thorn (from the early 1950s) Thom class. One has the following formula of Thom (from the early 1950s) relating classesWj wj E the StiefelLWhitney Stiefel-Whitney characteristic characteristic classes E Hj(I3;Z/2) Hj (Bj 71,/2) of of the the relating the vector bundle Q and the Steenrod squares Sqi: vector bundle Tl and the Steenrod squares Sq): = ti-lSq’C$(l).

Wj(v)

(3.10) (3.10)

Applied this Applied to to the the tangent tangent bundle bundle r(Mn) r(Mn) of of a a smooth smooth closed closed manifold manifold M”, Mn, this formula leads to a proof of the homotopy invariance of the Stiefel-Whitney formula leads to a pro of of the homotopy invariance of the Stiefel-Whitney classes the fact classes wi(Mn) wj(M n ) of of Mn. Mn. By By exploiting exploiting the fact that that the the tangent tangent bundle bundle r(Mn) r(Mn) is isomorphic to the normal bundle to the diagonal A c M” x in conis isomorphic to the normal bundle to the diagonal .1 c Mn X M”, Mn, in conjunction obtain explicit for Thom's formula formula (3.10), (3.10), one one may may obtain explicit expressions expressions for junction with with Thorn’s the classes wi(h/m) in terms of the Steenrod squares Sq’ acting on the cothe classes W j (Mn) in terms of the Steenrod squares S qi acting on the cohomology u’s formulae, of the 1950s). To homology ring ring H*(Mn; H*(Mnj Z/2) 71,/2) (W (Wu's formulae, of the early early 1950s). To obtain obtain Wu’s formulae one solves the equation (for each i = 1, . . , [n/2]) Wu's formulae one solves the equation (for each i 1,··· , [n/2]) Sqyz)

= z lci,

(3.11)

(3.11)

over the group II”-“(M”; (for all .zE Hnpz(Mn; Z/2)). over the group Hn-i(M''; Z/2) Z/2) for for the the unknowns unknowns K+ Iii (for ail z E Hn-i(M'\ Z/2)). We note that the equation (3.11) h as a solution since any homomorphism We note that the equation (3.11) has a solution sinee any homomorphism H”-“(Ad”; Z/2) --+ Z/2 (and so in particular the homomorphism Hn-i(Mn; Z/2) ---> Z/2 (and so in particular the homomorphism Sq” : fP-“(Ad”;

Z/2)

b

H”(M”;

Z/2)

Hn(M"j Z/2)

2 Z/2) 2;!

Z/2)

is obtained via Poincare duality by multiplying the elements of Hnpi by a is obtained via Poincaré duality by multiplying the elements of Hn-i by a single i-dimensional class. Wu’s formulae have the following form: single i-dimensional class. Wu's formulae have the following form:

Sq(w) = 6

Sq(W)

=

li,

w=l+w1+...+w,, W = 1 + Wl + ... K=l++l+...+F+/z],

li

1 + iiI

+ Wn ,

(3.12) (3.12)

+ ... + li[n/2J,

sq = 1 + sq1 + sq2 + .

Sq = 1 + Sq1 + Sq2 + .... These formulae yield, for instance, the result that all orientable 33manifolds These formulae yield, for instance, the result that al! orientable 3-manifolds are parallelizable. are parallelizable.

53. §3. Smooth Smooth manifolds manifolds and and homotopy homotopy theory theory

213 213

IfIf aa manifold manifold Mn Mn is is almost almost parallelizable, parallelizable, i.e. ifif Mn Mn with with one point point removed is is parallelizable, parallelizable, then then wi Wi = 0 for for i << n, n, and and from from this this together together with with (3.12) one may that ifif nn == 41c 4k then then in in H2”(Mn;Z/2) H2k(Mn; 7l/2) every every square is zero: zero: may deduce that .z~ z2 = 00 (modulo (modulo 2). This This implies implies that that the the scalar scalar square square (z2, [M4”]) [M4k]) = (z, (z, Z) z) is even even for for every every zE E H2k(M4k; H 2k (M4k; Z)/(Torsion). 7l)/(Torsion). Since Sinee the the quadratic quadratic form form (z, z) is even and unimodular, unimodular, itit then then follows that that 4k the signature r(M4”) is divisible by 8. The signature T(M~~) the signature T(M ) divisible by The T(M 4k ) is defined as as the the difference difference in in the the numbers of of positive positive and negative negative squares squares in the the (di(diagonalized) agonalized) quadratic quadratic form form (5, (x, y) y) given given by by the the intersection intersection index index of elements Z, y E H2k(M4k; Z) (or equivalently by (z, y) = (my, [M4k]), J: x, y E H2 k(M4k; 7l) (or equivalently by (x, y) (xy, [M4k]), x = Drc, Dx, yy = = Dy), Dy), although this quadratic form is non-degenerate only for closed manifolds. although this quadratic form is non-degenerate only for closed manifolds. ItIt can can be be shown shown that that for for closed closed manifolds manifolds the the signature signature is is invariant invariant under under 4k 4k 1 bordisms. if M4” = 6’W4k+1 for some orientable manifold-with-boundary bordisms: if M = âW + for sorne orientable manifold-with-boundary W4k+1, W 4k+1, then then T(M~~) T(M 4k ) = 0. O. (To (To prove prove this this one one first first observes observes that that ifif two two 21C2k4k 4k cycles in M4” are null-homologous in 4k+1, then their intersection W cycles in .M are null-homologous in W +l, then their intersection index index is is zero, zero, so so that that (( ,, )) is is zero zero on on Im lm i* i* where where i* i* is is the the inclusion inclusion homomorhomomor4 --+ H2k(M4k; Q). The desired conclusion 2k phism H2k ( W4k+1; Q) phism H (w4k+ 1; Q) ---> H2k (M k; Q). The desired conclusion then then follows follows by Im i* Q) is is exactly half the that the the dimension dimension of of lm i* cC H2”(M4’; H 2k (M4k; Q) exactly half the by showing showing that om this conclusions follow; for dimension dimension of of Hzk(M4”; H 2k (M4k; Q).) Q).) Fr From this fact fact important important conclusions follow; for So is isomorphic to Z with generator [CP”]. In $1 instance that the group f14 instance that the group n!f° is isomorphic to 7l with generator [CP2]. In §1 above we saw that the first Pontryagin class pi of @P2 2 satisfies above we saw that the first Pontryagin class Pl of CP satisfies (1 +pi) (1 + + 3u2 H*(@P2; 7l), Z), (1 + Pl) = (1 + u?)~ v?)3 == 11 + 3u2 EE H*(CP2; where of H2(CP2; It follows 3. Binee Since where u u is is a a generator generator of H 2(Cp2; Z) 7l) ”~ Z. 7l. It follows that that pi(@P2) Pl(CP2) = 3. +C.p2) = 1 (th is can be obtained directly from knowledge of the structure of T(CP2) 1 (this can be obtained directly from knowledge of the structure of H* (CP2; (@P2; Q)), and sinee since [CP [@P2] generates a:‘, that 2] generates H* Q)), and n!f°, itit follows follows that

1 +f4) = '$1, 3 P1 ,

(3.13)

T(M 4 )

(3.13)

for every orientable closed 4-dimensional manifold M4 (since 37(M4) - pl is for every orientable closed 4-dimensional manifold M 4 (since 3T(M 4) - Pl is an identically zero linear form on L’,so 2 Z). (This result is due to Rohlin and an identically zero linear form on n!f° ~ 7l). (This result is due to Rohlin and Thorn in the early 1950s.) Generalization of the formula (3.13) to higher diThom in the early 1950s.) Generalization of the formula (3.13) ta higher dimensions requires computation of the groups LQs” @Q; we shall be considering mensions requires computation of the groups nfo 0Q; we shaH be considering this problem in the sequel. this problem in the sequel. The signature T(M~~) of a manifold-with-boundary M4k has the property The signature r(M4k) of a manifold-with-boundary M 4k has the property of additiuity: if M’,4k , M$kk are two manifolds-with-boundary and VI, V2 &’ k of additivity: if Mt , Mi are two manifolds-with-boundary and VI, V2 ~ V are diffeomorphic connected components of their respective boundaries, V are diffeomorphic connected components of their respective boundaries, then for the manifold obtained by identifying VI and Vz one has (Rohlin and then for the manifold obtained by identifying VI and V2 one has (Rohlin and Novikov in the late 1960s): Novikov in the late 1960s): (3.14) T(M;” k u M”“) = T(M;“) + T(M;“).

T(Mt U Mik)

T(M{k)

+ T(Mr).

(3.14)

We remark that this additivity property is possessedalso by the PoincarkWe that thisofadditivity propertymanifolds is possessed also Poincaré Eulerremark characteristic even-dimensional M2”, but by by the no other char2k , but M by no other charEuler characteristic of even-dimensional manifolds acteristic. acteristic.

214 214

Chapter 4. Smooth Smooth Manifolds Manifolds Chapter

For any any group group G c c Om, O,, the the Thom Thorn space space TI) Tn of of aa universal universal vector bundle bundle For vector with fibre IRm IL?” and and base base BG BG will will be be denoted denoted by by M(G). M(G). More generally generally any with fibre More any representation G ---> --+ Om, O,, not not necessarily necessarily one~to~one, one-to-one, determines Thorn p :: G determines aa Thom representation space; for for instance instance the the double double caver cover space;

p : Spin,

+

SO,,

determines the Thorn space space M(Spin M(Spin,). m ). If If G G C C SOm, SO,, one one speaks speaks of of the the orioridetermines the Thom entable Thorn space M(G). In the case G = (1) c SO,, we obtain the entable Thom space M(G). In the case G {1} C SOm, we obtain the m-sphere: A&(G) ” S”. By (3.9) we have the following Thorn isomorphisms: m sphere: Al(G) sm. By (3.9) we have the following Thom isomorphisms: 4 a)

HJ(BG; Z) ---> + Hm+j H”+j(M(G); Z), 4>q5 :: Hj (BG; Z) (M(G); Z), qq.z) = = 2Z· Uu,,m , 4>(z)

b) b)

urn U m

G c SOm! SO,, Ge

= 4>(1); 4(l);

+ Hm+j(M(G);Z/2), H”+j(M(G); Z/2), 4 : Hj(BG;Z/2) Hj(BG; Z/2) ---> 4>:

4>(z) =

Z· U m ,

1t m

G c Om! O,, Ge

= 4>(1).

Classical concentrates (as have seen) seen) on on the the cases cases G G = = Classical cobordism cobordism theory theory concentrates (as we we have Further developments of that theory are based on the following 0, or SO,. Dm or SOm. Further developments of that theory are based on the following facts (established (established by Thorn): facts by Thom): (i) An (i) An element element zt ifold Wnem Mn, ifold wn~m cC Mn, that

by a closed submanE H,-m(Mn;Z/2) as re p resentable precisely if there exists a map f : M” + M(Om) M(0,) such precisely if there exists a map f : Mn ---> such

E Hn~m(Mn; Z/2) is representable by a closed subman-

that

f*(u,) j*(U m ) = Dz Dz EE H”(M”; Hm(Mn; Z/2), Z/2),

where u, = 4(l) is the Thorn class (see 3.9) of the Thorn space M(O,), where U m = 4>(1) is the Thom class (see 3.9) of the Thom space M(Om), and and D is the Poincare-duality D is the Poincaré-duality isomorphism. isomorphism. (ii) An element z E H,-,(Mn; Z), Mn orientable, is representable (ii) An element z E Hn_m(MrI; Z), Mn orientable, is representable by an orientable submanifold W”-”n m c Mn n if and only if there is a map c Jl.1 if and only if there is a map by an orientable submanifold w f : Mn + M(S0,) such that

f : Mn

--->

M(SOm) su ch that

f*(um) = Dz E Hm(Mn;iZ). (iii) The bordism groups L?,+ o , 0:” are canonically isomorphic to the corre(iii) The bordism groups are canonically isomorphic to the corresponding “stable” homotopy groups of M(0,) and M(S0,):

af, aro

sponding "stable" homotopy groups of M(Om) and M(SOm): 6’:

2 r~~+k(M(o~)),

k < m - 1, (3.15)

fq0 (Thorn,

early

N - ~,+,+(M(SW),

k < m - 1,

(3.15)

1950s).

(Thom, carly 1950s).

Note that if we replace SO, by the trivial group (1) in (3.15) we obif we replace SOm by by Pontryagin the trivial group (3.15) between we obtain Note the that isomorphism established (in the {1} late in1930s)

tain the isomorphism established by Pontryagin (in the late 1930s) between

53. Smooth Smooth manifolds manifolds and and homotopy homotopy theory theory §3.

215 215

nt

and the the group group QLr of of bordisms bordisms of of framed framed manimanim+k (sm) and ~m+k(M({ll)) == 7rriT,+k(Sm) folds, kIc < m -- 1. 1. folds, Statements (i), (i), (ii) (“)u and (iii) (iii) above above were were established established using using the the "theorem “theorem Statements transversal-regularity” (or t-t-“regularity”), according to to which which any any map map on transversal-regularity" (or "regularity") , according M” --> + X Xqq of smooth smooth manifolds manifolds can can be approximated approximated arbitrary arbitrary closely closely 9g : A1n by a map map f that that is is "t-regular" “t-regular” on any any prescribed prescribed submanifold submanifold yq-m Y’Jern cc XX4.q . by on (A map map f is t-regula'r t-regular on yq-m Y 4- m if for for every every point point y of of the the complete complete inverse inverse (A image f-l(Yq-“) c Mn, M”, the the image image of the the tangent tangent space space to Mn M” at y (under (under f-1 (yq-m) C image the induced induced map map of of tangent tangent spaces) spaces) together together with with the the tangent tangent space space to to yq-m Y’J+’ the at f(y), spans the the whole whole of of the the tangent tangent space space to to xX4q at at f(y).) f(y).) This This condition condition at f(y), spans ensures that that the the complete complete preimage preimage f-1 f-‘(Yqem) is aa smooth smooth submanifold submanifold of of (yq-m) is ensures n of dimension M” n m: A1 of dimension n - m: 7r m+k (M ( {1} ))

f-yyq-")

= p-m

c M”,

moreover such such that that the the restriction restriction map map iV”-” --+ yq-m YqPm extends extends naturally naturally to to moreover Nn-m --> a map of the respective normal bundles with fibre IR”. a map of the respective normal bundles with fibre IRm. To obtain obtain the cycles by by submanifolds submanifolds (Statements (Statements (i) (i) and and 1'0 the representability representability of of cycles q (ii)) one takes M(0,) (or M(S0,)) in the role of X4, and BO, c M(0,) (ii)) one takes M(Om) (or M(SOm)) in the role of X , and BO m C M(Om) (or in the the role of Yq-“; submanifold representing representing aa (or BSO BSO c C M(S0,)) M(SOm)) in role of yq-m; the the submanifold cycle is then fel(BOm) (or f-‘(BSO,)). To obtain Statement (iii) one one takes takes 1 cycle is then (BO m ) (or f-1(BSOm)). 1'0 obtain Statement (iii) XQ = 9. q x sq.

r

Remark. M(0,) and with the the natural natural maps maps Rema'rk. The The spaces spaces M(On) and M(S0,) M(SOn) together together with M(G)

-

M(On+l),

M(SOn)

-

M(SOn+l)

define (as objects objects in in the by MO MO and and 8pectm (as the appropriate appropriate category), category), denoted denoted by define spectra MS0 respectively. The “homotopy groups of the spectra” MO and MS0 M SO respectively. The "homotopy groups of the spectra" MO and M SO coincide with the above “stable” homotopy groups: coincide with the above "stable" homotopy groups: 7rkMo = 7rm+k(M(o,)), k < m - 1, ?rkMSO = xIT,+k(M(SO,)), k < m - 1. 7rkMSO 7f m+k(M(SOm)), k < m 1. The spectra MU, MSU, MSpin and MSp are defined in a similar manner. The spectra MU, MSU, MSpin and MSp are defined in a similar manner.

o A natural geometric interpretation of the homotopy groups of M(G), G c A natural geometric interpretation of the homotopy groups of M(G), Ge O,, is as follows: Define a G-manifold to be the normal bundle V, on a 0m' is as follows: Define aG-manifold to be the normal bundle l/m on a submanifold Wk of Sm+k (or of KC”+“) equipped with a G-structure. Two submanifold Wk of sm+k (or of IRm+k) equipped with aG-structure. Two G-manifolds WY, Wzm C Smfk are said to be G-bordant if there exists a c sm+k are said to be G-bordant if there exists a G-manifolds W 1 , G-manifold Vm+l c Sm+k x I, I = [a, b], normal to the boundaries: G-manifold vm+1 C sm+k X l, 1 = [a, b], normal to the boundaries: av=wl”lJw,m,

wr

aV = W1Uw:r\

Wlm c Sn+k x {u} , Wzmc Sn+k x {b} W 1 c Sn+k X {a}, c Sn+k x {b} (analogously to the framed bordism introduced earlier). (analogously to the framed bordism introduced earlier).

wr

216

Chapter Chapter 4. Smooth Smooth Manifolds Manifolds

ItIt turns turns out out that that the the equivalence equivalence classes classes of of G-manifolds G-manifolds are are in in natunatural one-to-one correspondence correspondence with with the the elements elements of of the the homotopy homotopy group group raI one&o-one lrm+l (M( G)). nm+l(M(G)). By By restricting restricting attention attention to to manifolds manifolds M” Mk cC Sm+lc sm+k having having normal normal bundles bundles with prescribed structure group, one is led to the following bordism with prescribed structure group, one is led to the following bordism rings rings corresponding to the the classical classical Lie Lie groups: groups: corresponding to L?:, t h e cclassical 1assical bordism early 1950s); 1950s); n,?, fi’,s” n~o ((the bordism rings, rings, introduced introduced in in the the early

ntr, nt

or d isms of introduced Of’, “4f’ % G = (1) ~ rq+*(Sn); lrq+n(sn); G {1} (b (bordisms of framed framed manifolds, manifolds, introduced in connexion with the study of the homotopy groups of spheres, the in connexion with the study of the homotopy groups of spheres, in in the late 1930s); late 1930s);

nf, n:

flu4’ flnsu. Uk , SUk SUk 4 u ’; Uk, grows); groups);

n:

Qp; p ; Spk SPk Q:pin;

c

C

c

C

SGzk and special S02k (the (the unitary unitary and special unitary unitary bordism bordism

So& S04k (the (the symplectic symplectic bordism bordism groups); groups);

p : Spink +

SGk spinor bordism groups). SOk (the (the spinor bordism groups).

n;?

Investigations into of the the G-bordism rings 0: for for G G == U, U, the structure structure of G-bordism rings Investigations into the SU, Sp, Spin, and their further generalizations, were initiated around 1960 SU, Sp, Spin, and their further generalizations, were initiated around 1960 (by For the the classical groups G, G-structure of the the (by Milnor Milnor and and Novikov). Novikov). For classical groups G, the the G-structure of stable normal bundle of a manifold is determined by the G-structure of the stable normal bundle of a manifold is determined by the G-structure of the tangent bundle, G == U U for for instance, instance, aa complex (or quasicomplex) quasi-complex) tangent bundle, so so that that for for G complex (or manifold M2” of real dimension 21c, determines a bordism class of Q$: 2k manifold M of real dimension 2k, determines a bordism class of

n:K:

A generic polynomial equation of degree n + 1 (in n + 1 homogeneous A generic polynomial equation of degree n + 1 (in n + 1 homogeneous co-ordinates) in cCP an SU-manifold M2n-2 representing a co-ordinates) defines de fines in iCpn an SU-manifold M2n-2 representing a class class [M2+2] 2n 2 E @f2.

[M

-

j

E

complex complex bordism bordism

n!J:!_2'

In the case n = 3 one obtains in this way the Kummer surface K4 (of 4 real In the case n = 3 one obtains in this way the Kummer surface K4 (of 4 real dimensions), which is almost parallelizable (i.e. the complement of a point is dimensions), which is almost parallelizable (Le. the complement of a point is parallelizable), simply-connected, and has the following invariants: parallelizable), simply-connected, and has the following invariants: x(K4)

= c2(K4)

X(K4) = c2(K 4 ) 7(K4) = 16, T(K 4 ) = 16,

= 24,

= 24,

pl(K4) 4 = 48, Pl(K ) 48,

c1(K4) = 0. cl(K 4 ) O.

(3.16) (3.16)

As noted in connexion with the definition of a characteristic class of a As noted in connexion with the definition of a characteristic class of a G-bundle, given in $1 above, each such characteristic class is uniquely deterabof oveH* , each such characteristic class is uniquely deterG-bundle, given in §1 $J mined by an element (BG), where BG is the base of the universal mined by an element ' l jJ of H*(BG), where BG is the base of the universal G-bundle. For a closed manifold M” k the characteristic numbers of Mk (alM the characteristic numbers (alG-bundle. For a closed manifold ready mentioned above) are then the values +([Mk]) ($J E H”(BG))of Mktaken ready mentioned above) are then the values 'ljJ([M k ]) ('ljJ E Hk(BG)) taken

$3. Smooth Smooth manifolds §3. manifolds and and homotopy homotopy theory theory

217

by of dimension dimension kk on fundamental homology class by characteristic characteristic classes classes of on the the fundamental homology class [iMk]. For G c SO, we have (as noted above) the Thorn isomorphism [Mk]. For G C SOm we have (as noted above) the Thom isomorphism q5: H”(BG;

Q) ” H”+“(M(Gm); f#l(z) 4>(z)

Q),

um .

== Z· z u,.

On by the Cartan-Serre theory (see (see Chapter Chapter 3, 3, §7), §7), for < On the the other other hand hand by the Cartan-Serre theory for kIc < m determines an an isomorphism isomorphism m -- 11 the the Hurewicz Hurewicz homomorphism homomorphism determines rm+k(M(G))

@Q ” H”+“(M(G);

Q).

From these two isomorphisms we conclude that that the the characteristic characteristic numbers From these two isomorphisms we conclude numbers provide collection of linear forms forms on the homotopy homotopy groups groups provide a a complete complete collection of linear on the

rm+k(M(G)) 8 Q. The numbers may viewed as as invariants invariants of G-bordism classes classes The characteristic characteristic numbers may be be viewed of G-bordism for any G c SO, provided k < m 1. It follows that the G-bordism classes for any G C SOm provided k < m - 1. It follows that the G-bordism classes of 6’: are determined up to torsion by the characteristic numbers. of are determined up to torsion by the characteristic numbers. As of Statement Statement (ii) (ii) of of Thorn’s theorem, itit can can be be shown shown that As aa corollary corollary of Thom's theorem, that for G = SO,, given any cycle z E H,-,(Mn;Z),n there is an integer X # 0 for G = SOm, given any cycle Z E Hn_m(M ;Z), there is an integer À 1= 0 such may be represented by a submanifold (this in fact such that that the the element element XZ Àz may be represented by a submanifold (this in fact holds even in nonstable dimensions). holds even in non-stable dimensions). For the simplest Lie groups (i.e. of small dimension) the Thorn spacesM(G) For the simplest Lie groups (Le. of small dimension) the Thom spaces M (G) are as follows: are as follows:

n;:

G = SOI

= {I},

{1},

M(SO~)

M(SOl)

= ~1;

G = O1 = Z/2, G = 01 = Z/2,

M(01) = lu=J; M(Ol) = Il~.poo;

G=S02=S1=U1, G = S02 SI

Ul ,

M(SO2)

M(S02)

= M(U1) = CP”. M(Ul) = Cpoo.

We seefrom this that the spacesM(SOi), M(Oi) and hl(SOz) = M(Ui) are We see from this that the spaces M(SOt), M(Ol) and M(S02) M(Ud are respectively the Eilenberg-MacLane spaces K(Z, l), K(Z/2,1) and K(Z, 2). the Eilenberg-MacLane spaces K(Z, 1), K(Z/2, 1) and K(Z,2). respectively It follows that all cycles in H,_1(M”;;Z),H,_1(M”;2/2) and H,-z(Mn; Z) It follows that aU cycles in Hn_ l (Mn;Z),Hn_ l (Mn;Z/2) and Hn_ 2 (M n ;Z) are realizable as submanifolds. are realizable as submanifolds. The ring OB” @Q has as generators the bordism classes[@Pzi] (the groups n!O trivial 0Q hasforasj generators bordism classes [CP2i] groups of0The @ring Q being # 4k). Asthe a linear basis for Q,&’ @ (the Q one may niO 0 Q being trivial for j 1= 4k). As a linear basis for nf2 0 Q one may take the set of bordism classesof the form take the set of bordism classes of the form

The Pontryagin numbers provide a complete set of invariants for The Pontryagin numbersr(M4”) provide a complete setinvariant; of invariants OB” @Q. The signature is also a bordism in factforr 4k ) is also a bordism invariant; in fact T n!O 0 Q. The signature T(M homomorphism 0:’ -+ Z with value 1 on the generators [@P2i]: homomorphism n!O ---+ Z with value 1 on the generators [CP2i]:

the ring the is a ring ring is a ring

218 218

Chapter Smooth Manifolds Chapter 4. Smooth Manifolds T(ikqk)

T(Mp)

= T(M,4” x iq), (3.17) (3.17)

7(@P2y

= 1.

How the signature signature r(M4’“) p ressed in terms of of the the Pontryagin Pontryagin numbers numbers How is is the T(M 4k ) ex expressed in terms of M4”? i.e. on group @O, Thorn-Rohlin formula of M4k? For For k == 1, Le. on the the group [li D, we we have have the the Thom-Rohlin formula $11, mentioned above. From From the formulae given in §1 $1 above above in in the the course course T == !pI, mentioned above. the formulae given in of our characteristic classes, one obtains obtains the following formulae formulae of our exposition exposition of of characteristic classes, one the following for the Pontryagin of the the manifolds @P4 and and CP2 @P2 xx @P2: for the Pontryagin classes classes of manifolds CP4 CP2; For For

@P4, Cp4,

p = = 1 1 +p1 P + Pl

+p2 P2 = = (1 (1 fU2)5,'u 2)5,

+

+

where ‘1~ is aa generator of H2(@P4; Z) ~ 2 Z; Z; hence where u is generator of H2(CP4; Z) hence pl = 5u2, Pl 5u 2 ,

4 p2 10u4, Pz = 10u ,

pi = pi

4 25~~. 25u .

For CPf (IX’: xx CP:}, (IX;, p == 11 ++ p1 = (1 + tLi)3(1 UT)“( 1 + + u~)3, $)“, For P Pl ++ p2 P2 = (1 + where ui generates H2(@Pf; Z); hence 2 where Ui generates H (CP?; Z); hence p; = pi

18&& 18uiu~,

p2 9uTu;. P2 = 9tLiu~.

The numbers for these manifolds manifolds are are shown shown in in the the following following table: table: The Pontryagin Pontryagin numbers for these

@P4 Cp4

@P2 xx Cp2 @P2 Cp2

7T

P? pt

P2 P2

11

25 25

10

1 1

18 18

99

10

Hence we infer the following expression for the signature r in terms of the Hence we infer the following expression for the signature T in terms of the Pontryagin numbers (on the group 62,““): Pontryagin numbers (on the group [li D ): 7 = ;(~Pz 1

T =

(3.18) (3.18)

- ~32

45 (7p2 - Pl)'

The general formula of this type, given by Hirzebruch in the mid-1950s is as The general formula of this type, given by Hirzebruch in the mid-1950s, is as follows: follows: (3.19) +f4k) = (-h(Pl, . ,Pk)r [M4”]), (3.19) where the polynomial Lk(pr,. . ,pk) is given by the homogeneous part of where the polynomial Lk(Pl"" ,Pk) is given by the homogeneous part of degree k of the series degree k of the series L = 1 + L1 + L2 + ‘. . + Lk + ‘. = fiN --!% Ui L = 1 + LI + L 2 + ... + Lk + ... = i=l -ui ,’ tanh tanh tLi i=l

II

N >> k, N» k,

expressed as a polynomial in the first k elementary symmetric functions expressed as a polynomial in the first k elementary symmetric functions

Smooth manifolds homotopy theory §3. Smooth manifolds and and homotopy theory Pj

=

up,

c

219 219

Uf>

il<...
This is a particular 51 with This particular case case of of the general method (noted (noted in § 1 in connexion connexion with discussion of obtaining multiplicative the discussion of characteristic characteristic classes classes given there) there) of obtaining multiplicative expressions in the Pontryagin charateristic classes classesby using a single single formal Pontryagin charateristic by using formaI expressions power series series Q(u) = 11 + + Q1U Q1u22 ++ Q2u4 Qgu4 f.. . . Q(u) = + .... For the Hirzebruch Hirzebruch formula formula above one takes the formal formaI series series

L(u2)== & tanhu'

L( 2)

(3.20) (3.20)

U

U

The values by any Q defined The values taken taken on on the the (CP” cpm by any multiplicative multiplicative characteristic characteristic Q defined in terms of the Pontryagin or Chern numbers on L?*sO @Q or fly respectively, in terms of the Pontryagin or Chern numbers on n~"o ® Q or n;' respectively, determine seriesQ(u) If one takes the the formaI formal series (a "generic “generic the series Q( u) completely: completely: If one takes series (a determine the series” for Q) the multiplicative multiplicative characteristic characteristic Q) series" for the L?Q(u)

=

c

Q ((=‘“> 2

QO

with = u, u, the general formula with formal formaI inverse inverse gGl(u): gQl(u): gg’(gQ(u)) gQl(gQ(u)) == the following following general formula (due to Novikov in the late 1960s) can be established (related the theory (d ue to Novikov in the late 1960s) can be established (related to to the theory of formal $9): of formaI groups groups - see see Chapter Chapter 3, 3, §9):

u Q(u)

(3.21 )

In the signature T, r, the the Hirzebruch Lk(p1,. . .pk) are In the case case of of the the signature Hirzebruch polynomials polynomials Lk(pl," 'Pk) are determined by the signature of the complex projective spaces: determined by the signature of the complex projective spaces: 7(@P2k) = 1,

T(cP2k+‘)

= 0.

We genusT, determined the Todd Todd genus T, determined We now now define define one one more more important important invariant, invariant, the by its values on the CP”: by its values on the Cpk: T(@P”)

= 1,

j 2 0,

and by the generic series ST(u): and by the generic series gT(U):

un + = - log(1 ST(U) = c -Un+l gT(U) = L --1 = -log(1 1

n>O + l - n n+ n_O >

-1 !?T 1

gr T(U) = &, u

T(u)

1 - e- u

'

u),

u),

= 1 - exp(-u), = 1 - exp( -u),

T(Mn) = (n 1 -“;-u, 1LMnl). j 1 _ e- l1j , [Mn]).

(JI J

(3.22)

(3.22)

220 220

Chapter Manifolds Chapter 4.4. Smooth Smooth Manifolds

By of eomplex complex algebraic By using using certain eertain general general categorical eategorieal properties properties of algebraie vavarieties and holomorphic fiber bundles, one can deduce from the Hirzebrueh rieties and holomorphie fiber bundles, one ean deduee from the Hirzebruch signature formula the Riemann-Roth-Hirzebruch Riemann-Roch-Hirzebruch theorem, theorem, as as follows. follows. signature formula (3.19) (3.19) the Let c be a holomorphic vector bundle over a complex algebraic variety Mn Let ç be a holomorphie vector bundle over a eomplex algebraie variety Mn (of the sheaf sheaf of (of complex eomplex dimension dimension n), n), and and denote denote by by 3~ Fç the of germs germs of of holomorphic holomorphie cross-sections of cross-sections of E. ç. Define Define a characteristic characteristic xX by by Hj(M”;3,). ç ). 2.)-l)j dim dimHj(Mn;F

x(Mn,E) X(Mn,ç) == c(-l)j j>O j"20

This invariant This invariant x( X( M”, Mn, <) ç) can can be be shown shown to to be be given given by by the the following following RiemannRiemannRoth-Hirzebruch Roch-Hirzebruch formula: formula:

x(M”,O ((ch ç)' 5) T(Mn), X(Mn,Ç) = = ((ch T(M n ), [M”I), [Mn]),

(3.23) (3.23)

where ch <ç is is the in (3.22), where ch the Chern Chern character eharacter of of I,ç, and and T(AP) T(M n ) isis as as in (3.22), with with nn = 2k. 2k. In the case where the vector bundle < is one-dimensional and trivial, one In the case where the vector bundle ç is one-dimensional and trivial, One obtains Todd genus, the “holomorphic obtains the the Todd genus, known known also also as as the "holomorphie Euler Euler characteristic” characteristic" of the variety AP. In the classical situation (the case of a complex of the variety Mn. In the classical situation (the ease of a complex curve curve Ml) Ml) aa holomorphic vector bundle [ is determined by the equivalence class of divisors holomorphie vector bundle ç is determined by the equivalence class of divisors D = C nixi, where E M1 and the ni are integers, so that the sheaf 3~ is D = L nixi, where xi Xi E Ml and the ni are integers, so that the sheaf :Fç is denoted instead by 3~. denoted instead by :FD. The The group group H”(M1; HO (Ml; 3~) :FD) then then consists consists of of the the functions functions having no poles outside the divisor D, and satisfying having no poles outside the divisor D, and satisfying

(f) D 2? 0, (1) + +D 0, where (f) = c mkyk, mk 2 0 is the divisor of the poles of the function f. where (1) = L mkYk, mk :::: 0 is the divisor of the poles of the function f. The classical Riemann-Roth theorem asserts that The classical Riemann-Roch theorem asserts that dim H”(M1; 3D) - dim H1(M1; 3D) = n(D) - g + 1, n = Cnj, n =

L

nj,

g = genus of the curve Ml.

9 = genus of the curve Ml.

This theorem may be cast in a different form by means of the equality This theorem may be cast in a different form by means of the equality dim H1(M1;

3D) = dim H1(M1;

3K-D),

where K is a divisor of the zeros and the poles of meromorphic forms. where K is a divisor of the zeros and the poles of meromorphic forms. The Todd genus (see (3.22) above) turns out, for reasons to be given beThe Todd genus (see (3.22) above) turns out, for reasons to be given below, to be integral-valued on the whole bordism group fl&. In the late 1950s low, to be integral-valued on the whole bordism group In the late 1950s Hirzebruch and Atiyah proved several “integrality theorems”, useful in many Hirzebrueh and Atiyah proved several "integrality theorems" in many calculations. The most important of these is as follows: From , useful the above forcalculations. The most important of these is as follows: From the above formula T = & u for the Todd genera we have mula T for the Todd genera we have 1 e- U

nSc.

T(u) = e”12&. T(u) = eU/2 . uj2 smh uj2

§3. Smooth Smooth manifolds and theory $3. manifolds and homotopy homotopy theory

221 221

Set Set

A J+~) u 2 = = * u/2 ( ) sinh sinh u/2 u/2

=

(3.24) (3.24)

e -“‘“T(u).

The characteristic determined (as (as described series: The characteristic determined described above) ab ove ) by by this this series: A(pl,

,pk, . .) =

Al=-5

Pl 24’ 24'

Al + . . + Ak + . . . ,

1 +

(3.25) (3.25)

A2= -&(-4~2

+ %:),

is out that ~ri(M~“)4k ) = = 0, 0, ~~(44~“) is called called the the A-genus. A-genus. ItIt turns turns out that provided provided wl(M w2(M 4k ) == 0, 0, integer for all k, k, and for kk odd. odd. It It follows follows the (Ak, [M4k]) [M4”]) is the number number (Ak' is an an integer for aH and even even for that Ak defines defines an an integer-valued integer-valued linear linear form form on that for for all all k, k, Ak on the the group group L$r, .of spinor bordisms. (Note that a manifold admits a spinor structure if and only spinor bordisms. (Note that a manifold admits a spinor structure if and only ifif itit is orientable (wi = 0) and also wz = 0. ) is orientable (Wl = 0) and also W2 O.) To give further theorems we we need need to to recall recall To give further applications applications of of these these integrality integrality theorems one of the most important consequences of Bott periodicity of the homotopy one of the most important consequences of Bott periodicity of the homotopy 7r- i (U) (which (which classify classify oriented and complex’bundles over groups 71;--i(SO), 1l'j-l(U) oriented and complex'bundles over groups 1l'j_l(SO), the sphere Sj). The generator of the group 7r4k(SO) corresponds to a “basic” the sphere Sj). The generator of the group 1l'4k(SO) corresponds to a "basic" Chern character (see Chapter Chapter 3, 3, §8) $8) oriented 174k over over S4” S4k with with Chern character (see oriented vector vector bundle bundle v4” = @p, where (p, [S4”]) = 1 and 4k ch 77 ch 17 ak/-L, where (/-L, [S4k]) 1 and

attnof

ak =

11 {1 22

for for for for

even even odd odd

k 2 2, k '2:2, k 2 1.

k?:l.

The Chern character of the “basic” U-bundle U-bundle 17c qc over over S4k S4k is The Chern char acter of the corresponding corresponding "basic" is ch 77~= CL.The Chern and Pontryagin classes of rl@ are as follows: ch 17c /-L. The Chern and Pontryagin classes of 17C are as follows: pk(rl@) = ak . (2k l)!P, Pk(17c) = ak . (2k -- 1)!/-L,

ck(q@) = Uk (k - l)!/L.

Ck(17c) (Note (Note Chern Chern

ak . (k

(3.26) (3.26)

1)!/-L.

that the formulas (3.26) follow from the inductive definition of the that the formulas (3.26) follow from the inductive definition of the character: character: ck = (k - I)! chk +

f(Cl,

. . ,ck-I).)

Remark. If Mn is an almost parallelizable manifold with stable normal Remark. If Mn is an almost parallelizable manifold with stable normal bundle V,JJ, then it is not difficult to show that there exists a “normal” map bundle VM, then it is not difficult to show that there exists a "normal" map f : Mn ---+ S” of degree one (i.e. mapping the fundamental class [Mn] to f[Sn]: Mn sn of degree one (Le. mapping the fundamental class [Mn] to in such a way that f*(c) ” VM, where 5 is some stable vector bundle [sn] in such (e) ~ Il M, where stable (and vector over the spherea way Sn). that This f* construction shows thatis sorne Pontryagin all bundle other) over the sphere sn). This construction shows that Pontryagin (and aH other) characteristic classes of almost parallelizable manifolds may be expressed in characteristic classes of classes almost ofparallelizable manifolds may beWe expressed in terms of characteristic vector bundles over spheres. note also termstheof class characteristic classes of vector over Pontryagin spheres. Weclass. note also 0 that pk(M4k) is the only possiblybundles nontrivial that the class Pk(M 4k ) is the only possibly nontrivial Pontryagin class. 0

e

222 222

Chapter 4. 4. Smooth Smooth Manifolds Manifolds Chapter

From this this remark remark itit follows follows that that the the stable stable normal normal (or (or tangent) tangent) bundle bundle From of any almost parallelizable manifold M4k is determined by a multiple of a multiple of of any almost parallelizable manifold M4k is determined the “basic” bundle q over the sphere S4”. In particular, the only nontrivial the "basic" bundle T) over the sphere S4k. In particular, the only nontrivial Pontryagin class class Pk(M pk(M4”)4 k) has the the form: form: Pontryagin pk(M4”)

= AM ak . p. (2k - l)! ,

(P>W4”1) = 1, for sorne some integer integer ÀM. AM. Let Let ÀXmin denote the least number AM over all almost for min denote the least number ÀM over all almost 4k . This parallelizable manifolds manifolds M M4”. This number number À Xmin determines the order of the parallelizable min determines the order of the previously mentioned mentioned group group J7r4k_l(SO) Jndk- i (SO) C C 7r4k_l(sn), n&- 1 (Sn), n12> > 4k, 4/c, which which may may be be previously realized by by me means of framings framings of of the the standard standard sphere sphere S4k-l S4”-l Cc jRn+4k-l: lP+4k-1: realized ans of

Xmin À min

= IIJr4k-l(SO)J. J7r 4k-l(SO)I·

(3.27) (3.27)

The A-genus A-genus of of sueh such aa manifold manifold M M4”4 k is is given given by by The

Aj(M4”)4k ) == 00 for for jj < < k, k, Aj(M &(M4”)

= a$+(2k

(3.28) (3.28)

- I)! ,

Œk so by integrality of the A-genus the number ak!“XM(2k - l)! must be even so by integrality of the A-genus the number ak (3k À M (2k 1)! must be even Pk for odd k. It follows that IJrr4k-i(SO)I IS ’ d ivisible by the denominator of the by the denominator of the for odd k. It follows that IJ7r4k-l(SO)1 is divisible fraction Bk/4k where Bk is the kth Bernoulli number, since fraction Bk/4k where Bk is the kth Bernoulli number, sinee ak -z Pk

Bk

2 . (2k)! ’

AdM4k)

= !$&

+ ....

2·(2k)!' (This result was proved by Milnor and Kervaire in the late 1950s.) (This result was proved by Milnor and Kervaire in the late 1950s.) Pl al = 2. Since the number Example a) k = 1. Here we have A i = -24, E:rample a) k 1. Here we have Al 24' al 2. Sinee the number Xmill. $ is even, it follows that the order of the group IJn3(SO) I is divisible by Àmin' 224 is even, it follows that the order of the group IJ7r3(SO) is divisible by 24. (Recall that in fact rs(SO) ” Z/24.) Since A1(M4) 4 is even and T = pi/3, 24. (Reeall that in fact 7r3(SO) 2:: /Z/24.) Since Al(M ) is even and T pI/3, the signature must therefore be divisible by 16 (Rohlin, in the early 1950s). the signature must therefore be divisible by 16 (Rohlin, in the early 1950s). 0 1

o

Remark. Algebraic considerations involving the intersection index Remark. Aigebraic considerations involving the intersection index quadratic form, in particular its even-valuedness and unimodularity, quadratic form, in particular its even-valuedness and unimodularity, only that the signature is divisible by 8. only that the signature is divisible by 8. Example b) k = 2. here we have a2 = 1 and E.mmple b) k 2. here we have a2 1 and A2 = &(7Pf

as a as a yield yield 0 0

- 4Pz),

so that Xmin & must in this case be integral. Hence (JTT(SO)I is divisible so that by 240. À min . 2!O must in this case be integral. Hence IJ7r7(SO) is divisible 0 1

~~.

0

53. Smooth Smooth manifolds manifolds and and homotopy homotopy theory theory §3.

223 223

Remark. The stable homotopy homotopy groups groups 7T7rlT,+j (Sn) have have been been computed using using n+j(sn) methods of Cartan-Serre-Adams Cartan-Serre-Adams type, type, significantly significantly beyond j = 7, 7, thanks to to the efforts efforts of of several several authors authors (in (in particular, particular, Toda). Toda). The The formula formula (3.27) (3.27) for for the J7rdk.-i(SO) for aIl all kk represents represents an an important important result in manifold manifold the order of J7T4k_l(SO) theory; in particular particular it it provides aa lower bound for the order of the stable theory; homotopy groups of spheres. spheres. This "geometric" “geometric” approach to the computation computation homotopy of the stable homotopy homotopy groups groups of spheres spheres may be be combined with with algebraic methods (the Cartan-Serre Cartan-Serre method method and and the the Adams spectral sequence) sequence)only only in in methods (the Adams spectral 0 the framework of extraordinary extraordinary cohomology cohomology theories. theories. the framework of D We mention one more application application of of the signature in in manifold manifold theory theory of of funfunWe mention one more the signature damental importance. By means of the signature formula, non-trivial smooth damental importance. By means of the signature formula, non-trivial smooth structures on on manifolds manifolds have have been been found, found, first first on on the the sphere sphere S7 S7 (by (by Milnor Milnor in in structures the late 1950s). The construction is as follows: the late 19508). The construction is as foIlows: Consider 4-dimensional vector bundle r]q of of Hopf Hopf type over the the sphere sphere Consider a a 4-dimensional vector bundle type over B = = S4. S4. The The corresponding corresponding spherical spherical bundle S(q) has has fiber fiber Sa, S3, and and structure structure B bundle S(r]) group SO4. We assume assumethat group S04' We that

= P, (II>[S41)= 1,

x4(7)

(3.29) (3.29)

where is aa generator that the the total Il is generator of of H4(S4; H 4 (S4; Z). Z). This This condition condition implies implies that total where p space of the bundle with fiber S3 is homotopy equivalent to S7; we shan shall space of the bundle with fiber S3 is homotopy equivalent to S7; we denote this total space by ML, where (Y is defined in terms of p and the first denote this total space by M~, where 0: is defined in terms of IL and the first Pontryagin class of q by Pontryagin class of r] by Pl(rl)

=

6%

wu,

[S”l) = 1.

Denote by Nt the total space of the corresponding disk bundle D(q) with Denote by N~ the total space of the corresponding disk bundle D(r]) with fiber D4. Note that 8Ni = I@:. It is not difficult to show that a: must be fiber D 4 . Note that 8N~ M~. It is not difficult to show that 0: must be even: Q: = 2k. The quaternionic Hopf bundle whose spherical bundle is the even: 0: 2k. The quaternionic Hopf bundle whose spherical bundle is the standard Hopf fibration S7 --+ S4 corresponds to the case k = 1. Milnor standard Hopf fibration S7 ---t S4 corresponds to the case k = 1. Milnor gave an explicit construction of these bundles for all even cr, at the same time gave an explicit construction of these bundles for ail even 0:, at the same Ume defining a Morse function on each such manifold iVl2 with exactly two critical defining a Morse function on each such manifold M~ with exactly two critical points (a maximum point and a minimum point). From this it follows that points (a maximum point and a minimum point). From this it follows that each M;I is homeomorphic (even piecewise linearly) to the standard sphere each M~ is homeomorphic (even piecewise linearly) to the standard sphere S7. Consider now the PL-manifold S7. Consider now the PL-manifold - 8

8

8

Na. = Na. U 87 D . Assuming that Nt is a smooth manifold, we compute its first Pontryagin class Assuming that N~ is a smooth manifold, we compute its first Pontryagin class and its signature: and its signature: pl(m;) = pl(Nz) = a = 2k; ~(fii) = T(N:) = 1. (3.30) (3.30) Hence by the Hirzebruch formula, Hence by the Hirzebruch formula,

457

+ (2k)2 7

Chapter 4. 4. Smooth Smooth Manifolds Manifolds Chapter

224 224

which is is not an an integer when k == 2 (i.e. acr = = 4). We We in infer that the manifold manifold which fer that fij is is not so that that MI Mi is is not diffeomorphic diffeomorphic to S7. S7. N] not smooth, so It follows from the topological topological invariance invariance of the Pontryagin Pontryagin numbers numbers that that It Ni is is not homeomorphic to any smooth smooth manifold; manifold; it is is in fact the manifold N] that N] @ is is aa PL-manifold. PL-manifold. Further Further development of the theory theory clear, however, that structures on spheres spheres and other manifolds will will be be discussed discussedin in the of smooth structures next section. next section. We now return return to cobordism theory theory and and the problem of realizing realizing cycles cycles by submanifolds. We have seen seen that that the computation of of bordism bordism rings rings via via by submanifolds. We have the computation the Thom-Pontryagin Thorn-Pontryagin construction construction leads leads to to the the study study of of homotopy homotopy properproperthe ties of of the the Thom Thorn spaces spaces of of the the univers universal vector G-bundles G-bundles (for (for G G = 0, 0, sa, SO, al vector ties U, ,537, Sp, Spin, and others). Adams’ method, on the other hand, involves U, Su, Sp, Spin, and others). Adams' method, on the other hand, involves the ca1culation calculation of the cohomology cohomology ring H*(M(G);Z/p) and the the action action on on the of the ring H* (M (G); 7l / p) and it of the Steenrod operations, i.e. the stucture of the cohomology ring as it of the Steenrod operations, Le. the stucture of the cohomology ring as aa module over over the Steenrod algebra algebra (see (see Chapter Chapter 3, 3, §6). $6). Except Except in in the the case case of of the Steenrod module H*(M(Spin); Z/2), the structure of these modules may be inferred from the H*(M(Spin); 7l/2), the structure of these modules may be inferred from the observation the embedding BG, + M(G,) of the the base base space space as as the observation that that the embedding ii :: BG M(G n ) of the n zero section, induces a monomorphism (of modules over the Steenrod algebra) zero section, induces a monomorphism (of modules over the Steenrod algebra) i* : H*(M(G,);

Z./p) --+ H*(BG,;

Z/p)

onto ideal generated generated by by i*(un), where Un U, is the Thorn class i* (un), where is the Thom class onto aa prime prime principal principal ideal in H*(M(G,); Z/p). Th is implies, in particular, that the product operator in H*(M(G ); 7l/p). This implies, in particular, that the product operator n

zz ~ t--+i*(u i*(un) ) . z z= = i*(z q5(1)) = i*qS(z) i*(z·. 4>(1)) = i*4>(z) n has has zero zero kernel kernel in in the the For G, = O,, SO, For Gn On, San

(3.31)) (3.31

appropriate of M(G M(G,) ). appropriate cohomology cohomology group group of n one obtains: one obtains: i*(un) = wn, (3.32)

H*(M(G,);

Z/2) 2 w, . H*(BG,;

H*(M(G n ); 7l/2)

~ Wn

(3.32)

Z/2).

. H*(BG n ;7l/2).

Note that if G, = U,, SU,, then n = 2k, and if G, = Sp, then n = 4k. One Note that if G n = Un, SUn then n = 2k, and if Gn = Spn then n = 4k. One has the following isomorphisms: has the following isomorphisms:

H*(M(S02k)j Z/P) 7l/p)

H*(M(S02k);

H*(MV2k);

VP)

H*(M(U2 k); 7l/p)

!:::<

z

x2k

Z/P), X2k .. H*(BS02k; H*(BS02k; 7l/p),

P > 2,

E !:::<

ck . H*(BU2k; z/p),

P > 2,

Ck . H*(BU2k; 7l/p) ,

H*(M(s!&k);

z/p)

2

ck

H*(M(Smk);

Z/P)

2

i*(u&)

H*(M(SU2 k); 7l/p) H*(M(Sp4k); 7l/p)

~

~

. H*(BSU2k;

p? 2,

z/P),

Ck . H*(BSU2k ;7l/p) , . H*(BSpa;

p> 2,

z/p),

i*(U4k) . H*(BSp4k; 7l/p),

P>

2,

(3.33)

(3.33)

p> 2,

p > 2.

p> 2.

Recall (see 51 above) that the cohomology rings of the BG, can be described Recall (see above) that the cohomology ringssymmetric of the BGpolynomials n can be described in terms of §1symbolic generators (namely the in the in terms of symbolic generators (namely the symmetric polynomials in the

53. Smooth §3. Smooth manifolds manifolds and and homotopy homotopy theory theory

225

generators of H*(Tn) where Tn G,:n : in tr, . . .,, tn t, mod 2 generators of H*(Tn) where T n isis a maximal maximal torus torus of of G in tb" mod for G, = O,, SO,, in terms of which w, = tl . . t,, and in the 2-dimensional for Gn = On, SOn, in ter ms of which W n = tl ... tm and in the 2-dimensional generators ~1, ... . Uk in other other cases, cases, in of which X2k == Ck Ck = = UI 211... . . . Uk). Uk). Uk in in terms terms of which X2k generators Ul, The isomorphism (3.32) implies that the cohomology ring H*(M(O,);Z/2) The isomorphism (3.32) implies that the cohomology ring H*(M(On); Z/2) is 5 2n the Steenrod Steenrod algebra with the the is in in dimensions dimensions ~ 2n a free free module module over over the algebra A2 A 2 with non-dyadic non-dyadic monomials monomials

4=Cqj
qj not not of the form form 21' 2r - 1, 1, qj of the

as generators, these generators generators on the Thom Thorn class class W w,n = = as generators, and and the the action action of of these on the

tltl .... tnt, isis ordinary of polynomials: ordinary multiplication multiplication of polynomials: p(w,)

=ptl...t,

E H “+“Pw?&

Z/2),

q= c qj.

(This of Thom.) Thorn.) This This result result allows allows (using Adams spectral spectral sese(This is is aa theorem theorem of (using the the Adams quence) the complete computation of the homotopy groups of the Thorn spaces quence) the complete computation of the homotopy groups of the Thom spaces M(0,) 5 2n), all stable stable homotopy groups of of M (On) (in (in dimensions dimensions ~ 2n), or, or, equivalently, equivalently, aIl homotopy groups the spectrum MO, which can be identified (via the Thorn-Pontryagin conthe spectrum MO, which can be identified (via the Thom-Pontryagin construction) bordism ring Qp = 1)1*. n,. As an immediate immediate application application of of struction) with with the the bordism ring ft;> As an this computation one has the following theorem: this computation one has the following theorem: Any < Any cycle cycle of of dimension dimension ~ may be realized by a closed may be realized by a closed of of aa closed closed manifold. manifold.

n/2 in in the the homology (mod 2) 2) of of aa manifold manifold M” n/2 homology (mod Mn submanifold, and, in all dimensions, as the image submanifold, and, in aU dimensions, as the image cl 0

For p > 2 and G = SO, U, SU, Sp the A,-modules H*(M(G);Z/p) turn For p > 2 and G = sa, U, SU, Sp the Ap-modules H*(M(G); Z/p) turn out to be direct sums of copies of the one-dimensional A,-module d,/(P) out to be direct sums of copies of the one-dimensional Ap-module A p /((3) (where p is the Bockstein operator and (/3) the two-sided ideal generated by (where (3 is the Bockstein operator and ((3) the two-sided ide al generated by ,D). The module d,/(P) is’ g iven by the relation p(z) E 0. (3). The module A p /((3) is given by the relation (3(z) == O. Remark. The latter statement also holds for H*(M(U);Z/2), with the Remark. The latter statement also holds for H*(M(U); Z/2), with the 0 Bockstein operator @ replaced by the operation Sql. Bockstein operator (3 replaced by the operation Sql. 0 The structure of the modules H*(M(G); Z/2) (as AZ-modules) for G = SO, The structure ofthe modules H*(M(G); Z/2) (as A 2-modules) for G sa, SU, Sp and Spin is somewhat different. They are also direct sums of oneSU, Sp and Spin is somewhat different. They are also direct sums of onedimensional AZ-modules, but the summands vary: For G = SO they are free or dimension al A2-modules, but the summands vary: For G = sa they are free or with the single relation Sq’(w) = 0 ( w h ere v is a generator); for G = SU with with the single relation Sql(V) = 0 (where v is a generator); for G SU with the same relation Sql(w) = 0 together with the identical relation Sq2(z) E 0 the same identical S q2(z) for every relation element Sql(v) z; for G =0 together Sp (the with most the difficult case)relation there are the two0 for every element z; for G = Sp (the most difficult case) there are the two identical relations Sq’(z) E 0, Sq2(z) =: 0 (f or all z) ; and finally for G = Spin q2(Z) identical relations Sql(Z) == 0, S == 0 (for aIl z); and finaIly for G = Spin there are summands with the two relations Sql(w) = 0, Sq’(v) = 0 (where w is are summands with the two(These relations Sql(v) (where vis athere generator of this AZ-module). results are due0, toSq2(V) Milnor 0and Novikov a(around generator of this A -module). (These results are due to Milnor and Novikov 1960), and in2 the caSe of Spin to Anderson, Brown and Peterson in (around 1960), and in the case of Spin to Anderson, Brown and Peterson in the mid-1960s.) theIt mid-1960s.) should be noted that there is a product structure (induced by the direct It should be noted that structurerings. (induced the direct product of manifolds) on there all of isthea product above bordism This by “geometric” product of manifolds) on aU of the above bordism rings. This "geometric"

226 226

Chapter 4. 4. Smooth Smooth Manifolds Manifolds Chapter

product structure structure determines determines an an algebraic algebraic "product" “product” on on the the Ap-modules dP-modules M M = = product via the the diagonal diagonal homomorphism homomorphism H*(M(G); 7l/p) H*(M(G); UP) via AIM-M@zlpM, ,1 :M M®z/pM, which in in turn turn determines determines aa multiplication multiplication on on the the the the cohomology cohomology groups groups which Ext;; (M, Z/p). Extj*(M,71/p). p After certain certain algebraic algebraic obstacles obstacles have have been been overcome, overcome, the the Adams Adams spectral spectral After sequence (see Chapter 3, §6,7) yields up the structures of the bordism rings sequence (see Chapter 3, §6,7) yields up the structures of the bordism rings Q:“, O,U (Milnor, Novikov). It should be mentioned that the 2-torsion of the [2:'0, [2;' (Milnor, Novikov). It should be mentioned that the 2-torsion of the ‘* QsU had been calculated ealier using different, purely geometric grows [2~0, 4 , [2~u + had been calculated ealier using different, purely geometric groups ideas (by (by Rohlin Rohlin and and Wall Wall in in the the late late 1950s 1950s in in the the case case of of [2~0, a:*, and by by and ideas Conner and Floyd in the mid-1960s for @‘; however the combination of Conner and Floyd in the mid--1960s, for [2~u; however the combination of these geometric methods with those of homological algebra turned out to be these geometric methods with those of homological algebra turned out to be feasible only only in in the the framework framework of of the the generalized generalized (extraordinary) (extraordinary) cohomology cohomology feasible (cobordism) theories Q;(.) and QG(.). (cobordism) theories [2û (-) and Oô (-). The results are as as follows: follows: None None of of the the bordism bordism rings rings [2: 0: (with (with G G as as before) before) The results are has p-torsion for p > 2, and in the case 0: for p 1 2. The rings 02 @ 7l/p Z/p ® has p-torsion for p > 2, and in the case n;, for p ~ 2. The rings and 0: @ Q are polynomial for p > 2. The ring 0: is polynomial (over and [2: ® Q are polynomial for p > 2. The ring 0;' is polynomial (over the integers) integers) in in even-dimensional generators, one one of of each each even even dimension. dimension. On On the even-dimension al generators, these generators [M2”] the Newton polynomials & = n! ch, take the following these generators [M 2n ] the Newton polynomials èn = n! ch n take the following values: values: 1 for n n#pi-1, =J pt - 1, p prime, [M2”] E 02”. (GL,[M2”1) = p1 for for n=pi-1, {{ p for n = pt - 1,

0:

Since (2;,, @P) = n + 1, it follows that only the elements [@P-l] the Sinee (è n , cpn) n + 1, it follows that only the elements [Cpp-l] satisfy satisfy the above condition. All other generators [M2n] may be constructed out of the above condition. AlI other generators [M 2n ] may be constructed out of the Milnor manifolds H,,, , which represent the cycles @P’ x (IX’-’ +@P’-i x @Pt Milnor manifolds Hr,t, which represent the cycles cpr X cpt-l +cpr-l x cpt in the product CP’ x @Pt. Linear combinations of the elements [HT,t] provide in the product cpr x cpt. Linear combinations of the elements [Hr,tl provide a full set of multiplicative generators of the ring 0,“. a full set of multiplicative generators of the ring n;'. The following theorem (Novikov, early 1960s) on the representability in The following theorem (Novikov, early 1960s) on the representability in the stable dimensions j < n/2 of integral cycles z E Hj(AP; Z) by oriented the stable dimensions j < n/2 of integral cycles Z E Hj(MTi; 7l) by oriented submanifolds (or, in any dimension, by continuous images of manifolds) is a submanifolds (or, in any dimension, by continuous images of manifolds) is a consequence of results concerning the stable homotopy structure of the comconsequence of results concerning the stable homotopy structure of the complexes M(SO,) (or the spectrum MSO): plexes M(SOm) (or the spectrum MSO): There exists an odd integer X such that the cycle Xz can be so represented; There exists an odd integer À such that the cycle Àz can be so represented; moreover if for all Ic > 1, p > 2, there is no p-torsion in the homology moreover if for all k ~ 1, P > 2, there is no p-·torsion in the homology groups H,(M”; Z) of d‘zmensions j = 2k(p - 1) - 1, then in fact X may be groups Hj(M"; 7l) of dimensions j = 2k(p - 1) - 1, then in fact À may be taken to be 1. taken to be 1. Remark. If there is p-torsion in the indicated groups, then the factor X may Remark. is p-torsion in the indicated groups, then the factor À may be taken as If thethere nroduct be taken as the product À

II p[2(P~1)],

P>2

p>2

54. Classification problems in the theory of smooth smooth manifolds manifolds §4. Classification problems in the theory of

227 227

which is the denominator of the general coefficient coefficient in HirzeHirzewhich is the formula formula for the denominator bruch’s bruch's polynomial polynomial L,(pl,.

. . ,p,),

when m =

[‘I i

.

o

In 1960s Buchstaber Buchstaber constructed constructed examples examples showing that this this result result In the the late late 1960s showing that cannot in general be improved. At the same time he showed that in the absence cannot in general be improved. At the same time he showed that in the absence of in certain certain intermediate intermediate groups, groups, the be improved. improved. of p-torsion p-torsion in the result result can ean be

$4. Classification in the the theory theory of of smooth smooth manifolds. manifolds. §4. Classification problems problems in The of immersions. immersions. Manifolds Manifolds with with the the homotopy homotopy type type The theory theory of of a sphere. Relationships between smooth and PL-manifolds. of a sphere. Relationships between smooth and PL-manifolds. Integral Pontryagin classes Integral Pontryagin classes The formulation of the problem problem of classification of of The general general formulation of the of the the homotopy homotopy classification immersions is as follows: immersions is as follows: Given two AP and and Nk, characterize the classes classes of of Given two manifolds manifolds Mn Nk, where whe're kk >> n, n, chamctenze the regularly homotopic immersions IF’ 4 Nk. k regularly homotopie immersions Mn ---+ N •

The of interest is that where Mn M” == 5’” and Nk = Iw’“. ConThe simplest simplest case case of interest is that where sn and N k = ]Rk. Consider the following two function spacesof immersions: sider the following two function spaces of immersions: a) The space Xk,, of immersions of the disc P equipped with a normal a) The space X~ s of immersions of the disc D n equipped with a normal field u, of dimension s, into lRk, where n + s < I;, under the condition that field I/s of dimensio~ s, into ]Rk, where n + s < k, under the condition that for each admissible immersion each point SOof the boundary dD of the disc for eaeh admissible immersion eaeh point So of the boundary aD of the dise should have a neighbourhood on which the immersion is “normalized”, in the should have a neighbourhood on which the immersion is "normalized", in the sensethat its image is essentially a region of a standard n-dimensional plane sense that its image is essentially a region of a standard n-dimensional plane in lRk equipped with a standard constant normal s-dimensional field of frames in ]Rk equipped with a standard constant normal s-dimensional field of frames in R”. in ]Rk. b) The space Y,“,, of immersions of the sphere S” equipped with a normal b) The space Y;'s of immersions of the sphere sn equipped with a normal field u,, in IKk, n + s < k, with the property that each immersion is “norfield I/s, in ]Rk, n + s < k, with the property that eaeh immersion is "normalized” on some neighbourhood U, of each point so of S”, i.e. the image of malized" on sorne neighbourhood Ux of each point So of sn, Le. the image of the neighbourhood U, is a region of a standard n-dimensional plane in lRk the neighbourhood Ux is a region of a standard n-dimensional plane in ]Rk equipped with a standard constant normal s-dimensional frame field in R”. equipped with a standard constant normal s-dimensional frame field in ]Rk. There is an obvious map There is an obvious map xi,,

-

YL,Sfl,

defined by taking the boundary S”-’ = dD” n of the disc together with an addefined taking the boundary sn-1 aD and of the disc together an additional by l-dimensional field normal to S”-l tangential to thewith interior of to sn-1 and tangential to the interior of ditional 1-dimensional field normal Dn. By an observation of Smale (made in the late 1950s) this map is the proDn. Bymap an observation of SmaleSince (made in the late thisallmap is the projection of a Serre fibration. therefore the 1950s) fibers are homotopically jection map of a Serre fibration. Since therefore the fibers are aH homotopically equivalent, it is not difficult to see that in fact they have the homotopy type equivalent, it isfurther, not difficult to see thatXk,, in fact they have thenj(Xk,,) homotopy of Y,,+. Since, the total space is contractible: = 0type for of Y;,s' Since, further, the total space X~,s is contractible: 1T'j(X~) 0 for

228 228

Chapter 4. Chapter 4. Smooth Smooth Manifolds Manifolds

j 2;:::: 0, we obtain an the appropriate groups we obtain an isomorphism isomorphism between between the appropriate homotopy homotopy groups of and base: of the the fiber fiber and base:

%(Y,,s)g “j+lwL,,+,).

(4.1) (4.1)

Now simply means identifying the Now classifying classifying the the immersions immersions of of S” sn in in IR”, ]Rk, simply means identifying the path-connected of Yk,s, , i.e. identifying 7rc(Yk,a). From the sepath-connected components components of Le. identifying 7To(Y:o)' From the seo quence from (4.1) (4.1) by quence of of isomorphisms isomorphisms obtained obtained f~om by letting letting jj vary, vary, we we have have immeimmediately diately m(Y,,I)> 2 c&g. (4.2) (4.2)

Y:

Since Yl Y& n is is just just the the Stiefel Stiefel manifold manifold with with points points the the non-degenerate non-degenerate fields fields of of Since n-frames on lRk: n-frame~ on ]Rk: y&

= Vik,

we infer Smale we infer Smale ‘s 's theorem: theorem:

The The particular particular case case kk == 2n 2n of of this this result result isis the the classical classical theorem theorem of of Whitney Whitney on the classification of immersions S” ----+ lf%2n. (The case n = 1 of on the classification of immersions sn ---+ ]R2n. (The case n 1 of Whitney’s Whitney's theorem is illustrated in Figure a); the illustrated in Figure 4.21 4.21 a); the invariant invariant of of the the homotopy homotopy class class theorem is of an immersion is in this case given alternatively by 1/27r $ of an immersion is in this case given alternatively by 1/27T f kk ds ds == n, n, the the integral of the curvature of the immersed circle over the whole circle.) As integral of the curvature of the immersed circle over the who le circle.) As another non-trivial consequence of Smale’s theorem we have that all immeranother non-trivial consequence of Smale's theorem we have that ail immersions S2 + lK3 are regularly homotopic, since 7r2(V2y3) s rz(SO3) = 0; this sions S2 ---+ ]R3 are regularly homotopie, since 7T2(V2~3) ~ 7T2(S03) = 0; this is obvious even is by by no no means means obvious even for for simple simple examples examples of of immersions: immersions: compare compare the the immersion S2 + lRP2 + IR3 with the standard embedding S2 + lR3. immersion S2 ]Rp2 ]R3 with the standard embedding S2 ]R3. Smale’s theorem generalizes without great difficulty to the situation of imSmale's theorem generalizes without great diffieulty to the situation of immersions S” --+ Mk of the sphere Sn in any manifold Mk ; here the remersions sn ---+ Mk of the sphere sn in any manifold Mk; here the result is that the homotopy classes of immersions are completely determined suIt is that the homotopy classes of immersions are completely determined (i.e. in onetoone correspondence with) the elements of the homotopy group (Le. in one-to-one eorrespondence with) the elements of the homotopy group TIT,(&k), where &k is the total space of the bundle of non-degenerate fields 7T (En ,k), where En,k is the total spaee of the bundle of non-degenerate fields ofn tangential n-frames on the manifold M”. Already in the case k = 2, for of tangential n-frames on the manifold Mk. Already in the ease k 2, for the surface Mg (sphere-with-g-handles) the fundamental group 7ri(Ei,z) of the surface Mg (sphere-with-g-handles) the fundamental group 7Tl (E1,2) of the manifold of “linear elements” (i.e. fields of l-dimensional subspaces of the the manifold of "linear elements" (Le. fields of 1-dimensional subspaees of the tangent space of Mg) is non-trivial; it is given by generators al, bi, . . . , a,, b,, c tangent space of Mg) is non-trivial; it is given by generators al, bl ,···, ag, bg, c with the defining relations

with the defining relations

fJg aibia,lb,l = 29-2, -lb-l 2g-2

II

b i=l ai iai i=l

i

-

C

,

Cai

=

CLiC,

cbi = bit.

(4.3) (4.3)

(The case g = 0 is illustrated in Figure 4.21 (b).) (The easeearly 9 = 01960s is illustrated in Figure 4.21 (b).) of Smale’s theorem was esIn the the ultimate generalization In the early 1960s the ultimate generalization Smale's theorem tablished by Hirsch, who showed that for any two of manifolds M”, Nk k (kwas> esn) tablished by Hirsch, who showed that for any two manifolds N ordinary (k > n) each class of immersions Mn --+ Nkk is uniquely determined Mn, by the eaeh class of immersions Mn ---+ N is uniquely determined by the ordinary

$4. Classification problems problems in the theory of §4. Classification the theory of smooth smooth manifolds manifolds

229

homotopy of maps maps of of tangent bundles induced by these immersions. homotopy class class of tangent bundles induced by these immersions. The proof makes crucial use use of the fact fact that, of an immersion, immersion, The proof makes crucial of the that, by by definition definition of no tangent n-frame at any point of M” becomes degenerate under map the map no tangent n-frame at any point of Mn becomes degenerate under the of spaces induced immersion ff : Mn N”. Consider the Mn -+ _ Nk. Consider the of tangent tangent spaces induced by by an immersion

(a) S1 +

R2, V$ N 9, f7

kds = n

2 El,2 r SOB, xl(SOs) (b) (b) S1 8 1 --t 8S2, 803,1rl(803) z Z/2; Z/2; , El.2 thus on going over from R2 to the sphere S2, thus on going over from JR2 to the sphere 82, the invariant in (a) modulo 22 the invariant in (a) becomes becomes reduced reduced modulo

Fig. Fig. 4.21 4.21 manifold G,(Nk) k ) whose whose points are the (subspaces) of of manifold Gn(N points are the n-dimensional n-dimensional planes planes (subspaces) the tangent spaces at the points of Nk. f induces a map k The immersion the tangent spaces at the points of N . The immersion f induces a map f : Mn Gn(Nk), k ), which covered by map of of principal bundles with with Mn ---+ Gn(N whïch is is covered by aa map principal bundles structure group GL,(IR); it is the homotopy classes of these bundle maps that structure group GLn(IR.); it is the homotopy classes ofthese bundle maps that determine the classes of immersions. determine the classes of immersions. A theory concerned with closed manifolds homotopy equivalent to spheres A theory concerned with closed manifolds homotopy equivalent to spheres was developed by Milnor and Kervaire around 1960. In order to describe was developed by Milnor and Kervaire around 1960. In order to describe

1:

o

o

Fig. 4.22

Fig. 4.22

this theory we need the construction called a “connected sum” Mr#M?J’ of this theoryMp, we need the construction calledthea "connected sum" Mf#M!J: manifolds MF, with respect to which set of (diffeomorphism classesof manifolds M!J:, with respect to whichn the (diffeomorphism classes of) smooth Mf, manifolds of each dimension formsset aofcommutative semigroup: of) smooth manifolds of each dimension n forms a commutative semigroup: From each of the manifolds Mr, MC a small disc is removed: From each of the manifolds Mf, M!J: a smaH disc is removed:

230

Chapter Chapter 4.4. Smooth Smooth Manifolds Manifolds 2 Cj(xj,)2 I:j(X~)2 << c2 ê

on on My, Mr,

2

on Mf,

I: j (yb)2 <

ê

in terms of of local on appropriate U,ex cC A/l;, VP cC Mf, MC, in terms local co-ordinates co-ordinates on appropriate charts ch arts U 1\1r, V{;I and then the resulting boundaries (each diffeomorphic to the standard sphere and then the resulting boundaries (each diffeomorphic to the standard sphere Sp-‘) are oriori8;"-1) are are identified identified (taking (taking orientation orientation into into account account ifif the the manifolds manifolds are ented) by means of the standard diffeomorphism; the resulting smooth maniented) by means of the standard diffeomorphism; the resulting smooth manifold the connected connected sum sum Mr#MF Mr#Mf (see (see Figure Figure 4.22). 4.22). fold isis the Example. exhaustive list (closed surfaces): We have have an an exhaustive list of of closed closed 2-manifolds 2-manifolds (closed surfaces): Example. We Mi ‘:. #Tg M; == ,T”# T 2 # ... #T 2,

T2 the torus, torus, T 2 the

'-..---'

g9 times times

N2

1>9 N'l,g

= K2#T2#...#T2 2 K 2 #1' #1'2, d' -#. .... ---' 9g - 11 times times

K2 the Klein K 2 the Klein bottle, bottle,

2 N2 RP2#T2#-#T2 Ni2>9•g == ffi.P2#1' ... #T 2 • -.' - # ..---' g9 times times

Hence T T2,2 , K2 generators for for the K2 and and lRP2 ffi.p2 are are generators the semigroup semigroup of of all aU closed closed surfaces, surfaces, Hence with defining relations with defining relations RP2#RP2#RP2

S lRP2#T2

K2#T2

= RP2#K2,

= K2#K2.

(Figures 4.24, 4.25, 4.26 illustrate these relations.) (Figures 4.24, 4.25, 4.26 illustrate these relations.)

Fig.

4.23.

The

sphere-with-g-handles

Fig. 4.23. The sphere-with-g-handles

A4,” (g = M; (g =

3 in the

figure)

3 in the figure)

Remark. The orientable closed surfaces form a free monoid on the single Remark.T2.The orientable surfaces a free S-manifolds monoid on the generator It would seemclosed that the closed form orientable also single form generator 1'2. It would seem that the closed orientable 3 manifolds also form

54. §4. Classification Classification problems problems in in the the theory theory of of smooth smooth manifolds manifolds

231

aa free free monoid; monoid; however however for for closed closed orientable orientable 4-manifolds 4-manifolds this this isis certainly certainly no no longer longer the the case. case. Note Note incidentally incidentaIly that that of of the the closed closed surfaces, surfaces, the the IV;,,,, Ni,g' g9 2~ 0, all aIl represent represent the the nonzero non-zero cobordism cobordism class class of of fig, while while all aU the the other other surfaces surfaces 00 represent the zero zero class. class. represent the

st?,

Fig. from the Fig. 4.24. 4.24. The The process pro cess of of deforming deforming aa handle handle so 80 that that itit goes goes from the outside out si de to to the the inside inside

Y-

Fig. 4.25

Fig. 4.25

@@ @ ::I->

Fig. 4.26

Fig. 4.26

Returning now to the topic of homotopy spheres, we first observe that the Returning now to the topie of homotopy spheres, we first observe that the oriented h-cobordism classes of homotopy n-spheres form a group (denoted oriented h-cobordism classes of homotopy n-spheres form a group (denoted by On) under the operation of taking the connected sum. (See 53 above for by en) under the operation of taking the connected sumo (See §3 above for the definition of h-cobordism.) Here the zero class consists of the oriented the definition of heobordism.) Here the zero class eonsists of the oriented manifolds AP bounding contractible manifolds-with-boundary IV+‘: manifolds Mn bounding contractible manifolds-with-boundary Wn+l: M” = ifqp+1, J/p+1 N 0. Mn ôW n+1 , Wn+l '" O. The inverse of (the class of) an oriented homotopy n-sphere AI; is (the class The inverse of (the class of) an oriented homotopy n-sphere M;:' is (the class of) the same manifold AL?_”with the opposite orientation. To see this, i.e. to see of) the same manifold M': with the opposite orientation. 1'0 see this, Le. to see that the connected sum Mg#M” is h-cobordant to S” consider the manifoldthat the connected sum M;:'#M': is h-cobordant to sn consider the manifoldwith-boundary Wnfl with dW”+l = MF#MTT, constructed as follows (see with-boundary Wn+l with ôWn+l M;:'#M':, construeted as follows (see Figure 4.27): From the cylinder MF x I remove the subset 0,” x I where cylinder x l E;remove the subset D~ x l where Figure 4.27): 0: c MT is aFrom small the open ball ofM;:' radius after appropriate “smoothing of D~ c M;:' is a small open bail of radius é; after appropriate "smoothing of corners” one is left with a smooth manifold-with-boundary corners" one is le ft with a smooth manifold-with-boundary Wn+’ = (MT x I) \ (D,n x I) = (MT \ D:) x I, w n + 1 = (M;:' x 1) \ (D~ x 1) = (M;:' \ D~) xI, for which dW”+l = MF# M”, and which is moreover contractible. for which ôW n+ 1 = M;:'#M"2, and which is moreover contractible.

232 232

Chapter Smooth Manifolds Chapter 4. Smooth Manifolds

sn

Fig. 0, M” N S” (homotopy (homotopy sphere) Fig. 4.27. 4.27. M,“#M_” M'+#M'::. N ,...., 0, Mn,...., sphere)

Smale’s important important “h-cobordism Smale's "h-cobordism theorem”, theorem" , stating stating that that every every h-cobordism h-cobordism Wn+’ simply-connected manifolds of dimension 5 is is W n + 1 between between simply-connected manifolds AJr, Mf, i$’ Mf of dimension nn 2 ~ 5 diffeomorphic to MT x I (whence IV? is diffeomorphic to IV,“), implies that diffeomorphic to Mf x 1 (whence Mf is diffeomorphic to M 2'), implies that for each n an exact classification of for each n > ~ 5 5 the the group group On en provides provides an exact classification of the the homohomotopy n-spheres, and hence of the distinct smooth structures definable topy n-spheres, and hence of the distinct smooth structures definable on on the the topological topological n-sphere. n-sphere. Of Of particular particular interest interest is is the the subgroup subgroup (denoted (denoted by by bP”+l) bPn+l) of of 0” en consisting consisting of the h-cobordism classes of boundaries of compact parallelizable of the h-cobordism classes of boundaries of compact parallelizable manifoldsmanifoldswith-boundary. shown how corresponding to each with-boundary. Milnor Milnor has has shown how to to construct, construct, corresponding to each even-dimensional, unimodular, integer matrix (aij) (as in (4.4)), a parallelizeven-dimensional, unimodular, integer matrix (aij) (as in (4.4)), a parallelizable sphere M M4k-1, 4k-l, able manifold-with-boundary manifold-with-boundary P4k p 4k with with boundary boundary a a homotopy homotopy sphere for which Hzk(P 4k; Z) is a free abelian group with a basis of cycles having for which H2k(p4k; Z) is a free abelian group with a basis of cycles having matrix (aij), and < 2k. matrix of of intersection intersection indices indices (aij), and T~(P~“) 7j(p4k) = 00 for for jj < 2k. The The minimal minimal signature of such a manifold is 8: rmin (P4k) = 8; here is a matrix A = (aij) signature of such a manifold is 8: 7m in(P4k) = 8; here is a matrix A (aij) yielding such a manifold:

yielding such a manifold:

A=

A=

/21000000 2 1 0 0 12100000 1 2 1 0 01210000 0 1 2 1 00121000 0 0 1 2 00012101 0 0 0 1 00001210 0 0 0 0 00000120 0 0 0 0 \00001002 0 0 0 0

0 0 0 0 0 0 1 0 2 1 1 2 0 1 1 0

0 0 0 0 0 1 2 0

0 0 0 0 1 0 0 2

(4.4 (4.4)

Note that the manifold P4k is a Spin-manifold (it is oriented and the Stiefel 4 is a Spin-manifold (it is oriented and the Stiefel Note class w2(that P4k)the = manifold 0). Thus pa knecessary condition for the boundary of P4k to class 0). - Thus a necessary condition for theofboundary of p4k to be theW2(p4k) ordinary (4k 1)-sphere S4k-1, is the integrality the a-genus of the be the ordinary (4k 1 )-sphere S4k-l, is the integrality of the Â-genus of the manifold

manifold

(see 53 above), given by

(see §3 above), given by

4k P4’ uaP4k D 4k P 4k U 8p4k D

54. §4. Classification Classification problems problems in in the the theory theory of of smooth smooth manifolds manifolds

233

1 /i[p4k

UaP4k

D4”]

=

,-(P4”)

’

22”fl(22”-1

-

1)’

Hence Hence in in this this situation situation the the signature signature r(P4”) T(p4k) must must in in fact fact be be divisible divisible by by p+y‘p-1

- 1).

Using Using the the fact fact that that the the signature signature is is divisible divisible by by 8, 8, one one has has that that the the order order of of the the group group bP4k bp4k is is divisible divisible by by 2-2(2-l

- 1).

(Recall (Recall that that for for even even Ic k the the a-genus Â-genus is is likewise likewise even, even, so so that that in in fact fact for for kk even even (22k-1 -- 1).) l).) the the order order of of the the group group bP4k bp4k is is divisible divisible by by 22”-1 22k-l(22k-l For 1) every N (N (N >> nn ++ 1) every homotopy homotopy sphere sphere M” Mn cc llP+jv JRn+N For large large enough enough N has trivial normal bundle Y. (For this one uses the classification of normal has trivial normal bundle v. (For this one uses the classification of normal fields fields of of N-frames N-frames on on M” Mn via via elements elements [v] [v] of of ~T~(SON), 1rn (SON), in in conjunction conjunction with with Bott periodicity for SO, the fact that pk(Mn) = 0 for n = 4k (a consequence Bott periodicity for SO, the fact that pdMn) = 0 for n = 4k (a consequence of and r(P) of Adams Adams of the the Hirzebruch Hirzebruch signature signature formula formula and T(sn) = 0), 0), and and aa result result of of the early 1960s implying that the classifying element [v] = 0 also of the early 1960s implying that the classifying element [v] = 0 also in in the the n view of this property, homotopy n-spheres cases where T~(SON) 2 Z/2.) I cases where 1rn (SON) ~ 7l/2.) In view of this property, homotopy n-spheres admit VN, and natural one-to-one one-to-one correspondence (see §3 $3 above) above) admit framings framings VN, and the the natural correspondence (see between framed normal bundles on n-manifolds in Wn+N and the elements between framed normal bundles on n-manifolds in JRn+N and the elements of TN+%(S~) (whereby of the the form VN) correspond to Of1rN+n(SN) (whereby framed framed manifolds manifolds of form (P, (sn,VN) correspond to e image under the Whitehead homomorphism the subgroup J(K~(SON)), th subgroup J(1r n(SON )), the image under the Whitehead homomorphism -thesee then yields see $3 §3 above) ab ove) then yields aa homomorphism homomorphism p : 0” --+ 7rpf+n(sN)/J?T,(so).

(4.5) (4.5)

The kernel of this homomomorphism coincides with the subgroup bP”+l The kernel of this homomomorphism coincides with the subgroup bPn+l of classes of homotopy n-spheres bounding parallelizable manifolds-withof classes of homotopy n-spheres bounding parallelizable manifolds-withboundary. A theorem of Milnor and Kervaire asserts that the homomorphism boundary. A theorem of Milnor and Kervaire asserts that the homomorphisrri p is an epimorphism provided n is not of the form 4k + 2, while for n = 4k + 2 f..L is an epimorphism provided n is not of the form 4k + 2, while for n = 4k + 2 the image under p has index at most 2. The kernel bP”+l is cyclic, and in the image under f..L has index at most 2. The kernel bPn+l is cyclic, and in fact trivial for even n. It is known further that for n of the form 4k + 1, the fact trivial for even n. It is known further that for n of the form 4k + 1, the has order at most 2, that bP” 2 Z/28, and that subgroup bP4k+2 c 04k+1 subgroup bP4k+2 c e 4k +1 has order at most 2, that bpoo ~ 7l/28, and that the order of bP4k increases as k -+ co. the order of bp4k increases as k -+ 00. A detailed technique for investigating homotopy spheres has been develA detailed technique for investigating homotopy spheres has been developed, involving “killing” the appropriate homotopy groups of framed manihomotopy groups oped, involving "killing" the appropriate in order to reduce themof toframed framedmanihofolds by means of “Morse surgery”, folds means of(This "Morse surgery", or der motopyby spheres. technique was inused, in to the reduee case n them = 2, to by framed Pontryaginhomotopy spheres. techniquewithwashis used, in the case n = 2, by Pontryagin in the late 1940s (This in connexion calculation of the groups 7r,+2(Sn); the late 1940s in connexion with his calculation of the in see $3 above.) The given framed manifold is first replaced groups by an 1rn+2(sn); equivalent see §3 above.) The given framed manifold is first replaced by an equivalent connected one, and this in turn by an equivalent simply-connected framed connected and one, soand in turn requiring by an equivalent framed manifold, on;this difficulties non-trivialsimply-connected analysis arise only in manifold, and on; odd difficulties requiringemerge non-trivial arisealthough only in dimension [n/2].so For n no invariants in this analysis dimension, dimension odd n no invariants in this although the proof of[n/2]. this For is non-trivial. For n ofemerge the form 4k, dimension, since the signature the proof of this is non-trivial. For n of the form 4k, sinee the signature

234 234

Chapter Smooth Manifolds Manifolds Chapter 4. Smooth

of is zero by Hirzebruch's Hirzebruch’s formula, one may always by formula, one of aa framed manifold manifold is means elementary Morse surgeries surgeries eliminate eliminate the n/2-dimensional me ans of elementary nj2-dimensional homotopy n/2 being in this case case cycles of dimension n/2 topy also, also, provided provided n > 4, the cycles realizable n/2-spheres, as as required for Morse surgery. For nn of realizable as as embedded embedded nj2-spheres, the form 4lc M4kf2 can also also be reduced reduced to aa manifold manifold 4k + + 2 the framed manifold manifold M4k+2 M4k+2 satisfying ~j’j(Mf~+~) for j :S: 5 21c. although the elements elements Mtk+2 satisfying 7rj(Mtk+2) = 0 for 2k. Here, although ) are, as in the previous case, realizable by of 1&k+i(&ff”t2) ” .rr2k+r(M;k+2 H 2k+1 (Mtk+2) ~ 7r2k+ 1 (Mtk+2) are, as in the previous case, realizable by k 2 means of embedded spheres S2”f1 c Mfk’2, these spheres may have nonmeans of embedded spheres S2k+1 C Mt + , these spheres may have nontrivial in Mfkf2, the non-trivial non-trivial element element [v] [v] trivial normal normal bundles bundles in Mik+2, corresponding corresponding to to the of the first non-stable homotopy group r2k(S&k+i): of the first non-stable homotopy group 7r2k(S02k+d:

[v] EE r2k(SOZk+l) 7r2k(S02k+l)

[v]

”~ 7lj2, 7744

# 0,1,3. 0,173. k/tZ -1

(These groups were earlier following following the discussion of of Bott Bott periperi(These groups were encountered encountered earlier the discussion odicity in 52 was remarked remarked that that itit is is their their non-triviality non-triviality that that §2 above, above, where where itit was odicity in gives the non-parallelizability non-parallelizability of of spheres spheres of of dimensions dimensions -11,3,7, # 1,3,7, and and gives rise rise to to the the non-existence of of division algebras of of dimensions # 2,4,8.) 2,4,8.) Thus Thus for for aIl all the non-existence division algebras dimensions -1 Ic # 0, 1,3, to each element k -1 0, 1, 3, to each element Z z) ~ 2 7r2k+1(M4k+2) 7T2k+@‘f4k+2) zE E i&k+l(M4k+2; H 2 k+1(M 4k+2; 7l) there element there corresponds corresponds an an element G(z)

=

[u]

E ~2k(S02k+l)

z

z/2,

the following "Arf “Arf identity" identity” the Arf Ar! function. function. The The following @(Zl + z2) = @(Zl) + @(z2) +

4>(Zl

+ zz)

4>(zd

+ 4>(Z2) + ZlZl 00 z2 Z2

(mod 2) (mod 2)

holds the Arf function @. For kk = 0, 1,3, the the Arf Arf function function may may be be defined defined holds for for the Arf function 4>. For ::::: 0,1,3, alternatively in terms of framed normal bundles on spheres S2”+l C M4k+2 c alternatively in ter ms offramed normal bundles on spheres S2k+l C M4k+2 C RN. (This approach was taken in $3 above, in the case k = 0.) Much as ]RN. (This approach was taken in §3 above, in the case k = O.) Much as described in $3 (see (4.8) there), the Arf function determines the A+invariant described in §3 (see (4.8) there), the Arffunction determines the ArJ-invariant @ : %+4k+2(SN)

4> : 7rN+4k+2(SN)

-

212.

7l/2. The “Arf-invariant problem” then consists in determining this homomorphism The "Arf-invariant problem" then consists in determining this homomorphism for the various k. It is nontrivial in the casesk = 0, 1,3, on the stable groups for the various k. It is nontrivial in the cases k 0,1,3, on the stable groups nN+2(SN), r~+s(S~), rN+14(SN). In the late 1950s Kervaire showed that 7rN+2(SN), 7rN+6(SN), 7rN+14(SN). In the late 1950s Kervaire showed that it is trivial when k = 2; this result then enabled him to construct a loit is trivial when k = 2; this result then enabled him to construct a 10dimensional PL-manifold not homotopy equivalent to any smooth manifold, dimension al PL-manifold not homotopy equivalent to any smooth manifold, and to show that bPl” g Z/2 (bP1’ c Og). For k = 0, 1,3 we have bP2 c and to show that bPlO ~ 7lj2 (bp 10 C 8 9 ). For k = 0,1,3 we have bP2 ~ bP6 E bP14 Z Z/2. It was then shown by Kervaire and Milnor that @ = 0 bp 6 ~ bP14 ~ 7lj2. It was then shown by Kervaire and Milnor that 4> a precisely if bP4kf2 2 Z/2. Somewhat later, in the mid-1960s it was proved precisely if bP4k+2 ~ 7lj2. Somewhat later, in the mid-1960s, it was proved by Anderson, Brown and Peterson that in fact the Arf-invariant @ is zero in fact proof the Arf-invariant is zero in by Anderson, all dimensionsBrown of the and formPeter n = son 8k +that 2; inintheir they used the4> idea (due aIl dimensions of the form n = 8k + 2; in their proof they used the idea (due to Brown and Novikov in the early 1960s) of extending the Arf-invariant @ to aBrown and Novikov in the early 1960s) group: of extending the Arf-invariant 4> to homomorphism of the SU-cobordism to a homomorphism of the SU-cobordism group: --->

§4. Classification theory of of smooth Classification problems problems in in the the theory smooth manifolds manifolds

.. su

.ft8k+2


--1>

235 235

2':: / 2.

homotopy groups of spheres The image of of the homotopy spheres in 0:” n~u turns turns out to be describable in quite quite simple terms, and itit is this this which which yields the result. For of the the form form 8k + 6, 6, this via ordinary ordinary cobordism theory 8k + this approach approach via cobordism theory For nn of does not succeed. However constructing a cobordism theory theory which which does not succeed. However by by constructing a special special cobordism maximally extends the Arf-invariant, Browder was able to show (in the late maximaUy extends the Arf-invariant, Browder was able to show (in the late 1960s) zero for all n form 22”8 - 2, 2, and that the the Arf-invariant Arf-invariant @ P is is zero for aU n not not of of the the form and 1960s) that moreover for n - 22 itit is is non-zero if and only ifif the elements that for n == 22”8 ~ non-zero if and only the elements moreover that 2 h;-, E hS-l E

8

2 2 Ext;;2(Z/2, Z/2) Ext.4; (2'::/2,2'::/2)

are cycles cycles with Adams differentials representing are with respect respect to to all aU the the Adams differentials (therefore (therefore representing nontrivial of the the group 7rN+2s-2(SN)). In In particular, in the the cases cases particular, in nontrivial elements elements of group KN+28_2(SN)). ss = = 5,6, in fact fact cycles cycles with respect to to all Adams 5,6, the the elements clements hz, ha, h$ hg are are in with respect aU Adams differentials, so bp3’ = 0, so that that bp30 0, bp6’ bp62 = 0.7 0. 7 differentials, There is another naturally occuring group rr”,n , related to en, On, obtained obtained as as related to There is another naturaUy occuring group a quotient of the group of connected components of the space Diff+(S+‘) a quotient of the group of connected components of the space Diff+(sn-l) of orientation-preserving diffeomorphisms diffeomorphisms ff :: S”-l S”-l under compoof orientation-preserving sn-l 4 ---> sn-l un der composition, by factoring by the normal subgroup of those diffeomorphisms that sition, by factoring by the normal subgroup of those diffeomorphisms ff that extend to diffeomorphisms 1c, of the disc D” bounded by S+‘: extend to diffeomorphisms 'Ij; of the disc Dn bounded by sn-l: II, : Dn +

Dn,

T,!~~.&-I= f.

We following sequence sequenceof whose composite composite we We thus thus have have the t.he foUowing of homomorphisms, homomorphisms, whose we denote by ji: denote by p: p : ~o(Diff+(S”-l))

+

r”

+

0” 5

7rn+N(SN)/Im

J.

(Here the homomorphism r” n + 0” is defined by associating with each diffeo(Bere the homomorphism r en is defined by associating with eaeh diffeomorphism f : Sp-’ 1 + S,“-‘,1 the homotopy n-sphere obtained by identifying morphism f : Sr- ---> S2- , the homotopy n-sphere obtained by identifying the boundaries of two discs Dy, Dg via that diffeomorphism.) the boundaries of two discs Dï, D 2 via that diffeomorphism.) It follows from results of Smale that rnn g 0” for n > 5, since for these n It follows from results of Smale that ~ en for n ~ 5, sinee for these n all homotopy spheres Mn can be constructed from discs Dy , 0; by identifyaU homotopy spheres Mn ean be eonstrueted from discs Dï, D 2 by identifying their boundaries by means of a diffeomorphism f : S;-’ 1 + ST-‘.1 The ing their boundaries by me ans of a diffeomorphism f : Sr- ---> S2- • The triviality of the groups rl, r22 is easy. That r33 = 0 follows from Smale’s result triviality of the groups rI, r is easy. That r = 0 follows from Smale's result (of the late 1950s) according to which Difff(S2) is homotopy equivalent to (of the late 1950s) aeeording to which Diff+(S2) is homotopy equivalent to SOs. That r44 = 0 follows from the (highly non-elementary) result of Cerf (of S03' That r = asserting 0 follows from the (highly non--elementary) result of Cerf (of the mid-1960s) that no(Diff+(S3)) = 1. On the other hand it folthe mid-1960s) asserting that KO (Di f f+ (S3)) = l. On the other hand it follows from Milnor’s example (see $3 above) that ~~o(Diff+(S”)) # 1. Putting lows from Milnor's example (see §3 above) that Ko(Diff+(S6)) :f. l. Putting these results together (with others) one has: these results together (with others) one has: rl = r2 = p = r4 = p = p = 0

r

rI

r2

r3

r7 = z/28,

r4

r5

rn = On,

r6

0

n # 3.

7The latter result is due to Barrat, Mahowald and J. D. S. Jones; they proved that hz due ta Barrat, Mahowald and J. D.in S. provedsequence. that h~ is a7The cycle latter of all result Adams isdifferentials by direct computation the Jones; Adams they spectral is of ail Adams differentials computation the cycles Adamsforspectral It ais cycle not known (as of 1994) whether byordirect not the elements hiin are j 2 6. sequence. It is not known (as of 1994) whether or not the elements are cycles for j 2: 6.

h;

236 236

Chapter Smooth Manifolds Chapter 4.4. Smooth Manifolds

The fiber The fiber bundle bundle projection projection

p : Diff+(Dq)

-

Diff+(SQ-l),

((4.6) 4.6)

and the inclusion inclusion map map and the K : Diff+(Dq,

sq-1)

--+ Difff(sy,

are fibre Fq are defined defined in in the the natural natural way. way. The The fibre Fq of of the the fibre fibre bundle bundle (4.6) (4.6) is is the the group diffeomorphisms fixing Sqel). One group of of diffeomorphisms fixing the the boundary: boundary: Fq Fq == Diff+(P, Diff+(Dq,Sq-l). One has of homotopy has the the usual usual induced induced homomorphisms homomorphisms of homotopy groups: groups: 7r@iff’(sq-l)) 8a :: 7rj(Dif f+ (Sq-l)) ----+ 7r_l(Diff+(Dq, 7rj-l (Dif f+ (Dq, S-l)), Sq-l)), K* fi. :: 7rj-@fff(lY, 7rj-l(Diff+(Dq, sq-1)) Sq-l)) ----+ 7r-lpiff+(Sq)). 7rj-l(Diff+(SQ)). Composing 2j Composing 2j times, times, we we obtain obtain aa homomorphism homomorphism

(fi* a 8) a ... a (fi* a 8),

À

x : ?rj(Diff+(s”-l))

-

7ro(Diff+(Sq+y).

For with the Milnor-Kervaire For the the composite composite of of X À with the Milnor-Kervaire above): above): p : 7ro(Diff+(Sq-l)) 7r&liff+(sq-l)) - rq+dSN)/Im il: ---+ 7rq+N(SN)jlm

homomorphism homomorphism ,G il (see (see J~,(SON), J7r q(SON),

the shown to stable homotopy the following following formula formula can can be be shown to hold ho Id in in the the ring ring of of stable homotopy groups of spheres (or in the cobordism ring L’F’ of framed manifolds) groups of spheres (or in the cobordism ring n!,r of framed manifolds) for for all aIl elements a E 7rc(Diff+(SN-‘)), b E 7rq(SO~): elements a E 7ro(Diff+(SN-l)), b E 7r (SON):

q

p(X(ub~-~)) 1

p(À(aba-

))

= p(a) o J(b)

= p(a) a J(b)

ah-1

(modJn,+N(SON)),

(modJ7rq+N(SON )),

E 7rq(Difff(P-1)).

(4.7) (4.7)

In certain dimensions the structure of the ring In certain dimensions the structure of the ring

n!:r =

L 7rN+j(SN) j?::O

is such that the right-hand side of (4.7) turns out to be non-zero (for approis such that the right-hand side of (4.7) turns out to be non-zero (for appropriate a,b); this occurs for instance when q = 1,n = 8 - see the table at occurs for q from 1, n this 8 that see at priate the enda,ofb);$7this of Chapter 3. instance It can by when inferred for the suchtable N the the end of §7 of Chapter 3. It can by inferred from this that for such N the component of the identity of the space Diff+(SN-‘) is not contractible to component of the identity of the space Dif f+(SN-l) is not contractible to the subgroup SON (Novikov, in the early 1960s).8 the subgroup SON (Novikov, in the early 1960s).8 The general classification theory of closed, simply-connected-manifolds of The general closed, simply-connected-manifolds of dimension n 2 classification 5, and then theory of the of non-simply-connected ones, depends on dimension n .:::: 5, and then of the non-simply-connected ones, depends on certain important, although elementary, properties of maps certain important, although elementary, properties of maps 8Proved

slightly

later

also by Milnor.

BProved slightly later also by Milnor.

54. Classification Classification problems problems in the the theory theory of smooth smooth manifolds manifolds §4.

237 237

f:Mî-M 2 of degree degree one one between between closed closed manifolds. manifolds. Since Since for for such such maps maps (of (of degree degree one) one) the the of 1 homomorphism f*D-‘f*D (where D is the Poincarkduality isomorphism) homomorphism f*D- f* D (where the Poincaré-duality isomorphism) is the identity map on the homology of M,“, that homology homology admits admits aa standard standard the identity map on the homology of Mî, that decomposition decomposition H,(Ml)

E (D-‘~*D)(H,(Mz))

@Ker

f*,

(4.8) (4.8)

which orthogonal with with respect respect to to the the intersection of cycles cycles (and (and analogously analogously which is is orthogonal intersection of for the cohomology of MF). The direct summand Ker f* of the homology of for the cohomology of Mï)· The direct summand Ker f* of the homology of M,” is closed under the Poincar&duality operator, and in general behaves Mï is closed under the Poincaré-duality operator, and in general behaves as represented the of sorne some manifold. Thus if if the the kernel kernel as ifif itit itself itself represented the homology homology of manifold. Thus 7rj 7rj Ker rj(MF) of the induced map map f*(n’) ) of of the homotopy group group is is of the induced the jth jth homotopy Ker fiaJ) ) cC 7rj(Mï) trivial for for jj < < k Ic -- 1, 1, then then for for the the kth kth such such kernel kernel the the analogue analogue of of Hurewicz' Hurewicz’ trivial theorem valid: theorem is is valid: Ker fLRk) E Ker fiH’), ((4.9) 4.9)

fi

fi

and the the induced the respective respective homotopy homotopy groups: groups: induced homomorphisms homomorphisms between between the and f* : ri(M;)

-

ri(M;),

are isomorphisms for ii < < Ic. result is is true also for are isomorphisms for k. This This result true also for non-simply-connected non-simply-connected manifolds provided the homology groups are considered as Z[7r]--modules Z[n]-modules manifolds provided the homology groups are considered as where rl(MF) 2 nl(M;) E 7r. Under the same assumption as before, namely where 7rl(Mï) ~ 7rl(Mr) ~ 7r. Under the same assumption as before, namely that the kernels of the induced homomorphisms of the 7ri should be trivial for that the kernels of the induced homomorphisms of the 7ri should be trivial for i < k - 1, one has the isomorphism i < k - l, one has the isomorphism Ker fLHk) ?Z Ker fLRk) @~I~] Z

(4.10) (4.10)

(provided the induced homomorphism of the fundamental groups 7rr has trivial (provided the induced homomorphism of the fundamental groups 7ri has trivial kernel). kernel). The classification theory of manifolds of given homotopy type was develThe classification theory of manifolds of given homotopy type was developed by Novikov in the early 1960s and of all homotopy types of closed smooth oped by Novikov in the early 1960s, and of aU homotopy types of closed smooth manifolds by Browder and Novikov (also in the early 1960s). In the course manifolds by Browder and Novikov (also in the early 1960s). In the course of solving these classification problems one encounters the situation where a of solving these classification problems one encounters the situation where a homotopy type of manifolds is represented by a particular CW-complex X. In homotopy type of manifolds is represented by a particular CW-complex X. In the most general context, as formulated by Browder, X need not be a manthe most context, as formulated Browder, not be manifold and general may have geometric dimension bydifferent fromX need the given n, aand it ifold and may have geometric dimension different from the given n, and is required only that there exist a fundamental class [Xn] E H,(Xn; Z) suchit is required that there a fundamental class [xn] E Poincare Hn(xn; Z) such that the caponly operation a --+ exist a n [Xn] defines an appropriate isomorthat the cap operation a an [xn] defines an appropriate Poincaré isomorphism from homology to cohomology; Cl&‘-complexes with this property are phism Poincare’ from homology to cohomology; -complexes this property are called complexes. It turns outCW that a Poincarewith complex X admits caUed Poincaré complexes. It turns out that a Poincaré complex X admits a unique stable “normal spherical bundle” vx which behaves like the usual a unique stable "normal l/x which behaves the usual stable normal bundle of aspherical manifold, bundle" and in the case where X is like a smooth (or stable normal bundle of a manifold, and in the case where X is a smooth (or PL-) manifold vx is just the stable homotopy class of its standard normal P L-) manifold l/x is just the stable homotopy class of its standard normal bundle. If there exists a homotopy equivalence f : Mn -+ X, where M” is a bundle. If there exists a homotopy equivalence f : Mn - X, where Mn is a

238 238

Chapter 4. Smooth Smooth Manifolds Chapter 4. Manifolds

manifold, then then there there must must exist exist aa vector vector bundle bundle ç c over over X X which which is is homotopy homotopy manifold, equivalent to the the stable stable normal normal spherical spherical bundle bundle Vx vx (if (if 9g :: X X ---t -+ Mn IW’ is aa hohoequivalent motopy inverse inverse of of J, f, then then ç 5= = g*VM). g*vM). Thus Thus the the classification classification problems problems above above motopy lead to to the the consideration consideration of of aa Poincaré Poincare complex complex X X together together with with aa vector vector lead bundle ç < over over X X which which is is homotopy homotopy equivalent equivalent to to the the stable stable normal normal spherical spherical bundle bundle vx. VX. However However when when X isis aa smooth smooth closed closed manifold, manifold, the the vector vector bundle bundle bundle homotopy equivalent equivalent to the the ordinary ordinary normal normal bundle bundle vx, vx, and and the the problem problem ç< is homotopy of determining determining the the latitude latitude in in such such vector vector bundles bundles ç c can can be be solved solved effectively effectively of (Novikov, early early 1960s). 1960s). (Novikov, Assuming that that there there is is aa vector vector bundle bundle ç 6 over over aa given given Poincaré Poincard complex complex Assuming X, a degree one map f : Mn + X is called normal if the normal bundle UM X, a degree one map J : Mn ---t X is called normal if the normal bundle VM is equivalent (as a vector bundle) to f*E, i.e there is a commutative diagram: is equivalent (as a vector bundle) to 1* ç, i.e there is a commutative diagram:

(4.11) (4.11)

M --'-- X In the kernel kernel Ker Ker J* f* behaves with respect respect to to both both homology homology this situation situation the behaves with In this and homotopy as in the case of parallelizable manifolds, which circumstance and homotopy as in the case of parallelizable manifolds, which circumstance permits natural analogue analogue of of the Milnor-Kervaire technique to to permits the the use use of of aa natural the Milnor-Kervaire technique reduce the kernels of the induced maps of homotopy groups by means of reduce the kernels of the induced maps of homotopy groups by means of Morse surgeries. For a given manifold (or, more generally, Poincare complex) Morse surgeries. For a given manifold (or, more generally, Poincaré complex) X” bundle ç ( over over Xn, one defines bordism classes classes of of xn and and aa given given vector vector bundle X n , one defines normal normal bordism normal maps f : M” + X”, f*< = UM, covered by bundle maps f^ : UM + 5, normal maps J : }YIn xn,1*f, VM, covered by bundle maps VM ---t ç, where VM is the stable normal bundle of M” C Rn+N, as follows: A normal where VM is the stable normal bundle of Mn c JR.n+N, as follows: A normal bordism is a normal map g covered by a map 4 of bundles, as indicated: bordism IS a normal map 9 covered by a map g of bundles, as indicated:

J:

(4.12) (4.12) T

f

N n+l

where the manifold Nn+l where the manifold Nn+l of lRWn+N x I transversely, of JR.n+N X 1 transversely,

L

c

M;: C

x I

c lRnfN x I, I = [a, b], approaches the boundary C JR.n+N X l, 1 [a, b], approaches the boundary and itself has boundary and itself has boundary aN”+l = M,” u MC,

âNn+l

MF

X”

IRn+N x {a},

JR.n+N

X

= Mna u Mnb'

{a},

M;

c

IRWnfN x {b} ,

MT: C JR.n+N x {b},

on each component of which there is given a normal map of degree one. We on eachthe component there is classes given a bynormal We denote resulting ofsetwhich of bordism N(Xn, map 0. of It degree can beone. shown denote the resulting set of bordism classes by N(xn,ç). It can be shown

§4. Classification Classification problems problems in the the theory theory of smooth smooth manifolds manifolds §4.

239

without difficulty difficulty that that the the set set N N(Xn, I) can can be realized realized as the the subset subset of those those without (X n , ç) elements (Y of the the homotopy homotopy group group 1r T~+N(T[) of the Thorn space of the bundle elements CI' of +N(Tç) of the Thom space of the bundle n <, homo homologous to the the fundamental fundamental class class [X [X”]n ] (see above), above), Le. i.e. such such that that Ç, logo us to

(4.13) where where H: 1rj(Tç)

Hj(Tç;Z) is and riqS H,(Xn; n ; Z) Z) --+ -+ Hn+N(Tç; Hn+~(T[; Z) is is is the the Hurewicz Hurewicz homomorphism, homomorphism, and > : Hn(X Z) the Thorn isomorphism. isomorphism. Cartan-Serre Cartan-Serre theory theory shows shows that that the the stable stable group group the Thom r,+N(T<) 1r n +N(Tç) has has the the form form --+

(4.14) (4.14) where fl(Xn, [) Ji! (X n , ç) where normal bordism normal bordism lows: lows:

is finite finite abelian; with respect respect to this decomposition decomposition the the above above is abelian; with to this classes of normal maps of degree one are represented as folclasses of normal maps of degree one are represented as folo! E N(xn,<), Cl'EN(Xn,ç),

a + a, aEJi!(Xn,Ç). a E Ai(X”,<). (4.15) CI' = 1 l+a, (4.15) In situation where where Xn is a manifold and < is is its its stable stable In the the situation Xn = = IP Mn is a smooth smooth manifold and ç normal bundle UM (M” c lRn+N ), the identity element 1 of ~TT,+N(T<) normal bundle VM (Mn c JR.n+N), the identity element 1 of 1rn+N(Tç) correcorresponds of identity sponds to to the the pair pair of identity maps maps id

id VM VM VM - -..... , VM

and assertions are and the the following following assertions are valid: valid: 1. Suppose fl : Mr -+ Mn, f2 : MF + Mn are maps of degree one which 1. Suppose h : Mf --+ Mn, 12 : Mlf --+ Mn are maps of degree one which together with the induced maps of normal bundles determine one and the same together with the induced maps of normal bundles determine one and the same normal bordism class. If in addition each of the maps fi, f2 is a homotopy normal bordism class. If in addition each of the maps h, 12 is a homotopy equivalence, then there exists a homotopy n-sphere En E bPnfl such that equivalence, then there exists a homotopy n-sphere En E bPn+l such that (4.16) M; ” Cn#M;. (4.16) It follows that for n 2 5 the manifolds MT- and MC are homeomorphic, with It follows that for 11, ?:: 5 the manifolds Mf and Mlf are homeomorphic, with the punctured manifolds obtained by removing a point from each actually difthe punctured manifolds obtained by removing a point from each actually diffeomorphic, and that for even n > 5, Mr and MC are in fact diffeomorphic feomorphic, and that for even 11, > 5, Mjl and Mlf are in fact diffeomorphic (Novikov). (Novikov). 2. The following result applies to an arbitrary fixed vector bundle < over 2. The following result applies to an arbitrary fixed vector bundle ç over Mn: For odd n > 5 each normal bordism class Q E n/(M”,[) is realized by Mn: For odd 11, ?:: 5 each normal bordism class E N (Mn, ç) is realized by TheCI' following condition on the some homotopy equivalence f : M’ + M”. some homotopy equivalence f : M' --+ Mn. The following condition on the Pontryagin classes of the vector bundle [:

Pontryagin classes of the vector bundle L(pl(J),

.

, Pi

ç:

= r(Mn)

240 240

Chapter 4. Smooth Smooth Manifolds Manifolds Chapter

necessary and sufficient suficient for be true also also in the the case case n == 4k 4k is both necessary for this to be (Browder, Novikov). Novilcov). (Browder, Prom the first first result one one obtains immediately immediately the corollary corollary that that for each each From given homotopy type there exist only finitely many pairwise non-diffeomorphic homotopy type only finitely non-diffeomorphic closed, simply-connected simply-connected n-manifolds n-manifolds (n ;::: 2 5) 5) with with the same same Pontryagin Pontryagin closed, 4 classes pj E H4j(Mn. Q). I nvoking the topological invariance of the Pontryaclasses Pj E H j (Mn; Invoking topological invariance Pontryagin classes, one infers’from this in turn the finiteness of the number of distinct gin classes, one infers from this in turn the finiteness of the number of distinct smooth structures that exist on each closed, simply-connected topological smooth structures that exist on each closed, simply--connected topological manifold of 2 5. 5. Since Since this this theory theory is is valid valid also also for for PL-manifolds, PL-manifolds, manifold of dimension dimension nn ;::: one obtains also the finiteness of the number of distinct PL-structures definone obtains also the finiteness of the number of distinct PL-structures definable on any such topological manifold. able on any such topological manifold. Remark. Note Note in in this this connexion connexion that that in in the the analogue analogue of of the the first first result result Remark. for PL-manifolds, one obtains an incidental strengthening of the conclusion, for PL-manifolds, one obtains an incidental strengthening of the conclusion, namely to to the the effect effect that the manifolds manifolds M; and Ml] I$? will will be be in in fact fact P PLthat the Mf and Lnamely homeomorphic, in view of the fact that all homotopy n-spheres are PLhomeomorphic, in view of the fact that all homotopy n-spheres are P Lhomeomorphic homeomorphic to to Sn. 0I3

sn.

An exact classification of diffeomorphism classes classesof of manifolds manifolds satisfying satisfying An exact classification of the the diffeomorphism the hypotheses of Result 1 above (i.e. of the same homotopy class, and with with the hypotheses of Result 1 above (i.e. of the same homotopy class, and maps to M” whose extensions to the stable normal bundles define the same maps to Mn whose extensions to the stable normal bundles define the same normal is obtained obtained by by taking the set N(Mn, of elements normal bordism bordism class) class) is taking the set N (Mn, /Jv,) n ) of elements of the form 1 + a E KN+~(Tv,) modulo the subgroup G(Mn, vn) n of those those of the form 1 + a E 7rN+n(T/Jn ) modulo the subgroup Çj(M , /Jn ) of elements representing homotopy classes of automorphisms of degree i.e. elements representing homotopy classes of automorphisms of degree one, one, Le. of self-maps f : M” + M” of degree one, preserving the stable normal of self-maps f : Mn Mn of degree one, preserving the stable normal bundle: f*v, = u,. Thus the diffeomorphism classesof manifolds satisfying bundle: f*/Jn = /Jn . Thus the diffeomorphism classes of manifolds satisfying the above Result are in in natural natural one-to-one correspondence the hypotheses hypotheses of of the the above Result 1 1 are one-to-one correspondence with the cosets of with the cosets of (4.17) N(Mn, bL)/G(M”, &I)

(4.17)

There is a smaller group J(M”, vn), a subgroup of the automorphism group There is a smaller group :J(Mn, /Jn ), a subgroup of the automorphism group O(Mn, vn), consisting of those automorphisms of the bundle V, inducing the Çj (Mn, /Jn), consisting of those automorphisms of the bundle /Jn ind ucing the identity map on the base Mn. The orbit set identity map on the base Mn. The orbit set (4.18) (4.18) is relevant to the following question: Given a normal homotopy equivalence is relevant to the following question: Given a normal homotopy equivalence f : MT- + M” of degree one, when can f be deformed to a diffeomorpism i ; Mf ---> Mn of degree one, when can f be deformed to a diffeomorpism (more precisely, to a diffeomorphism up to forming the connected sum with (more precisely, to a diffeomorphism up to forming the connected sum with a Milnor sphere C” from the group bP n+l)? The answer is that a normal a Milnor sphere En from the group bPn+1)? The answer is that a normal bordism defined by such a map f is so deformable if and only if its projection bordism defined by such a map i is so deformable if and only if its projection to the coset (4.18) is the same as the projection of the distinguished element to the coset (4.18) the same the+projection of the distinguished (represented by the isidentity mapasMn Mn). Futhermore if fi : MF element + Mn the identity map Mn ---> Mn). Futhermore if fI : Mf Mn (represented by and f2 : MF + M” are normal maps of degree one (representing normal and h :classes), Ml] ---> then Mn they are normal mapsto of normal bordism correspond thedegree same one coset(representing in (4.18) if and only then they correspond to the same coset in (4.18) if and only bordism classes), if f1fi-l 1 is homotopic to a diffeomorphism (modulo bP”+‘). if id:; is homotopie to a diffeomorphism (modulo bP n+ 1 ).

54. §4. Classification Classification problems problems inin the the theory theory ofof smooth smooth manifolds manifolds

241 241

In In the the situation situation where where the the given given vector vector bundle bundle <ç isis the the stable stable normal normal bundle bundle of of M” Mn cc sN+n, sN+n, 5ç == v,, Vn , the the Thorn Thom space space TV, TVn has has particular particular geometric geometric signifsignificance: é, let let U, UE; be be the the E-neighbourhood é-neighbourhood of of the the manifold manifold icance: For For aa suitably suitably small small E, M” Mn cC SN+n; SN+n; then then by by definition definition we we have have TVM = SN+“/(SN+“\UE), the N+n obtained the quotient quotient space space of of SsN+n obtained by by identifying identifying to to aa point point the the complecomplement of the neighbourhood U, (h ere N >> r~). The associated natural map ment of the neighbourhood Us (here N » n). The associated natural map SN+n sN+n --+ --; TV, TVn thus thus restricts restricts to to the the identity identity map map on on U,; Us; itit has has degree degree one one and and represents the manifold Mn. For n of the form n = 4k 1('n+N (Tvn )) the manifold Mn. For n of the form n 4k ++ 22 itit represents (in (in r,+N(Tv,)) turns where aa ranges turns out out that that the the elements elements of of x,+N(Tv,) 1('n+N(Tvn ) of of the the form form 11 ++ aa where ranges n v,) of index 1 or 2, are over over aa certain certain subgroup subgroup BB of of N(M”, N(M , vn ) of index 1 or 2, are precisely precisely those those corresponding corresponding to to n-manifolds n-manifolds of of the the same same homotopy homotopy type type as as M”. Mn. (The (The spaces spaces TV, TVn are are often often useful useful for for carrying carrying out out calculations calculations for for particular particular manifolds.) manifolds.) The above-described theory has been generalized to the The above-described theory has been generalized to the situation situation of of manifolds-with-boundary by Go10 and Wall (in the mid-1960s). manifolds-with-boundary by Golo and Wall (in the mid-1960s). Around authors9 9 demonstrated usefulness of Around the the mid-1960s mid-1960s several several authors demonstrated the the usefulness of the concept of “S-duality”, which has a more general categorical character. the concept of "S-duality", which has a more general categorical character. Spanier Spanier and and G.W. G.W. Whitehead Whitehead defined defined the the S-dual S-dual DK DK of of aa CW-complex CW-complex K K embedded in a sphere S” of sufficiently high dimension, to embedded in a sphere sm of sufficiently high dimension, to be be simply sim ply the the complement \ K; thus as follows: follows: complement S” sm\K; thus the the operator operator D D acts acts on on CW-complexes CW-complexes K K as D:KwSm\K. In S-dual of of aa sphere sphere is is aa sphere sphere (up to homotopy In particular particular the the S-dual (up to homotopy equivalance). equivalance). On (homotopy classes of) maps f : K ---+ L the S-dual operator reverses the On (homotopy classes of) maps f : K --; L the S-dual operator reverses the arrows (this is clear if f is an embedding, and the general case is reduced to arrows (this is clear if f is an embedding, and the general case is reduced to this by means of the mapping cylinder):

this by means of the mapping cylinder):

Df:DL--+DK, Df: DL --; DK,

[K, Lls = PL, DKls, [K, Lls [DL, DKls, where [ , Is denotes the set of stable homotopy classes of maps. It was observed where [, 1s denotes the set of stable homotopy classes of maps. It was observed by Atiyah in the early 1960s that the S-dual of the Thorn space of the stable by Atiyah in theof early 1960s that theMnS-dual the Thom of the vector stable normal bundle a closed manifold is theofThorn space space of a trivial normal bundle of a closed manifold Mn is the Thom space of a trivial vector bundle EQ over Mn: bundle éq over Mn: D(TvM) = C(MT) = TE’, D(TvM) = E(M~) = Té q , where MT = M” u {*} (the disjoint union with the one-point space). The wheref M+ u {*}turns (the out disjoint union with the map : SN+= Mn ----+ TVM to have as S-dual theone-point map Df :space). TE’J --+The SQ map f : sN+n --; TVM turns out to have as S-dual the map Df : Té q sq from the Thorn space of the trivial vector bundle e’J to Sq, of degree one on from fibre, the Thom the point trivialof vector bundle éq to sq, of degree one on each varyingspace with ofthe the base as parameter.

each fibre, varying with the point of the base as parameter. gA. Schwarz, 9 A.

Novikov,

Sullivan

Schwarz, Novikov, Sullivan.

242 242

Chapter 4. 4. Smooth Smooth Manifolds Manifolds Chapter

Denote by SG SG,q the the semigroup semigroup of of maps maps sq SQ---+ SQof of degree degree one. one. We We have, have, Denote - t sq each normal normal bordism bordism class class0:o E E N(Mn, N(M”, 0, t), aa map map from from the the corresponding to each base manifold Mn M” to the semigroup semigroup SG SG,:q : base D-f : M” +

SG,,q , SG

(q (4 --- t (0). m).

The image image under under this this map map in in the the quotient quotient SGqjSO SG,/SO then then constitutes constitutes an an obobThe struction to to the the existence existence of of aa diffeomorphism diffeomorphism between between the the various various manifolds manifolds struction Mp in in the the class class0:. o. Note Note that that as asqq --f 00, CQ,the the homotopy homotopy properties properties of of SG SG,q (tak(takMf ing aeeount account of of its its structure structure as as an an H H-space) stabilize. The The natural natural embeddings embeddings -spaee) stabilize. ing give ri rise to aa spectrum spectrum give se to SG,q C c SG SG,+l ... SG q+1 Cc ... with limit limit SG, SG, say. say. We We also also have have embeddings embeddings with SG > SPL SPL ::J > SO, SO, SG::J where SPL the group group of of germs germs of of origin-fixing origin-fixing and and orientationorientationwhere S P L denotes denotes the preserving PL-automorphisms of a disc of sufficiently high dimension. The preserving P L-automorphisms of a dise of sufficiently high dimension. The induced monomorphism rj (SO) --+ rj (SG) of homotopy groups coincides induced monomorphism 1['j(SO) - - t 7rj(SG) of homotopy groups coincides with the (see §3 53 above) above) once once one one makes makes the the with the Whitehead Whitehead homomorphism homomorphism JJ (see identification afforded by the isomorphism identification afforded by the isomorphism TV

= TV+#‘~),

N>j+1.

One gives to the the universal universal base (i.e. with with One gives meaning meaning to base BSG BSG for for spherical spherical bundles bundles (i.e. spherical fibres) by defining equivalence of such bundles in terms of bundle spherlcal fibres) by defining equivalenee of su ch bundles in terms of bundle maps the projections, and of of degree degree one one on on the the fibres. Almaps commuting commuting with with the projections, and fibres. Although the space SG, clearly has the homotopy type of @(S’J), the connected though the space SG q clearly has the homotopy type of [2q (sq), the connected component of the identity of the q-dimensional loop space, so that component of the identity of the q-dimensional loop space, so that nj(flq(Sq))

E nq+JSq) ” rj(SG),

the multiplicative structures of these two spacesare different. Note that the the multiplicative structures of these two spaces are different. Note that the space BSG does not have the homotopy type of any (q - 1)-loop space space BSG does not have the homotopy type of any (q ~ l)-loop space m-l(Y). [2q-l(y). This theory transfers to the context of PL-manifolds without significant This theory transfers to the context of PL-manifolds without significant change. The tangent and stable normal bundles of a manifold, equipped change. The tangent and stable normal bundles of a manifold, equipped with the group of germs of origin-fixing PL-automorphisms of discs of unwith the group of germs of origin-fixing P L-automorphisms of discs of unspecified dimensions, are called, following Milnor, microbundles. A theorem specified dimensions, are called, following Milnor, microbundles. A theorem proved by several authors in the early 1960s asserts that a PL-manifold proved by several authors in the early 1960s asserts that aPL-manifold admits a compatible smooth structure if and only if the stable normal (or admits compatible smooth if and only if the (or tangent) a microbundle reducesstructure to a vector O-bundle, 0 cstable PL normal (or vector tangent) mlcrobundle reduces to a vector O-bundle, 0 c PL (or vector SO-bundle, SO c SPL, in the orientable case). The relative homotopy SO-bundle, c SPL, in the orientable relative homotopy groups nj(SPL,SO SO), representing obstructionscase). to theThe reducibility of a PL7rj(SPL, SO), representing obstructions to the reducibility of aPj-l: PLgroups microbundle to a smooth one, are isomorphic to the respective groups microbundle to a smooth one, are isomorphic to the respective groups r j - 1 : (4.19) 7rj(SPL, SO) = 7rj(SPL/SO) = I+l. (4.19)

§4. Classification Classification problems problems in in the the theory theory of of smooth smooth manifolds manifolds §4.

243 243

From this this isomorphism isomorphism one one obtains obtains the the obstructions obstructions to From to the the existence existence of of a smoothness stucture stucture in in a neighbourhood neighbourhood of of the the j-dimensional skeleton smoothness j-dimensional skeleton of of a PL-manifold M M as the the cohomology classes in in Hj(M; PL-manifold cohomology classes Hj(M; rj). rj). The The theory theory of of PLPLmanifolds was was developed developed starting starting from manifolds from the the 1930s 1930s by by many many authors authors (Newman, (Newman, Whitehead, Zeeman, Munkres, Mazur, Mazur, Hirsch Hirsch and and others). Whitehead, Zeeman, Munkres, others). On open region region of of a PL-manifold where bundle is trivOn a small small open PL-manifold where the the tangent tangent bundle trivial, a “local” "local" smooth smooth structure structure can ial, can always always be defined, defined, and and it it is this this fact fact which which allows the the natural natural transference transference to the context of the the entire allows to the context of of PL-manifolds PL-manifolds of entire above-described apparatus of of the the classification theory of of smooth smooth above-described technical technical apparatus classification theory manifolds of of a given given homotopy type, since since the the Morse surgeries eliminating manifolds homotopy type, Morse surgeries eliminating the the kernels of the appropriate appropriate homotopy groups (see above) kernels of the homotopy groups above) are are carried carried out out over over small parallelizab!e parallelizable regions. regions. small into account the fact that in LHowever one one needs needs to However to take take into account the fact that in general general a P PLvector bundle bundle has has more more automorphisms automorphisms acting identically on the base than a vector acting identically on the base than smooth (or SO-) bundle. Hence in the approach via smooth (or SO-) bundle. Hence in the approach via S-duality S-duality the the obstructions obstructions to the existence of of P L-isomorphisms between between manifolds manifolds are by to the existence PL-isomorphisms are represented represented by homotopy classes classes of of maps homotopy maps

Mn --) ----> SGjSPL, Mn SGISPL, rather than maps to to SGjSO. In 1966 it and Wagoner rather than maps SG/SO. In it was was shown shown by by Sullivan Sullivan and Wagoner that the groups 7rj(SGjSPL) (or have that the groups n,-(SG/SPL) (or the th e isomorphic isomorphic groups groups 7rj(GjPL» rj(G/PL)) have and obey obey a nice 4-periodicity: simple structure simple structure and nice 4-periodicity:

~

~ ~/2 z

nj(SG/SPL) .,(SGISPL) = TQ(G/PL) .,(GI PL) = { !& i

0

j j ; j j

=

4k 4k

= 4k + 1 1;; 1; = 4k + 2

= 4k 4k + + 3.

Subsequently, using the the homotopy homotopy structure and Subsequently, using structure of of the the space space SGjSPL,lO SG/SPL,10 and its connexion connexion with Bott periodicity, periodicity, together Sullivan its with Bott together with with other other properties, properties, Sullivan has been been able to establish establish certain certain important important general however up up to has able to general facts, facts, however to the the present (1994) (1994) complete complete proofs of Sullivan's results have have not present proofs of Sullivan’s results not appeared appeared in in print. print. the homotopy are isomorphic As we we have mentioned, the As have mentioned, homotopy groups groups 7rj(SPLjSO) rj(SPL/SO) are isomorphic j - l . The results above imply that that the inclusion homoto the groups rYj-l. to the groups The results above imply that that the inclusion homomorphism morphism 7rj(SO) +----> 7rj(SPL) 7Lj(SO) 7rj(SPL) is in in fact fact an an isomorphism isomorphism for one has the image is for j ::; < 6. For For j = 7 one has that that the image of of 7r7(SO) QT~(SO) in 7r7(SPL) has order order divisible latter result established in xT(SPL) has divisible by by 7. The The latter result may may be established without difficuty by going going over over to to the the inclusion B SO -+ -+ BSPL B S P L of of the without difficuty by inclusion BSO the bases bases of the the universal universal SOSO- and Lbundles, and using the fact that the Pontryagin of and SP SPL-bundles, and using the fact that the Pontryagin class of of a PL-manifold may be fractional with denominator class PL-manifold may fractional with denominator 7 (see (see §3). $3). From From this divisibility divisibility result result one may infer infer the this one may the following following corollary: corollary: lOThe homotopy properties properties of of the SG/SPL are completely described described in “The homotopy the spaces spaces G/PL, G/PL, SGISPL are completely in the book book by by Madsen, Madsen, Milgram, Milgram, Classifying Spa ces in in Surgery and of Manifolds, the Classifying Spaces Surgery and Cobordism Cobordism of Manifolds, Princeton University Press, Annals of mathematics mathematics studies, no. Princeton University Press, 1979, 1979, Annals studies, no. 92. 92.

244

Chapter Smooth Manifolds Chapter 4. Smooth Manifolds

There r-torsion in There exists exists aa CW-complex CW -complex M M (a (a smooth smooth manifold) manifold) having having 7-torsion in its its cohomology group H’(M; Z), and two SO-vector bundles ql, q2 over M, cohomology group H8(M; 2), and two SO-vector bundles 1]1,1]2 over M, such su ch that that 6) ~2(771) (i) P2(r7d #=f. PZ(VZ), P2(1]2), and and (ii) the 71, 772 (ii) the vector vector bundles bundles 1]1, 1]2 are are PL-isomorphic. PL-isomorphic. This implies that integer Pontryagin This result result implies that the the torsion torsion parts parts of of integer Pontryagin classes classes are are not early 1960s). 1960s). not in in general general PL-invariant PL-invariant (Milnor-Kervaire, (Milnor-Kervaire, early As early as 1940s itit had the 1940s had been been established established (by (by Whitney) Whitney) that that provided provided As early as the nn #=f. 2, 2, under under suitable, suitable, simple simple conditions conditions an an immersion immersion S” sn + ---t M2” M 2n can can be be regularly deformed into a smooth embedding. This fails in the exceptional regularly deformed into a smooth embedding. This fails in the exceptional case case underlies 2; this this exceptional exception al case underlies the the cardinal cardinal technical technical difficulties difficulties of of case nn == 2; the theory of simply-connected 4-dimensional manifolds. From the the theory of simply-connected 4-dimensional manifolds. From the end end of of the the 1950s throughout the 1960s 1950s throughout the 1960s various various topologists topologists (Haefliger, (Haefiiger, Stallings, 8tallings, Levine, Levine, and others) developed a theory of embeddings of simply-connected and others) developed a theory of embeddings of simply-connected manifolds manifolds (starting (starting with with homotopy homotopy spheres) spheres) into into Euclidean Euclidean spaces, spaces, using, using, among among other other things the entire apparatus described above for classifying manifolds. things the entire apparatus described above for classifying manifolds. We We shall shaU not expound the this theory the results results of of this theory here; here; aa separate separate essay essay in in the the not however however expound present present series series is is to to be be devoted devoted to to this this topic, topic, and and also also to to the the deep deep theory theory of of 3-manifolds ]R3. 3-manifolds and and knots knots in in R3.

55. in topology. of §5. The The role role of of the the fundamental fundamental group group in topology. Manifolds Manifolds of low dimension (n an open low dimension (n == 2,3). 2,3). Knots. Knots. The The boundary boundary of of an open manifold. The topological invariance of the rational Pontryagin manifold. The topological invariance of the rational Pontryagin classes. The The classification classes. classification theory theory of of non-simply-connected non-simply-connected manifolds of dimension 2 5. Higher signatures. Hermitian manifolds of dimension 2 5. Higher signatures. Hermitian K-theory. K-theory. Geometric Geometrie topology: topology: the the construction construction of of non-smooth non~smooth homeomorphisms. Milnor’s example. The annulus conjecture. homeomorphisms. Milnor's example. The annulus conjecture. Topological and P&structures Topological and PL-structures The fundamental group plays a singularly important role in topology; it The fundamental group plays a singularly important role in topology; it is involved in all of the technical apparatus of the subject, and likewise in is involved in all of the technical apparatus of the subject, and likewise in all applications of topological methods. In fact for low-dimensional manifolds aU applications of topological methods. In fact for low-dimensional manifolds (i.e. of dimension 2 or 3) the fundamental group underlies essentially all non(i.e. of dimension 2 or 3) the fundamental group underlies essentially all nontrivial topological facts. For instance, as the reader will recall, the classification trivial topological facts. For instance, the reader the classification of the self-homeomorphisms of a closedas surface M2,will its recall, homotopy and isotopy of the self-homeomorphisms of a closed surface M2, its homotopy and isotopy classes (i.e. classes of homotopic homeomorphisms), all reduce, according to aU reduce, according classes (i.e. classes of homotopie homeomorphisms), Nielsen, to consideration of the automorphism group of 7ri(M2) 2 taken moduloto Nielsen, to consideration of the automorphism of KI (Mto ) composites taken modulo the subgroup of inner automorphisms, and sogroup ultimately of the subgroup of inner automorphisms, and so ultimately to composites of “elementary” automorphisms. For the orientable surface iVi of genus g, the "elementary" automorphisms. For the orientable surface !vIi of genus g, the automorphism group of ni(Mi), presented as usual in terms of 2g generators automorphism group by of KI (Mi), presented as usual in terms of 2g generators and a single relation and a single relation by

55. The The role of the the fundamental fundamental group §5. role of group in topology topology

245 245

99 al, b1 ,···, a b II aibia;lb;l al,bl,...,ag,bg; 22% z == 1,1 n a.b.aT1bY1 g, g;

i=l i=l

is generated automorphisms (Dehn, 1920s): is generated by by the the following following elementary elementary automorphisms (Dehn, 1920s): ai :

bi H biai;

h H bk,

Pi

ai H aibi;

ak

:

“li :

-

ak7

k # 6 k #

4

a, c) a,;

1 I i 5 g;

b, H b,;

1 < i 5 g;

bi ++ aL&biai;

bi+l H bi+lbia,‘b,‘ai+l;

b, +-+ b,,

4 # i, i + 1;

(5.1)

ai+i H a~~lbiaib,‘ai+lbia,lb~lai+l; ak

5:

-

ak,

aj H bg+l-j;

l
k#i+l,

bj H a,+l-j.

Note CY~,(Ji, pi, yi while 156 reverses reverses it. it. that ai, '"Yi preserve preserve orientation, orientation, while Note that The fundamental group also determines the theory of knots, i.e. of of embedembedThe fundamental group also determines the theory of knots, i.e. dings S1 -+ S3; the knot group determined by such an embedding is dings SI '---t S3; the knot group determined by such an embedding is defined defined as 7Tl 7ri(S3\S1). element a, namely the the class of as (S3\S1). This This group group has has aa distinguished distinguished element a, namely class of the loop S,’ obtained by moving the embedded circle S1 a small distance E in the loop S: obtained by moving the embedded circle SI a small distance € in to S1 in such such aa way way that that the aa direction direction normal normal to SI cC S3 S3 at at each each point, point, in the resultresulting circle Si has zero linking coefficient with the original knot S1 The ing circle S: has zero linking coefficient with the original knot SI cC S3. S3. The boundary of the tubular s-neighbourhood of the knot S1 c S3 is a torus T2,2 boundary of the tubular t:-neighbourhood of the knot SI C S3 is a torus T , and it can be shown that for non-trivial knots the inclusion and it can be shown that for non-trivial knots the inclusion T2 it

(S3\S’)

determines in canonical fashion a subgroup isomorphic to Z @ Z with canondetermines in canonical fashion a subgroup isomorphic to Z El) Z with canonically defined generators (one of which is the above-defined element a). It is ically defined generators (one of which is the above-defined element a). It is likely that the knot group together with this canonically defined subgroup likely that the knot group together with this canonically defined subgroup (gZ@Z) withd is t inguished basis, affords a complete invariant of the knot. Z El) Z) with distinguished basis, affords a complete invariant of the knot. In particular the group of a knot is isomorphic to Z if and only if the knot In particular the group of a knot is isomorphic to Z if and only if the knot is trivial, i.e. the embedding S1 L) S3 is isotopic (i.e. deformable by means is trivial, i.e. the embedding SI '---t S3 is isotopie (i.e. deformable by means of self-diffeomorphisms of S3) to the trivial embedding (Papakyriakopoulos’ of self-diffeomorphisms of S3) to the trivial embedding (Papakyriakopoulos' theorem, proved in the late 1950s). theorem, proved in the late 1950s). However difficulties of a group-theoretical kind prevent one from obtaining difficulties of a group-theoretical prevent one afrom by However these means an algorithm for determining kind whether or not givenobtaining knot is by these means an algorithm for determining whether or not a given knot is trivial. Such an algorithm was constructed by Haken in the early 1960s using trivial. 8uch algorithmwith wasthe constructed Haken early 1960s different ideas,an however drawback bythat it isinanthealgorithm onlyusing “in different ideas, however with the drawback that it is an algorithm only principle”, in the sense that the vast number of steps in the algorithm rule "in it principle", in thepurposes. sense that the vast number of steps in the algorithm rule it out for practical out for practieal purposes.

246 246

Chapter Smooth Manifolds Manifolds Chapter 4. Srnooth

There are various other use (the most practically practically useful There other knotinvariants knot-invariants in use of which is the “Alexander polynomial") polynomial”) aIl all ultimately ultimately determined by the the "Alexander of which group of is convenient for some purposes to consider consider invariants invariants of the the knot. knot. ItIt is sorne purposes occuring Z[Z]-module homology (see (see Chapter Chapter 3, 3, §5) $5) of the canonical occuring in in the Z[Z]-module Z-covering 6 + of the complement S3\S1 U. Z-covering Û -- U U of the knot knot complement 8 3 \8 1 = U. There is a class of relatively simple knots for which the knot knot complement complement There is a class of relatively simple knots for which the 1 U is a fibre bundle over S1 with fibre a punctured suface of genus g genus U Is bllnclle 8 with a pllnctured sllface genus 9 (the genus of the knot): of the knot): F” ~ i$\{*}. u + s’, U --+ 81, F M;\ {* } . (The of an an arbitrary arbitrary knot knot is is the least genus genusof surface in in U U having having the the (The genus genus of the least of a a surface knot as boundary.) It can be shown in the case of a knot whose complement U knot as boundary.) It can be shown in the case of a knot whose complement U is a fibre bundle (as above), that the knot is defined by an algebraic equation is a fibre bundle (as above), that the knot is defined by an algebraic equation 3 in c2 > 8 S3: in C 2 :-J : 2 = & > 0, f(z, = 0, /2j22 + + Iwl /WI2 f(z, w) tu) = 0, Izl = E > 0, where at zz = tu w = 0; 0; in in general general where f(z, f (z, W) tu) is is a a polynomial polynomial satisfying satisfying fifz = fW f w = 00 at such an equation determines a link in S3, 3 consisting of several embedded such an equation determines a link in 8 , consisting of several embedded circles, but often the the solution solution set connected and and so so represents knot 8S’1 C c but often set is is connected represents aa knot circles, S3. 3 In terms of f the fiber bundle U is given essentially by 8 . ln terms of f the fiber bundle U is given essentially by (z, w) H arg f = 4,

f = peid,

p2 = E.

By a theorem of Papakyriakopoulos (of the late 195Os), if M33 is any maniBy a theorem of Papakyriakopoulos (of the late 1950s), if M is any manifold for whieh which 1T2(M3) 7rz(M3) of. # 0, then there there exists exists an an embedded embedded 2-sphere 2-sphere 8S2 M33 2 cC M 0, then fold for not homotopic to zero. It follows that for the complement S3\S’ of a knot, one not homotopie to zero. It follows that for the complement 8 3 \8 1 of a knot, one has 7ri(S3\S’) 0 for i 2 2, so that all homotopy invariants are, in principle, = a for i ~ 2, 80 that ail hornotopy invariants are, in principle, has 1Ti(8 3 \8 1 ) = determined by ~1. determined by 1TI' Every finitely presented group (i.e. presentable by means of a finite number Every finitely presented group (Le. presentable by me ans of a finite number of generators and relations) can be realized by means of a standard construcof generators and relations) can be realized by me ans of a standard construction as the fundamental group ri(K) of a finite Cl&complex of dimension tion as the fundamental group 1T1 (K) of a finite CW -complex of dimension < 2, or as the fundamental group rri(AJ) of a manifold A4 of dimension 5 4. ::::: 2, or as the fundamental group 1TI (M) of a manifold M of dimension::::: 4. From results of P. S. Novikov, Adyan, and Rabin (of the early and midFrom results of P. S. Novikov, Adyan, and Rabin (of the early and mid1950s) on the algorithmic undecidability of various questions in the theory 1950s) on the algorithmic undecidability of various questions in the theory of finitely presented groups (in particular the question of whether or not a of finitely presented groups (in particular the question of whether or not a given arbitrary presentation presents the trivial group) it is not difficult to given arbitrary presentation presents the trivial group) it is not difficult to infer the algorithmic undecidability of the question as to whether or not an infer the algorithmic undecidability of the question as to whether or not an arbitrary given finite CW-complex of dimension 2 2 is homotopically equivaarbitrary given finite CW -cornplex of dimension?: 2 is homotopically equivalent to some standard simply-connected CW-complex, and likewise whether to an sorne standard simply-connected -complex, and whether lentnot or arbitrary given closed manifoldCW of dimension > 4 likewise is homotopically or not an arbitrary given closed manifold of dimension ~ 4 is hornotopically equivalent to somestandard simply-connected manifold. This observation was sorne standard manifold. This observation was equivalent made by A.to A. Markov (in simply-connected the late 195Os), who also showed (this is more A. A. Markov (in the late 1950s), who also showed (this is more made by difficult) that it is also not algorithmically decidable for every manifold of difficult) is also ornotnotalgorithmically decidable for every manifold or of dimensionthat 2 4 it whether it is homeomorphic (or PL-homeomorphic, ~ 4 whether or not it is homeomorphic (or P L-homeomorphic, or dimension diffeomorphic) to a standard simply-connected manifold. Contractibility of diffeomorphic) simply-connected manifold. Contractibility of a CW-complex tois aanstandard algorithmically unrecognizable property in dimensions a CW -cornplex is an algorithmieally unrecognizable property in dimensions

$5. The The role role of of the the fundamental fundamental group group in in topology topology §5.

247 247

2 3, and and the question question of of whether whether or or not not aa closed closedmanifold manifold of of dimension dimension nn ?: 2 55 ?: is homeomorphic to the n-sphere n-sphere sn S” is is also also undecidable; undecidable; these these undecidability undecidability is results were were established established by S. S. P. P. Novikov Novikov in the early 1960s 1960susing using homologhomologresults ical algebra; algebra; it is is likely likely that that the latter latter question remains remains undecidable undecidable also also for for ical 4. Note Note that, that, on on the the other other hand, hand, the the homeomorphism homeomorphism problem problem for for finite finite nn = 4. 22dimensional CW Cl&complexes and the problem problem of deciding deciding whether or or not aa 2-dimensional -complexes and given 3-manifold-with-boundary 3-manifold-with-boundary is homeomorphic homeomorphic to to aa contractible contractible manifold, manifold, given is are in alllikehood all likehood algorithmically algorithmically soluble. soluble. The latter latter problem reduces, reduces, modulo are calculation of of the the genus genusof of the the boundary boundary of of the the 3-manifold-with-boundary, S-manifold-with-boundary, to calculation to that of recognizing the homotopy homotopy type type of of S3. S3. The simplest simplest algorithmic algorithmic probthat lems of of the the theory theory of of closed closed 3-manifolds 3-manifolds and and knots knots are, are, it it would would seem, seem, also also lems soluble, although, although, as as the the problem problem of of recognizing recognizing the the trivial trivial knot knot shows shows (see (see soluble, above), notwithstanding the existence of an algorithm, a problem may still be above), notwithstanding the existence of an algorithm, a problem may still be intractable in practice. For the recognition problem for the sphere S3 there intractable in practice. For the recognition problem for the sphere S3 there is an an algorithm algorithm solving solving it it in in the the class class of of Heegaard Heegaard diagrams diagrams of of germs genus gg = 22 is (Birman, Hilden, in the late 196Os).i’ (Birman, Hilden, in the late 1960s).11 It is is possible some 3-dimensional 33dimensional problems problems there there are are efficient, efficient, sufsufIt possible that that for for some ficiently fast, algorithms, for instance for recognizing the trivial knot or the ficiently fast, algorithms, for instance for recognizing the trivial knot or the sphere S3, in the sense that, although not always strictly applicable, they sphere S3, in the sense that, although not always strictly applicable, they nonethetheless give aa quick resolution of of the the problem “practically always"; always”; nonethetheless give quick resolutioll problem "practically such algorithms would for practical purposes be more useful than those apsuch algorithms would for practical purposes be more useful than those applying always but only “in principle”. It is of interest to mention in this regard plying always but only "in principle" . It is of interest to mention in this regard a experiment (recorded (recorded in literature) carried carried out out in in the biophysical biophysical literature) a numerical numerical experiment in the connexion with an investigation of the properties of, for instance, substances connexion with an investigation of the properties of, for instance, substances possessinglong, closed molecules (%atenated”) in the configuration of knots possessing long, closed mole cules ("catenated") in the configuration of knots and links (Frank-Kamenetskii and others, in the 1970s); similar structures and links (Frank-KamenetskiI and others, in the 1970s); similar structures have been encountered also in other areas of physics. In conclusion we note have been encountered also in other areas of physics. In conclusion we note that up to a large number of crossings knots can be effectively and uniquely that up to a large number of crossings knots can be effectively and uniquely recognized through their Alexander polynomials.12 recognized through their Alexander polynomials. 12 The Poincarit conjecture to the effect that S3 is the only simply-connected, The Poincaré conjecture to the effect that S3 is the only simply-connected, closed S-manifold, remains unproven. In the late 1950s Milnor showed that closed 3-manifold, remains unproven. In the late 1950s Milnor showed that from the above-mentioned theorem asserting the existence of a homotopically from the above-mentioned theorem asserting the existence of a homotopically ‘lIn his recent article The sol&ion to the recognition problem for S3,3 Haifa, Israel, his H. recent article The the recognition problem for for recognizing 8 , Haifa, the Israel, MayIl In1992, Rubinstein claims solution to have toconstructed an algorithm 3May 1992, H.sphere. Rubinstein claims ofto thehave constructed dimensional The author present survey an doesalgorithrn not know forif recognizing the argumentsthe 3in dimensional sphere. The verified. author of the present survey does not know if the arguments in this article have all been this have1980s ail been verified. “In article the late V. Jones discovered remarkable new polynomial invariants of knots, 121n the 19808ofV.certain Jones classical discovered rernarkable polynomial of knots, allowing the late solution problems of knot new theory. Here theinvariants idea derives from allowing the solution of certain classical problerns of knot theory. Here the idea derives from representations of equivalence classes of knots by conjugacy classes in different groups equivrepresentations of knots byofconjugacy classes in different equivalent via “Markov of equivalence moves”, and classes representations braid groups arising in the groups Yang-Baxter aient via "Markov moves", and representations braid groups arising in the Yang-Baxter theory of mathematical physics, yielding precise ofsolutions of certain 2-dimensional models of mathematical solutions certain 2-dimensional models oftheory statistical and quantum physics, physics.yielding Followingprecise Jones’ work of several topologists, algebraists, of statistical Following the Jones' several (See topologists, algebraists, and mathematical and quantum physicists physics. have developed theoryworkfurther. for example Kauffand rnathematical physicists have developed further. example man, Louis H., ‘Knots and physics”. Singapore:the theory World Scientific, (See 1994.for Series on Kauffknots man,everything Louis H., or "Knots and physics". Singapore: ‘Lectzlres World Scientific, Series on knots and the lectures of Dror Bar-Natan on Vassiliev1994. &variants”). This and everything or thereformulated lectures of Dror This theory has now been in theBar-Natan more modern"Lectures setting onof Vassiliev “topological invariants"). quantum field theory has llOW been in the more (See modern of "topological quantum field theory”, invented by reformulated A. Schwarz and Witten. also setting the Appendix.)

theory", invented by A. Schwarz and Witten. (See aIso the Appendix.)

248 248

Chapter Manifolds Chapter 4. Smooth Smooth Manifolds

2 c M3 in manifolds M33 with 7rz(M3) non-trivial, # 0, 0, non-trivial, embedded 2-sphere 8S2 C M3 M with 7(2(M 3) f. one can deduce the uniqueness, up to homotopy equivalence, of the decompothe uniqueness, homotopy sition as aa connected sum (see (see §4 $4 above) sition of aa S-dimensional 3-dimensional closed closed manifold manifold as of 3-manifolds of the following three elementary types: of 3···manifolds of the following elementary

Type with 7(1 ~1 finite; finite; Type I:1: those with Type Type II: II: those those with with 7r2 7(2 = 0; O', Type Type III: III: M3 M3 = = S2 8 2 xX S1. 8 1. Moreover Poincare conjecture is true true then then this this decomposition decomposition is is topoMoreover ifif the the Poincaré conjecture is topologically unique. logieally unique. Various can be be constructed by factoring by the the action action Various Type Type lI manifolds manifolds can constructed by factoring by of finite subgroups of SU2 or SO4 acting freely (i.e. in such a way that of finite subgroups of 8U2 or 804 acting freely (Le. in such a way that no no non-trivial elements fixes fixes any any point) point) on on S Similarly, manifolds manifolds of of Type Type II II 8 33.. Similarly, non-trivial elements may be obtained as the orbit spaces of actions of discrete groups G of momay be obtained as the orbit spaces of actions of dis crete groups G of mo3 G c Oa,r, acting freely and tions Lobachevskian space space L L3, , Ge 0 3 ,1, acting freely and tions of of 3-dimensional 3-dimensional Lobachevskian with compact fundamental region. Over the period from the the mid-1960s mid-1960s till till with compact fundamental region. Over the period from the early 1980s V. S. Makarov produced several infinite series of such manthe early 1980s, V. S. Makarov produced several infinite series of such manifolds; examples have have not studied in in detail. detail.13 An 13 An ifolds; however however these these examples not yet yet been been studied extensive program of investigation of 3-dimensional manifolds undertaken extensive program of investigation of 3-dimensional manifolds undertaken by by Thurston several ago, is is complete complete only for the the rather rather narrow class of of Thurston several years years ago, only for narrow class Haken manifolds. It has issued in the “Geometric Conjecture”, which, if true, Haken manifolds. It has issued in the "Geometrie Conjecture" , whieh, if true, would have as corollary the following elegant result: If a closed S-manifold would have as corollary the following elegant result: If a closed 3-rnanifold M and 7(l(M) nl(M) without subgroups other other M has has 7rz(M) 7(2(M) = = 00 and without non-trivial non-trivial abelian abelian subgroups than Z, then M is homotopy equivalent to a compact 3-manifold of constant than Z, then M is homotopy equivalent to a compact 3-manifold of constant negative curvature, so that its fundamental group is isomorphic to a discrete negative curvature, so that its fundamental group is isomorphie to a dis crete group of isometries of Lobachevskian 3-space, with compact fundamental dogroup of isometries of Lobachevskian 3-space, with compact fundamental domain. Relevant to this program is the conjecture that if x1 (M3) is infinite and main. Relevant to this program is the conjecture that if 7(1 (M3) is infinite and n2(M3) = 0, then the manifold M3 is homotopy equivalent to a 3-manifold 7(2(M3) 0, then the manifold M3 is hornotopy equivalent to a 3-manifold of constant negative curvature, and therefore obtainable as the orbit space of of constant negative curvature, and therefore obtainable as the orbit space of the action of a discrete group of motions of L3. the action of a discrete group of motions of L 3 . We now turn to problems concerning the topology of higher-dimensional We now turn to problems concerning the topology of higher-dimensional manifolds (of dimension n > 5). The problem of the topological invariance manifolds (of dimension n ;:.:: 5). The problem of the topological invariance of the Pontryagin classespk E H4”(Mn; Q) of rational or real cohomology, of the Pontryagin classes Pk E H 4 k(Mn; Q) of rational or real cohomology, i.e. of the invariance of integrals of the classespk over cycles under homeoi.e. of the invariance of integrals of the classes Pk over cycles under homeomorphisms (assuming the Pontryagin classesrepresented as differential forms, morphisms (assuming the Pontryagin classes represented as differential forms, perhaps expressed in terms of the curvature tensor relative to some Riemanperhaps expressed terms the ofcurvature to some nian metric), wouldin on the offace it seemtensor to be relative not at all relatedRiemanto the nian metrie), would on the face of it seem to be not at aU related to the fundamental group; in fact in solving the problem one may assume without fundamental group; in fact in solving the problem one may assume without r31n the mid-1980s Fomenko and Matveev constructed, with the help of a computer, the mid-1980s Matveev the helpcurvature, of a computer, nice families of closed Fomenko 3-manifolds andwith constantconstructed, (normalized) with negative using nice families of closed 3-manifolds with constant negative curvature, Matveev’s “complexity theory” for 3-manifolds. One (normalized) of these manifolds turned out to using have Matveev's 3-manifolds. One of these out to have volume less "complexity than that oftheory" a certain for 3-manifold conjectured by manifolds Thurston toturned be minimal.

volume less than that of a certain 3-manifold conjectured by Thurston to be minimal.

§5. 55. The The role role of of the the fundamental fundamental group group in topology topology

249 249

loss Thus loss of of generality generality that that the the manifold manifold Mn M” is is simply--connected. simply-connected. Thus at at first first glance the the (affirmative) (affirmative) solution solution of of the the problem problem (by (by Novikov Novikov in in the the mid~1960s) mid-1960s) glance by resorting resorting to to the the consideration consideration of of non-simply-connected non-simply-connected toroidal regions regions by toroidal and techniques techniques specifie specific to to non-simply-connected non-simply-connected manifolds, appears appears artificial. artificial. manifolds, and However no alternative alternative proof proof has hitherto hitherto been found. found. Moreover Moreover the the device device However of introducing introducing toroidal toroidal regions regions in order order to reduce reduce problems problems concerning concerning homehomeof omorphisms to auxiliary auxiliary problems problems concerning concerning the the smooth smooth or or P P&topology of omorphisms L-topology of non-simply-connected manifolds, has undergone undergone substantial substantial development development in in non-simply-connected manifolds, the first first place place by by Kirby Kirby as a means means for for proving proving the the so-called so-called "Annulus “Annulus ConConthe jecture” (finally (finally settled settled by Siebenmann, Siebenmann, Hsiang, Hsiang, Shaneson, Shaneson, Wall Wall and and Casson Casson jecture" in the the late late 1960s), 196Os), and and has has subsequently subsequently been been further further refined refined to to aa tool tool for for in solving topological problems problems about about purely purely topological topological manifolds manifolds and and homeohomeosolving topological morphisms between between them them (by (by Kirby, Kirby, Siebenmann, Siebenmann, Lashof Lashof and and Rothenberg). Rothenberg). morphisms Apart from from the the signature signature formula, formula, there there exist exist for for simply-connected simply-connected maniApart manifolds essentially no other homotopy-invariant relations among the Pontryagin folds essentially no other homotopy-invariant relations among the Pontryagin classes numbers. (By (By theorems theorems of of Thom Thorn and and Wu Wu concerning concerning the the classes classes classes and and numbers. Wi and pl, respectively, the only possible homotopy invariants are certain Wi and Pk respectively, the only possible homotopy invariants are certain characteristic numbers modulo modulo 22 (for (for the the classes wi) and and modulo modulo 12 12 (for (for the the characteristic numbers classes Wi) classes pk), and certain other characteristic numbers. The fact that the ratioclasses Pk), and certain other characteristic numbers. The fact that the rational classes Pk pl, are are not general homotopy-invariant homotopy-invariant is aa relatively relatively nal Pontryagin Pontryagin classes not in in general is straightforward consequence (as was observed by Dold in the mid-1950s) of straightforward consequence (as was observed by Dold in the mid1950s) of the on the finiteness of the groups 7rn+s(Sn)n ) for for ni: n # 4. 4. the SerreRohlin Serre--Rohlin theorem theorem on the finiteness of the groups 11"n+3(8 The simplest examples are afforded by the family of SOs-vector bundles with The simplest examples are afforded by the family of S03-vector bundles with base fibre 8S2, determined by by aa class class Pl pl EE H4(S4; Z) (~ (2 7l) Z) (or (or 2 , each and fibre each determined H4(8 4;7l) base S4 8 4 and by an element of 7rs(SOs) E Z). It can be inferred from the finiteness of the by an element of 11"3(803) 7l). It can be inferred from the finiteness of the image image under’the under' the homomorphism homomorphism J:r3(SO3)

-

7rtj(S3)G

&2,

that the total spaces of these vector bundles fall into only finitely many hothat the total spaces of these vector bundles faIl into only finitely many homotopy types (determined by the class of the bundle to within “fiberwise motopy types (determined by the class of the bundle to within "fiberwise homotopy equivalence”, as defined by Dold). Since in fact there are infinitely homotopy equivalence", as defined by Dold). Since in fact there are infinitely many such bundles with different Pontryagin classes pl, it follows that at best many such bundles with different Pontryagin classes Pl, it follows that at best only pi(mod 48) may be a homotopy invariant in the class of manifolds obonly Pl(mod 48) may be a homotopy invariant in the cIass of manifolds obtained as total spaces of these bundles. Theorems of Browder and Novikov (see tained as total spaces ofthese bundles. Theorems of Browder and Novikov (see 54 above) show that even for a particular simply-connected manifold there §4 ab ove ) show that even for a particular simply-connected manifold there are no homotopy-invariant relations among the Pontryagin classes apart from are no homotopy-invariant relations among the Pontryagin classes apart from the Hirzebruch-Thorn-Rohlin formula for the signature. the Hirzebruch-Thom-Rohlin formula for the signature. Thorn, Rohlin and Schwarz showed in the late 1950s that the rational PonThom, Rohlin Schwarz showed in the late 1950s that and the rational Pontryagin classes pl, and are invariant under PL-homeomorphisms, on this basis tryagin classes Pk are invariant under P L~homeomorphisms, and on this basis were able to propose the following combinational definition of those classes. wereeach ablecycle to propose the following combinational definition M4k of those For z E H4k(Mn; Q) realizable as a submanifold C M”classes. with 4 For each cycle Z E H4 k(Mn; IQ) realizable as a submanifold M k C Mn with trivial normal bundle trivial normal bundle

250 250

Chapter Smooth Manifolds Manifolds Chapter 4. Smooth M4k

x Ip-lk

L)

M?z

,

(5.2) (5.2) i*([M4”]) 4k ]) = i*([M

2, z,

we set (cf. the Hirzebruch formula (3.19)) Hirzebruch formula (Lk(Pl,. . ,Pk), ) 2) ) (L ( ... ·,Pk,Z kPl,

=

+f4”). 4k). r(M

(5.3) (5.3)

show that that the values General categorical categorical properties properties may now now be be invoked to to show values of L Lk(pl, pk) at at such cycles zz determine determine its values on on all all of of H H4k(Mn; a). such cycles its values of k (Pl,'" ,,Pk) 4k(Mn j Q). Now from form of polynomials, we we have Now from the the form of the the Hirzebruch Hirzebruch polynomials, have L k

= 22k(22k-1 - 1)Bk (2k)!

Pk

(5.4) (5.4)

+~k(Plr,Pk-I),

where Bk is the kth kth Bernoulli it follows follows that that the the class classPk pk may where Bk is the Bernoulli number, number, whence whence it may be terms of L1, ... . ,, L Lk: : be expressed expressed in in terms of LI, k Pk

=Pk(Ll,...,Lk).

(5.5) (5.5)

On the the basis definition of of the pk one one can can give give On basis of of this this “signature” "signature" definition the classes classes Pk combinatorial treatment of their their properties. properties. If If Mn is aa P PL-manifold and treatment of Mn is L~manifold and aa combinatorial ff :: M” simplicial map of sufficiently sufficiently fine fine triangulations, triangulations, then then the the Mn + Sn-4k sn-4k a a simplicial map of complete inverse (x) of of any any interior of aa simplex simplex O'n-4k 0n-4k of of complete inverse image image f-l f-l(x) interior point point xz of Sne41c of maximal dimension, has the form sn-4k of maximal dimension, has the form f-l(x) = M M4’,4k , f-l(X) =

f-‘(Int gn-4k) == Int Int O'n-4k cP-4k Xx M4’, f-l(Int O'n-4k) M 4k ,

where PL-submanifold Mn. We now define where M4” M4k is is aa P L-submanifold of of Mn. We now define (Lk(P1,

.

,pk),

z)

=

7(M4”),

where z is the element of H4k(Mn; Q) represented by the PL-submanifold where Z is the element of H 4k(Mn Q) represented by the PL-submanifold M4”4k c M”, and thence we define the Pontryagin classespk as before, using M C Mn, and thence we define the Pontryagin classes Pk as before, using the formula the formula Pk = Pk(L1, . . , Lk). Pk = pk(L 1 , ... , Lk)' As earlier this can be shown to determine pk on H4k(Mn; a), and so affords As earlier this can be shown to determine Pk on H 4 k(M n Q), and so affords a PL-invariant definition of the rational Pontryagin classespk. We see that aPL-invariant definition of the rational Pontryagin classes Pk. We see that underlying this definition is the natural and simple analogue for PL-maps of underlying this definition is the natural and simple analogue for P L-maps of the property of transversal regularity for smooth maps. the property of transversal regularity for smooth maps. Note that this definition is non-local, by contrast with the smooth case, Note that this definition is non-local, by contrast with the smooth case, where the classes pk are expressed to terms of the curvature tensor. The are expressed to ofterms of the pl, curvature where theofclasses existence a localPkPL-representation the classes has beentensor. proved The (by existence of a local P L-representation of the classes Pk has been proved (by I.M. Gel’fand, Losik and Gabrielov in the mid-1970s), but so far an effective I.M. Gel'fand, Losik anddefinition Gabrielovis in the mid-1970s), but so far an effective (i.e. constructive) local lacking. (Le.We constructive) local definition is lacking. now sketch the proof of the topological invariance of the rational PonWe now sketch proof ofM”.the topological invariance of the rational PonTo begin with, note that different smooth tryagin classes of athe manifold tryagin classes of a manifold Mn. '1'0 begin with, note that different smooth j

j

55. group in topology §5. The The role role of the fundamental fundamental group

251 251

(or PL-) single continuous manifold manifold Mn M” must, by definition P L-) structures structures on a a single definition of homeomorphism, all aU possess possess the same same underlying underlying collection collection of open sets sets of Mn Mn (each P L-) manifold with respect to each (each of which which is aa smooth (or PL-) manifold with each M”). The rational rational of the the various superimposed superimposed smooth or PL-structures PL-structures on Mn). Pontryagin classes classes pk Pk of aa manifold Mn, Mn, or, more precisely, their Pontryagin their integrals the individual individual cycles of M”, Mn, turn turn out to be representable as over the as homotopy homotopy invariants of of certain certain open “toroidal” "toroidal" regions Mn and their their covering spaces, invariants regions of M” spaces, this one one infers their their topological topological invariance invariance for the whole manifold and from this manifold Mn. (The definition of the pl, Pk via the signature is Mn. (The above definition is used.) used.) to begin, consider aa fixed embedding Thus, to 1~ xx T Tn-4k-1 n - 4k lR

Rn-4k 1 L-t '--;0 lR n - 4k

neighbourhood of a torus, and aa corresponding "toroidal" Mn of aa neighbourhood “toroidal” region of M” u” = M4” x Tn-lk-l

x R ~-f M”.

(5.6)

With respect to aa smooth (or P L-) structure With PL-) structure on Mn, M”, this region need need not be aa smooth direct direct product; product; what what is here is is important important here is that that the region is is homotopy homotopy 4k) equivalent manifold M4k M4k xx Tn-4k-l, equivalent to the manifold Tn-4k-1, and that that the signature T(M 7(M4”) homotopy invariant may be expressed expressed as as a homotopy invariant of this region or of its covering spaces spacesin terms of their their cohomology rings. It It may be assumed assumedwithout without loss loss of 4k are simply-connected, M4” that k > 1, 1, that the manifolds M”, Mn, M simply-connected, and that generality generality that 81c.By By using methods of differential differential topology it may be be shown that that for aa n < < 8k. topology it manifold with free abelian fundamental fundamental group the following following assertion is is valid: valid: manifold with If W IV”,m , m > such a manifold manifold (i.e. ,i(Wm) free abelian) on which there If 2: 6, is such (Le. 7rl (W m ) is free action of the group /Z Z with with the property property is defined a free continuous continuo us or smooth action m that orbit space space w Wm/Z with the homotopy homotopy type type of aa that the orbit j/Z is a closed closed manifold manifold with m is + S’ with base S1, then then the manifold manifold W Wm is diffeomorphic fibre bundle bundle Vm v m ---> SI with base SI, diffeomorphic to the direct product the direct product of aa closed closed manifold manifold and the real line: W”

= Iv-l

x Et.

(In suffices that that the manifold manifold W W”m with with free free Z-action, this conclusion itit suffices /Z-action, (In fact fact for this homotopy-equivalent to a finite finite CW -complex, and that the orbit orbit space be homotopy-equivalent CW-complex, and that space m j/Z be compact; moreover the w the condition Wm/Z condition on 7rl(wm) rl(Wm) may be weakened weakened to requirement that that the Grothendieck Grothendieck group Kc(7ri) KO(7rl) (see (see below) be trivial: the requirement trivial: KO(7rl) == 0.) O.) Ko(m) From this this lemma the topological invariance of the rational Pontryagin topological invariance rational Pontryagin classes is deduced deduced as classes as follows: We first apply apply the lemma to the region un U” (i.e. n) endowed endowed with with any smooth structure, structure, taking taking (Le. we take take Wm W m = U”, un, m = Tt) Z /Z to act in the the natural natural way: T : (x, (z, t) ---> - (x, (LC,t + + 1), l), T:

where it: xE M 4k xX Tn-4k-l, E R. lR. Since E M4k Tna41cp1,t E Since 7rl(M4k) ~I(M~~) == 0, we have that that 7rl(Un) ,1(V) is so that by the above lemma there is is aa diffeomorphism is free abelian, so that by diffeomorphism

Chapter Chapter 4. Smooth Smooth Manifolds Manifolds

252 252

U” 2 v-1

x Kc,

(5.7) (5.7)

where where Vnpl V n - 1 isis homotopy homotopy equivalent equivalent to to M4” M 4k x X Tn-lkwl. T n - 4k - 1 . Since Sinee the the direct direct dedecomposition composition (5.7) (5.7) isis smooth, smooth, we we have have for for the the Pontryagin Pontryagin classes classes p.j(U”)

= pj(T1),

j = 1,2,. . . .

Consider Consider the the Z-covering Z-covering space space of of Vnel V n - 1 determined determined by by the the Z-covering Z~covering of of the the torus: torus: Tn-4k-2 x y& 2 Tn-4k-1, Tn-4k-2 x lR -!. T n - 4k - 1 ,

v

en-1n

- 1 5~ vn-1; V n - 1.

,

n - 4k - 2 . Applying here Tn-4k-2. here en-l Vn - 1 has has the the homotopy homotopy type type of of M4” M4k xX T Applying the the procedure procedure 1 n once cn-’ - in W”,m , we once again, again, this this time time with with V in the the role role of of W we obtain obtain p-1

”

p-2

pj(Vn-2)

V”-2

x R ,

= pp-1)

N

j,,f4”

= pj(u”),

x y-4k-2

,

j 2 1.

(5.8)

Iterating Iterating this this argument argument we we finally finally arive arive at at p4k+l

gP”)

g

= &(Pkfl)

v4k

x R,

= “. = &(U”),

j > 1,

(5.9) (5.9)

where V4k4k is M4”. The 4 where V is aa smooth, smooth, closed closed manifold, manifold, homotopy-equivalent homotopy-equivalent to to M k. The signature formula gives signature formula gives T(M~~)

= &(M4”)

= s&(V4”)

= (&@I,.

,pk), [M4”]),

where the Pontryagin classes are defined in terms of any smooth structure on where the Pontryagin classes are defined in terms of any smooth structure on the manifold Un. From this the topological invariance of the rational Pontryathe manifold un. From this the topological invariance of the rational Pontryagin classes quickly follows. gin classes quickly follows. For n > 81cthe above argument yields a smooth embedding V4”4k x lRnp41c For n 2: 8k the above argument yields a smooth embedding V X lR n - 4k '--> U”, endowed however with a possibly different smooth structure, whence one un, endowed however with a possibly different smooth structure, whenee one obtains a diffeomorphism obtains a diffeomorphism 1/4”

x an-4k

-

M4k

x Rn-4k

7

where the latter manifold is endowed with any smooth structure. Following where the latter manifold is endowed with any smooth structure. Following soon after the appearance of the above proof, Siebenmann was able to show soon after the appearanee of the above proof, Siebenmann was able to show by means of a more careful analysis of the argument that this conclusion holds by me ans of a more careful analysis of the argument that this conclusion holds also in the non-stable dimensions n < 8lc. the non-stable dimensions < 8k. alsoForin nonsimply-connected closedn manifolds there are non-trivial homotopyFor non~simply-connected closed manifolds there are non-trivial homotopyinvariant relations between integrals of the rational Pontryagin classes over invariant relations between examples integrals of of such the rational classes over certain cycles; the simplest relations Pontryagin are as follows: certain cycles; the simplest examples of such relations are as follows:

§5. §5. The The raIe role of of the the fundamental fundamental group group in topology topology

253 253

Example 1. 1. Let Let nn = 4k 4k + + 11 and and zz EE H4k(Mn). H4k (yn). Consider Consider aa /Z-covering Z-covering space space Example projection p : Mn M” ---> + Mn M” such such that that P* p*rl(M,) consists just just of of those those cycles cycles 1y projection 'TrI (Mn) consists having zero zero intersection intersection index index with with the the cycle cycle z: having p*7r&i?P)

0 z = 0.

There is is aa cycle cycle Z i E E H H4k(Mn) definable in homotopically homotopically invariant invariant fashion, fashion, There 4k (Mn) de fin able in T

Mi such such form form

Mi+1

Fig. 4.28 4.28 Fig.

that ring H’(A@;Q) we have the bilinear bilinear that p,i == z. z. In In Jhe the cohomology cohomology ring H*(Mn; Q) we have the defined on Hzk(M”;Q) by 2k defined on H (Mn; Q) by

P.z

k&Y) == (xy,z) (XY,4 = (Y,X), (x,y) (y,x),

(5.10) (5.10)

which has carrier (Le. (i.e. the elewhich has finite-dimensional finite-dimensional carrier the subspace subspace annihilating annihilating every every element has finite codimension). Denoting by r(Z) the signature of this bilinear ment has finite codimension). Denoting by r(:Z) the signature of this bilinear form, one has the following formula: form, one has the following formula: (Lk(Pl,...,Pk)rz)

=7(z).

(5.11)

(5.11)

Example 2. If a smooth manifold V 4k+q has the homotopy type of a product Example 2. If a smooth manifold V 4k + q has the homotopy type of a product of the form M4k4k x TQ: of the form M x ']'q: v4”+4 wM4k xTq, V 4k +q "" M 4k X Tq, then the following formula (due to Novikov in the mid-1960s) may be estabthen the following formula (due to Novikov in the mid-1960s) may be est ablished by means of the technique used to prove the topological invariance of lished by me the ChSSeS Lkans (pi, of. the , pk):technique used to prove the topological invariance of the classes Lk(PI, ... ,Pk): Lk(v4”+‘)

= ‘+!f4”).

0

o

The general conjecture concerning the “higher signatures” consists in generalAsconjecture the Hopf "higher consists in the The following: was shownconcerning already by (in signatures" the early 1940s) every the following: As was shown already by Hopf (in the early 1940s) every CW-complex K has a distinguished set of cohomology classes determined CW -complex K (K)has, anamely distinguished set under of cohomology determined completely by ~1 the image the inducedclasses homomorphism completely by 'TrI (K), namely the image under the induced homomorphism

254 254

Chapter Chapter 4.4. Smooth Smooth Manifolds Manifolds

f*f* :: H*(K(n, H*(K(7r, 1)) 1)) +~ H*(K) H*(K) (with (with any any coefficients) coefficients) of of the the canonical canonical map map ff :: KK +~ K(n, 1). (Hopf described these in the 2-dimensional K (7r, 1). (Hopf described these in the 2-dimensional case.) case.) An An algebraic the homotopy homotopy types types of of the the spaces spaces K(r, K (7r, 1) 1) and and algebraic characterization characterization of of the their H*(K(7r, 1)) 1)) (with (with respect respect to to arbitrary arbitrary coefficients, coefficients, includincludtheir cohomology cohomology H*(K(r, ing Z[n]-modules) was given by Eilenberg and MacLane the mid-1940s; mid-1940s; ing Z[7r]-modules) was given by Eilenberg and MacLane in in the they they then then defined defined the the cohomology cohomology H*(T) H* (7r) of of the the group group 7r, 7r, as as being being given given by by H*(K(-/r, 1)). (Slightly later, but independently, D. K. Faddeev also gave H*(K(7r, 1)). (Slightly later, but independently, D. K. Faddeev also gave defidefinitions algebraic nitions of of these these algebraic algebraic objects, objects, motivated motivated by by considerations considerations from from algebraic number theory.) In fact this concept lay at the base of the homological algebra number theory.) In fact this concept lay at the base of the homological algebra extended Grothendieck and context extended by by Cartan, Cartan, Eilenberg, Eilenberg, Serre, Serre, Grothendieck and others others to to the the context of of modules modules and and more more general general abelian abelian categories. categories. (One (One such such generalization generalization to -was used to the the category category of of A-modules A-modules -was used in in 56, §6, 7, 7, 9 9 of of Chapter Chapter 33 above, above, as as part part of of the the algebraic algebraic apparatus apparatus of of stable stable homotopy homotopy theory.) theory.) Of closed manifold Of course course one one has has the the corresponding corresponding class class of of cycles cycles (of (of aa closed manifold Mn) Mn) given given by by Df*H*(qQ)

c H,(AP;Q).

The “conjecture concerning The "conjecture concerning the the higher higher zz in Df*H*(r; Q) of dimension 4k the the in D f* H* (7r; Q) of dimension 4k class over that cycle: class over that cycle:

signatures” is that for for each each cycle cycle signatures" is then then that integral of the Pontryagin-Hirzebruch integral of the Pontryagin-Hirzebruch

should 4k + and arbitrary arbitrary 7r be homotopically homotopically invariant. invariant. For For 72 n = 4k + 11 and 7r = should be 7ri(M4k+1) the distinguished cycles z E H~~(IM~~+‘; Q) of codimension one 7rl(M4k+l) the distinguished cycles Z E H 4k (M4k+I;Q) of codimension one are precisely those for which the formula (5.11) holds with respect to the are precisely those for which the formula (5.11) holds with respect to the bibilinear form defined intersections of have this linear form defined by by (5.10), (5.10), and and intersections of such such cycles cycles also also have this property. case that that 7r free abelian all cycles of this this form.) form.) The The the case 7r is is free abelian aIl cycles are are of property. (In, (In the conjecture concerning higher signatures for such intersections of cycles of codiconjecture concerning higher signatures for such intersections of cycles of codimension one, was settled by Rohlin in the caseof intersections of two cycles (in mension one, was settled by Rohlin in the case of intersections of two cycles (in the second half of the 196Os), and then (at the end of the 1960s) by Kasparov, the second half of the 1960s), and then (at the end of the 1960s) by Kasparov, Hsiang and Farrell for intersections of any number of cycles: Hsiang and Farrell for intersections of any number of cycles: z = D(yl

A...A

yk),

yj E H1(Mn;Q).

An analytic proof of this theorem using the theory of elliptic operators was An analytic proof of this theorem using the theory of elliptic operators was obtained in the early 1970s by Lusztig, who also established the conjecture obtained in the early 1970s by Lusztig, who also established the conjecture for certain cycles in the case where n is a discrete subgroup of the group of for certain cycles in the case where 7r is a discrete subgroup of the group of motions of a symmetric space of constant negative curvature. For abelian 7r motions of a symmetric space of constant negative curvature. For abelian 7r he made use, for the first time, of a family of elliptic complexes associated he made use, for the first time, of a family elliptic complexes associated with finite-dimensional representations (and oftheir characters) of the group; with finite-dimensional representations (and their characters) of the group; for non-abelian 7r he was able to exploit infinite-dimensional “Fredholm” repnon-abelian ableXitoofexploit infinite-dimension al "Fredholm" repfor resentations pi :7r7rhe+ wasAut 7r in rings of unitary operators on Hilbert resentations Pi : 7r Aut 'Hi of 7r in rings of unitary operators on Hilbert spaces Xi. (A pair of unitary representations pi, ~2, together with a Noethe'Hi' (A Fpair unitary representations together with a Noethespaces rian operator : ‘Hiof -+ 7-12,is called a FredholmPl, P2, representation if Fpl - p2F operator F : 'Hl ~ 'H2, is called a Fredholm representation if F Pl - P2F rian is a compact operator; cf. end of Chapter 1.) It turns out that, by exploitis a analogues compact operator; cf. of end of Chapter 1.) It turns out by index exploiting of formulae Atiyah-Hirzebruch-Singer type that, for the of ing analogues of formulae of Atiyah-Hirzebruch-Singer type for the index of

§5. group in 55. The The role role of of the the fundamental fundamental group in topology topology

255 255

elliptic operators operators with with coefficients coefficients in in aa Fredholm Fredholm representation, representation, constructed elliptic constructed using the the geometry geometry of compact compact manifolds manifolds of of positive positive curvature curvature to to define define quanquanusing tities like like "the “the signature signature with with coefficients coefficients from from aa representation" representation”, , the the higher~ highertities signatures conjecture conjecture can can be fully fully established established for for aIl all groups groups 7r7r realizable realizable as signatures discrete groups groups of of motions, motions, with with compact compact fundamental fundamental region, region, of of symmetric symmetric discrete spaces (Mishchenko, (Mishchenko, early 1970s 1970s ). This This result result was was generalized generalized by by Solov'ev Solov’ev spaces early and others others in the the 1970s; 1970s; it it now now appears appears that that these these methods methods have have evolved evolved to and the extent extent of yielding yielding a proof proof of of the the conjecture conjecture for aIl all dis discrete subgroups of of the crete subgroups Lie groups groups (Kasparov, (Kasparov, in the the early early 1980s). 1980s). 14 l4 Lie It is is interesting interesting to to note note in in this this connexion connexion that that the the above above mentioned mentioned "Fred“FredIt holm representations” were first introduced by Atiyah in the late 1960s for the holm representations" were first introduced by Atiyah in the late 1960s for the algebras C*(iVP) of complex-valued functions, in the algebra of bounded opalgebras C*(Mn) of complex~valued functions, in the algebra of bounded operators on Hilbert space; in this context a Fredholm representation is a triple erators on Hilbert space; in this context a Fredholm representation is a triple pi, ~2, F :: 7-ti ?t2, as asdefined defined above, above, with with in in addition addition Pi pi(f) = Pi pi(f+). Every Pl, P2, F HI + H2, U) = U+). Every pseudodifferential elliptic operator D of order m = 0 determines a Fredholm pseudodifferential elliptic operator D of or der m 0 determines a Fredholm representation (Pl, (pi, D, D, P2), pz), where where PiU) pi(f) is is the the operation operation of of multiplication multiplication by by ff representation on the sections of bundles vi: on the sections of bundles 'T)i:

Since Df is a compact operator, the triple triple (pl,D,p2) (pi, D, ~2) is is indeed indeed aa I 6, attention: Under what conditions does a smooth open manifold W , m ::::: 6, admit a simply-connected boundary, i.e. under what conditions is W” m realadmit a simply-connected boundary, Le. under what conditions is W realizable as the interior of a smooth manifold-with-boundary mm with simplyizable as the interior of a smooth manifold~with~boundary Wm with simply~ connected boundary awm = V”-lm 1 (nl(VmP1) = {l}, W” m = Int mm)? If connected boundary aWm = V - (7rl(Vm-l) {1}, w = Int Wm)? If it is assumed that the purely continuous version of this problem, i.e. for the it is assumed that the purely continuo us version of this problem, Le. for the class of topological manifolds-with-boundary, is settled in some sense, then c1ass of topological manifolds~with-boundary, is settled in sorne sense, then the problem as formulated can be shown to reduce to the construction of a the problem as formulated can be shown to reduce to the construction of a diffeomorphism between a manifold M” homeomorphic to the direct product diffeomorphism between a manifold Mn homeomorphic to the direct product of a certain closed topological manifold with the real line, and the product of of a certain c10sed topologicai manifold with the realline, and the product of 141n this connexion many deep results have been obtained over the last decade using both 14In this connexion deep results have beenNovikov’s obtained conjecture over the last using both topological and analyticalmany methods. In particular hasdecade been established topological and analytical been established for the “hyperbolic groups” methods. introduced In particular by Gromov, Novikov's and for conjecture certain otherhasclasses of groups, for Cohn, the "hyperbolic Gromov, for certain of groups, by Pedersen, groups" Gromov, introduced Rosenberg byand several and others. Some other beautifulclasses applications Cohn, Pedersen, Gromov, Rosenberg and several others. Sorne applications ofby cobordism theory to the theory of manifolds with positive scalar beautiful curvature have been of cobordism theory toon the manifoldsKreck with and positive have been found and elaborated by theory Gromov, of Lawson, Stolz. scalar Here, curvature in the non-simplyround and case, elaborated on by Gromov, and higher Stolz. signatures, Here, in thebutnon-simplyconnected characteristic numbers Lawson, analogous Kreck to the with the connected case, characteristic the higher signatures, but with the L-genus replaced by the A-genus, numbers play ananaJogous important to role.

L-genus replaced by the A-genus, play an important role.

256

Chapter 4. Smooth Manifolds Chapter Smooth Manifolds

a smooth line. If If on the other smooth closed closed manifold manifold with with the the real realline. on the other hand hand the the solution solution of the following conthe continuous continuous version version is is not not assumed assumed beforehand, beforehand, then then the the following conof ditions are open manifold ditions are imposed imposed instead: instead: the the open manifold Wm W m should should have have only only finitely finitely many at infinity infinity in in the the direction direction of of each each of of many ends, ends, and and be be simply-connected simply-connected at these ends, and furthermore the homotopy type at infinity in the direction of these ends, and furthermore the homotopy type at infinity in the direction of every every end end should should be finite. finite. The The theorem theorem of of Browder-Levine-Livesay Browder-Levine-Livesay asserts asserts that under these assumptions (once made precise) that under these assumptions (once made precise) the the open open manifold manifold Wm w m isis the interior of a compact manifold-with-boundary Wm with each component the interior of a compact manifold-with--boundary Wm with each component of of its its boundary boundary simply-connected: simply-connected: v-1 vm-l = = Uj U. vjm-l, Vm - 1 J

J

'

7rlpy1) = 0. 1rl(Vjm-l) o. In certain cases conditions at infinity on In certain cases the the conditions at infinity on W” W m may may be be replaced replaced by by global global restrictions Of especial such restrictions of w m . Of especial interest interest among among such restrictions are are those those of restrictions on on W”. an W” m have an algebraic algebraic character, char acter , for for instance instance the the condition condition that that w have the the homotopy homotopy type finite CW-complex, admit aa free discrete group type of of aa finite CWcomplex, and and admit free action action of of aa discrete group G G of smooth (or continuous) transformations with compact fundamental of smooth (or continuous) transformations with compact fundamental region. region. ItIt was of modification was precisely precisely this this sort sort of modification of of the the above ab ove problem problem concerning concerning the the boundary of an open manifold Wm (in the particular case G xl(Wm) boundary of an open manifold wm (in the particular case G == Z, Z, 1rl(Wm) free in the the proof free abelian) abelian) that that proved proved to to be be of of technical technical importance importance in proof of of the the topological invariance of the rational Pontryagin classes (see above). topological invariance of the rational Pontryagin classes (see above). ItIt seems seems that that at at the the same same time time as as the the Browder-LevineeLivesay Browder-Levine-Livesay theorem theorem was was being proved, Siebenmann was independently investigating the non-simplybeing proved, Siebenmann was independently investigating the non-simplyconnected of that although publication of his connected version version of that theorem; theorem; however, however, although publication of his results results was delayed over a protracted period, it has emerged (if one goes by was delayed over a protracted period, it has emerged (if one goes by his his work work published in the late 1960s) that he considered only a restricted case of the published in the late 1960s) that he considered only a restricted case of the problem, namely that of the representability of a smooth manifold in the form problem, namely that of the representability of a smooth manifold in the form v x Et. VxR In the non-simply-connected case an obstruction to the representability of In the nonsimply-connected case an obstruction to the representability of a manifold W in the form W = V x LR, where V is closed and of dimension a manifold W in the form W V x lR., where V is closed and of dimension 2 5, is given by an element of the Grothendieck group iYe( K = 7rr (V), :::: 5, is given by an element of the Grothendieck group K O(1r), 1r = 1rl(V), which by definition (cf. Chapter 3, 58) consists of the stable classes of finitewhich by definition (cf. Chapter 3, §8) consists of the stable classes of finitedimensional, projective Z[7r-modules qi (i.e. direct summands of free moddimensional, projective Z[1r]-modules Tli (Le. direct summands of free modules), where ~1,772 are said to be equivalent if ules), where 171, Tl2 are said to be equivalent if

Tli EB NI ~ Tl2 EB N 2 for some finite-dimensional free modules Ni , N2. (For the ring C*(X) of funcfor some NI, N 2 .group (For the ring C* (X) of functions on afinite-dimensional compact space X,free themodules Grothendieck Ks(C*(X)) consists of tions on a compact space X, the Grothendieck group Ko(C*(X)) consists of classes of modules of cross-sections of non-trivial vector bundles over X (the classes of modules of cross-sections of non-trivial vector bundlesand over X (the free modules corresponding in this case to the trivial bundles), coincides free modules corresponding in this case to the trivial bundles), and coincides with K”(X, *).) Taking into account the involutary operation in the group with KO(X, *).) Taking into account the r,involutary operationnaturally in the group ring Z[K] defined by inversion in the group one can associate with ring Z[1r] defined by inversion in the group 1r, one can associate naturally with each Z[n]-module 7 the dual n* = HomA (Q, A), A = Z[rr], which is projective Tl the dual Tl* = HomA (Tl, A), A Z[1r], which is projective each Z[1r]-module if q is projective. Taking the dual defines an involution on Kc(n): if Tl is projective. Taking the dual defines an involution on KO(1r):

55. The role the fundamental fundamental group §5. role of the group in topology

257 257

IfIf W is is known to to have the homotopy finite CW CW-complex, it homotopy type type of a finite -complex, then it can be shown that can be shown that 7j + + (-l)nn* 1] (--I)n1]* = 0 in Kc(n), 11' 7r = 1I'1(W). xl(W). in the the group group Ko(lI'), At around the same time obstruction to homotopy finiteness finiteness At around the same time Wall Wall found found an an obstruction to homotopy of CW-complexes L (under the assumption that L is embeddable as retract of CW-complexes L (under the assumption that L is embeddable as aa retract in some homotopically finite CW-complex X, i.e. that there exists map in sorne homotopically finite CW-complex X, Le. that there exists a a map ff :: X --+ L, L c X, such that f]~ = 1~). Th is obstruction is also given X L, LeX, such that fiL IL)' This obstruction is also given by by an element of Kc(r), 7r = nr(L). It turns out that that for for aa manifold W, Wall’s an element of Ko (11') , 11' = 1I'1(L). It tums out manifold W, Wall's obstruction to W is just 1] n&q* is the the NovikovNovikovto homotopy homotopy finiteness finiteness of of W is just ±1]* where where 1]Q is obstruction Siebenmann obstruction to the existence of a smooth direct decomposition Siebenmann obstruction to the existence of a smooth direct decomposition W V xx Iw. the case 7r '2! 2 Z/p, p prime, prime, the Ka [7r] may interpreted = V R In In the ca..'le 11' Z/p, P the group group Ko[lI'] may be be interpreted W = as the group of Kummer classes of ideals of the algebraic number field Q[ jP], as the group of Kummer classes of ideals of the algebraic number field Q[fi], and along with with the duality operator operator *, *, in in the and has has been been computed computed in in detail, detail, along the duality the context of algebraic number theory. Using these results, Golo, in the second context of algebraic number theory. Using these results, Golo, in the second half of the 1960s constructed constructed a of examples examples of of specific open manifolds manifolds half of the 1960s, a series series of specifie open W not decomposable in the form V x Et. W not decomposable in the form V x R The of the open manifold manifold W W as as The general general problem problem of the realizability realizability of of aa given given open the interior of a compact manifold-with-boundary (where the boundary is the interior of a compact manifold-with-boundary (where the boundary is no longer necessarily simply-connected) was considered by Brakhman (in the no longer necessarily simply-connected) was considered by Brakhman (in the early 1970s) under the assumption that W is a regular covering space of some early 1970s) under the assumption that W is a regular covering space of sorne closed manifold. The special case where the manifold W is homotopically closed manifold. The special case where the manifold W is homotopically equivalent to a manifold of the form M’J x Y-4 (n > 5), with rr(Mq) free equivalent to a manifold of the form Mq x Tn-q (n ;::: 5), with 11'1 (Mq) free abelian, was considered earlier. (If I@ is a torus, then of course the universal abelian, was considered earlier. (If Mq is a torus, then of course the universal covering space of this manifold is IWn.) covering space of this manifold is IRn.) As indicated earlier, for closed manifolds V” with the homotopy type of As indicated earlier, for closed manifolds vn with the homotopy type of the Pontryagin classespi(P) are all trivial. By using a torus, V” N Tn, a torus, vn cv T n , the Pontryagin classes Pi (V n ) are aU trivial. By using Adams’ theorem (of the early 1960s) on the injectivity of the Whitehead Adams' theorem (of the early 1960s) on the injectivity of the Whitehead homomorphism homomorphism J : 7rJ(SO) ---+ 7Qr+j(SN),

N > j + 1,

for j # 4k - 1, together with the homotopy equivalence of the suspension CTnn for j of- 4k - 1, together with the homotopy equivalence of the suspension ET with a bouquet of spheres, it can be shown that all homotopy tori are stably with a bouquet of spheres, it can be shown that all homotopy tori are stably parallelizable (Novikov, in the mid-1960s). Further applications of homotopy parallelizable (Novikov, in the mid-1960s). Further applications of homotopy tori will be described below. toriRecall will be described below. (from Chapter 3, 85) that in connexion with the concept of “simple Chapter 3, §5)bythat in connexion with the concept of "simple RecaIl (from homotopy type”, as defined Whitehead, elements of K1 (rr) arise as obKI (11' ) arise as obhomotopy type", as defined by Whitehead, elements of structions to simple homotopy equivalence between CW-complexes known to ta simple homotopy between CW -complexes known to structions be homotopy equivalent in the equivalence ordinary sense. For manifolds this obstruction be homotopy equivalent in the ordinary manifolds this obstruction (actually in the Whitehead group K~(x)/sense. & rr)For arises in connexion with the the Whitehead group KI (11')/ ± 11' ) arises in connexion (actually in generalization to the non-simply-connected case of Smale’s theoremwith on the the generalization to the non-simply-connected case of Smale's theorem on the triviality of h-cobordisms between simply-connected manifolds of dimension triviality of h-cobordisms between simply-connected manifolds of dimension

258 258

Chapter Chapter 4. 4. Smooth Smooth Manifolds Manifolds

>2: 5. An An h-cobordism h~cobordism Wn+’ W n +1 between between non-simply-connected non~simply~connected manifolds manifolds may may be if Wn+l be non-trivial non~trivial even even for for n 22: 5; even even if W n + l is contractible contractible onto onto one one of of its its boundary components (nj(Wn+‘, P)n ) == 00 for 2 0), 0), there boundary components (11"j(W n +l , V for j 2: there may may nonetheless nonetheless be obstruction to triviality, given to triviality, given by by an an element element be an an obstruction a(Wnfl

, V”)

E Wh(x)

= K1(n)/

f n.

(5.12) (5.12)

In fact for Wh(n) In fact for each each nn >2: 55 every every element element aa of of W h( 11") is is realizable realizable as as such such an an n n is necessary obstruction, and the vanishing of the obstruction a(Wnfl, Vn) obstruction, and the vanishing of the obstruction a(W +1, V ) is necessary and sufficient for h-cobordism: Wn+l and sufficient for triviality triviality of of the the h-cobordism: W n+ l = Vn vn xx 1I (Mazur, (Mazur, first first half of the 1960s; the proof was brought to completion by Barden and Stallings half of the 1960s; the pro of was brought to completion by Barden and Stallings in in the the middl96Os). mid~ 1960s). The The appropriate appropriate algebraic algebraic definition definition of of Kz K 2 for for associative associative rings rings with with an an identity element was discovered by Milnor and Steinberg in the second identity element was discovered by Milnor and Steinberg in the second half half of 1960s and analogues Kj constructed around the 1960s, and the the higher higher analogues Kj were were constructed around the the turn turn of the of several authors of that that decade decade by by several authors (Quillen, (Quillen, Volodin, Volodin, Gersten, Gersten, Karoubi, Karoubi, VillaVillamayor); the equivalence equivalence of of the the various various definitions definitions proposed proposed was was established established mayor); the somewhat later. The topological realizations of these groups (as considered by by somewhat later. The topological realizations of these groups (as considered Wagoner), of great shall not Wagoner), especially especially of of Kz, K2' are are of great interest; interest; however however we we shaH not pursue pursue the the topic topic here, here, not not least least because because this this line line of of investigation investigation remains rernains far far from from complete. complete. ItIt is of aa ring is noteworthy noteworthy that that for for the the Laurent Laurent extension extension A[t, A[t, t-l] Cl] of ring A, A, besides besides the obvious projectors the obvious projectors Ko(A[t, t-l]) ;=: = Ko(A), Ko(A), Ko(A[t, Cl])

Kl(A[t,t-‘I) = K1(A), Kl(A), K 1 (A[t,C l ]);=: determined by the natural ring epimorphism determined by the natural ring epimorphism A[t,t-‘1 A[t, t~ 1] 4- - t AA and and sion A --+ A[t, t-l], there is a non-trivial projector B (the Bass sion A - - t A[t, t~l], there is a non-trivial projector B (the Ba88 discovered in the mid-1960s): discovered in the mid 1960s): Ko(A)

the incluthe incluprojector, projecto'T',

l!.. KI (A[t, Cl])

Ko(A) z!!- K1(AIW1l),

Ko(A)

KI(A[t, Cl]),

satisfying BB = 1, with the kernel of B given satisfying BB 1, with the kernel of B given this projector is as follows: If q is a projective this projector is as follows: If T) is a projective module for some ii. The infinite direct sum module for sorne fi. The infinite direct sum

M= M

explicitly. The construction of explicitly. The construction of A-module, then v + q is a free A-module, then 17 + fi is a free

n<+m n<+oo c tn(r7+ii) =~(%+Irln) i7n) n>-CCtn(T)+i}) 2:=(T)n+ n

2:=

n>-oo

n

is then a free A[t,t-‘l-module. We define B(q) as the automorphism of the is then a free A[t, t~l]~module. A-module A4 commuting with t,We t-l, define given B(T)) by as the automorphism of the A~module M commuting with t, Cl, given by

55. The The role role of of the the fundamental fundamental group in in topology topology §5. group

259 259

B(rJ) :

(5.13) (5.13)

The definition definition of B B is as follows: follows: Each Each automorphism automorphism X of of the the free free A[t, A[t, tt-l]The of À module FN of rank N is, after being multiplied by a sufficiently large power module FN of rank N is, after being multiplied by a sufficiently large power n of by a matrix matrix over over A[t] A[t] relative relative to to some some basis basis el, er, ... . . ,, eN eN of of of t, representable representable by FN. Denoting by F& the “positive part” of FN: F N · Denoting by Ft; the "positive part" of F N :

F$ = { cajej

1 aj E A[t]}

,

we set set we

l?(X) = F,$//tY

= 77.

(5.14) (5.14)

It is to verify verify the the projectivity projectivity of module ri, q, and and then then that that It is straightforward straightforward ta of the the module BB = 1 on Ko(A). BB = 1 on Ko(A). It follows follows that one always always has It that one has

Kl(A[t,

t-l])

” &,(A)

(5.15) (5.15)

CBICI(A) @P.

In the 7r] with abelian, Bass showed P = O. 0. Since Since In the case case A A = Z[ Z[71"] with n71" free free abelian, Bass showed that that P for such 7r (g Z @ . $ ‘Z) every finitely-generated Z[r]-module has a free for such 71" (<::: Z ED ... ED Z) every finitelygenerated Z[71"]-module has a free acyclic resolution and Ka(7r) one obtains obtains by KO(71") == 0, 0, one by induction induction acyclic resolution of of finite finite length, length, and Bass’ theorem free abelian TTT: Bass' theorem for for free abelian 11": Wh(7r) It follows

that

in the situation

= Kl(T)/

It 7r = 0.

of free abelian

fundamental

group

the above

It follows that in the situation of free abelian fundamental group the above mentioned invariant of non-simply-connected manifolds is trivial. mentioned invariant of non-simply-connected manifolds is trivial. A natural continuation of this sort of problem is that of representing a A natural continuation of this sort of problem is that of representing a manifold W” n as a smooth fiber bundle over the circle 5” (IV” n + S1), on manifold W as a smooth fiber bundle over the circle SI (w SI), on the assumption that the given manifold W” has the homotopy type of such the assumption that the given manifold W n has the homotopy type of such a bundle. If 7ri (Wn) cz Z and n > 6, and the fiber of the fibration is simplya bundle. If 71"1 (W n ) ~ Z and n 2: 6, and the fiber of the fibration is simplyconnected, then the problem has an affirmative solution (Browder, Levine, connected, then the problem has an affirmative solution (Browder, Levine, Livesay, in the midp1960s). In the more general situation, where the fiber Livesay, in the mid-1960s). In the more general situation, where the fiber is not simply-connected, obstructions arise involving both &-a(r) and Kr(r), is not simply-connected, obstructions arise involving both KO(71") and KI (71"), where 7r is the fundamental group of the fiber given as the kernel Ker f* of the where 71" is the fundamental group of the fiber given as the kernel Ker f* of the homomorphism f* : 7r --+ Z, induced from the appropriate map f : W” ---+ homomorphism f* : 71" ........... Z, induced from the appropriate map f : W n ........... S1, a candidate for being a smooth projection. (The solution of the problem SI, a candidate for being a smooth projection. (The solution of the problem was obtained in the second half of the 1960s by Hsiang and Farrell.) was obtained in the second half of the 1960s by Hsiang and Farrell.) We turn now to the general classification theory of nonsimply-connected to the general classification theory of non-simply-connected We turnofnow manifolds dimension n 2 5. This theory follows closely the Browdern 2: 5. This theory follows closely the Browdermanifolds of dimension Novikov scheme for classifying the simply-connected manifolds of dimension Novikov scheme for classifying the simply-connected manifolds of dimension n 2 5 (see $4 above); however rather than yielding a final classification then 2: 5 the (see theory §4 above); however rather than yielding a final theorem, reduces the problem to the existence of classification certain algebraic the theory reduces the problem to the existence of certain algebraic orem, obstructions, which we shall now describe. All of the Browder-Novikov appaobstructions, we ofshaH now maps describe. AH ofone the ofBrowder-Novikov ratus (described which in $4) normal of degree closed manifolds, appaand ratus (described in §4) of normal maps of degree one of closed manifolds, and

260

Chapter 4. 4. Smooth Smooth Manifolds Manifolds Chapter

of bordisms bordisms between between them, them, carries carries over over to to the the non--simply-connected non-simply-connected case. InInof case. voking the the non-simply-connected non-simply-connected analogue of of Hurewicz' Hurewicz’ theorem, theorem, concerning concerning voking analogue the kernels kernels of of the the homomorphisms homomorphisms induced from from normal normal maps maps f : Ml &Ii ----7 --f M M of of the induced degree one, one, one one carries carries out out Morse Morse surgeries surgeries transforming transforming such a map map into into one one degree such inducing isomorphisms isomorphisms between the the respective respective homotopy homotopy groups groups up up to to and and inducing between including dimension dimension [nJ2] [n/2] - 1. 1. For For even even n n (= 2k) 21c) analysis analysis of of the the homotopy homotopy including kernel in dimension dimension kk (in terms of of universal universal covering covering manifolds) manifolds) reveals reveals that that kernel in (in terms this kernel kernel is is a finitely-generated free free stable stable Z[7r]-module, Z[n]-module, on on which which there there is is this a finitely-generated defined aa n-invariant scalar product product given given by by the the intersection intersection index index of of cycles cycles defined 7r-invariant scalar lifted onto onto the universal covering manifold: manifold: lifted the univers al covering (z,w) (z, w)

= 2:)z -JT(z 0 aw) E qr]. 0 O"w) E Z[7r]. UEK aE7r

(5.16) (5.16)

Thus this this scalar scalar product takes its its values values in in Z[7r], iZ[n], on on which which there there is is an an involuinvoluThus product takes tion ii, where tion uu H t---> ü, where

u= c aigi, CLi EZ, giET, ii= c Wi-l, 2

(5.17) (5.17)

i

with of an an involution: involution: with the the usual usual properties properties of im = vü, vu, uv =

iYi = u. ü = u.

(5.18) (5.18)

Remark. for non-orientable the definition definition of of the invoRemark. Note Note that that for non-orientable manifolds manifolds the the involution needs to be adjusted as follows: lution needs to be adjusted as follows: U ++

ti

=

C(Sgn

gi)aigtrl,

(5.19)

(5.19)

where sgn g2 = -1 if the action of gi is orientation-reversing and sgn gi = +I where sgn gi -1 if the action of gi is orientation-reversing and sgn gi + 1 0 otherwise. otherwise. 0 The scalar product (5.16) is Hermitian for k even and co-Hermitian for k The scalar product (5.16) is Hermitian for k even and co-Hermitian for k odd: odd: (uz, w) = u(z, w), (6 w) = *tw, 4, (uz, w) = u(z, w), (z, w) = ±(w, z), (5.20) (5.20) (z, uw) = (z, w)u,

(z, uw)

= (z, w)ü,

and is non-degenerate in the sense that relative to a free basis {ei} for the and is non-degenerate in the sense that relative to a free basis {ei} for the Z[r]-module M = Ker firk) (n = 2k), the matrix Z[7r]-module M = Ker fi 7rk ) (n = 2k), the matrix B = (hj) = ((ei, ej)) is invertible. (Recall that we are already in the situation where Ker fisj) = 0 is invertible. that we are in the situation wheretoKer fi1f'j) this0 for j < k.) In (Recall invariant form (i.e. already expressed without reference a basis) formthe(Le. expressed reference an to isomorphism a basis) this for j < k.) In invariant non-degeneracy means that scalar productwithout ( , ) determines non-degeneracy means that the scalar product ( , ) determines an isomorphism

55. The The role role of of the the fundamental fundamental group group in in topology topology §5.

h:M-+M* HomA(M,A), h: M ---> M* = HomA(M,A),

‘261 261

Z[r], AA = Z[1T],

given by U u f-? ++ h(u), h(u),

(h(u),v) @(‘L~),v) == ('u,v). (u,~).

(5.21)) (5.21

(Note that that the the above-mentioned above-mentioned involution involution allows allows M* M’ to to be be endowed endowedwith with the structure of a Z[rr-module.) The scalar product has the property of “evenstructure a Z[1T]-module.) scalar product has the property of "evenness”, which in matrix terminology (given that the module M is free) means ness", which in matrix terminology (given that the module M is free) me ans that the the above above matrix matrix B B is is expressible expressible in in the the form: form: that B = v cl3(-l)kv+, T where V+ = V VT, T denoting denoting the the operation operation of of taking taking the the transpose. transpose. For For odd odd kk where V+ ,T there also exist Z/2-invariants (analogous to the Arf invariant) which we shall there also exist Zj2-invariants (analogous to the Arf invariant) which we shaH not, however, describe here. If Mn is neither a smooth nor PL-manifold but not, however, describe here. If Mn is neither a smooth nor PL-manifold but only a CW-complex whose Z[r]-homology and cohomology satisfy Poincare only a CW-complex whose Z[1T]-homology and cohomology satisfy Poincaré 7rk duality, then the the module module M Ker fin’) will be merely a projective Z[n]M = Ker ) will be merely a projective Z[1TJduality, then module rather than stably free. In this case one may may proceed proceed geometrically, geometrically, module rather than stably free. In this case one adding more generators by attaching handles of index k in order to make aa stastaadding more generators by attaching handles of index k in order to make bly free module out of the projective module M. The obstruction then appears bly free module out of the projective module M. The obstruction then appears as before in Hermitian and and co-Hermitian groups K~(Z[1T]), Kt(Z[x]), Ki”h(Z[r]). In h (Z[1T]). In as before in the the Hermitian co-Hermitian groups Kô this context the role of the trivial element is taken by the module (denoted this context the role of the trivial element is taken by the module (denoted by basis of cycles whose with respect to the Hg) having having a a canonical canonical basis of cycles whose behavior behavior with respect to the by Hg) scalar product is analogous to that of the standard cycles on the surface scalar product is analogous to that of the standard cycles on the surface of of genus g with respect to the intersection index: genus 9 with respect to the intersection index:

fi

al ,..., (ai,aj) = 0,

ag,

(bi, bj) = 0,

Under the operation of forming Under the operation of forming tions Hg N 0 for all g imposed, tions Hg '" 0 for aIl 9 imposed, Grothendieck groups: Grothendieck groups:

bl,...,

b,,

(ai,bj) = Sij = &(bj,ai).

the direct sum of modules, with the relathe direct sum of modules, with the relaone obtains the following analogues of the one obtains the following analogues of the

K~(Z[*IT]) t

K6 (Z[1T])

k even, the Hermitian case; k even, the Hermitian case;

Kth(;Z[7r]) h

k odd, the co-Hermitian case, k odd, the co-Hermitian case,

Kô (Z[1T])

(5.22) (5.22)

where Z[r] is equipped with the involution (5.17) (or (5.19) in the nonoriwhere Z[1T] is equipped with the involution (5.17) (or (5.19) in the nonorientable situation). entable situation). The algebraic formulation of the obstruction theory to Morse surgeries was The algebraic the obstruction Morse surgeries was given in the caseformulation n = 2k by of Novikov and Wall,theory in thetomid-1960s and was the case n 2k by Novikov and Wall, in the mid-1960s, and given in extended to odd n by Wall at the end of the 1960s. It turns out that the was obextended to odd nto by Wall atofthe of the 1960s. It(kturns that the obstructions reduce elements theend groups KF(Z[7r]) even),out and Kih(Z[r]) structions reduce to elements of the groups (k even) , and K: hclasses (Z[1T]) (k odd), constructed on the analogy of the Kf(Z[1T]) group Ki(n), using stable (k odd), constructed on the analogy of the group K 1 (1T), using stable classes of automorphisms of the canonical modules defined by (5.22), preserving the the canonical modulesare defined by (5.22), preserving the of automorphisms scalar product. Theof Whitehead relations imposed on these classes (i.e. scalar product. The Whitehead relations are imposed on these classes (Le.

262 262

Chapter Chapter 4. 4. Smooth Smooth Manifolds Manifolds

direct direct sums sums of of such such automorphisms automorphisms are are identified identified with with their their corresponding corresponding composites) composites) yielding yielding commutativity commutativity together together with with aa further further relation, relation, conveconveniently fo11ows: any any automorphism automorphism XÀ :: Hs Hg --+ -- Hg Hg for for which which niently formulated formulated as as follows: x(aj) g, isis set À(aj) == aj, aj' jj == 1,. 1, .... ,,9, set equal equal to to the the trivial trivial one. one. These These groups groups of of obobstructions structions toto surgeries, surgeries, defined defined via via automorphisms automorphisms of of free free modules modules over over the the ring ring Z[n], Z[7f], are are called ca11ed the the Wall Wall groups groups of of R, 7f, denoted denoted by by L,(X): L n (7f): Lo ” K,h,

L1 E K;,

L2 ” Kih,

L3 ” Kih.

(5.23) (5.23)

As As noted noted earlier, earlier, we we shall sha11 not not enter enter here here aa into into discussion discussion of of the the Z/2%structure Z/2~structure of of these these groups groups and and the the associated associated analogues analogues of of the the Arf Arf invariant. invariant. There of that that described There isis the the alternative alternative approach approach (along (along the the lines Hnes of described prior prior to to the the preceding preceding paragraph) paragraph) to to the the definition definition of of the the groups groups Kk, K6', Kk, K{', Klh, Kôh, Kfh or Kî h via via projective projective modules modules equipped equipped with with aa non-degenerate non-degenerate Hermitian Hermitian or co-Hermitian the requisite requisite property property of of “evenness” "evenness" and and co-Hermitian scalar scalar product product (with (with the the an isomorphism the structure structure carried carried by by the the Arf Arf invariant) invariant) determining determining an isomorphism hh :: M involution. (In above M 2 M*, M*, and and with with values values in in aa ring ring with with an an involution. (In the the abovc topological Z[rr] with involution (5.17) the ring ring was was Z[7f] with the the involution uu +---> 6iL as as in in (5.17) topological context context the or terms of direct The groups groups E-,h K~ and and J?ih Kô h are are defined defined as as usual usual in in terms of direct or (5.19).) (5.19).) The sums of such relations equating modules of sums of such modules, modules, with with the the relations equating the the projective projective modules of the form A4 @ N* N) with one, imposed: the form M == NN œ N* (N** (N** E’ ~ N) with the the trivial trivial one, imposed: (N, N) = (N*, (N*, N*) N*) = 0, 0, (N, N) N@N*-0, NœN*rvO,

h:N@N*+(N@N*)*, h:NœN*--(NœN*)*,

(5.24) (5.24)

where

isomorphism is given by the pair of canonical isomorphisms: where the the isomorphism hh is given by the pair of canonical isomorphisms: h: h: The

algebraic

definition

of 2:

N* 5

N*

N* N* N r,,, { N N** and kfh

(5.25)

(5.25 )

1s based not on automorphisms

(as

The algebraic definition of Kr and Kî h is based not on automorphisms (as above) but rather on the concept of a Lagrangian submodule L c N 63 N* of a ab ove ) but rather on the concept of a Lagrangian submodule L c N œN* of a canonical projective module N @ N*, defined by the conditions that h\L = 0 canonical projective module N œN*, defined by the conditions that hl L = 0 and that L be a direct summand satisfying and that L be a direct summand satisfying M=N@N*-L@L*,

M=NœN*""LœL*, The

group

operation

is given

by the direct

L**=L.

L**

(5.26)

L.

sum of pairs

(5.26) (M, L), with

those

The group operation is given by the direct sum of pairs (M, L)\ with those pairs taken to be trivial for which the projection A4 + N (N* ---) 0) restricts pairs taken to be trivial for which the projection M ---> N (N* ---> 0) restricts to an isomorphism L + N. The stable equivalence classes of pairs (M, L) to an isomorphism L ---> N. The stable equivalence classes of pairs (M,L) then constitute the groups Kk and l?ih, the Novikov- Wall groups. then constitute the groups Kr and Kîh, the Novikov- Wall groups. Without considering the possible variants or entering into any detail, we Without considering the possible variants or entering into any detail, we note that the higher analogues p: and j?ih h can be defined for any ring A note that the higher analogues K~ and Kt can be defined for anycases. ring A with an involution u + G (UV = a~) in both the free and projective It with an involution u ---> fi (uv = vü) in both the free and projective cases. It turns out that, for .instance, Rk - h = klh - sh and l?zh - sh = k,$. - h turns out that, for mstance, K 2 = Ko and K 2 Ko.

§5. §5.

The role of of the the fundamental group in topology The role fundamental group topology

263 263

An algebraic theory theory of Z[a], Kt Kbhh (>9 @ Z[+] was formulated formulated An algebraic of the the groups groups K? Kr @ (>9 Z[H Z[~I was by Novikov (in the late 1960s) 1960s) in of rings rings with and by Novikov (in the late in the the category category of with involution, involution, and included an of the projectors (see (see included an algebraic algebraic construction construction of of analogues analogues of the Bass Bass projectors above): above): K,h(A[t,

t-l])

+

i?B BB = 1.

K;+l(A),

(5.27) (5.27)

The K,” 8(>9 Z[i] furnish aa periodic (of period period 4) theory in in the the The groups groups KI' Z[ ~ 1 furnish periodic (of 4) homology homology theory category of rings with involution, in which the Hermitian analogues of the Bass category of rings with involution, in which the Hermitian analogues of the Bass projectors subsume the the co-Hermitian in view view of the isomorphisms isomorphisms projectors (5.27) (5.27) subsume co-Hermitian case case in ofthe K3”h ” Ksh K;$, K J j+2'

h K;hsh ” K K3”+2. K J j+2'

(5.28) (5.28)

Note this connexion the isomorphism Note in in this connexion the isomorphism (modulo (modulo tensoring tensoring with with Z[i]) Z[~]) K,h+l(A[t,t-l])

g K;(A)

@ Kjh+l(A),

(5.29) (5.29)

contrasting the corresponding corresponding situation in ordinary ordinary K K-theory where contrasting with with the situation in -t.heory where there is a non-trivial kernel N. there is a non-trivial kernel N. The above as formulated by Novikov, Novikov, uses uses analysis analysis of of The above Hermitian Hermitian K-theoy, K -theory, as formulated by geometric realizations of the above algebraic objects together with analogues geometric realizations of the above algebraic objects together with analogues of concepts of Hamiltonian formalism arising in connexion with analytical and of concepts of Hamiltonian formalism arising in connexion with analytical and quantum mechanics, as intensively by Maslov in in the 1960s to to the the quantum mechanics, as applied applied intensively by Maslov the early early 1960s construction of short-wave asymptotics in the theory of hyperbolic equations construction of short-wave asymptotics in the theory of hyperbolic equations (and developed further by many authors). Once translated into an algebraic (and developed further by many authors). Once translated into an algebraic context, the standard terminology of symplectic geometry and Hamiltonian context, the standard terminology of symplectic geometry and Hamiltonian formalism turns out to be extremely useful; in particular it clarifies the ideas formalism turns out to be extremely useful; in particular it clarifies the ideas behind the algebraic constructions (and sometimes even suggests the approbehind the algebraic constructions (and sometimes even suggests the appropriate ideas), and makes more explicit the mechanism underlying the algebraic priate ideas), and makes more explicit the mechanism underlying the algebraic analogues of Bott periodicity. Of course once the algebraic theory has been analogues of Bott periodicity. Of course once the algebraic theory has been precisely formulated, it ceases to be dependent on its source in the analogy precisely formulated, it ceases to be dependent on its source in the analogy with Hamiltonian formalism. with Hamiltonian formalism. It was shown by Mishchenko (around 1970) that tensoring by Z[;] elimiIt was shown by Mishchenko (around 1970) that tensoring by Z[~l eliminates the differences between the various proposed Hermitian K-theories. At nates the differences between the various proposed Hermitian K-theories. At the same time he constructed a homotopy invariant 7x7r of an orientable manithe same time by he constructed homotopy(where invariant of an orientable manifold Mj, given an element ofa K~(Z[T]) x = T rin/lj), analogous to the fold Mj, given by an element of Kj(Z[11"]) (where 11" = 11" l Mj), analogous to the signature, and determining a homomorphism from each bordism group of the signature, and determining a homomorphism from each Hermitian bordism group of the Eilenberg-MacLane space K(T, 1) to the corresponding K-theory: Eilenberg-MacLane space K(11", 1) to the corresponding Hermitian K-theory: To : L’,S’(K(q For normal maps f : Mr --+ For normal maps f :theory Ml' --; general classification of general classification theory of

1)) -

K,h(l+r])

~3 Z[$].

M” of degree one (figuring centrally in the Mn of degree one (figuring manifolds of dimension > 5), centrally obstructionsin the to manifolds of dimension 2: 5), obstructions to

264 264

Chapter 4. 4. Smooth Smooth Manifolds Manifolds Chapter

Morse surgeries surgeries are given given by by the the formula formula Morse T”(MT’)

- T”(bP)

E K,h(qr])

@IZ[$].

Remark. Note Note that that even even for for simply-connected simply-connected manifolds the the calculation calculation of of Remark. manifolds such obstructions obstructions without dividing dividing by by 2 was was impossible impossible owing owing to to a difficulty difficulty such without involving the the Arf Arf invariant. invariant. involving 00 Novikov’s conjecture conjecture on the higher higher signatures signatures concerns concerns the the existence existence of of a Novikov's purely algebraic algebraic analogue analogue of of the the Chern Chern character: character: purely ch : K,h(Z+r])

8 Q -

H,(q

Q),

with the the property property that that for for aa closed closed manifold manifold Mn M” the the following following formula formula should should with hold: hold: (z, ch o ?(AP)) = &(AJP), &5*(z)). (5.30) (5.30) Here 7r 7r = 7rl(Mn), 7rr(Mn), Mn -+ K(7r,l) K(K, 1) is is the the canonical canonical map, map, Dcp*(z) D+*(z) E cp4 :: Mn E Here H4k(Mn; Q), and Lk is the Hirzebruch polynomial in the Pontryagin classes of H4 k(Mn; Qi), and Lk is the Hirzebruch polynomial in the Pontryagin classes of Mn. It It would that this formula, if if true for any any group group 7r, 7r, would would embrace embrace Mn. would appear appear that this formula, true for all possible relations between the Pontryagin classes for a given homotopy type aIl possible relations between the Pontryagin classes for a given homotopy type of the manifold M”. In all cases where the general conjecture on the higher of the manifold Mn. In aIl cases where the general conjecture on the higher signatures the existence of such such an an algebraic algebraic analogue existence of analogue signatures has has been been established, established, the of the Chern character (satisfying (5.30)) has also been shown. of the Chern character (satisfying (5.30)) has also been shown. In the second half of the 1960s it was observed by I. M. Gel’fand and In the second half of the 1960s it was observed by 1. M. Gel'fand and Mishchenko that for rings C*(X) of complex-valued functions on compact Mishchenko that for rings C* (X) of complex-valued functions on compact spaces X, one has spaces X, one has K;(c*(x)) % K;(x). Kg(C*(X)) ~ Kg(X). In fact such an isomorphism exists also for Kr, and therefore for all the K,h of In fact such an isomorphism exists also for K l, and therefore for aIl the K~ of such function algebras with involution (the involution on C*(X) being given such function algebras with involution (the involution on C* (X) being given by f ---f f). In the case X = Tn,n use of the Fourier transform yields the by f -> 1). In the case X = T , use of the Fourier transform yields the following isomorphism modulo tensoring by Z[ :]: following isomorphism modulo tensoring by Z[~l: (” K;(Tn)). (5.31) K,h(C[Z x . . . x Z]) ” K,h(C*(Tn)) (5.31 ) Hence the ordinary Chern character of a vector bundle over Tn affords an Hence the ordinary Chern char acter of a vector bundle over Tn affords an algebraic analogue of the Chern character for 7r = Z x . . x Z (n times), algebraic analogue of the Chern character for 7r Z x ... x Z (n times), although this method of obtaining such an analogue is obviously not algebraic. although this method of obtaining such an analogue is obviously not algebraic. The algebraic definition of the analogues of the Bass projectors in K,hThe algebraic definition of the analogues of the Bass projectors in K~­ theory for the rings C*(X), yields a suspensionisomorphism and an approach theory the rings different C*(X), yields suspension and an approach to Bott for periodicity from athat affordedisomorphism by the ordinary K-theory of to Bott periodicity different from that afforded by the ordinary K-theory of Atiyah and others. Note that the presence of an imaginary i (i22 = -1) in that the presence of an imaginary i (i -1) in Atiyah and others. Note the ring C*(X) allows the Hermitian theory to be transformed to the cothe ring C*so (X) the Hermitian theory to imaginary be transformed to we thehave coHermitian, thatallows for algebras A with such an an element Hermitian, so that for algebras A with such an an imaginary element we have K;(A) = Ksh(A), and the periodicity is shortened to 2. K~(A) = K!h(A), and the periodicity is shortened to 2. By sorting out the various algebraic constructions one arrives at the signifiBy sorting out thefrom various algebraic constructions one arrives at thehowever significance of periodicity the point of view of Hamiltonian formalism; canee of periodicity from the point of view of Hamiltonian formalism; however

§5. 55.

The role role of of the the fundamental fundamental group group in in topology topology The

265 265

this this theory theory (as (as conceived conceived by by Novikov Novikov in in the the late late 1960s) 1960s) concerns concerns the the tensor tensor product @I%?$]. A complete formalization of Hermitian algebraic K-theory product K~ I2lZ[ A complete formalization of Hermitian algebraic K -theory (called (called algebraic algebraic L-theory), L-theory), avoiding avoiding tensoring tensoring by by Z[i], Z[~], was was achieved achieved by by RanRanicki in the first half of the 1970s by recasting the theory over the integers; icki in the first half of the 1970s, by recasting the theory over the integers; for Z[1f] the the corresponding corresponding groups groups are are denoted denoted by by L,(n). L n (1f). These These for aa group group ring ring Z[rr] groups were studied by several people throughout the 1970s and into groups were 8tudied by several people throughout the 19708 and into the the 1980s 1980s (Pedersen, (Pedersen, Beck, Beek, Sharpe). Sharpe). In In the the first first half half of of the the 1970s 1970s Wall Wall and and Beck Beek develdeveloped for computing computing these these objects objects in in the the situation situation where where oped detailed detailed techniques techniques for 7r is finite and the modules are free, and actually carried out the 1f is finite and the modules are free, and actually carried out the calculations calculations for for various various particular particular groups. groups. On the other not available available until On the other hand hand although although the the full full algebraic algebraic theory theory was was not until the end of the 1960s it was possible earlier on to obtain complete the end of the 1960s, it was possible earlier on to obtain complete informainformation tion about about the the groups groups L,(n) L n (1f) in in certain certain special special cases cases (although (although noneffectively) noneffectively) without having to hand the details of the algebraic without having to hand the details of the algebraic definition, definition, for for instance instance x Z. In the case n = Z the when . . x Z. In the case 1f is free free abelian, abelian, rr1f = ZZ xx .... Z the theory theory was was when r1f is formulated mid~1960s) as as aa geometric geometric obstruction obstruction theory theory formulated by by Browder Browder (in (in the the mid-1960s) to theory was to reducibility reducibility to to the the simply-connected simply~connected situation. situation. In In this this form form the the theory was then case 1f 7r = ZZ xx ... . . xx ZZ by by Shaneson Shaneson (in then extended extended to to the the more more general general case (in the the late 1960s). Independently of the various algebraic definitions of the groups late 1960s). Independently of the various algebraic definitions of the groups L,(T), Ln (1f), there there exists exists an an isomorphism isomorphism

!].

L(n

x Z)

”

Lx(~)

@ -Lfl(~lT),

(5.32) (5.32)

which established geometrically, geometrically, and isomorphism allows allows the which can can be be established and this this isomorphism the strucstructure of the groups L,(Z x . x Z) to be inferred (non-effectively) from ture of the groups Ln(Z x ... x Z) to be inferred (non-effectively) from that that of the groups L,(l) of the groups Ln(1) == L,, Ln, which which are are as as follows: follows: ifn=O (mod4), ifn===O (mod4), if n E 2 (mod 4), if n 2 (mod 4), otherwise. otherwise. (These isomorphisms were already known from simply-connected surgery the(These isomorphisms were already known from simply-connected surgery theory.) Knowledge of the groups L,(Z x . x Z) leads to a complete classification ory.) Knowledge of the groups Ln (Z x ... X Z) leads to a complete classification of the homotopy n-tori, i.e. of the n-manifolds having the homotopy type of of the homotopy n-tori, i.e. of the n-manifolds having the homotopy type of the torus T”,n n 2 5. In particular, it turns out that such a manifold always the torus T , n ::::: 5. In particular, it turns out that such a manifold always has a finite-sheeted covering manifold diffeomorphic to the standard torus T”.n has a finite-sheeted covering manifold diffeomorphic to the standard torus T . L, z Ln ~

z Z Z/2 Z/2 0 {1 o

Remark. We note that all surgery obstructions (i.e. the elements of the Remark. We note that aH surgery obstructions (i.e. the elements of the are realized geometrically as invariants distinguishing two grows L+I (n)) groups Ln+ 1 (1T)) are realized geometrically as invariants distinguishing two manifolds in the same homotopy class as the given n-manifold. The construcclass as n-manifold. The construcmanifolds in the same tion is as follows: If twohomotopy normal maps fi the : MFgiven + Mn are homotopy equivation is as follows: If two normal maps fi : Mf Mn are homotopy equivalences and lie in the same normal bordism class, then there exists a bordism lences and lie in the same normal union bordism class, exists a bordism Wn2+i with boundary the disjoint of Mr andthen Mg there (6W’“+1 = Mr U M$) n + 1 with boundary the disjoint union of Mï and Ml} (âW n +1 = MïUMl}) W together with a normal map F : Wnfl ---+ Mn x I, whose restrictions to n+1 ~ Mn X I, whose restrictions to together with acomponents normal map : W MF the boundary MFF and are fi, fz respectively. Denote by and ft, hUnder respectively. pthe : Mboundary x I + Mcomponents the projectionMï on theMl} first are factor. the aboveDenote conditionby p: Mx 1 ~ M the projection on the first factor. Under the above condition

266 266

Chapter 4. Smooth Smooth Manifolds Manifolds Chapter

n +1 ----+ the map map FI Fl = = ffcl 1’ 0opo p o F affords affords aa normal retraction retraction W Wnfl ---f Mf n/r? onto onto the the the boundary component Mf IVY (see (see§4 $4 above). Starting Starting with with aa given given manifold manifold Mf &IF boundary n + 1 every one can can realize geometrically geometrically by me means such bordisms bordisms w Wn+’ every element element one ans of such of the surgery obstruction group, just as in the simply-connected case. surgery obstruction group, just as simply-connected case. Thus of n +1 corresponding one may construct (starting with MT) such a bordism Wn+’ corresponding construct (starting with Mf) such a w one to any prescribed element of of the group Ln+l L,+r(r)(1f) ~ Z J{~+1 K,h+,(n). 0 (-71'). D

For homotopy homotopy tori tori there there is is aa subgroup subgroup An A, of of finite finite index: index: .E
t M

-

f

M --'--

t Tn+l n T +1

$5. role of the fundamental group in topology §5. The role fundamental group

267 267

trivial ZP+l: trivial over T” Tn CC Tn+l:

f-l@‘“)

“T”

vMITn = f*~~l~n

c W+‘,

= o.

By cutting embedded torus Tn Tn one By cutting Mnfl Mn+! along the embedded one then obtains the desired desired manifold a, and with with each each of its two boundary boundary compomanifold realizing realizing the element 0:, nents diffeomorphic Tn. diffeomorphic to to Tn. Given two CW-complexes or manifolds one can distinguish them, Given two CW-complexes or manifolds one can often often distinguish them, i.e. show that they are not homeomorphic (or not PL-homeomorphic, or not not i.e. show that they are not homeornorphic (or not PL-homeomorphic, or diffeomorphic), by means of algebro-topological invariants. However if all condiffeomorphic), by means of algebro-topological invariants. However if aU conceivable invariants coincide coincide for these two spaces, then then one one is is forced forced to to resort resort ceivable invariants for these two spaces, finally to seeking some means or other of actually constructing a homeomorfinally to seeking sorne means or other of actually constructing a homeomorphism is interesting interesting to to consider consider the the various various methods methods that that phism between between them. them. ItIt is have been employed to this end over the last two or three decades. have been employed to this end over the last two or three decades. In In differdifferential relied either producing the approach approach to to this this problem problem has has relied either on on producing ential topology topology the an appropriate map directly by analytic means, or else on using completely an appropriate map directly by analytic means, or else on using completely finitistic selection and and analysis analysis of of sequences of finitistic procedures procedures involving involving careful careful selection .sequences of attachments of handles (for example, by means of Morse surgeries, when the attachments of handles (for example, by means of Morse surgeries, when the topology function alters as its value passes topology of of the the level level sets sets of of aa Morse Morse function alters as its value passes through through a critical point). In carrying out sequences of Morse surgeries, their on a critical point). In carrying out sequences of Morse surgeries, their effects effects on homotopy, the algebraic obstructions to realizing them that arise, and various homotopy, the algebraic obstructions to realizing them that arise, and various reductions simplifications that (for instance that occur occur (for instance the the mutual mutual cancellacancellareductions and and simplifications tion of a pair of handles) are all investigated. These basic techniques, tion of a pair of handles) are ail investigated. These basic techniques, which which have been in use for a very long time, underwent intensive elaboration in have been in use for a very long time, underwent intensive elaboration in the 1960s in parallel with the remarkable developments in the topology of the 1960s, in parallel with the remarkable developments in the topology of manifolds of that era. We shall not however consider them further here; they manifolds of that era. We shall not however consider them further here; they have been mentioned at various points earlier on in this book. What other have been mentioned at various points earlier on in this book. What other tools does topology have at its disposal appropriate to the direct investigatools does topology have at its disposai appropriate to the direct investigation of homeomorphisms? In low dimensions (n < 3) every compact family of tion of homeomorphisms? In low dimensions (n :S 3) every compact family of homeomorphisms of manifolds can be approximated by PL-homeomorphisms, homeomorphisms of manifolds can be approximated by P L-homeomorphisms, and by diffeomorphisms, and the whole group of self-homeomorphisms of a and by diffeomorphisms, and the whole group of self-homeomorphisms of a manifold has the same local and global homotopy properties as the group manifold has the same local and global homotopy properties as the group of PL-homeomorphisms (Moise, in the early 195Os), and even the group of of P L-homeomorphisms (Moise, in the early 19508), and even the group of self-diffeomorphisms. However in higher dimensions this is certainly no longer self-diffeomorphisms. However in higher dimensions this is certainly no longer the case. Among the best-known of the early results relevant to this sort of the case. Among the best-known of the early results relevant to this sort of question, obtained by elementary visual means, is that of Mazur and Brown question, obtained by elementary visual means, is that of Mazur and Brown (of around 1960): Given a ‘flat” embedding of a sphere, i.e. an embedding of (of arollnd 1960): Given a "fiat" embedding of a sphere, i.e. an embedding of a whole neighbourhood: a who le neighbourhood: F : S”-l

x I ----) S”,

F: Sn-l xl

sn,

I = [-I, 11, 1= [-1,1],

the image F(S”-1 x (0)) of the “middle” sphere Snpl x (0) bounds a closed the image F(sn-l x {O}) of the x"middle" sphere sn-l X {O} bounds a closed disc Dnn c P: dD” n = F(S”-1 (0)) (the “g eneralized Schoenflies conjecdise D C sn: ôD F(sn-l x {O}) (the Schoenflies conjecture”). Following hard on the appearence of "generalized the Brown-Mazur theorem, the ture"). Following hard on the appearence of the Brown-Mazur theorem, the “Annulus Conjecture” was formulated (by various people): "Annuills Conjecture" was formulated (by various people):

268 268

Chapter 4. Smooth Smooth Manifolds Chapter 4. Manifolds

Given embeddings FI Fl and F22 of of sn-l Pm1 in in sn Sn:. Given two two non-intersecting non~intersecting flat fiat embeddings and F F&3”-’

x (0)) n F&Y’

x (0))

=

0,

is it true true that closed region of sn Sn between between them them is homeomorphic homeomorphic to an is it that the the closed region of annulus S”-l xxl? I? ann~tlus sn-l For n 5:S 33 this this conjecture for elementary elementary reasons, reasons, and and for for n n 2:: > 6 6 it it For n conjecture holds holds for holds under the assumption of smoothness by virture of Smale’s h-cobordism holds under the assumption of smoothness by virture of Smale's h-cobordism theorem. higher dimensions, the Annulus Annulus Conjecture Conjecture can can be be est estabtheorem. In In fact, fact, in in higher dimensions, the ablished without difficulty under the assumption that the embeddings Fl, F22 lished without difficulty under the assumption that the embeddings FI, F are smooth at at least one point. (We shall return to the discussion of the are smooth at at lcast one point. (We sha11 return to the discussion of the Annulus Annulus Conjecture Conjecture below.) below.) We now another result result of Mazur (from (from the the early early 1960s) 1960s) which, which, of Mazur We now describe describe another although itit concerns concerns smooth smooth manifolds, manifolds, uses uses entirely entirely elementary elementary techniques, techniques, although and allows to establish establish that two CW-complexes known to to be and a110ws one one to that aa certain certain two CW -complexes known be combinatorially inequivalent, are (Milnor, also also in in combinatorially inequivalent, are nonetheless nonetheless homeomorphic homeomorphic (Milnor, the early 1960s). is as as follows: the early 1960s). The The theorem theorem is follows: Iff normal homotopy homotopy equivalence equivalence between two If f ::M,n Ml' +---> MC Mf is is aa normal between two (1) MTf*(vE’) uN , where where v;’ normal Ml' and and Mz, Mf! i.e. i. e. J* (vfP) == v~)! v~) denotes denotes the the normal MF c lRN+n, iV 2 n + 1, then MI" with with respect respect to to an an embedding embedding MF MI" c]R +n! N 2:: n + 1! then products products M;xlRN and M; x RN

manifolds manifolds bundle of of bundle the direct direct the

are are difleomorphic. difJeomorphic. The the proof as follows: follows: Consider Consider embeddings embeddings The idea idea of of the pro of is is as M;xDN~M;xRN Mf x D N <-+

Mf

x]RN

and and

MF Mf

x DN L-) My x RN x D N <-+ Mf x ]RN

(5.33) (5.33)

respectively “approximating” the homotopy equivalence f : Mp + Mp, and respectively "approximating" the homotopy equivalence f : Ml ---> 1I1f, and a homotopy inverse g : M; -+ Mp (f o g N 1, g o f N 1). The normal bundles a homotopy inverse 9 : M!); ---> Ml (J 0 9 "-' 1, 9 0 f "-' 1). The normal bundles of MF c M; x RN and M; c M,” x lRN (where the inclusions are obtained of Mf C Ml X ]RN and Ml C Mf x ]RN (where the inclusions are obtained by restricting the above embeddings to Mr x (0) and MF x (0)) are trivial in by restricting the above embeddings to Ml x {a} and Mf x {a}) are trivial in view of the assumption f *(v@)) = Y!‘. From the pair of embeddings (5.33) N view of the assumption J* (v~)) = v~). From the pair of embeddings (5.33) we then obtain an expanding sequenceof regions we then obtain an expanding sequence of regions M~xD~cM;xD;cM;xD;c... , Mf x D~ c Mf x D!j c Mf x Dr: c . .. , where the radii of the balls Dy c RN increase, all of the embeddings where the radii of the balls c ]RN increase, a11 of the embeddings

Dr

Mf x D~+l <-+ Mf X ]RN are standard, and the embeddings of MF x DE, MC x DE+2, etc. represent are standard, and the and embeddings of M!); xThe D~,assertion Mf X D~+2' represent successive extensions are all isotopic. of the etc. theorem now and are a11 isotopie. The assertion of the theorem now successive extensions follows readily. follows readily. We now describe Milnor’s example. Note first that every 3-dimensional oriWe now describe Milnor's example. Note 3-dimensional orientable, closed manifold is parallelizable. Letfirst Lil, that L& every be two lens spaceswith entable, closed manifold is parallelizable. Let L~l' L~2 be two lens spaces with

§5. The The role role of of the the fundamental fundamental group group in in topology topology 55.

269 269

fundamental fundamental group group isomorphic isomorphic to to Z/p Z/p and and invariants invariants ql, ql, q2 q2 E Z*, Z*, satisfying satisfying q1 ql == X2q2 )..2q2 mod mod p. p. ItIt is is aa classical classieal result result that that this this condition condition guarantees the the homotopy homotopy equivalence of of the the two two lens lens spaces, spaces, but but that that they they may be chosen chosen so as as to to have different different Reidemeister Reidemeister torsion, torsion, which whieh implies implies that that they they are not not PL-homeomorphic PL-homeomorphie (See (See Chapter Chapter 3, $5). §5). By By Mazur’s Mazur's theorem theorem the the manifolds Lil RN L 3ql xx]RN

and

Lil IWN L 3ql xx]RN

are diffeomorphic diffeomorphie for for some sorne N, N, so that that the the Thorn Thom spaces spaces (which (which are CWCWcomplexes) of of the the trivial trivial bundles Kl = (Lil x 0”)

/ (LZ1 x P-1)

K2 = (L& x 0”)

/ (Li* x P-l)

)

are they are are not not combinatorially combinatorially equivalent equivalent (Mil(Milare also also homeomorphic. homeomorphic. However However they nor). combinatorial neighnor). To To see see this this one one first first observes observes that that the the boundary boundary of of aa combinatorial neighbourhood has the form Lii SNml, and and then the singular singular point point of of Ki Ki has the form L~i xx SN-l, then that that bourhood of of the the relative Reidemeister torsion of Ki (with respect to the singular point) is the relative Reidemeister torsion of Ki (with respect to the singular point) is well-defined and equal to the ordinary Reidemeister torsion of the lens space well-defined and equal to the ordinary Reidemeister torsion of the lens space Lii. L~i' The The result result now now follows. follows. Thus the Huuptwemnutung in general of dimendimenThus the Hauptvermutung is is false false in general for for CW-complexes CW-complexes of sion 3 + N. Since N can be reduced to 3, this applies in fact to CW-complexes sion 3+N. Since N can be reduced to 3, this applies in fact to CW-complexes of 2: 6. 6. of dimension dimension > We the topie topic of necessarily smooth) homeomorphisms. We now now return return to to the of (not (not necessarily smooth) homeomorphisms. Recall that in the early 1960s Stallings established, independently Smale of Smale Recall that in the early 1960s Stallings established, independently of and Wallace, a continuous version of the higher-dimensional Poincare conand Wallace, a continuous version of the higher-dimensional Poincaré conjecture, showing that a PL-manifold homotopy equivalent to the sphere jecture, showing that aPL-manifold homotopy equivalent to the sphere S” (n 2 5) is homeomorphic to the sphere S*. As part of the proof he showed (n 2: 5) is homeomorphie to the sphere As part of the proof he showed that every “locally flat” embedding 5’” c Snfk can be reduced, by means that every "locally fiat" embedding sn C Sn+k can be reduced, by means of a homeomorphism of the sphere Sn+k, to a standard embedding provided to a standard embedding provided of a homeomorphism of the sphere Sn+k, n > 3, Ic 2 3. (In the caseIc = 2 one requires also that 7~~(5’“+~\ Sn) E Z.) An n 2: 3, k 2: 3. (In the case k 2 one requires also that ?Tl (sn+2 \ sn) ~ Z.) An embedding F : S” 4 Snfk is called locally fiat if for each point x of 5’” there embedding F : sn ----+ Sn+k is called locally fiat if for each point ;1: of sn there is a neighbourhood 0: such that the restriction of F to that neighbourhood is a neighbourhood D~ such that the restriction of F to that neighbourhood extends to an embedding extends to an embedding D,“xI-Snfk, I = [-I, 11, D~ xl ----+ Sn+k, 1 [-1,1],

sn.

sn

where 0: is identified with 0,” x (0). where D~ is identified with D~ x {O}. Technically more complex is the proof of the following theorem of CherTechnically more complex is the proof of the following theorem of Chernavskii (of the mid 1960s): navskii (of the mid 1960s): The group of self-homeomorphisms of any closed topological manifold, or of The group of self-homeomorphisms of any closed topological manifold, or of RN, is locally contractible. RN, is locally contractible. In the spaceof all self-maps of a given manifold endowed with the Co-topology In space of aIl self-maps of a given manifold endowed the CO-topology in contrast with with the diffeomorphisms, thethe homeomorphisms are “unstable”, the homeomorphisms are "unstable", in contrast with the diffeomorphisms,

270 270

Chapter 4. 4. Smooth Smooth Manifolds Manifolds Chapter

with respect respect to to the the Cl-topology Cl-topology form form an an open open subset subset in in the the space spaceof of ail all which with self-maps. However However aa diffeomorphism diffeomorphism arbitrarily arbitrarily close closeto to the the identity identity smooth self-maps. map with with respect respect to the the CO-topology, Co-topology, may may still be be very very complex, complex, and and extremely extremely map difficult to deform to to the identity identity map map within within the class classof homeomorphisms; homeomorphisms; difficult as will will be be indicated indicated below, this fact fact has has played played an an important important role role in in topology. topology. as Chernavskii constructs such such isotopies isotopiesof of aa diffeomorphism diffeomorphism to to the the identity identity map map Chernavskii using essentially essentially elementary, elementary, but but extremely extremely complicated complicated techniques. techniques. using At the end end of the 1960s 1960sKirby Kirby found an an approach to the proof proof of the the Annulus Annulus At Conjecture via via reduction reduction to to the the situation situation of of smooth smooth (or (or P PL-) homotopy tori, tori, Conjecture L-) homotopy and almost immediately thereafter the crucial problems concerning homotopy and almost immediately thereafter the crucial problems concerning homotopy tori associated associated with with this this approach approach were were solved solved by by Wall, Wall, Siebenmann, Siebenmann, Hsiang, Hsiang, tori Shaneson and Casson We shall now describe Kirby’s idea. Shaneson and Casson We shaH now describe Kirby's idea. In the the theory theory of of homeomorphisms homeomorphisms ]Rn Iw” --+ ]Rn lRn the the concept concept of of aa stable stable homehomeIn omorphism is important; this is a homeomorphism h : R” + R” that ]Rn that omorphism is important; this is a homeomorphism h:]Rn is a composite h = hl 0 . 0 hk of homeomorphisms hi each of which reis a composite h = hl 0 . . . 0 hk of homeomorphisms hi each of which restricts to the identity on some non-empty open region Ui. Diffeomorphisms stricts to the identity on sorne non-empty open region Ui . Diffeomorphisms and PL-homeomorphisms are stable, stable, as as are are homeomorphisms homeomorphisms a,pproximable approximable and PL-homeomorphisms are by PL-homeomorphisms. If a homeomorphism agrees with a stable one on on by P L-homeomorphisms. If a homeomorphism agrees with a stable one some open non-empty region, then it must itself be stable. It therefore makes sorne open non-empty region, then it must itself be stable. It therefore makes senseto the pseudogroup pseudogroup of of stable stable homeomorphisms homeomorphisms between between open open to speak speak of of the sense regions of Iw” and thus of the class of stable manifolds determined by this regions of ]Rn and thus of the class of stable manifolds determined by this pseudogroup, where the co-ordinate transformations are restricted to being pseudogroup, where the co-ordinate transformations are restricted to being stable (Brown, and others, others, in 1960s). From From the the stability stability of of stable (Brown, Gliick Glück and in the the early early 1960s). orientationpreserving homeomorphisms of R”, the Annulus Conjecture for orientation-preserving homeomorphisms of ]Rn, the Annulus Conjecture for spheres and for ]Rn lRn follows easy. We shall now now sketch sketch the the proof proof of of follows relatively relatively easy. We shall spheres and for stability. stability. In the Connell made made the observation that homeomorIn the early early 1960s 1960s Connell the useful useful observation that a a homeomorphism h : R” --f R” for which the distance between z and its image h(z) is phism h : ]Rn ---. ]Rn for which the distance between x and its image h(x) is bounded above ([h(z) - ICI < M) is stable. From this it follows that all homebounded above (Ih(x) xl < M) is stable. From this it follows that aU homeomorphisms of a torus are stable; for if a homeomorphism h : T” + Tnn fixes omorphisms of a torus are stable; for if a homeomorphism h : Tn ---. T fixes a point and is homotopy equivalent to the identity map, then the universal a point and is homotopy equivalent to the identity map, then the univers al covering map h : IRn + R” satisfies covering map h : ]Rn ---. ]Rn satisfies Ii(z) - ~1 < M Ih(x) xl<M

for all 5 E lRn, for aIl x E]Rn,

whence it follows that in fact h : Tn + Tnn is stable. The general case of whence it follows that in fact h : Tn ---. T is stable. The general case of a homeomorphism h of Tnn is reduced to this special case by composing h a homeomorphism h of T is reduced to this special case by composing h with suitable affine transformations. There is a resemblance between Kirby’s with suitable affine transformations. There is aand resemblance Kirby's strategy for proving the Annulus Conjecture, the proof between of the topologistrategy for proving the Annulus Conjecture, and the proof of the topological invariance of the Pontryagin classes,in that at a certain point in Kirby’s Pontryagin classes, in thathowever at a certain pointdifferently: in Kirby's cal invariance of the scheme a “toroidal region” of LR” is introduced, somewhat scheme a "toroidal region" of]Rn is introduced, however somewhat differently: one considers an immersion 4 : Tnn \ {*} + R”, which, although such immer4> : T \ {*}n-manifold, ]Rn, which, immerone considers immersion sions exist for an any open, parallelizable mayalthough here be such constructed sions exist for any open, parallelizable n-manifold, may here be constructed by elementary direct means. (In fact its precise form is unimportant.) Difby elementary direct means. (In+factRnitsinduce precisepotentially form is unimportant.) Different homeomorphisms g : lRn different smooth ferent homeomorphisms 9 : ]Rn ---. ]Rn induce potentially different smooth

§5. §5. The The role role of of the the fundamental fundamental group group in in topology topology

271 271

structures structures on on the the image image of of Tn rpn \\ {*} {*} inin IP ~n by by mapping mapping this this region to other other

regions ~n. regions of of Iw”. Now every every smooth open manifold manifold Mn Mn of of dimension n >2': 6, with with a neighbourhood Iw, is realizable bourhood of of infinity infinity homeomorphic to to S-l sn-1 xx~, realizable as as the interior interior of of aa smooth the homotopy homotopy type type smooth manifold-with-boundary manifold-with-boundary whose whose boundary boundary has has the of e earlier of S”-l sn-1 (see (see th the earlier discussion discussion of of the the problem problem of of Browder-Levine-Livesay). Browder-Levine-Livesay). Hence spheres of Henee by by using using the the result result of of Smale Smale and and Wallace Wallace that that all aIl homotopy homotopy spheres dimension compactify 5 are are PL-homeomorphic P L-homeomorphic to to S” sn (see (see above), above) , we we may may compact ify dimension nn 22': 5 such such an an image image region region Tn \ {*I with 4 with its its new new smooth smooth structure structure (induced (induced from from aa homeomorphism homeomorphism 9g :: IWN ~N ---'> n endowed IWN) by means of a single point *’ to obtain a homotopy torus V” ~N) by means of a single point *' to obtain a homotopy torus V endowed with with aa PL-structure PL-structure (Wall, (Wall, Hsiang, Rsiang, Shaneson, Shaneson, Siebenmann, Siebenmann, Casson): Casson): V”\

{d} ” Tn \ {*}.

If the V”n is Tnn then homeoIf the new new closed closed manifold manifold V is PL-isomorphic P L-isomorphic to to T then we we have have a a homeomorphism T” -+ Tn, and since such homeomorphisms are, as observed above, morphism Tn ---'> Tn, and sinee such homeornorphisrns are, as observed ab ove , always stable, it follows that the original homeomorphism g : Iw” + is ~n is always stable, it follows that the original horneornorphisrn 9 : ~n ---'> llF also stable. The proof is completed by showing that it suffices to construct also stable. The proof is completed by showing that it suffices to construct aa PL-homeomorphism covering spaces spacesV pnn 4 Fn, n , P L-horneornorphism between between some sorne finite-sheeted finite-sheeted covering ---'> rather than between V” and Tn. Hence all self-homeomorphisms of Iw” are n n rather than between V and T . Rence aIl self-horneornorphisrns of ~n are stable, the Annulus Annulus Conjecture Conjecture follows. follows. stable, and and the Using results about smooth or piecewise linear homotopy tori, it can be Using results about srnooth or piecewise linear homotopy tori, it ean be shown that for n > 5 there exists a PL-homeomorphism h : Tnn -+ Tnn not shown that for n 2': 5 there exists a P L-homeornorphism h : T ______ T not isotopic obstructions to isotopies, in in general, general, lylyisotopie to to the the identity identity map, map, with with obstructions to such sueh isotopies, ing in Z/2. (In fact there is a PL-homeomorphism not even “pseudo-isotopic” ing in 2/2. (In fact there is a P L-homeornorphism not even "pseudo-isotopie" to the identity map.) Lifting such an h to a PL-homeomorphism i: ?’n + Fnn to the identity map.) Lifting sueh an h to a P L-homeomorphism h: ---'> of a covering spacewith an odd number of sheets, we can, provided the number of a eovering space with an odd number of sheets, we ean, provided the nurnber of sheets is large enough, deform the latter PL-isomorphism (via a topologiof sheets is large enough, deform the latter P L-isomorphism (via a topological isotopy) to a self-homeomorphism arbitrary close, in the Co-topology, to cal isotopy) to a self-homeomorphism arbitrary close, in the CO-topology, to the identity map, but still not PL-isotopic to it in view of the fact that the the identity map, but still not P L-isotopic to it in view of the fact that the obstruction to such a PL-isotopy (with values in Z/2) prevails in odd-sheeted obstruction to such a P L-isotopy (with values in 2/2) prevails in odd-sheeted covering spaces.However by Chernavskii’s theorem, since the homeomorphism covering spaces. However by Chernavskii's theorem, since the horneomorphism h is Co-close to the identity map, it must in fact be topologically isotopic to h is CO-close to the identity map, it must in fact be topologically isotopic to it. The upshot of this delicate discrimination between PL- and topological it. The upshot of this delicate discrimination between P L- and topological homeomorphisms is another counterexample to the Hauptvermutung, and an is another counterexample to the Hauptvermutung, and an homeornorphisms example of a topological manifold not admitting a compatible PL-structure example of a topological manifold not admitting a compatible PL-structure (Kirby and Siebenmann, end of the 1960s). (Kirby and to Siebenmann, of thepeople 1960s).(Lees, Lashof, Rothenberg, at the Thanks the work ofend several Thanks to the work of several people (Lees, Lashof, ofRothenberg, at the end of the 196Os), it is now known that such properties closed topological end of the 1960s), it is now known that such properties of closed topological manifolds of dimension 2 5 as those of admitting a compatible PL- or smooth as those of admitting a compatible L- or(roughly smooth manifolds of structure, aredimension equivalent~to5 the reduction of the structure groupPTOP structure, are equivalent to the reduction of the structure group TOP (roughly speaking, the group of germs of homeomorphisms of an open disc) of the the group of of horneomorphisms of 0an(even openin disc) of the speaking, tangent microbundle, to germs its subgroup PL c TOP or to non-stable

r

r

tangent microbundle, to its subgroup PL

r

c TOPor to 0 (even in non-stable

272 272

Chapter Smooth Manifolds Chapter 4.4. Smooth Manifolds

dimensions These investigations, together with dimensions of of these these bundles). bundles). These investigations, together with those those of of Kirby and Siebenmann, Siebenmann, yield yield the Kirby and the following following structures structures of of the the relative relative homotopy homotopy groups 7ri(TOP/PL): groups rri(TOP/PL):

0 7ri(TOP/PL) ~ { '1./2

for i -=J 3, for i 3.

Knowledge Knowledge of of these these groups groups renders renders the the results results of of the the above above authors authors concernconcerning (i.e. admissibility of aaPL-structure) PL-structure) effective. An ing triangulability triangulability (Le. admissibility of effective. An upper upper bound definable on bound for for the the number number of of distinct distinct PL-structures PL-structures definable on aa topological topological manifold manifold Mn Mn (where (where nn >2 55 for for closed closed manifolds, manifolds, and and nn 22 66 for for manifoldsmanifolds~ with-boundary), is afforded Z/2). Note with-boundary), is afforded by by the the order or der of of the the group group H3(Mn; H 3 (M n ; '1./2). Note that here simpleconnectness is not involved. (Complete proofs of all of that here simple-connectness is not involved. (Complete proofs of all of the the results of this results of this theory theory depends depends on on some sorne results results of of the the Sullivan Sullivan theory, theory, which, which, as as noted noted earlier, earlier, isis as as yet yet incomplete.) incomplete.) ItIt is noteworthy that that 7r3(PL/TOP) rra(PL/TOP) g~ Z/2, which alone is noteworthy that the the fact fact that '1./2, which alone disdistinguishes PL-structures on manifolds from continuous ones, is ultimately tinguishes PL-structures on manifolds from continuolls ones, is ultimately aa consequence of Rohlin’s that the the signature signature 7T of of every consequence of Rohlin's theorem theorem to to the the effect effect that every almost parallelizable manifold M4 by 16, while in all dimensions 4 is divisible almost parallelizable manifold M is divisible by 16, while in all dimensions of form 4/c almost parallelizable PL-manifolds with of the the form 4k >> 44 there there exist exist almost parallelizable closed closed PL-manifolds with 7T = 88 (Milnor; see $3 above). (Milnor; see §3 above). Noteworthy among the the more more recent recent successful constructions of Noteworthy among suc cess fui constructions of homeomorhomeomorphisms are the following two: phisms are the following two: 1. The of aa homeomorphism 1. The direct direct construction construction of homeomorphism C2M3

--+ S5

from the double suspension of any S-dimensional manifold M3 satisfying from the double suspension of any 3-dimensional manifold M3 satisfying H1(M3;Z) = 0 (Ed wards, in the late 1970s). Prom this there follows the Hl (M3 j '1.) = 0 (Edwards, in the late 1970s). From this there follows the existence of a triangulation of S5 with respect to which it is not a PL-manifold, existence of a triangulation of S5 with respect to which it is not aPL-manifold, and which is not combinatorially equivalent to the standard triangulation. and which is not combinatoriaLly equivalent to the standard triangulation. 2. The direct (and not exceptionally complicated) construction of a homeo2. The direct (and not exceptionally complicated) construction of a homeomorphism between any smooth S-connected closed d-manifold and the sphere morphism between any smooth 2-connected closed 4-manifold and the sphere S4 (Freedman, around 1980). On the other hand with respect to diffeomorS4 (Freedman, around 1980). On the other hand with respect to diffeomorphisms the d-dimensional analogue of the Poincare’ conjecture remains open. phisms the 4-dimensional analogue of the Poincaré conjecture remains open. (Note that in dimension 4 PL- homeomorphisms are in this context equivalent (Note that in dimension 4 P L- homeomorphisms are in this contextequivalent to smooth.) to smooth.)

55. role of of the the fundamental group in in topology §5. The role fundamentai group

273 273

Concluding Remarksks Conel uding Remar In this survey ideas and methods of topology In survey of of the ideas topology far from all topics topies have been considered. Little has been mentioned here of 3-dimensional topology been considered. Little has been topology and the theory of Kleinian groups as developed by Thurston and others, or the theory Kleinian as Thurston and of the developments in 4-dimensional topology due to Donaldson, Freedman, of the developments in 4-dimensional topology due to Donaldson, Freedman, and surveyed here qualitative theory theory of of foliations: foliations: and others. others. Neither Neither have have we we surveyed here the the qualitative notably the theorems of Reeb, Haefliger and Novikov on analytic foliations notably the theorems of Reeb, Haefliger and Novikov on analytie foliations and and on foliations with compact sheet; the techniques for establishing and and on foliations with compact sheet; the techniques for establishing the existence existence of foliations and geometric structures open manifolds the of foliations and other other geometrie structures on on open manifolds (Gromov), of foliations of codimension one of odd-dimensional spheres (Law(Law(Gromov), of foliations of codimension one of odd-dimensional spheres son, Tamura, and others), and of compact manifolds of arbitrary dimensions son, Tamura, and others), and of compact manifolds of arbitrary dimensions (Thurston); the the theory theory of of characteristic characteristic classes classesof of foliations (Bott, Godbillon, Godbillon, foliations (Bott, (Thurston); Vey, Bernshtein, and others); the classification theory of foliations Vey, Bernshtein, and others); the classification theory of foliations (Haefliger (Haefliger and of Lie Lie algebras of vector fields on on manmanand others); others); the the cohomology cohomology theory theory of algebras of vector fields ifolds (I. M. Gel’fand, Fuks); the theory of singular points of complex hyifolds (1. M. Gel'fand, Fuks); the theory of singular points of complex hypersurfaces (Milnor, Brieskorn); real algebraic curves and surfaces (Gudkov, persurfaces (Milnor, Brieskorn); real algebraic curves and surfaces (Gudkov, Arnol’d, Viro). Arnol'd, Rohlin, Rohlin, Kharlamov, Kharlamov, Kirilov, Kirilov, Viro). We have also omitted from our survey such areas areas of as the the thetheWe have also omitted from our survey such of topology topology as ory of embeddings of manifolds and higher-dimensional knots (created by ory of embeddings of manifolds and higher-dimensional knots (created by several people, beginning with Whitney and including Haefliger, Stallings and sever al people, beginning with Whitney and indu ding Haefliger, Stallings and Levine), and and and functions" functions” the theory theory of of “typical "typieal singularities singularities of of maps maps and Levine), as as also also the (Whitney, Pontryagin, Thorn, Boardman, Mather, Arnol’d). (Whitney, Pontryagin, Thom, Boardman, Mather, Arnol'd). The author of the present essay has preferred - and this is in full conThe author of the present essay has preferred ~ and this is in full conciousness- to omit discussion of topological results of a general categorical to omit discussion of topologieal results of a general categorical ciousness and abstract nature, nothwithstanding their usefulnessand even necessity as and abstract nature, nothwithstanding their usefulness and even necessity as far as the intrisic logic of topological concepts is concerned. far as the intrisic logic of topologieal concepts is concerned. There is also missing from the survey a summary of the homology theory There is also missing from the survey a summary of the homology theory of general spaces, in particular of subsets of IWn, developed in the 1920s by of general spaces, in particular of subsets of ]Rn, developed in the 1920s by several authors, including P. S. Alexandrov, Tech, Pontryagin, Kolmogorov, several authors, including P. S. Alexandrov, Cech, Pontryagin, Kolmogorov, Steenrod, Chogoshvili, Sitnikov, Milnor, among others. Steenrod, Chogoshvili, Sitnikov, Milnor, among others. There is also missing a discussion of the topological properties of manifolds There is also missing a discussion of the topologieal properties of manifolds with various differential-geometric properties (global geometry). with various differential--geometric properties (global geometry). Finally, it has not proved possible to discussthe substantial applications of Finally, it has not proved possible to discuss the substantial applications of topology that have been made over recent decades to real physical problems, that have been made over recent decades to real physical problems, topology and have transformed the apparatus of modern mathematical physics. We apparatus modern mathematical physics. We and have hope that transformed this lack will the be made good of in other essays of the series. hope that this lack will be made good in other essays of the series.

274 Appendix. Appendix. Recent Recent Developments Developments in in the the Topology Topology of of 3-manifolds 3-manifolds and and Knots Knots 274

Appendix Appendix Recent Developments Developments in the the Topology Topology Recent in of 3-manifolds 3-manifolds and and Knots Knots of 51. Introduction: Introduction: §l. Recent developments developments in in Topology Topology Recent The survey of topology was was in in fact fact written written in in 1983-84 1983-84 and and published published The present present survey of topology (in Russian) in 1986. Over the intervening decade several very beautiful new (in RU8sian) in 1986. Over the intervening decade several very beautiful new ideas have appeared in the pure topology of 3-manifolds and knots, and it ideas have appeared in the pure topology of 3-manifolds and knots, and it is to these that the present appendix is devoted. It should however be noted 18 to these that the present appendix i8 devoted. It should however be noted that impressive impressive developments developments have have occurred occurred also also in in other other areas areas of of topology: topology: that in symplectic topology, with the creation of “Floer homology” and its subsein symplectic topology, with the creation of "Floer homology" and its subsequent development by several mathematicians (Gromov, Eliashberg, Hoffer, quent development by several mathematicians (Gromov, Eliashberg, Hoffer, Salamon, and others); others); in in the theory of SOz-actions on smooth manithe theory of 80 Salamon, McDuff, McDuff, and 2 -actions on smooth manifolds, including analysis on loop spaces (Witten, Taubes), originating in quanquanfolds, including analysis on loop spaces (Witten, 'l'aubes), originating in tum mechanics, and “elliptic genera” (Oshanine, Landweber) allowing formal tum mechanics, and "elliptic genera" (Oshanine, Landweber) allowing formaI groups and other techniques of complex complex cobordism theory to to be applied to to the the be applied groups and other techniques of cobordism theory theory of such actions (already mentioned in the body of this survey in connectheory of such actions (already mentioned in the body of this survey in connection with with work by the Moscow school 1970); in in the the so-called so-called tion work carried carried out out by the Moscow school around around 1970); “topological quantum field theories” (whose beginnings were also mentioned "topological quantum field theories" (whose beginnings were also mentioned above in connection with the construction of Reidemeister-Ray-Singer torsion above in connection with the construction of Reidemeister-Ray-Singer torsion via a functional integral, by A. Schwarz in 1979-80) as developed by Witten via a functional integral, by A. Schwarz in 1979-80) as developed by Witten in the late 1980s following a suggestion of Atiyah, and by others in the 1990s. in the late 1980s following a suggestion of Atiyah, and by others in the 1990s. The present author is of the opinion that these conformal and topological The present author is of the opinion that these conformaI and topological quantum field theories constitute a new kind of analysis on manifolds, hithquantum field theories constitute a new kind of analysis on manifolds, hitherto remaining, however, without rigorous foundation. Nonetheless these new erto remaining, however, without rigorous foundation. Nonetheless these new methods have led to the discovery (by physicists such as Candelas and Vafa, methods have led to the discovery (by physicists such as Candelas and Vafa, among others, with subsequent more-or-less rigorous justification by variamong others, with subsequent more-or-less rigorous justification by various mathematicians) of subtle features of the structure of rational algebraic ous mathematicians) of subtle features of the structure of rational algebraic curves on certain symplectic manifolds and algebraic varieties (for instance curves on certain orsymplectic manifolds and algebraic varieties (for instance on “Calaby-Yau” “toroidal” manifolds). on "Calaby-Yau" or "toroidal" manifolds). In a similar manner deep facts about the topology of moduli spaces have In a similar manner deep facts about the topology of moduli spaces have been obtained by Kontzevich using the technique of “matrix models” borthe technique "matrix(Gross, models" borbeen obtained by Kontzevich rowed from statistical mechanics,using by means of which of physicits Migdal, rowed from statistical mechanics, by means of which physicits (Gross, Migdal, Brezin, Kazakov, Douglas, Shenker, and others) were led in 1989-90 to the disdisBrezin,ofKazakov, Douglas,results, Shenker, were led in 1989-90 covery some beautiful in and part others) topological, in string theory,to the finding part topological, in string theory, finding covery of some beautiful results, in in particular a connection between that theory and the celebrated integrable in particular a connection between that theory and the celebrated integrable models of soliton theory. models of soliton theory. Thus over the last two decades (since the discovery of instantons in the midThus the last two decades (sinee discoverymentioned of instantons in the 1970s by over Belavin, Polyakov, Schwarz andtheTyupkin, in the bodymidof 1970s by Belavin, Polyakov, Schwarz and Tyupkin, mentioned in the body this survey), several very interesting developments in topology have been initi-of this survey), several very interesting developments in topology have been initi-

$2. and modern modern approaches approaches §2. Knots: Knots: the the classical classical and

275 275

ated by quantum outcome has been quantum field physicists. One out come of this this has been that that some sorne firstfirstrank physicists have gone physics proper to working rank go ne from doing research research in physics working on problems of abstract and mathematical abstract mathematics mathematics using using the the approach approach and mathematical on problems of techniques of from their Altechniques of quantum quantum field field theory theory familiar familiar from their training training in in physics. physics. AIthough standard among among physicists, scarcely known though the the latter latter technique technique is is standard physicists, itit is is scarcely known to the community to the community of of pure pure and and applied applied mathematicians, mathematicians, partly partly because because some sorne of the of the of the basic basic notions notions and and techniques techniques of the theory, theory, although although certainly certainly mathemathematical in of many nature (contrary (contrary to to the the opinion opinion of many mathematicians mathematicians that that they they matical in nature belong to easily amenable treatbelong to physics), physics), are are not not easily amenable to to rigorous rigorous mathematical mathematical treatment. In ment. In fact fact the the current current rigorous rigorous version version of of quantum quantum field field theory theory formulated formulated in terms of 20th century pure mathematical analysis, although highly nonnonin terms of 20th century pure mathematical analysis, although highly trivial from the viewpoint of analysis, seems impoverished trivial from the viewpoint of functional functional analysis, seems rather rather impoverished by The interaction interaction of of topology topology and and by comparison comparison with with its its physical physical prototype. prototype. The algebraic geometry with modern quantum physics over the last two decades decades algebraic geometry with modern quantum physics over the last two has proved it is also to is also to be be hoped hoped that that has proved extremely extremely fruitful fruitful in in pure pure mathematics; mathematics; it ultimately it will bear fruit also outside pure mathematics. ultimately it will bear fruit also outside pure mathematics.

$2. §2. to to the the

Knots: the Knots: the classical classical and and modern modern approaches approaches Alexander polynomial. Jones-type Alexander polynomial. Jones-type polynomials polynomials

We modern knot We shall shall sketch sketch here here some sorne of of the the ideas ideas of of modern knot theory, theory, inaugurated inaugurated in the mid-1980s by the discovery of the “Jones polynomial”. We begin begin by by in the mid-19S0s by the discovery of the "Jones polynomial". We recapitulating the relevant basic notions of knot theory. recapitulating the relevant basic notions of knot theory. We consider only an object is a knots: such such an object is a smooth, smooth, closed dosed We shall shaH consider only classical classical Icnots: nonselfinterskcting curve K in Euclidean space lR3or in the 3-sphere S3, with nonselfintersecting curve K in Eudidean space ]R3 or in the 3-sphere 8 3 , with everywhere nonzero tangent vector. A linlc is a union of such curves, pairwise everywhere non zero tangent vector. A link is a union of such curves, pairwise nonintersecting: nonintersecting:

K=UKi;

K, n Kj = 0 for i # j.

By projecting a knot or link orthogonally onto a plane (or “screen”) in the By projecting a knot or link ort.hogonally onto a plane (or "screen") in the direction of some suitable vector 7, we obtain the diagram of the knot or direction of sorne suitable vector 7], we obtain the diagmm of the knot or link, consisting of a “generic” collection of plane curves (where by generic link, consisting of a "generic" collection of plane curves (where by generic we mean that the tangent vector is everywhere non-zero, all intersections we mean that the tangent vector is everywhere non-zero, all intersections ( LLcrossings”)are transversal, and there are no triple intersection points); the ( "crossings") are transversal, and there are no triple intersection points); the diagram should also include, for each crossing, the information as to which of diagram also is indude, each crossing, thetoinformation to which of two curveshould segments “above”forthe other relative the screen.asThe diagram is "above" the other relative to the screen. The diagram two curve segments is said to be oriented if each component curve of the knot or link is directed, is said to be oriented if each component curve of the knot or link is directed, otherwise unoriented. unoriented. otherwise Classical knot theory is concerned with the space S33 \ K = M, an open Classical There knot theory is concerned with of thethe space an open \ K 3-manifold. is a natural embedding torus8 T2 in M,M,namely as 2 3-manifold. There is a natural embedding of the torus T in M, namely the boundary of small tubular neighbourhood of the knot K. Similarly, for asa the link boundary we obtain ofa small disjointtubular union neighbourhood of 2-tori in M. of the knot K. Similarly, for a link we obtain a disjoint union of 2-tori in M.

276 Appendix. Appendix. Recent Developments in the Topology of 3-manifolds and Knots Knots Recent Developments the Topology of 3-manifolds The principal topological invariant of a knot knot K the fundamental fundamental group group The principal topological invariant of K is the rr(M) complement M of K, K, with the natural natural 'TrI (M) of of the the complement M of with distinguished distinguished subgroup subgroup the 2 image of 7rr(T2), T22 cC M M2,2 , with standard basis. The classical classical image of 'Trl (T ), T with the the obvious obvious standard basis. The theorem of Papakyriakopoulos Papakyriakopoulos of the the 1950s 1950s asserts asserts that that a knot equivalent knot is equivalent theorem of of to is abelian. abelian. ItIt was shown by by Haken Haken to the the trivial trivial one one ifif and and only only ifif nr(A4) 'Tr l (M) is was shown in the early early 1960s 1960s that there is an algorithm algorithm for for deciding whether or or not not that there is an deciding whether in the any is equivalent knot. However, However, while it appears appears to to have have any knot knot is equivalent to to the the trivial trivial knot. while it been (by Waldhausen and others others in 1960s and 1970s) that that two two been established established (by Waldhausen and in 1960s and 1970s) knots are topologically if and and only only ifif the corresponding fundamental fundamental the corresponding knots are topologically equivalent equivalent if groups subgroups are are isomorphic, the existence existence of of an an groups with with labelled labelled abelian abelian subgroups isomorphic, the appropriate for deciding such equivalence remains an an open open question. question. appropriate algorithm algorithm for deciding such equivalence remains The complexity of knot goup goup 'Trt nr(M) led to the search search for for more more The complexity of the the knot (M) has has led to the effectively invariants to distinguish knots knots and and links. links. The The first first such such effectively computable comput able invariants to distinguish non-trivial was the polynomial. We We shall shall non-trivial invariant invariant to to be be discovered discovered was the Alexander Alexander polynomial. now invariant in in terms terms of of now give give an an alternative alternative elementary elementary definition definition of of this this invariant the oriented diagram of the knot or link, following early articles of Alexander the oriented diagram of the knot or link, following early articles of Alexander himself. rediscovered by by Conway Conway around 1970.) This This himself. (This (This definition definition was was rediscovered around 1970.) approach to the definition is based on the very strong additive property approach to the definition is based on the very strong additive property of of Alexander respect to to an an operation operation eliminating eliminating crossings. crossings. Alexander polynomials polynomials with with respect (The Jones, HOMFLY, and Kauffman Kauffman polynomials have similar similar properties (The Jones, HOMFLY, and polynomials have properties - see see below.) below.) Consider Consider the three oriented diagrams of Figure A.l, differing differing the three oriented diagrams of Figure A.l, from one another only in a small neighbourhood of a single crossing. The first from one another only in a small neighbourhood of a single crossing. The first

y(x)-g-( x)=(x) D n,+

%,-

k-1

Fig. A.l. Defining the Alexander and HOMFLY

polynomials

Fig. A.l. Defining the Alexander and HOMFLY polynomials

two diagrams D,,* have exactly n crossings and identical projections on the two diagrams D n ,± have exactly n crossings and identical projections on the screen; moreover their oriented diagrams coincide exept for one crossing. The screen; moreover their oriented diagrams coincide exept for one crossing. The third diagram D,-1 has n - 1 crossings (one fewer then the other two), being third diagram Dn-l has n - 1 crossings (one fewer then the other two), being obtained from either D,,* by replacing a neighbourhood of the crossing at obtained from either Dn,± by replacing a neighbourhood of the crossing at which these differ by the configuration indicated in Figure A.l, in natural which these differ by the configuration indicated in Figure A.1, in natural agreement with the orientation. The Alexander polynomials corresponding to agreement with the orientation. The Alexander polynomials corresponding to the respective diagrams are related as follows: the respective diagrams are related as follows: zp&-,

cz)

=

pD,,,+

(z)

-

PO,&,-

(2).

This will serve as a defining condition once supplemented by the conditions This willAlexander serve as apolynomial defining condition the conditions that the should beonce zerosupplemented for any diagramby with 2 or more that the Alexander polynomial should be zero for any diagram with 2 or more connected components, and should be equal to 1 for the trivial one-component connected(i.e. components, should be equal to 1 for the trivial one-component diagram for a simpleandclosed curve). diagram (i.e. for a simple closed curve).

$2. Knots: the classical modern approaches approaches §2. Knots: the classical and and modern

277

This polynomial as a polynomial in the the variable variable Zj z; This defines defines the the Alexander Alexander polynomial polynomial in the standard version is then obtained by means of a substitution: the standard version then obtained by means of substitution: A(t, t-l) 1 ) = P(z), A(t,tP(z),

l 2 - t-1’2, I 2 zZ=t = t1’2 / _C / ,

yielding finally the well-defined up multiyielding finally the Alexander Alexander polynomial polynomial A(t, A(t, t-l), Cl), well-defined up to to multiplication by a power of t. plication by a power of t. The polynomial H(x, is defined using the same three three diagrams diagrams The HOMFLY HOMFLYpolynomial H(x, y) y) is defined using the same but with the general condition but with the general condition ~HD,-,(x,Y) XHDn_l

YHD_,+(x,Y) (x, y) -- Y-~HD,,-(x,Y), (x, y) == yHDn.+ y-l HDn,_ (x, y),

together the condition that H for the Note that together with with the condition that H = 11 for the trivial trivial knot. knot. Note that the the most this type, type, involving involving three three variables, variables, can be normalized most general general formula formula of of this can be normalized to by aa suit suitable to this this form form by by multiplying multiplying by able factor. factor. Setting yy = 1, yields the Alexander polynomial. polynomial. Putting Putting xx = yl/2 y1j2 1, xx = zz yields the Alexander Setting -1/2 yields celebrated Jones Jones polynomial of which, which, therefore therefore the Yy-I/2 yields the the celebrated polynomial J(y), J(y), of the HOMFLY polynomial is a natural generalization. We shall call all of these HOMFLY polynomial is a natural generalization. We shall call aIl of these polynomials “Kauffman polynomial" polynomial” defined defined below) below) Jones-type polynomials (including (including the the "Kauffman Jones-type polynomials. polynomials. We describe Kauffman’s method, based on unoriented diagrams, We shall shaH now now describe Kauffman's method, based on unoriented diagrams, of constructing polynomial invariants constituting a different generalization of constructing polynomial invariants constituting a different generalization of the Jones polynomials. For four unoriented diagrams Dn,*, I&.r,r, D+r,z of the Jones polynomials. For four unoriented diagrams Dn,±, Dn-l,l, D n - I ,2 differing only in a small neighbourhood of one crossing, as depicted in Figure differing only in a small neighbourhood of one crossing, as depicted in Figure A.2, we first define a 2-variable state polynomial Q via the following recurrence A.2, we first define a 2-variable state polynomial Q via the following recurrence

(s)( (X) x)=j(=)-( ,()I (X) z[(x) ()()] Fig. A.2. Defining the state polynomial and the Kauffman polynomial Fig. A.2. Defining the state polynomial and the Kauffman polynomial relation: relation:

QLL,,

- QD,&,- =

~QD,-~,~ - QLA-~.~)~

together with together with (a-a-1)+1 l ) (a-az +1 z for the trivial one-component diagram Do, and the requirement that Q change for the trivial one-component diagram Do, and the requirement that Q change by a factor A*’ with each Reidemeister move (see Figure A.3). The Kauflman by a factor A±l with each Reidemeister move (see Figure A.3). The Kauffman polynomial of a diagram D, is then defined by polynomial of a diagram Dn is then defined by QDOzp=

QDo = IL =

KD_

= a -“(Drb)Q~,L

(a, z),

KD n = a-w(Dn)QD n (a , z) , where the where the defined as defined as

“writhe number” w(D) of "writhe number" of the algebraic sum w(D) of signs the algebraic sum of signs

an oriented knot or link diagram D is an link diagram D is fl oriented attached knot to theor crossings according ±1 attached to the crossings according

278 Appendix. Appendix. Recent Developments Developments in in the the Topology Topology of 3-manifolds 3-manifolds and and Knots Knots 278 Recent

(x)=+)>

(X)=61(-)

Fig. A.3. A.3. The The first first Reidemeister Reidemeister move move and and the the state state polynomial polynomial Fig.

to the to the vector vector (Thus (Thus

orientation of the 2-frame determined determined by each each crossing, crossing, with the first first orientation of the 2-frame by with the of the frame given by the (directed) upper branch (see Figure A.4). of the frame given by the (directed) upper branch (see Figure A.4). the orientation of the the diagram diagram D is is significant significant only for for the the number number the orientation of D only

x x -1 -1

+1 +1

Fig. A.4. A.4. The determined by the orientations orientations of the crossings crossings Fig. The writhe writhe number, number, determined by the of the W(O).) completes the the definition definition of Kauffman polynomial, an invariant invariant w( D).) This This completes of the the Kauffman polynomial, an of the knot or link in question. of the knot or link in question. For any oriented oriented diagram D the Jones polynomial of D D is given by by For any diagram D the Jones polynomial of is then then given KD(z zz t-l14 - t1f4, a = t314) J () _= K D (zt- 1/ 4 t1/4, a t 3 / 4 ) JD(t) - t-112 /-;::;2,-------'t - - - ' - - - t - :t1/2-/-;::;2,---t---:-

. D

1

1

On the other, hand by a theorem of Jaeger the Kauffman of a diaOn the other hand by a theorem of J aeger the Kauffman polynomial polynomial of a diagram can be expressed as a linear combination (with appropriate coefficients) gram can be 'expressed as a linear combination (with appropriate coefficients) of HOMFLY polynomials of certain diagrams associated with the given diaof HOMFLY polynomials of certain diagrams associated with the given diagram. gram.

(x)=(x)+(

)( )T( O)=d

Fig. A.5. Defining the bracket polynomial and the Jones polynomial Fig. A.5. Defining the bracket polynomial and the Jones polynomial Kauffman also defines a polynomial S D, via the recurrence relations (see Kauffman also defines a polynomial BDn via the recurrence relations (see Figure A.5)

Figure A.5)

with

with

Son(A,B;d)

=

ASD,,+,

+

BSD~L--~,~,

BDn (A, B; d) = ABDn_l,,, + BBDn_l,l"

diagram Do. Upon substituting d for the trivial diagram Do. Upon substituting -d = A2 + A-2, B = A-‘, B A--l, -d A 2 +A- 2 ,

SD,, = d for the trivial BDo =

one obtains

the Jones polynomial

in the form

one obtains the Jones polynomial in the form

52. Knots: Knots: the the classical classical and and modern modern approaches approaches §2.

(1)

/ b -

(11) (II)

X I ~ -I( ) (

(III) (III)

;'<::::;\ ~ , , / ,

f-l-1 -

279 279

\ d

‘X x, ’ x x /" >èY Fig. A.6. The The Reidemeister Reidemeister moves Fig. A.6. moves

J(A4, A-4) = (-A)-3”(D)S(A,

B; d),

t = A-4.

(The "bracket" “bracket” polynomial polynomial S will appear appear later later on in the definition of of the the (The S will on in the definition Turaev-Reshetikhin invariant of 3-manifolds using Kauffman’s ,approach.) T'uraev-Reshetikhin invariant of 3-manifolds using Kauffman's .approach.) Thus Kauffman Kauffman provides provides two two different different ways at the Jones polynoThus ways of of arriving arriving at the Jones polynomial.

mial. The invariance of of aU all of of the above polynomials can be be est estabThe topological topological invariance the above polynomials can ablished in a purely combinatorial manner using “Reidemeister moves”, which lished in a purely combinatorial manner using "Reidemeister moves", which are elementary changes (of three types) of diagrams, realizing equivalences of are elementary changes (of three types) of diagrams, realizing equivalences of knots and links (see Figure A.6). knots and links (see Figure A.6). Thus one has merely to check that each polynomial remains unaffected by Thus one has merely to check that each polynomial remains unaffected by Reidemeister moves; the appropriate recurrence relation may then be used to Reidemeister moves; the appropriate recurrence relation may then be used to calculate the polynomial. (It should be noted that the identification of these calculate the polynomial. (It should be noted that the identification of these elementary moves was non-trivial in the era of Reidemeister and Markov (the elementary moves was non-trivial in the era of Reidemeister and Markov (the 1930s and 194Os), but that the introduction of the concept of “generic prop1930s and 1940s), but that the introduction of the concept of "generic properties” by Whitney, Pontryagin and Thorn (from about 1935 till the 1950s) erties" by Whitney, Pontryagin and Thom (from about 1935 till the 1950s) simplified the problem to the point of becoming an exercise for students: simplified the problem to the point of becoming an exercise for students: “Consider a generic deformation and decompose it into elementary topologi"Consider a generic deformation and decompose it into elementary topological moves.“) cal moves.") In older treatments of knot theory (in particular in ‘(Modern Geometry”, In older treatments of knot theory (in particular in "Modern Geometry", Part II, by Dubrovin, Fomenko and the present author) the definition of the Part II, by Dubrovin, Fomenko and the present author) the definition of the Alexander polynomial is couched in terms of the somewhat unintuitive opAlexander polynomial is couched in terms of the somewhat unintuitive operation of “differentiation in the group ring”. It has been pointed out (by L. group ring". It has been pointed out L. eration of "differentiation in theseminar) Alania in the author’s Moscow that this “differentiation” can (by be arAlania in the author's Moscow seminar) that this "differentiation" can be arrived at via the following topological route: Starting with the oriented diagram via the foUowing route: Starting in with orientedmanner diagrama rived of the atknot or link K on topological the plane, one calculates thethe standard of the knot or link K on the plane, one calculates in the standard manner a presentation of the group ni(M) of the knot (A4 = S3 \ K), obtaining one the group '/Tl (M) of the knot (M S3 \ K), obtaining one presentation of generator for each edge of the diagram (see Figure A.7) and a pair of relagenerator for crossing. each edgeSince of the Figure a pair of relations for each onediagram relation (see of each suchA.7) pair and simply equates the tions for each crossing. Since one relation of each such pair sim ply equates the pair of generators corresponding to the edgesforming the upper branch of the pair of generators corresponding to the edges forming the upper branch of the crossing, the presentation reduces immediately to the standard one involving crossing, the presentation reduces immediately to the standard one involving the same number of generators and relations. The 2-complex L with exactly the same number of generators and relations. The 2-complex L with exactly

280 280 Appendix. Appendix. Recent Recent Developments Developments in in the the Topology Topology of of 3-manifolds 3-manifolds and and Knots Knots

Fig. the fundamental fundamental group group Fig. A.7. A.7. Calculating Calculating the one one O-cell, O-cell, and and with with l-cells 1-cells labelled labelled by by generators generators and and 2-cells 2-cells labelled labelled by by the the relations, is then a deformation retract of M. Lifting to the universal cover relations, is then a deformation retract of M. Lifting to the univers al coyer we we obtain obtain aa boundary boundary operator operator on on aa complex complex of of free free Z[7ri]-modules; Z[1I"1]-modules; which which takes takes the the form form of of aa square square matrix matrix with with entries entries from from this this group group ring, ring, and and itit isis this this matrix which is related to the above-mentioned “differentiation”, matrix which is related ta the above-mentioned "differentiation", as as follows. follows. Denoting ai and rj, one the generators generators by byai and relators relators by by Tj, one defines defines the the operator operator a,, oai Denoting the by by ax&j) = &j, &* (bc) = &%(b) + &i (c)i the matrix in question then has entries qij given by the matrix in question then has entries qij given by 4ij

=

aa,

CTj)

Mapping each generator ai to t we obtain a complex of modules over the Mapping each generator ai to t we obtain a complex of modules over the ring of integer Laurent polynomials, with boundary operator the correspondring of integer Laurent polynomials, with boundary operator the correspond~ ing square matrix now with Laurent polynomials as entries. The determinant ing square matrix now with Laurent polynomials as entries. The determinant of this matrix turns out to be zero, and the highest common factor of its of this matrix turns out to be zero, and the highest corn mon factor of its cofactors, after multiplication by a suitable power of t, turns out to be just cofactors, after multiplication by a suitable power of t, turns out to be just the Alexander polynomial A(t). the Alexander polynomial A(t). The homological treatment of the Alexander polynomial by Milnor in the The homological treatment of the Alexander polynomial by Milnor in the 196Os, formulated in terms of the torsion of the first homology module over 1960s, formulated in terms of the torsion of the first homology module over the Euclidean ring Q[t,t-l] of the Z-covering of L, may be given along these the Euclidean ring Q[t, rI] of the Z-covering of L, may be given along these lines. A note by the author in Soviet Math. Doklady contains the following lines. A note by the author in Soviet Math. Doklady contains the following more attractive exposition. Consider all one-dimensional representations p : more attractive exposition. Consider representations in suchP a: ~1(M) + @, and the homology Hf(M;all one-dimensional @) with local coefficients 11"1 (M) ~ e, and the homology Hf(M; C) with local coefficients in such a representation. In fact we may take M to be any manifold, replace @ by C”, representation. In fact we may MRep,(ni) to be anyofmanifold, replace e ofby~1en, and consider more generally thetake space all representations in and consider more generally the space RePn (11"1) of aIl representations 11"1 in GL,(C), an algebraic variety over Z. The rank hi(p) of the homologyofgroup GLn(C), an then algebraic variety over Z. The rank bi(p)onofthis the algebraic hornologyvariety, group Hf(M;P) defines an integer-valued function Hf(Mj en) then de fines an integer-valued function on this algebraic variety, which is constant almost everywhere (i.e. outside certain “‘jumping” algebraic which is constant almost everywhere (Le. outside certain "jumping" algebraic

$2. Knots: Knots: the the classical classical and §2. and modern modern approaches approaches

281 281

subvarieties Wjj cC RePn(?Tl) Rep,(ri) over Z); Z); the value of bi(p) hi(p) jumps on each each such such subvarieties W subvariety, further on their so on. subvariety, jumps further their pairwise intersections, and so 3 \\ K, In case we we were considering, Le. i.e. where M = 8S3 K, K K aa In the the particular particular case knot, and n = 1, we obtain the variety Repi(7ri) = C\ (0) = @*. One has n obtain the variety Repl (?Tl) C \ {O} C*. has the knot, following result: following result: For the jumping which turn turn For ii = 1 the jumping subvarieties subvarieties constitute constitute aa finite finite set set of of points points which out the roots roots of of the out to to be be just just the the Alexander Alexander polynomial. polynomial.

For have Hi(M) Rep,(ri) = For a a link link with with klc components, components, we we have H1 (M) = Z” Zk and and RePn(?TI) (C*)“, and one obtains corresponding subvarieties, and and aa "generalized “generalized (C*)k, and one obtains corresponding jumping jumping subvarieties, Alexander in Ic variables. A A modern modern elementary elementary combinatorial combinatorial Alexander polynomial” polynomial" in k variables. treatment of the multi-variable Alexander polynomial was given few years years treatment of the multi-variable Alexander polynomial was given aa few ago by H. Murakami. ago by H. MurakamL The algebraic geometry picture is homotopy invariant, invariant, but but may may be be The algebraic geometry in in this this picture is homotopy complicated, and even in the case A4 = S3\K, K a link, is not well understood. complicated, and even in the case M = S3\K, K a link, is not weIl understood. The present author conjectured that that one should be be able extract the The present author had had conjectured one should able to to extract the Jones and HOMFLY polynomials directly from the structure of the Jones and HOMFLY polynomials directly from the structure of the jumping jumping subvarieties in in the = 2. 2. However However a years ago Tu, in in Moscow, subvarieties the case case n n ::::: a few few years ago Le Le Tu, Moscow, investigated these subvarieties in the interesting case of the 2-bridge knots, investigated these subvarieties in the interesting case of the 2-bridge knots, and discovered that their structure is far more complex than expected. Thus and discovered that their structure is far more complex than expected. Thus the problem of finding a classical algebraic-topological treatment of the Jones the problem of finding a classical algebraic-topological treatment of the Jones polynomial open. polynomial remains remains open. We now give a third definition of the Alexander, Jones, and HOMFLY We now give a third definition of the Alexander, Jones, and HOMFLY polynomials, B,. We the definition of these these the braid braid groups groups Bn. We recall recall the definition of polynomials, involving involving the groups: groups: Consider the group with generators gi, i E Z, and relations Consider the group with generators (Ji, i E Z, and relations (JWj

(Jj(Ji,

li jl > 1,

(Ji(Ji+l(Ji = (Ji+l(Ji(Ji+l·

The nth braid group B, is then the subgroup generated by n consecutive The nth braid group Bn is then the subgroup generated by n consecutive generators Q+i, . , gk+n. There is thus a natural embedding of B, in &+I. generators (Jk+l, . .. , (Jk+n' There is thus a natural embedding of Bn in Bn+l' (It is more standard to take B, to be the subgroup generated by ~1,. . . , cm, (It is more standard to take Bn to be the subgroup generated by (JI, ... ,(Jn, i.e. to take Ic = 0.) Le. to take k O.) By a theorem of Markov (Jr.) of the 194Os,the set of equivalence classes By a theorem of Markov (Jr.) of the 1940s, the set of equivalence classes of knots and links is in natural one-to-one correspondence with the classes of knots and links is in natural one-to-one correspondence with the classes of braids under the equivalence relation determined by the following two elerelation determined by the foIlowing two eleof braids operations under the equivalence mentary (Markov moves): mentary operations (Markov moves): 1. Conjugation of any braid a E B, by any element c E B,: a N cat-I. 1. Conjugation of any braid a E Bn by any element c E Bn: a caC 1 . 2. Multiplication of any braid a E B, by g;:i E B,+l: a N aazil. 2. Multiplication of any braid a E Bn by (J!~l E B n + l : a a(J!~l' f'V

f"V

Two braids are then equivalent if one can be transformed into the other by Two braids are then equivalent if one can be transformed means of a finite sequence of elementary Markov moves. into the other by means of a finite sequence of elementary Markov moves.

282 282 Appendix. Appendix. Recent Recent Developments Developments inin the the Topology Topology of of 3-manifolds 3-manifolds and and Knots Knots

The The idea idea behind behind this this correspondence is is both both simple and natural. natura!. Consider Consider the the circle circle S1 Sl cC IK3 ]R3 given given by by 2’ x2 + + y2 y2 = = 1, z == 0, 0, and and aa tubular tubular neighbourhood neighbourhood T of of small small radius: lR3 ]R3 > :::) T > :::) S1. Sl. A A transverse knot knot or link link is is then then defined to to be be a simple closed closed curve curve (or (or collection collection of of such) such) in in T with with tangent tangent vector vector everywhere to the the a-balls 2-balls orthogonal orthogonal to to the the circle. circle. A A braid braid is is then then everywhere transverse transverse to obtained obtained from from aa transverse transverse knot knot or or link link by by cu.tting cutting the the tubular tubular neighbourhood neighbourhood orthogonally orthogonally to to the the circle circle at at any any point. point. ItIt was was in in fact fact Alexander Alexander who who first first showed that any any knot knot or or link link in in lR3 ]R3 is is isotopic isotopie to to aa transversal transversal one; one; Markov Markov showed that corrected corrected aa deficiency deficiency in in the the argument. argument. Any of the the nth nth braid braid group group B, En in in GL,(A), GLn(A), where where Any linear linear representation representation pp of A A is is any any commutative commutative and and associative associative ring, ring, has has character character (trace (trace function) function) invariant the first first elementary elementary Markov Markov operation operation above: above: invariant under under the

xp(a) Xp(a) = = T+(a)l, Tr[p(a)], a aE E Bn En. To To obtain obtain such such aa representation representation to to start start with with (for (for all aIl nn simultaneously) simultaneously) one one may try to represent the generators (pi, i E Z, by matrices satisfying the the may try to represent the generators CTi, i E Z, by matrices satisfying above above defining defining relations. relations. Jones Jones obtained obtained a a class class of of representations representatioris in in this this way way using using the the technique technique for for solving solving certain certain integrable integrable models models of of statistical statistical physics physics and quantum field theory; thus he was able to obtain from the Yang-Baxter Yang-Baxter and quantum field theory; thus he was able to obtain from the equation equation aa special special “local, "local, translation-invariant” translation-invariant" class class of of representations representations of of all aIl braid groups simultaneously. braid groups simultaneously. Let V be be a a vector vector space space and and Let V

S:V®V-tV®V a with the a linear linear transformation transformation with the property property that that on on the the tensor tensor product product V@V@V V ®V ®V of three copies of V, we have of three copies of V, we have ‘%2s23sl2

=

s23%2s23,

where Stj coincides with S on the tensor product of the ith and jth copies where Sij coincides with S on the tensor product of the ith and jth copies of V in V @ V @V, and restricts to the identity map on the remaining copy. of V in V ® V ® V, and restricts to the identity map on the remaining copy. (Note that in the theory of solvable models of statistical mechanics one may (Note that in the theory of solvable models of statistical mechanics one may find the Yang-Baxter equation written more generally for a map of the form find the Yang-Baxter equation written more generally for a map of the form R = TS, the composite of S : Vi @ Vj + Vi @ Vi with the permutation map R = TS, the composite of S : Vi ® Vi - t Vi ® Vi with the permutation map T : vi @vj + vj @vi, where now the factors of Vi @Vj @Vk may be different, T : Vi ® Vj Vj ® Vi, where now the factors of Vi ® Vj ® Vk may be different, in the form in the form Defining Defining

P(ai)

P(CTi)

=

Si,i+l,

Si,i+1,

we obtain a representation p of the infinite braid group by linear transformawe obtain representation p of the of infinite braidmany groupcopies by linear tions of thea infinite tensor product countably of V,transformarestricting the infinite tensor product of countably many copies V, restricting; tions of to a representation pn of the nth braid group B, by linear of transformations to a representation Pn .of the nth+braid group En bythe linear transformations of the space V @IV @ .@ V ((n 1) factors). Thus image ~(a,) of each of the space V ® V ® ... ® V ((n + 1) factors). Thus the image P(CTi) of each

52. Knots: Knots: the classical classical and modern modern approaches approaches §2.

283 283

generator O'i gi is is determined determined by by its its action action "locaIly" “locally” on on Vi V, 0@ Vi+! V,+r (since (since it it acts acts generator identically on all other factors), and is “translation-invariant” in the sense "translation-invariant" in the sense identically aIl other factors), and that for for aU all i, i, P(O'i) p(ai) acts acts in in the the same same way way as as S S acts acts on on V V 0@ V. V. that (One may easily imagine variants of this construction using different (pe(pe(One may easily imagine variants of this construction using different riodic) partitions of Z; however to date no examples of interest have been riodic) partitions of Zi however to date no examples of interest have been found.) found.) A is said said to to have have the the Markov Markov property property if if there there exists exists aa A representation representation pP is constant A independent of n such that for every n and every braid B, constant A independent of n such that for every n and every braid aa EE Bn one has one has Tr[p,(a)lA = Tr[Pn+l Tr[p,+l(aa,+l)l. Tr[Pn(a)]A = (aO'n+l)]' Then according to Jones, given given such such aa representation quantity Then according to Jones, representation p, P, the the quantity

O't:

A-“(“)Tr[p,(a)], 0:; ... . . .C$Z E E Bn, B,, A-w(a)Tr[Pn(a)], aa == O'f: where w(a) w(a) = CT=“=, affords aa topological topological invariant. invariant. where I:~~1 j,, jk, affords We shall give below some important examples of such such representations representations yieldWe shaH give below some important examples of yielding as invariants the Alexander, Jones and HOMFLY polynomials. However ing as invariants the Alexander, Jones and HOMFLY polynomials. However before we give give aa direct geometric explanation explanation of of the connection bebedirect geometric the connection before doing doing so so we tween knots and solutions of the Yang-Baxter equation. Let R$ and R$ be tween knots and solutions of the Yang-Baxter equation. Let R~~ and Rt be two the Yang-Baxter the indices indices aU all run run over over the the two solutions solutions of of the Yang-Baxter equation, equation, where where the same basis for for the the vector vector space space V. V. Given Given aa diagram diagram D, D, one same set set I1 indexing indexing aa basis one lalabels each crossing with a copy of the solution R or x according as the crossing bels each crossing with a copy of the solution R or R according as the crossing is “positive” or “negative” in the earlier sense. Each incoming edge is labelled is "positive" or "negative" in the earlier sense. Each incoming edge is labelled with the appropriate lower indices (cd or kl) and each outgoing edge with the with the appropriate lower indices (cd or kt) and each outgoing edge with the appropriate upper indices. On carrying out a total tensor contraction of the appropriate upper indices. On carrying out a total tensor contraction of the product of all Rs and %% associated with all crossings of the diagram, one product of aU Rs and Rs associated with aIl crossings of the diagram, one obtains a number (DIIR,??) depending on the diagram D and the solutions R obtains a number (DIIR, R) depending on the diagram D and the solutions R and R. and The number (D/IR,R) gives a topological invariant of the corresponding The number (DIIR, R) gives a topological invariant of the corresponding knot or link under two more additional conditions. First, the number (011 R,??) knot or link under two more additional conditions. First, the number (DIIR, R) is invariant with respect to the second and the third Reidemeister moves prois invariant with respect to the second and the third Reidemeister moves provided the following “channel unitarity” and “cross-channel unitarity” equavided the following "channel unitarity" and "cross-channel uni tarit y" equations hold: tions hold: pfRij = pbb. ZJ

cd

c

d,

-jd

RZ.?R.zcjd = ,j”Sb. R36ia ll 8ca [;bd jb ie = c d' the restrictions

Secondly, let R, R satisfy Secondly, let R, iL satisfy the restrictions

ab Acb R- ab A-1cb R ad = ud' 'ad = ud' Then the number A-w(D)(DII R, R) is invariant with respect to the first ReiThen the number (DIIR, R) isainvariant respect to the first Reidemeister move andA -w(D) therefore affords topologicalwith invariant. demeister move and therefore affords a topological invariant. (Conditions of this type, together with the associated terminology, come of this type, with the associated terminology, from(Conditions the interpretation by together Yang, Zamolodchikov, and others, of the come entithe interpretation by Yang, Zamolodchikov, and others, of the entifrom ties R,R in terms of the “factorizable scattering amplitude” in integrable ties R, R in terms of the "factorizable scattering amplitude" in integrable

284 284 Appendix. Appendix. Recent Recent Developments Developments in in the the Topology Topology of of 3-manifolds 3-manifolds and and Knots Knots models models of of 2-dimensional 2-dimensional quantum quantum field field theory. theory. The The “Baxter-type” "Baxter-type" integrable integrable 2-dimensional lattice models of statistical mechanics furnish different 2-dimensionallattice models of statistical mechanics furnish different interpreinterpretations. that in in both both cases cases the the actual actual Yang-Baxter Yang-Baxter solutions solutions are are tations. Note Note however however that much mu ch more more complicated, complicated, depending depending on on an an additional additional ‘(spectral "spectral parameter” parameter" .).) ItIt isis convenient convenient to to introduce introduce “cup” "cup" and and “cap” "cap" matrices matrices Mab Mab and and Mcj M ij satsatisfying isfying MaiMib MaiMib == 6;. 8b· The The pair pair of of solutions solutions of of the the Yang-Baxter Yang-Baxter equation equation related related by by --ab ~1'ce dh R,, cd == MceR%M

-Rab

hb hb "v- Rea M ,,

then then satisfy satisfy both both channel channel and and cross-channel cross-channel unitarity. unitarity. These These matrices matrices arise arise from eliminating aa single the diagrams diagrams obtained obtained from from diagrams diagrams of of braids braids by by eliminating single from the crossing; they are are associated associated with with the the extrema extrema of of the the Morse Morse function function on on the the crossing; they zz-plane xz-plane which which projects projects the the diagram diagram on on the the z-axis z-axis (see (see Figure Figure A.8). A.8). (Here (Here we an n-braid as consisting we are are assuming assurning the the standard standard definition definition of of an n-braid as consisting of of nn nonintersecting curve segments segments within within the cylinder nonintersecting and and nonself-intersecting nonself-intersecting curve the cylinder ~?+y~
a b .>(, kb a

C

c

b

1

ad b

a

X X i U ab R abab R

dd

c c

d d

R-abab Rab

M””

(-)

n

a

a

bb

Ma,

Mab ab Fig. A.8. Defining R-matrices and cup and cap matrices ab

Fig. A.8. Defining R-matrices and cup and cap matrices

defines the Marlcov trace by defines the Markov trace by

T(D) = Tr[1J Tr[rl@ ~8.. . @wn(B)l, 01J 0 ... 01JPn(B)],

r(D)

where p is the representation obtained from the solution R of the Yang-Baxter where p is the representation obtained from the solution R of the Yang-Baxter equation, and equation, and 7; = MaiMb”.bi 1Jab = M ai M . Consider the explicit solution R given by Consider the explicit solution R given by RF; = AMabMcd + A-%,a&, b - AMabMcd + A- 1 8ca8d' R ab cd -

$2. Knots: Knots: the the classical classical and and modern modern approaches approaches §2.

1

22

.1 "2

nn

...

1 1

22

nn

., .

\

_1.... =)-3-J

11

1

'\1 \

1‘

x

z=o z=o

,

1 \

i

n

285 285

0

\ 1

1 1

1---1~

~

,

z=1 z

=

l

112 2 nn

Fig. A.9. A.9. The The diagram diagram of of aa closed closed braid braid on on the the plane. plane. A A closed closed braid braid in in the the cylinder cylinder Fig.

where ii22 = = -1, -1, the the matrix matrix M M is is of of the the form form where

M= (-$

M and the the index and index set set lI is considered is considered here here determined by the by the determined formula formula

iA) k”), o '

= (+, (+, --).). The The diagram D of of the braid B B in in question question = diagram D the closed closed braid

to be unoriented. However D D has has aa natural natural orientation orientation to be unoriented. However direction of the z-axis in the (z, 2) plane. This yields the the direction of the z-axis in the (x, z) plane. This yields J(A4, AA-4)4 ) = (-A)-3”‘D’~(D) J(A\ = (_A)-3w(D)T(D)

for polynomial discussed discussedabove. above. for the the Kauffman-Jones Kauffman-Jones polynomial In the case of oriented diagrams one has an solution R R given given by In the case of oriented diagrams one has an explicit explicit solution by the formula the formula

RF2= ((4- 4-l) 1a < b] + q[a =

b]) 6;s; + [a # b]S~S,b,

where [P] is defined to be 1 if the proposition P is true and 0 if P is false, and where [Pl is defined to be 1 if the proposition P is true and 0 if P is false, and the index set I = 1, = (-n, -n + 2,. . . , n). The associated solution x may the index set l In (-n, -n + 2, ... ,n). The associated solution R may be obtained by replacing q by q-l and a, b by -a, 4 in this formula. These be obtained by replacing q by q-l and a, b by -a, -b in this formula. These Yang-Baxter solutions yield the following topological invariant of a given knot Yang-Baxter solutions yield the following topological invariant of a given knot or link with oriented diagram D: or link with oriented diagram D:

-w(D) (DIIR,??). Phn) = ( q(n+l) ) -w(D) (DIIR, R). The HOMFLY polynomial may then be obtained by taking the index set I The HOMFLY polynomial may then be obtained by taking the index set l of cardinality n + 1 (where n > 1) and by making the substitution y = qn+l. of 1 (where n ?:: 1) qand by making the substitution y = qn+ l . In cardinality the case n =n + 1 the substitution = t112 gives the Jones polynomial. Note In the case n = 1 the substitution q = t l / 2 gives the Jones polynomial. Note that the substitution q = 1 does not yield any new information. that the substitution q = 1 does not yield any new information. Consider now “tangles”, i.e. oriented diagrams D1 in the strip 0 5 z < 1 thelines stripz0 =S 0z and S 1 Consider now "tangles", i.e. output orientededge, diagrams with one input edge and one endingDlatinthe with one input edge and one output edge, ending at the lines z = 0 and z = 1 respectively. For the index set I = (-, +) we always label both the z = 1and respectively. For the set l +. = (-, +) we always label both the input output edges with index the index One then obtains the Alexander input and output edges with the index +. thenplane obtains the Alexander polynomial for the closed oriented diagram DOne in the by cutting it along the closed oriented diagram D in the plane by cutting it along polynomial for an (arbitrarily chosen) external edge. an (arbitrarily chosen) external edge.

286 Appendix. Appendix. Recent Recent Developments Developments in in the the Topology Topology of of 3-manifolds 3-manifolds and and Knots Knots Following Following Jaeger Jaeger and and Kauffman, Kauffman, one one may may obtain obtain the the Alexander Alexander polynomial polynomial from the following following R-matrix: R-matrix: taking taking the the index index set set I1 == (fl, (+1, -l), -1), one one sets sets from the R = (q - q-l)X+-

+ qX++ - q-IX--

+ Y,

and then obtains obtains ETI by by substituting substituting q-l q-l for for qq and and X-+ and then tensors X,, are given by tensors X sp are given by 11 ifa=c=sandb=d=p, if a c sand b Kx~ (Xsp)~~ == {{ 00 otherwise, otherwise, and the tensor tensor Y Y by by and the

d

for for X+X+_ ;; here here the the

p,

Y,“d” = [a # b]S$S,b,

where a, b, b, c, c, dd == *l. ±1. where a, The Alexander The Alexander polynomial polynomial may may also also be be obtained obtained from from the the “Burau "Burau reprerepresentation” of the braid group on the direct sum of one-dimensional spaces sentation" of the braid group on the direct sum of one-dimensional spaces Iwi lRi == IL?, lR, given given by by 1 0 0 OTO; P(ui) = pra;) [ 0 0 1 I here here 1 1 denotes denotes the the identity identity matrix matrix of of arbitrary arbitrary size size and and T T is is the the 22 xx 2 2 matrix matrix

[~~

T=

n

[ Y (&)]

acting on Iw, @Iwi+1. The Alexander polynomial is then given by acting on lRi EB lRi+!' The Alexander polynomial is then given by ALL = (det)*(p(a) - L), where D, is the relevant closed braid obtained from the braid a, and (det)’ where Da is the relevant closed braid obtained from the braid a, and (det)* denotes any (n- 1)-minor of the matrix p(a) - 1,. As obtained by these means denotes any (n l)-minor of the matrix p(a) ln. As obtained by these means the Alexander polynomial is well-defined up to multiplication by fP, m m any the Alexander polynomial is well-defined up to multiplication by ±t , m any integer. integer. It is worthwhile pointing out the following interesting identity for the exIt is worthwhile pointing out the following interesting identity for the exterior powers of the linear transformation M: terior powers of the linear transformation M: det(M - Xl,)

det(M -

À1n)

j=n

2:

= c Trace((-X)kAkM);k = j=o Trace(( _À)k A M); j=ü

this leads to a (non-Markov) trace, and to Yang-Baxter solutions R, ?? essenthis leads to a (non-Markov) trace, and to Yang-Baxter solutions R, essentially equivalent to those used above in constructing the Alexander polynomial tially equivalent to those used above in constructing the Alexander polynomial using tangles. using tangles. There are three nice algebraic objects that have turned out to be very There are modern three nice algebraic objects that have turned out algebra, to be very useful in the theory of knots and 3-manifolds: the Hecke the the modern theory of knots and 3-manifolds: the Hecke algebra, useful in Temperley-Lieb (or Jones) algebra, and the Birman- Wenzl algebra. They the are Temperley-Lieb (or Jones) algebra, and the Birman- Wenzl algebra. They are defined as follows: defined as follows:

52. Knots: approaches §2. Knots: the the classical classical and and modern modern approaches

287 287

A He&e algebra is generated by the elements elements (J'i, cri, i E satisfying the braid Hecke algebra E Z, satisfying relations (see above), together with relations of the form relations (see together with of the (Ti - CT-1 = z,

where z may be either either a number or in the centre of of the the algebra. The algebra (TL) (TL) over the ring Z[A, A-Il A-‘] is is given by gengenThe Temperley-Lieb Temperley-Lieb algebra ring Z[A, erators vi, ii E the relations erators Vi, E Z, Z, satisfying satisfying the relations v,?Jj = vjvi, ViVj VjVi,

> 1, 1, liIi -jl jl > VjV~flV~ = vi. ViVi±1 Vi = Vi·

Here may be some number. There is associated abstract trace, Here AAmay be taken taken as as sorne number. There is an an associated abstract trace, i.e. a linear map Le. a linear map ff:: TL TL --; -+ Z[A, A-‘], Z[A,A-l], satisfying If(uv) = for all all U, w. Note Note also representasatisfying = If(vu) for u, v. also that that one one obtains obtains a a representation p of the braid group in TL in the general form tion p of the braid group in TL in the general form ,o(ai)

= awi + b,

for appropriate numbers for appropriate numbers a, a, b. b. The Jones algebra is the same as TL, however considered rather as given in The Jones algebra is the same as TL, however considered rather as given in terms of generators ei = 8-IVi, 6-l~i, where -6 == A2 A2 + Ap2; the ei are projectors: terms of generators ei where -8 + A-2; the ei are projectors: ef = ei. eT = ei· The Birman- Wenzl algebra is also generated by the pi, with the additional The Birman- Wenzl algebra is also generated by the (J'i, with the additional relations relations ci - ai1 = Z(TJ(- 1). (J'i - (J'il Z(Vi - 1). This is easily seen to be realizable as a subalgebra of TL. This is easily seen to be realizable as a subalgebra of TL. From braids and the Yang-Baxter equation the theory of “quantum groups” From braids and the Yang-Baxter equation the theory of "quantum groups" has been constructed (by Drinfeld, Jimbo, Sklyanin, Kulish, Faddeev, Takhtadhas been constructed (by Drinfeld, Jimbo, Sklyanin, Kulish, Faddeev, Takhtadjan, Reshetikhin, Voronovich, and others). This represents a pleasing systemjan, Reshetikhin, Voronovich, and others). This represents a pleasing systematization of the behaviour of the most familiar solutions of the Yang-Baxter atization of the behaviour of the most familiar solutions of the Yang-Baxter equation in terms of certain algebras (shown by Drinfeld to be Hopf algebras equation in terms of certain algebras (shown by Drinfeld to be Hopf algebras equipped with a nice additional structure called a “quantum double”. (In his equipped with a nice additional structure called a "quantum double". (In his terminology this is “quantum deformation” of the enveloping algebra of a terminology this is "quantum deformation" of the enveloping algebra of a semisimple Lie algebra.) semisimple Lie algebra.) This striking terminology derives, in fact, merely from the property that This striking terminology derives, in fact, merely from the pro perty that to a first approximation this deformation is determined by a certain Poisson to a firstonapproximation determined by a certain Poisson bracket the Lie group,this so deformation that, on the isface of it, it appears to represent bracket on the Lie group, so that, on the face of it, it appears to represent some sort of quantization of the group as of a phase space. (Such brackets sornemade sort an of appearance quantizationearlier of theingroup as ofwith a phase space. brackets had connexion certain (not (Such all) integrable had made an appearance earlier in connexion with certain (not aH) integrable models of the theory of solitons.) However this procedure does not in fact models ofany the quantization theory of solitons.) However thisalgebra procedure does not in fact represent of the group or Lie as objects measuring represent any quantization of the group or Lie algebra as objects measuring the symmetry of some physical system to be quantized. theSeveral symmetry of sorne physical system be quantized. examples show that this veryto appealing construction is more like Several examples show that this very appealing construction is more like a “discretization” , in a certain subtle, nontrivial sense, than a “quantization” a "discretization", in a certain subtle, nontrivial sense, than a "quantization"

288 288 Appendix. Appendix. Recent Recent Developments Developments inin the the Topology Topology ofof 3-manifolds 3-manifolds and and Knots Knots

as as this this term term isis usually usually employed employed in in the the context context of of symmetry symmetry in in analysis analysis (and (and in in fact fact similar similar nontrivial nontrivial “q-deformations "q-deformations of of analytic analytic calculus” calcul us" can can be be traced traced back back to to the the 19th 19th century). century). However However the the term term “quantum "quantum group” group" sounds sounds imimpressive, and lends lends a, perhaps perhaps somewhat somewhat spurious, spurious, air air of of excitement excitement to to the the pressive, and topic. topie. (A (A first-rate first-rate quantum quantum physicist physicist at at the the Landau Landau Institute Institute who who had had himhimself self discovered discovered many many nontrivial nontrivial solutions solutions of of the the Yang-Baxter Yang-Baxter equation, equation, once once told told the the present present author author (in (in the the late late 1980s) 1980s) that that he he did did “not "not know know what what ‘quan'quanturn tum group’ group' means, means, but but itit sounds sounds very very attractive”. attractive". ItIt seems seems however however that that he he himself of himself had had in in effect effect already already applied applied quantum quantum groups groups to to the the discretization discretization of the the Schrodinger Schrodinger operator operator in in a magnetic magnetic field.) field.) ItIt seems imseems likely likely that that the the technique technique of of quantum quantum groups groups can can be be used used to to improve invariants. of the the known known results results concerning concerning the the above above polynomial polynomial invariants. prove some sorne of However aside from hitherto nothing nothing along along these these lines lines has has been been achieved achieved aside from a However hitherto systematization of known results, although the approach continues to systematization of known results, although the approach continues to look look promising. promising. The The Jones Jones polynomial polynomial has has led led to to some sorne beautiful beautiful new new results. results. One One such such result result is as follows. In the late 19th century the physicist Tait, following,a is follows. In the late 19th century the physicist Tait, following,a suggestion suggestion of either Maxwell Maxwell or or Thompson Thompson (Lord (Lord Kelvin), Kelvin), began began to to study study knots. knots. Having Having of either compiled a table of the simplest nontrivial knots, he observed in particular compiled table of the simplest nontrivial knots, observed in particular various various properties properties of of the the “alternating” "alternating" knots knots in in his his table, table, i.e. i.e. those those where where undercrossings alternate with overcrossings as one traces out the oriented undercrossings alternate with overcrossings as one traces out the oriented diagram diagram of of the the knot. knot. Using Using the the Jones Jones polynomial polynomial itit was was shown shown by by Kauffman Kauffman and Murasugi in the late 1980s that, provided such an oriented is and Murasugi in the late 1980s that, provided such an oriented diagram diagram is irreducible (i.e. is not obtainable by joining two nontrivial diagrams together irreducible (i.e. is not obtainable by joining two nontrivial diagrams together by segments (see then the the by means me ans of of aa pair pair of of nonintersecting nonintersecting line line segments (see Figure Figure A.lO)), A.lO)), then number of crossings is a topological invariant of the knot. (In fact the number number of crossings is a topological invariant of the knot. (In fact the number of crossings irreducible alternating diagram is the difference of crossings of of an an irreducible alternating diagram is equal equal to to the difference between the largest and smallest powers oft occuring in the Jones polynomial between the largest and smallest powers of t occuring in the Jones polynomial J(t, J(t, t-l).) Cl ).)

Fig. A.lO. A.lO. A reducible diagram Fig. A reducible diagram

Although had never never been been considered considered aa major major question question of knot theory, theory, it it Although this this had of knot is significant that the methods of classical topology had seemed inadequate for is significant that the methods of classical topology had seemed inadequate for settling (and other results) provide provide evidence evidence of the great great importance importance settling it. it. This This (and other results) of the of the discovery of the Jones polynomial. of the discovery of the Jones polynomial.

§3. $3. Vassiliev Vassiliev Invariants Invariants

289 289

We mention mention in passing passing also also the the following following nice property property of the the Alexander Alexander We in nice of polynomial (known for sorne some time): time): For For a ribbon ribbon knot knot (Le. (i.e. one one bounding bounding an an polynomial (known for immersed 2-disc in in ]R3) R3) there there exists exists a polynomial polynomial f(t) such that that immersed 2-disc f(t) such

A(t, t-‘)

= f(t)f(t-‘).

$3. Vassili Vassilievev Invariants Invariants §3. The of the Alexander and Jones Jones polynomials polynomials have aa very very The additive additive properties properties of the Alexander and have attractive interpretation in terms of “Vassiliev invariant?. The theory of these attractive interpretation in terms of "Vassiliev invariants" . The theory of these invariants appeared over the last 5 5 years years or or so; so; chief chief among among its its exponents exponents the last invariants has has appeared over are V. Vassiliev, D. Bar-Natan, J. Birman , X-S. Lin, M. Kontzevich. are V. Vassiliev, D. Bar-Natan, J. Birman, X-S. Lin, M. Kontzevich. Consider the space space Fk of all all immersions Gk of of the the circle circle S1 S’ in in ]R3 R3 with with Consider the Fk of immersions Gk the property that the image has exactly k double points (crossings), Pi, Pi the property that the image has exactly k double points (crossings), Pi, Pi EE S’, k, at each of of which which the image curve curve meets meets itself itself transversely. transversely. SI, ii = 1,. 1, .... ,,k, at each the image By perturbing the image near one such intersection point we obtain By perturbing the image near one such intersection point we obtain aa pair pair of immersions of SI S1 each each with (k -- 1) 1) crossings, crossings, and therefore G;-1, Gt-l' Gi-i G k- 1 of immersions of with (k and therefore belonging (see Figure 11). Consider Consider aa topological topological invariant belonging to to the the space space FkFk-1r (see Figure A. A.Il). invariant

x-x x

Fig. A.1 1. Defining the Vassiliev derivative Fig. A.Il. Defining the Vassiliev derivative fk of the space Fk, i.e. any function the set of components of Fk to some !k of the space Fkl Le. any function from From the set of components of Fk to sorne abelian group A:

abelian group A:

.fk

: ro(Fk)

+

A.

Thus in particular when k = 0 we obtain precisely the topological invariants Thus in particular when k 0 we obtain precisely the topological invariants of knots. We call such an invariant fk a Vassiliev derivative of some invariant of knots. \\Te calI sueh an invariant fk a Vassiliev derivative of sorne invariant f&i if the following equation holds:

fk-1 if the following equation holds:

fk-l($,)

- fk-l(Gi-l)

= fk(Gk).

A topological invariant of knots is then called a Vassiliew k-invariant if its A topological invariant of knots is then ealled a Vassiliev k-invariant if its (k + l)st Vassiliev derivative is zero. (k + 1)st Va'isiliev derivative is zero. By a result of Bar-Natan, all coefficients of the Alexander polynomial are polynomial are By a result of Bar-Natan, all coefficients of theLin,Alexander Vassiliev invariants. A few years ago Bar-Natan, and Birman were also Vassiliev invariants. A few years ago Bar-Natan, Lin, and Birman were also able to deduce from the additive properties of the HOMFLY polynomial that the additive properties of the HOMFLY polynomial that able to deduee from after performing the substitution.

after performing the substitution. y = exp(Nz),

y

z = exp(G)

exp(Nx) , x

- exp(-G),

290 Appendix. Appendix. Recent Recent Developments Developments in in the the Topology Topology of of 3-manifolds 3-manifolds and and Knots Knots 290 the coefficients coefficients of of the the HOMFLY HOMFLY polynomial as as aa series series in zt are are Vassiliev Vassiliev the polynomial invariants. invariants. There is an attractive attractive formula formula due to to Kontzevich Kontzevich expressing expressing aIl all Vassiliev Vassiliev There invariants analytically in terms of multiple integrals, assuming that the knot knot invariants analytically in terms of multiple integrals, assuming that the or link link diagram diagram comes comes with with some some generic generic Morse Morse function function (for (for instance instance the the projection of the planar diagram on the y-axis). projection of the planar diagram the y-axis). There is is also also aa purely purely combinatorial combinatorial characterization of aIl all possible possible VasVasThere characterization of siliev invariants, which, however, is at present computationally impractical siliev invariants, which, however, is at present computationally impractical for calculating calculating specifie specific invariants. invariants. (In (In fact fact there there are are at at present present no no known known spespefor cific Vassiliev invariants other than the above-mentioned coefficients of the cifie Vassiliev invariants other than the above-mentioned coefficients of the Alexander and HOMFLY polynomials of knots.) Alexander and HOMFLY polynornials of knots.)

u

=

i

Fig. A.12. A.12. The The diagram diagram U U Fig.

This combinatorial description is is as follows. Define Define aa chord chord diagmm diagram to to be be This combinatorial description as follows. aa circle circle with 21c distinct points labelled Pj, Qj,j = 1,2,. . , k, marked on it. with 2k distinct points labelled Pj, Qj,j = 1,2, ... , k, marked on it. (In fact only the cyclic order of the points around the circle will be significant (In fact only the cyclic order of the points around the circle will be significant here.) We the following following relations relations on on the the free free abelian abelian group group freely freely We now now impose impose the here.) generated by all chord diagrams (with 2k points): generated by aIl chord diagrams (with 2k points): 1. A chord diagram is set equal to zero if it contains a pair Pj, Qj of points 1. A chord diagram is set equal to zero if it contains a pair Pj , Qj of points not “linked” with any other pair, i.e. if for any i one has Pj < Pi < Qj in not "linked" with any other pair, Le. if for any i one has Pj < Pi < Qj in the cyclic order round the circle, then also Pj < Qi < Qj. the cyclic order round the circle, then also Pj < Qi < Qj. 2. Given any (k - 2) pairs Pj, Qj, j = 1,. . . , k - 2, of points and three 2. Given any (k - 2) pairs Pj, Qj, j = 1, ... , k - 2, of points and three further points Al, AZ, A3 on the circle, all distinct, one may construct a further points Al, A2' A3 on the circle, aIl distinct, one may construct a chord diagram with 2k points as follows: First replace one of the three chord diagram with 2k points as follows: First replace one of the three points Al, AZ, A3 by two adjacent points (close to the chosen Ai) labelled points Al, A2' A3 by two adjacent points (close to the chosen Ai) labelled Pk-l,Pk, and relabel one of the remaining points Qk-1 and the other Qk; Pk-l, Pk, and relabel one of the remaining points Qk-l and the other Qki since this relabelling can be carried out in two distinct ways, there are two since this relabelling can be carried out in two distinct ways, there are two chord diagrams, C+ and C- say, obtainable in this way; thus if Al is the chord diagrams, C+ and C_ say, obtainable in this way; thus if Al is the point first chosen, then C+ and C- are given by

point first chosen, then C+ and C_ are given by

c, : A2 = Qk-1,A3 = Qk, C+ : A 2 Qk-l, A3 = Qk,

c- : A2 = Qk, A3 = Qk-1. C_ : A 2 = Qk, A3 = Qk-l' One then imposes the relations ensuring that the difference C+ - C- is One then imposes relations thetriple difference independent of the the choice of the ensuring first point that of the Al, Aa,C+A3.- C_ is independent of the choice of the first point of the triple Al, A2' A3' The resulting quotient group Bk then has encoded in it all Vassiliev kThe resulting quotient group Bk then has encoded in it aIl Vassiliev kinvariants. invariants.

$4. New topological invariants 3-manifolds. §4. New topological invariants for for 3-manifolds.

291 291

The graded sum The graded sum

B

I:Bk, k>O k2:0

can given the structure of of aa graded Hopf ring ring by defining multiplication multiplication can be be given the structure graded Hopf by defining via the connected connected sum sum of of chord diagrams along along arcs arcs disjoint from the via the chord diagrams disjoint from the marked marked points. multiplication is of the the above (That this this multiplication is well-defined well-defined is is aa consequence consequence of above points. (That defining of the the Bk.) Bk') defining relations relations of We the formula formula of of Kontzevich Kontzevich from from which follows easily easily that that We now now describe describe the which itit follows each Bk does to aa Vassiliev Vassiliev invariant. Let each element element of of the the Bk does indeed indeed correspond correspond to invariant. Let K be an oriented knot in Iw3 given via co-ordinates z = .z(t)(= x(t) +iy(t)), t. K be an oriented knot in ]R3 given via co-ordinates z = z(t)( = x(t) + iy(t)), t. (Here t may be regarded as a generic Morse function on K.) Consider the (Here t rnay be regarded as a generic Morse function on K.) Consider the following (of "Kniznik-Zamolodchikov “Kniznik-Zamolodchikov monodromy”): following sum sum (of monodrorny"): Z(K)

I:-1-J. ..J m

tmiu
(-l)Nll\dlog(zj-z])DN; ,.tn
j

here is taken taken with with respect the variables here the the integral integral is respect to to the variables t,. tB • t,t one has the set of points (z(ts),ts) on the knot K, and ) one has the set of points (z(ts), t on the knot K, and B s chooses (the choice choice being being indicated indicated by chooses (the by AS) AS) m, ms pairs pairs (zj(b),b),

(+(t,),t,),

For each each value of For value of from this set one from this set one

j = 1,. . ,msr

where m. The The associated associated chord is then the circle circle where C, I:s m, ms = m. chord diagram diagram is then just just the with this collection of pairs of points. The symbol N J denotes the number with this collection of pairs of points. The symbol N l denotes the number of descending chords with respect to t. Writing s = SK for the number of of descending chords with respect to t. Writing S = SK for the number of minimum points of the knot K, and y := Z(U) where U is as in Figure A.12, Z(U) where U is as in Figure A.12, minimum points of the knot K, and "( one has the following formula of Kontzevich : one has the following formula of Kontzevich : ypSZ(K)

"(-s Z(K)

is a topological (Vassiliev) invariant. is a topological (Vassiliev) invariant.

$4. New topological invariants for 3-manifolds. §4. New topological invariants for 3-manifolds. Topological Quantum Field Theories Topological Quantum Field Theories We shall now describe some applications of these ideas to the construction We shaH now describe some applications of these ideas to the construction of new topological invariants of 3-manifolds. These invariants were discovered of new topological invariants of 3-manifolds. These invariants were discovered by Witten as natural non-abelian generalizations of the Schwarz represenWitten natural non-abelian generalizations Schwarz represenby tation of theas Reidemeister-Ray-Singer invariant, in of thetheform of a functional tation of the Reidemeister-Ray-Singer invariant, in the form of a functional integral of “Chern-Simon? type, i.e. a multi-valued action functional on integral i.e. ofa Yang-Mills multi-valuedfields action on the spaceofof "Chern-Simons" gauge-equivalent type, classes (i.e.functional differentialthe space of gauge-equivalent classes of Yang-Mills fields (i.e. differentialgeometric connexions on principal G-bundles where G is any compact Lie geometric group (e.g.connexions G = SUz)). on principal G-bundles where G is any compact Lie group (e.g. SU2 )). function” defined on the class of 3-manifolds of the There is G a “partition There is a "partition defined theof): class of 3-manifolds of the following form (provided function" this can be made on sense following form (provided this can be made sense of):

292 292 Appendix. Appendix. Recent Recent Developments Developments in in the the Topology Topology of of 3-manifolds 3-manifolds and and Knots Knots 2(M3) Z(M 3 ) =

JJ

DA(z) DA(x) exp exp {ikS(A)} {ikS(A)} , i2 i 2 = -1. -l.

Here Here Ic k isis an an integer integer by by virtue virtue of of the the “topological "topological quantization quantization of of the the coupling coupling constants” (as formulated for multi-valued functionals by the present author constants" (as formulated for multi-valued functionals by the present author in in 1981, 1981, by by Deser-Jackiv-Templeton Deser-Jackiv-Templeton in in the the special special case case of of the the Chern-Simons Chern-Simons action action (1982), (1982), and and by by Witten Witten in in 1983), 1983), which which in in turn turn is is aa consequence consequence of of the requirement that the “Feynman amplitude” exp {i&5’} should the requirement that the "Feynman amplitude" exp {ikS} should be be singlesinglevalued. that the the CS-action CS-action has has the the following following local local form: form: valued. Note Note that

2 3) . S(A) S(A) == Tr(&Ai Tr(8i A i -- t3,Aj 8j Aj ++ ;A3). 3A Using along some Using aa Heegaard Heegaard decomposition decomposition of of the the 3-manifold 3-manifold along some surface, surface, Witten Witten developed developed aa beautiful beautiful “Hamiltonian” "Hamiltonian" approach approach to to topological topological quantum quantum field field theory, to reduce reduce the the problem problem of of defining defining and and calculating calculating the the theory, enabling enabling him him to quantity quantity Z(M3) Z(M 3 ) ttoo certain certain problems problems of of 2-dimensional 2-dimensional conformal conformaI field field theory theory on on this this surface. surface. A A perturbation perturbation series series for for Z(M3) Z(M 3 ) in in the the variable variable i +---> co, 00, constructed Singer (and special case case by by Kontzevich) Kontzevich) constructed by by Axelrod Axelrod and and Singer (and in in aa special yielded the properties properties and and possible possible yielded some some interesting interesting topological topological quantities; quantities; the applications hitherto not investigated. applications of of these these has, has, however, however, hitherto not been been investigated. The of giving an exact, topologThe problem problem of giving an exact, rigorous, rigorous, nonperturbative, nonperturbative, purely purely topological definition of this invariant was solved by Reshetikhin and Turaev. The ical definition of this invariant was solved by Reshetikhin and Turaev. The theory of this type of invariant has been further developed by several peotheory of this type of invariant has been further developed by several people, including Viro, Lickorish. We We shall shall now describe some some now describe ple, including Viro, Lawrence, Lawrence, and and Lickorish. of these invariants, as elaborated in the work of Lickorish of the late 1980s. of these invariants, as elaborated in the work of Lickorish of the late 1980s. (Our is largely largely taken Kauffman’s book book ‘(Knots and (Our account account of of this this work work is taken from from Kauffman's "Knots and Physics” .) Physics" .) One can can obtain obtain any any 3-manifold 3-manifold Mi performing surgery surgery along along aa suit suitable One Ml ofby by performing able link or knot L in S3. This realization a 3-manifold was used by Lickorish, link or knot L in S3. This realization of a 3-manifold was used by Lickorish, Zieschang and the present author in the early 1960s in the construction of a Zieschang and the present author in the early 1960s in the construction of a non-singular 2-foliation on an arbitrary closed 3-manifold, starting from the non-singular 2-foliation on an arbitrary closed 3-manifold, starting from the Reeb fibration of S3 and performing a “tubulization” operation along some Reeb fibration of S3 and performing a "tubulization" operation along sorne suitable transversal knot or link, i.e. along some appropriate closed braid suitable transversal knot or link, i.e. along some appropriate closed braid in S’ x O2 c S3. It seems likely that Dehn had already been aware of in SI x D2 C S3. It seems likely that Dehn had already been aware of the possibility of obtaining an arbitrary 3-manifold via surgery along a knot the possibility of obtaining an arbitrary 3-manifold via surgery along a knot or link in S3; the reduction to closed braids is certainly due to Alexander or link in S3; the reduction to closed braids is certainly due to Alexander (see above). At some time in the 1980s it was pointed out by Kirby that the (see above). At some time in the 1980s it was pointed out by Kirby that the latitude in this construction can be characterized via two standard elementary latitude in this construction can be characterized via two standard elementary operations figuring in the multi-dimensional surgery theory used by Smale and operations figuring in the multi-dimensional surgery theory used by Smale and Wallace in the proof of the n-dimensional Poincare conjecture for n > 4. The Wallace in the proof of the n-dimensional Poincaré conjecture for n > 4. The two operations in question are as follows (where we now assume that a link two in question as follows (where we now assume that a link comesoperations with a “frame” - see are below): cornes with a "frame" see below): 1. The first Kirby operation consists in the addition of a further unknotted 1. The first Kirby operation consists in the addition of a further unknotted component to the link, separated from the original link by a “wall” Iw2 C component to the link, separated from the originallink by a "wall" JR;2 C &X3, and with frame of linking number *l (see below). JR;3, and with frame of lin king number ±1 (see below).

*

54. New New topological topological invariants invariants for 3-manifolds. 3-manifolds. §4.

293 293

2. The second second Kirby Kirby operation operation is as follows: follows: Choose Choose two two components components Li, Li, Lj, Lj, 2. The Lf the parallel shift of one one ofthem of them (Li)' (Li). Join Li Li and and Lj Lj bya by a and denote by Li Jordan arc equipped with with aa normal frame compatible compatible with with normal frames frames Jordan and Lj Lj not meeting the other other components. components. Form the "connected “connected on Li Li and sum” Lj zi of of Li and Lj Lj along along this this framed framed arc, arc, analogous analogous to to the the "connected“connectedLi and sum" sum” construction of the the theory theory of of framed framed surgeries surgeries (as (as for for instance instance in in the the sum" construction of calculation of ns(S2) aeeording according to to Pontryagin's Pontryagin’s scheme). Finally, replace replace calculation of 1I"3(S2) scheme). Finally, the pair pair Li, Lj by pair Lf,zj (the remaining remaining components components being being the Li, Lj by the the new new pair LI, Lj (the left as as before). left before).

Fig. A.13. 2r-twist Fig. A.13. A A 21T-twist (Instead the second second operation, often prefer prefer to use aa (Instead of of the operation, 3-manifold 3-manifold theorists theorists often to use combination of the two operations. Thus for instance if the link has the form combination of the two operations. Thus for instance if the link has the form of an unknotted framed circle with linking number 4~1 (see below) bounding of an unknotted framed circle with linking number ±1 (see below) bounding a disc, together with a number of other components all intersecting this disc a dise, together with a number of other eomponents aIl intersecting this dise transversely, then one removes this unknotted component and gives a “21rtransversely, then one removes this unknotted component and gives a "211"twist” to the others (see Figure A.l3).) A s mentioned above, in the present twist" to the others (see Figure A.13).) As mentioned above, in the present context the given knot or link is assumedto come equipped with a frame (i.e. a context the given knot or link is assumed to come equipped with a frame (Le. a normal field of unit vectors in IR3). A full topological invariant of such a frame normal field of unit vectors in JR.3). A full topologieal invariant of such a frame is afforded by the linking number of the original curve (component) S1 with is afforded by the linking number of the original curve (component) SI with its shift a small distance in the direction of the normal field. The canonical its shift a smaU distance in the direction of the normal field. The canonical normal frame is the one for which this linking number is zero: { S1, S:} = 0. normal frame is the one for whieh this linking number is zero: {SI, S~ } Q. A surgery along a closed Jordan curve Si in a 3-manifold M3, with trivial A surgery along a closed Jordan curve SI in a 3-manifold M3, with trivial normal bundle T(S’) = S1 x D22 c M3, is determined completely by the normal bundle T(Sl) = SI X D C M3, is determined completely by the linking number of a framing of the curve: the shift S: is to be null-homotopic linking number of a framing of the curve: the shift S~ is to be null-homotopie after the surgery, in the new manifold A42 (and if the surgery is along a link, the surgery, in the new manifold (and if the surgery is along a link, after then the same applies to each component of the link). then the same applies to each component of the link). For a framed link L one has a linking matrix, a symmetric n x n matrix, For na framed link L one has a linking The matrix, a symmetrie n x n matrix, where is the number of components. index i(L) of the link is then where n is the number of eomponents. The index i( L) the link is then defined to be the number of strictly negative entries in theofdiagonalization of defined to be the number of strictly negative entries in the diagonalization of the linking matrix. theAlinking matrix. blackboard frame is the normal vector field to a planar diagram of a link frame of is a the normal vector field to aAplanar link A blackboard L, obtained by means generic plane projection. shift ofdiagram the knotofora link L, obtained by me ans of a generie plane projection. A shift of the knot or link L a small distance in the direction normal to the plane of projection yields a small knot distance in the normal to of shall projection yields aL parallel or link. If Ddirection is any diagram of the the plane link, we understand a parallel knot or link. If D is any diagram of the link, we shaH understand

Ml

294 294 Appendix. Appendix. Recent Recent Developments Developments in in the the Topology Topology of of 3-manifolds 3-manifolds and and Knots Knots that 1,2,. ... , nn in that the the same same diagram diagram DD with with the the nn components components (numbered (numbered 1,2, in some il, ... , i,, in, represents represents aa new new link link with with i,iq sorne order) order) labelled labelled with with integers integers ir, components components parallel parallel toto the the qth qth component component (labelled (labelled iq), i q ), and and close close to to one one another. Thus this new link has altogether ir + . + i, components. another. Thus this new link has altogether il + ... + in components. For write I,Ir = (0, {O, 1, 1, ... ,,rT - 2}, 2}, and and denote denote For each each fixed fixed positive positive integer integer Tr write by c an arbitrary function from the set of components of the link (numbered by c an arbitrary function from the set of components of the link (numbered 1,2,. 1,2, ... ,, n) n) to to I,. Ir. Given Given any any numbers numbers X0, À o, Xl,. À 1 , ... ,, Xr--2, Àr-2' we we define define aa number number ((15)) (where L denotes the link in question) by ((L)) (where L denotes the link in question) by

(W)) ((L)) = CL(l) I::>-C(l)

xx

A-‘; 6); ... xx X,++!~D,,(A, Àc(n)SDL(A,A-\b);

c

here (as in in Kauffhere the the summation summation isis over over all all such such functions functions c, c, 6b = -A2 _A 2 - Ae2 A-2 (as Kauffman’s construction of the Jones polynomial-see earlier), and DL denotes man's construction of the Jones polynomial-see earlier), and DL denotes the the diagram diagram of of the the link link L. L. The The formula formula TR(@) TR(MfJ == riy(L)) b-i(L)((L))

Ml

then the 3-manifold (the TR TR or or TuraevTuruevthen gives gives aa topological topological invariant invariant of of the 3-manifold Mi (the Reshetikhin 3, A A = exp {i7r/2r}, and the X, Reshetikhin invariant) invariant) provided provided rr :2: -: 3, exp {i7r /2r}, and the numbers numbers Àq satisfy the linear system satisfy the overdetermined overdetermined linear system 6’j =

b =

i=r-2 i=r-2

C I: i=o

XiTi+j,

j 2 0,

ÀiTi+j, j ::::-: 0,

i=O

where Tj(j So, (A, A-l; 6) determined diwhere TJ(j EE Z) Z) denotes denotes the the quantity quantity SD;(A,A-l;O) determined by by the the diagram copies of of the elementary diagram Dj consisting consisting of of jj distinct distinct parallel parallel copies the elementary diagram agram Dj with just one crossing (see Figure A.14). Although of considerable intrinsic inwith just one crossing (see Figure A.14). Although of considerable intrinsic interest, this invariant has not so far found use in the solution of any topological terest, this invariant has not so far found use in the solution of any topological problem.

problem.

i

Fig. A.14. The elementary diagram and its parallel copies Fig. A.14. The elementary diagram and its paraUel copies The important notion of a “topological quantum field theory” (TQFT) has The important notion of a "topological quantum field theory" (TQFT) has emerged in recent years, after it began to be realized that TQFT represents emerged in recent years, after it began to be realized that TQFT represents more than the mere construction of certain topological invariants of knots more than the mere construction of certain topological invariants of knots and manifolds via functional integration or the more rigorous combinatorial combinatorial and manifolds via functional integration or the more rigorous approach. We shall now describe an axiom system for TQFT, devised by Segal We shall now describe an axiom system for TQFT, devised by Segal approach. and Atiyah. (This system may perhaps be regarded as analogous to the welland Atiyah. (This system may perhaps be regarded as analogous to the known Eilenberg-Steenrod system of axioms for homology, which provedwellto known Eilenberg-Steenrod system ofaxioms for homology, which proved to be particularly useful in connexion with generalizations of homology such as be particularly useful incobordism connexiontheory.) with generalizations of homology such as K-theory and complex

K-theory and complex cobordism theory.)

54. New New topological topological invariants for 3-manifolds. 3-manifolds. §4. invariants for

295 295

An n-dimensional n-dimensional TQFT is postulated postulated to be be a category category of of closed, closed, oriented oriented An TQFT to (n l)-dimensional manifolds V, each equipped with a Hilbert space Hvv (n 1 )-dimensional manifolds each equipped with Hilbert space H (usually finite-dimensional). The morphisms correspond to the oriented n(usually finite-dimensional). The morphisms correspond to the oriented nmanifolds W with boundary V partitioned into two parts (the “inside” and manifolds W with boundary V partitioned into two parts (the "inside" and “outside”): : "outside") dW = = V v = = VI v, U u V v2, 2,

aw

where Vz2 carry induced orientation. orientation. Any such cobordism W determines determines where VI, VI, V carry the the induced Any such cobordism W aa linear map linear map fw :: Hv, Hv, ....... --+ HV f+,, ' fw 2

(where the the bar the opposite which may may be be considered considered (where bar indicates indicates the opposite orientation) orientation) which as an element of Hvluv, : as an element of Hv,uv2 :

Hv, ®H-v* 2 , in view of the the axiom in view of axiom

Hv = H+ H"v = H~' v'

(here the indicates the the adjoint adjoint space). space). Clearly for aa cylinder cylinder with the (here the asterisk asterisk indicates Clearly for with the natural orientation, the map fw will be the identity map. Thus a morphism natural orientation, the map fw will be the identity map. Thus a morphism is determined element fw fw E E Hv W with with boundary V. is determined by by an an element Hv for for any any manifold manifold W boundary V. If the manifold V happens to be the boundary of two manifolds WI, W,, If the manifold V happens to be the boundary of two manifolds W 1 , W 2 , we shall have two associated elements

we shall have two associated elements fw,,fw,

E Hv.

We then have that the number (f wl, fw,) is a topological invariant of the We then have that the number (fWl' fW2) is a topological invariant of the closed manifold WI U Wz. closed manifold W 1 U W 2 . In classical topology there were only two examples of such invariants, In classical topology there were only two examples of such invariants, namely the Euler characteristic of even-dimensional manifolds, and the signanamely the Euler characteristic of even-dimensional manifolds, and the signature of manifolds-with-boundary of dimension 41c; the “Additivity Lemma”, ture of manifolds-with-boundary of dimension 4k; the "Additivity Lemma", proved by Rohlin and the present author in the mid-1960s shows that the proved by Rohlin and the present author in the mid-1960s, shows that the latter example satisfies the axiom system. In these examples the spaces Hv latter example satisfies the axiom system. In these examples the spaces Hv are all one-dimensional, and the morphisms amount to multiplication by are ail one-dimensional, and the morphisms amount to multiplication by exp(aT(W)), where r is the signature (or the Euler characteristic in the simexp(aT(W)), where T is the signature (or the Euler characteristic in the simpler case). However we now have many new examples where the dimensions of pler case). However we now have many new examples where the dimensions of the spaces Hv exceed 1; these represent highly nontrivial TQFTs, and yield the spaces Hv exceed 1; these represent highly nontrivial TQFTs, and yield the above invariants. theIn above invariants. the case n = 2 any cobordism can be achieved by iterating the elIn the case n = 2i.e.anythecobordism canwithbe boundary achieved by iteratingofthe elementary “trousers”, 2-manifold consisting three ementary "trousers", Le. the 2-manifold with boundary consisting of three circles VI, Vz, V3, on which there is a Morse function definable with exactly circles VI, Vpoint on which is a Morse function with 2 , 17 3 , and one critical constant there on each of the two parts definable VI U Vz and 7s exactly of the one critical point and constant on each of the two parts V U V and V 3 of the 2 boundary. As a morphism in the appropriate TQFT this l surface determines morphism in the surface determines aboundary. tensor C;kAsona the space Hv of theappropriate circle V = TQFT S1; in this fact its components are a tensor Cjk on the space Hv of the circle V SI; in fact its components just the structural constants of a certain “Frobenius algebra”, i.e. a commuta-are just the of a certain algebra" a commutative and structural associative constants algebra with identity "Frobenius element, and with ,aLe. scalar product

tive and associative algebra with identity element, and with a scalar product

296 Appendix. Developments in 3-manifolds and Appendix. Recent Recent Developments in the the Topology Topology of of 3-manifolds and Knots Knots qij ( ,, )),, satisfying satisfying (fg, h) = (f, gh). The scalar product 1]ij = ( (fg,h) (f,gh). The scalar product qij 17ij corresponds corresponds to to the circles carrying the cylinder cylinder S1 SI xx lI with with boundary boundary circles carrying the the same same orientation. orientation. This This construction construction leads leads to to the the description description of of the the values values of of the the “correlation "correlation functions” expectations for the vacuum vacuum expectations for products products of of fields fields in in terms terms of of functions" (or (or the TQFT, as follows. TQFT, which which are are constant constant by by definition) definition) as follows. Let Let {ei} {ei} be be aa basis basis of of the the corresponding derive the corresponding Frobenius Frobenius algebra, algebra, then then one one can can derive the following following properties: properties: (a) (a) (b) (b)

the function the O-loop O-loop correlation correlation function the O-loop correlation function the O-loop correlation function cijk

of qij; of pairs pairs is is equal equal to to 1]ij; of triples is equal to of triples is equal to

=

%s cjk

;

(c) (c) any any O-loop O-loop correlation correlation function function may may be be expressed expressed in in the the form form (eiej

.

ek,

1);

(d) any k-loop may be expressed in (d) any k-Ioop correlation correlation function function may be expressed in the the form form (eiej , . . ek, H”), where is the appropriate element of the Frobenius algebra under Hk = qi’eiei 1]i j eiej is the appropriate element of the Frobenius algebra under where Hk consideration. consideration. In In the the most most interesting interesting cases cases one one has has families families of of such such theories theories (i.e. (i.e. families families of Frobenius algebras) depending on several parameters, and here “axiomthe "axiomof Frobenius algebras) depending on several parameters, and here the atization” of such families becomes very important. (A suitable axiom system atization" of such families becomes very important. (A suitable axiom system was formulated by Witten, Dijgraaf, and the Verlinde brothers.) It should be was formulated by Witten, Dijgraaf, and the Verlinde brothers.) It should be noted that the space of parameters satisfies certain “soliton-type dispersionnoted that the space of parameters satisfies certain "soliton-type dispersionless integrable hierarchies” or “hierarchies of hydrodamic type” (in the sense less Integrable hierarchies" or "hierarchies of hydrodamic type" (in the sense of Dubrovin and the present author, involving the Hamiltonian formalism of Dubrovin and the present author, involving the Hamiltonian formalism determined by a certain flat metric on the space of parameters). determined by a certain fiat metric on the space of parameters). Remark. There are some aesthetically very pleasing cases when the space Remark. There are some aesthetically very pleasing cases when the space Hv, V = Si, turns out to be the even part of the cohomology ring of certain H v, V = SI, turns out to be the even part of the cohomology ring of certain Calaby-Yau manifolds, and the associated Frobenius algebras correspond to Calaby-Yau manifolds, and the associated Frobenius algebras correspond to certain “quantum deformations” of the cohomology ring. A rigorous treatment certain "quantum deformations" of the cohomology ring. A rigorous treatment of this case, at least in part, has been given by Yau, Kontzevich, Manin, Tian, of this case, at leastGivental, in part, has been given by Yau, Kontzevich, Manin, Tian, Ruan, Piunikhin, Salamon, MacDuff and others, using techniques Ruan, Piunikhin, Givental, Salamon, using techniques 0 from symplectic topology and algebraic MacDuff geometry and andothers, topology. 0 from symplectic topology and algebraic geometry and topology. Following Witten, Dijgraaf, the Verlinde brothers, Dubrovin, Krichever and Following Dijgraaf, Verlinde others, we Witten, shall describe anthe axiom systembrothers, for the Dubrovin, particular Krichever case of theand 2others, we shaH axiomofsystem for the particular of the in2dimensional TQFTdescribe in the an absence a gravitational field. This case approach dimensional TQFT in the absence of a gravitational field. This approach involves a space with co-ordinates ti, and a function F(t) satisfying volves a space with co-ordinates ti, and a function F(t) satisfying qk”aiaj8kF(t)

= Ctj(t),

(4.1) ( 4.1)

$4. New New topological topological invariants invariants for for 3-manifolds. 3-manifolds. §4.

297 297

where the metric 7] q is is constant constant in in the the t-coordinates, t-coordinates, and C(t) C(t) is made made up up of of where the metric and the structural constants of of a family family of of time-dependent time-dependent Frobenius algebras (as(asthe structural constants Frobenius algebras sociated with with the the same same metric) metric) at at each each value value of oft.t. We We also also have have the the equations equations sociated Qj

=

(4.2)

alaiajF(t),

since the vector corresponds to the the identity identity in these these Frobenius Frobenius alsince the first first basic basic vector corresponds to in algebras. The (4.1), (4.2) (4.2) are are called called the the associativity equations. (Note (Note The equations equations (4.1), associativity equations. gebras. that the equations (4.1) are are the difficult part of this this system system to to study.) study.) that the equations (4.1) the most most difficult part of These provide geometric description of the the so so called called Frobenius Frobenius geometry of of These provide aa geometric description of the space with with the co-ordinates the space the co-ordinates tt’.i . The following very pleasing pleasing description description of the the Frobenius Frobenius geomThe following aesthetically aesthetically very of geometry Assuming the metric is such such that that the the tensor tensor 7]isCjk r/isC,Sk etry isis due due to to Dubrovin. Dubrovin. Assuming the metric 7]7 is is symmetric, consider the the operators operators & - ~6’~~ depending depending on the the parameter parameter is symmetric, consider Oi on k zz and defining aa connection connection on our our space. space. It It turns turns out out that that the the curvature curvature of and defining on of this zero provided provided the associativity equations (4.1), (4.2) (4.2) ho holdId this connection connection is is zero the associativity eqllations (4.1), associative and the the tensor tensor (signifying (signifying that that the the Frobenius Frobenius algebras algebras C(t) C(t) are are associative and field with the of some function F(t) Cijk coincides coincides with the third third derivative derivative of sorne function F(t) which which may may field C’ijk be “free energy" energy” for the corresponding corresponding TQFT). be interpreted interpreted as as the the "free for the TQFT). Following scheme of of Dijgraaf Dijgraaf and one may may use use fiat flat cocoFollowing the the scheme and Witten Witten (1991) (1991) one ormal series series in the variable 2) to define the “disperordinates ( as fformaI hj,m (t) (as in the variable z) to define the "disperordinates hi,,(t) sionless and so-called “O-loop gravitation continuation” (in this sionless hierarchy” hierarchy" and so-called "O-loop gravitation continuation" (in this TQFT), as follows: TQFT), as follows:

zCl

00

L

hi= hj = ghj,,(t), hj,m(t), m=O

j=1,2 j = 1,2,,..., ... ,

m 0, m 2 2: 0,

hj,o = tivij,

hj,o = ë7]ij,

m=O

Hj,m = j-h&t(X))

dX.

Here the Hj,m may be interpreted also as commuting Hamiltonians (of the Here the Hj,m may be interpreted also as commuting Hamiltonians (of the hydrodynamic type) for this hierarchy, with “times” Tj),, and X = T1lo, hydrodynamic type) for this hierarchy, with "times" Tj,m, and X = Tl,O, tj = Tj>‘. The Hamiltonian formalism here is determined by the flat metric q il Tj,o. The Hamiltonian formalism here is determined by the fiat metric 7] in the sense of Dubrovin and Novikov (early 1980s; corresponding examples in the sense of Dubrovin and Novikov (early 1980s; corresponding examples may be found in the book Modern Geometry, II, by Dubrovin, Fomenko, may be found in the book Modern Geometry, II, by Dubrovin, Fomenko, Novikov). In particular, the “free energy” O-loop was defined there as a special Novikov). In particular, the "frcc cncrgy" O-loop was dcfined there as a special solution F(X, T). The partial derivatives with respect to the variables Tjlm of solution F(X, T). The partial derivatives with respect to the variables Tj,m of the solution F(X, T) coincide with the “O-loop correlation functions” in this the solution F(X, T) coincide with the "O-loop correlation functions" in this TQFT. The following initial conditions are imposed on the solution F(X, T):

TQFT. The following initial conditions are imposed on the solution F(X, T): F(X,T) = F(t),

F(X, T)

=

F(t),

where Tj,m-l = 0 for m > 1, Tj,O = tjj and X = tl; tl., whcrc '[j,m-l 0 for m > 1, Tj,O t and X the condition for a “dispersionless string”:

the condition for a "dispcrsionless string":

1

+2

-11'

·T"·0·0 t J , .,

'ltJ

298 298 Appendix. Appendix. Recent Recent Developments Developments inin the the Topology Topology of of 3-manifolds 3-manifolds and and Knots Knots the the O-loop O-loop relation relation for for the the corresponding corresponding correlation correlation functions: functions: 3 8(t,m),(J,n),(k,p) F .

2 3 = (8(t,m-I),(l,D) F 17ls (8(s,D),(),n),(k,p) F). .

There There are are some sorne simple simple cases cases when when itit isis natural natural toto conjecture conjecture the the exisexistence of a “/c-dimensional loop gravitation extension” for all values tence of a "k-dimensional loop gravitation extension" for aIl values of of Ic. k. Here Here the the dimension dimension nn= I 1, , and and the the “free "free energy” energy" function function satisfies satisfies the the celecelebrated brated KdV KdV hierarchy hierarchy with with the the Schrodinger Schr6dinger potential potential uu having having initial initial value value U(X) = (ax)‘F = X, and vanishing for all other “times”. u( X) = (8 X )2 F = X, and vanishing for aIl other "times". ItIt turns turns out out that that the the Schrodinger Schr6dinger potential potential may may easily easily be be obtained obtained as as aa D formal power series with integral coefficients, in the variables Tm, To = X. X. formaI power series with integral coefficients, in the variables Tm, T The coefficients of this formal power series are equal to the Chern numbers The coefficients of this formaI power series are equal to the Chern numbers of the corresponding corresponding moduli moduli spaces spaces of certain certain holomorphic holomorphie vector vector bundles bundles over over the of algebraic curves (of arbitrary genus and with certain points of algebraic curves (of arbitrary gemls and with certain points labelled); labeIled); this this result due to to Kontzevich Kontzevich (1992), (1992), who who also also proved proved the the above above conjecture conjecture of of result isis due Witten. Witten. AA nice niee exposition exposition of of the the differential-geometrical differential-geometrical approach approach to to the the subject subject may be found in the recent lectures “Geometry of 2-dimensional may be found in the recent lectures "Geometry of 2-dimensional topological topological field field theories” theories" (SISSA (SISSA preprint, preprint, 1994) 1994) given given by by Dubrovin. Dubrovin. We conclude with a few more words on the associativity equations equations for for the the We conclude with a few more words on the associativity function F(t): we note that in the case ~11 = 0, 7712 = 1, 7722 = 0, the function function F(t): we note that in the case 1711 = 0, 1]12 = 1, 1722 = 0, the function F(t) (for n =c 2) has the form: F(t) (for n =2) has the form:

F(t) F(t) =

~t~t2 ++ f(t2), f(t2),

$32

where f is an arbitrary where f is an arbitrary function. function. There There are are very very interesting interesting 2-dimensional 2-dimensional examples of TQFT in this case which are significant for physics. examples of TQFT in this case which are significant for physics. Since this topic is quite recent, not all of the interesting observations that Since this topic is quite recent, not aIl of the interesting observations that have been made concerning it have been given rigorous proof. It remains have been made concerning it have been given rigorous pro of. It remains an intensely active area of research. The present author is not aware of the an intensely active area of research. The present author is not aware of the existence of a good survey article on the subject. existence of a good survey article on the subject. For this brief survey the book ‘(Knots and Physics” by Kauffman, and lecFor this brief survey the book "Knots and Physics" by Kauffman, and lectures of Bar-Natan on Vassiliev invariants were the main sources; in addition tures of Bar-Natan on Vassiliev invariants were the main sources; in addition a number of recent papers and the above-mentioned lectures by Dubrovin a number of recent papers and the above-mentioned lectures by Dubrovin were consulted. Most of the remaining material was gleaned by the author were consulted. Most of the remaining material was gleaned by the author from private conversations, lectures and seminars.

from private conversations, lectures and seminars.

Bibliography Bibliography

299 299

Bibliography Bibliography ’1 The The items items listed listed below below have have not not been been cited cited in in the the text. text. This This bibliography bibliography is is intended the text text -- likewise likewise making making no no pretensions pretensions as as to to comcomintended as as aa supplement supplement to to the pleteness study inin more more basic basic and and detailed detailed fashion fashion pleteness -- by by means means of of which which readers readers may may study the the areas areas of of topology topology and and its its applications applications surveyed surveyed in in the the text. text. The The books books and and ararticles the author author considers considers methodologically methodologically the the best best - in in the the pedagogical pedagogical ticles which which the sense that, as as itit seems seems to to him, him, they they expound expound the the important important parts parts of of topology topology and and sense that, adjacent adjacent disciplines disciplines in in aa way way that that is is especially especially intuitive, intuitive, clear clear and and convenient convenient for for the the reader reader -- are are indicated indicated by by an an asterisk asterisk (*). (.). We the items items of of the the bibliography bibliography under under appropriate appropriate headings headings as as indicated. indicated. We group group the (The (The date date of of an an item item isis that that of of its its first first appearence.) appearence.)

I.I. Popular Popular Books Books and and Articles Articles on on Geometry, Geometry, Topology Topology and and Their Their Applications Applications

1.1. I.l. Books Books of of the the 1930s 1.9805 Aleksandrov, P.S., Efremovich, V.A.: The Basic Topology: an Outline. Aleksandrov, P.S., Efremovich, V.A.: The Basic Concepts Concepts of of Topology: an Outline. Moscow Leningrad: ONTI 1936, 94 pp. Moscow Leningrad: ONTI 1936,94 pp. Hilbert, Berlin: Springer 310 Hilbert, D., D., Cohn-Vossen, Cohn-Vossen, S.: S.: Anschauliche Anschauliche Geometrie. Geometrie. Berlin: Springer 1932, 1932, 310 pp. Zbl. 5,112 pp. ZbI. 5,112 1.2. I.2. Modern Modern Popular Popuiar Geometric-Topological Geometrie-Topoiogieai Books Books Boltyanskij, V.G., Efremovich, V.A.: Descriptive Topology. Moscow: Moscow: Kvant, 21, Boltyanskij, V.G., Efremovich, V.A.: Descriptive Topology. Kvant, No. No. 21, 1982, 149 pp. [German transl.: Braunschweig 19861. Zbl. 606.57001 1982,149 pp. [German transI.: Braunschweig 1986]. Zbl. 606.57001 Chinn, Steenrod, N.: Concepts of New York: York: Random Random House House Chinn, W.G., W.G., Steenrod, N.: First First Concepts of Topology. Topology. New 1966, 115 pp. Zbl. 172,242 1966, 115 pp. Zbl. 172,242 1.3. Recent Popular Articles Written by or with Physicists J.8. Recent Popular Articles Written by or with Physieists *Volovik, G.B., Mineev, V.P.: Physics and Topology. Moscow: Znanie 1980, 63 pp. *Volovik, G.B., Mineev, V.P.: Physics and Topology. Moscow: Znanie 1980, 63 pp. Mineev, V.P.: Topological Objects in Nematic Liquid Crystals. Addendum to the Mineev, Topological in Nematic Liquid Crystals. to the book: V.P.: Boltyanskij, V.G., Objects Efremovich, V.A.: Topology. Moscow:Addendum Kvant, No. 21, book: Boltyanskij, Efremovich, V.A.: Topology. Moscow: Kvant, No. 21, 1982, 148-158, Zbl. V.G., 606.57001 1982, 148-158, Zbl. 606.57001 *Frank-Kamenetskij, M.D.: The Most Important Molecule. Moscow: Kvant 1983, *Frank-Kamenetskij, M.D.: The Most Important Molecule. Moscow: Kvant 1983, 159 pp. 159 pp. II.

Text-Books

on Combinatorial

and

Algebraic

Topology

II. Text-Books on Combinatorial and Aigebraic Topology 11.1. 1930s J1.1. 1.980s Lefschetz, S.: Algebraic topology. Am. Math. Sot. Coll. Publ. 27, 1942, 389 pp., Zbl. Lefschetz, 36,122 S.: Aigebraic topology. Am. Math. Soc. Coll. PubI. 27, 1942,389 pp., Zbl. 36,122 H., Threlfall, W.: Lehrbuch der Topologie. *Seifert, Leipzig Berlin: Teubner 1934, *Seifert, W.: Lehrbuch der Topologie. Leipzig Berlin: Teubner 1934, 353 S., H., Zbl.Threlfall, 9,86 353 S., ZbI. 9,86 11.2. 1940s and 1950s JJ.2. 19408 and 1950s Aleksandrov, P.S.: Combinatorial Topology. Moscow, Leningrad: OGIZ-Gostekhizdat Aleksandrov, P.S.:Zbl. Combinatorial Topology. Moscow, Leningrad: OGIZ-Gostekhizdat 1947, 660 pp., 37,97 1947, 660 pp., ZbI. 37,97 ‘For the convenience of the reader, references to reviews in Zentralblatt fur 1 For the convenience of the using reader,thereferences to reviews Zentralblatt Mathematik (Zbl.), compiled MATH database, and inJahrbuch iiber für die Mathematik (ZbI.), compiled using MATH and Jahrbuch über die Fortschritte der Mathematik (Jbuch.)the have, as database, fas as possible, been included in Fortschritte der Mathematik (Jbuch.) have, as fas as possible, been included in this bibliography. this bibliography.

300 300

Bibliography Bibliography

Pontryagin, of Combinatorial Combinatorial Topology. Topology. Moscow: Moscow: OGIZ-GostekhOGIZ-GostekhPontryagin, L.S.: L.S.: Foundations Foundations of izdat izdat 1947, 1947, 1st Ist edition edition 143 143 pp. pp. [English [English transl.: transI.: Rochester Rochester 19521. 1952]. Zbl. ZbI. 71,161 71,161 Pontryagin, Combinatorial Topology. Pontryagin, L.S.: L.S.: Foundations Foundations of ofCombinatorial Topology. Moscow: Moscow: Nauka Nauka 1976, 1976, 2nd 2nd edition, edition, 136 136 pp., pp., Zbl. ZbI. 463.55001 463.55001 Eilenberg, Eilenberg, S., S., Steenrod, Steenrod, N.: N.: Foundations Foundations of of Algebraic Aigebraic Topology. Topology. Princeton, Princeton, New New Jersey: Princeton Univ. Univ. Press Press 1952, 1952, 328 328 pp., pp., Zbl. ZbI. 47,414 47,414 Jersey: Princeton Godement, 1252, Godement, R.: R.: Topologie Topologie algebrique algébrique et et theorie théorie des des faisceaux. faisceaux. Actual. Actual. Sci. Sci. Ind. Ind. 1252, Publ. 13, Paris: 1958, 283 Pub!. Inst. Inst. Math. Math. Univ. Univ. Strasbourg, Strasbourg, No. No. 13, Paris: Hermann Hermann 1958, 283 pp., pp., Zbl. Zbl. 80,162 80,162 *Steenrod, The Topology Topology of of Fibre Fibre Bundles. Bundles. Princeton, Princeton, New New Jersey: Jersey: Princeton Princeton *Steenrod, N.: N.: The Univ. Univ. Press Press 1951, 1951, 224 224 pp., pp., Zbl. ZbI. 54,71 54,71 11.3. II. 3. 1960s 19608 and and mid-1970s mid-19708 Dold, A.: Lectures on Algebraic Topology. Dold, A.: Lectures on Aigebraic Topology. Berlin Berlin Heidelberg Heidelberg New New York: York: Springer Springer 1972, 377 pp., Zbl. 234.55001 1972, 377 pp., Zbl. 234.55001 Hilton, Hilton, P.J., P.J., Wylie, Wylie, S.: S.: Homology Homology Theory. Theory. New New York: York: Cambridge Cambridge Univ. Univ. Press Press 1960, 1960, 484 pp., Zbl. 91,363 484 pp., Zbl. 91,363 Spanier, 1966, 528 Spanier, E.H.: E.H.: Algebraic Algebraic Topology. Topology. New New York: York: McGraw McGraw Hill Hill 1966, 528 pp., pp., Zbl. Zbl. 145,433 145,433 11.4. 11.4. Late Late 19’70s 19708 and and 1980s 19808 Borisovich, Yu.G., Bliznyakov, N.M., Izrajlevich, Fomenko, T.N.: T.N.: An An IntroIntroBorisovich, Yu.G., Bliznyakov, N.M., Izrajlevich, Ya.A., Ya.A., Fomenko, duction to Topology. Moscow: Vysshaya Shkola 1980, 296 pp., Zbl. 478.57001 duction to Topology. Moscow: Vysshaya Shkola 1980, 296 pp., Zbl. 478.57001 *Dubrovin, Modern Geometry. The Methods Methods of of *Dubrovin, B.A., B.A., Novikov, Novikov, S.P., S.P., Fomenko, Fomenko, A.T.: A.T.: Modern Geometry. The Homology Theory. Moscow: Nauka 1984, 344 pp., Zbl. 582.55001. [English transl.: Homology Theory. Moscow: Nauka 1984, 344 pp., ZbI. 582.55001. [English transI.: Grad. Heidelberg: Springer 1990, 416 Grad. Texts Texts Math. Math. 124. 124. New New York York Berlin Berlin Heidelberg: Springer 1990, 416 pp.] pp.] Postnikov, M.M.: Lectures in Algebraic Topology. Foundations of Homotopy Theory. Postnikov, M.M.: Lectures in Algebraic Topology. Foundations of Homotopy Theory. Moscow: Zbl. 571.55001 571.55001 Nauka 1984, 1984, 416 416 pp., pp., Zbl. Moscow: Nauka Postnikov, M.M.: Lectures in Algebraic Topology. Homotopy Theory of Cell Spaces. Postnikov, M.M.: Lectures in Aigebraic Topology. Homotopy Theory of CeU Spaces. Moscow: Zbl. 578.55001 578.55001 Moscow: Nauka Nauka 1985, 1985, 336 336 pp., pp., Zbl. Rokhlin, V.A., F&s, D.B.: Beginner’s Course in Topology. Geometric Chapters. Rokhlin, V.A., Fuks, D.B.: Beginner's Course in Topology. Geometrie Chapters. Moscow: Nauka 1977, 488 pp., Zbl. 417.55002 [English transl.: Berlin Heidelberg Moscow: Nauka 1977, 488 pp., Zb!. 417.55002 [English transI.: Berlin Heidelberg New York: Springer 19841 New York: Springer 1984] Massey, W.S.: Homology and Cohomology Theory. New York Basel: Marcel Dekker, Massey, W.S.: Homology and Cohomology Theory. New York Basel: Marcel Dekker, Inc. 1978, 412 pp., Zbl. 377.55004 Inc. 1978, pp., Zbl.Topology: 377.55004An Introduction. Massey, W.S.: 412 Algebraic New York Chicago San FranMassey, W.S.: Algebraic Topology: An Introduction. Newpp., York cisco Atlanta: Harcourt, Brace& World, Inc. 1967, 261 Zbl.Chicago 153,249San Francisco Atlanta: Harcourt, Brace & World, Inc. 1967, 261 pp., ZbI. 153,249 Switzer, R.M.: Algebraic Topology-Homotopy and Homology. Berlin Heidelberg New Switzer, Topology-Homotopy York: R.M.: SpringerAigebraic 1975, 526 pp., Zbl. 305.55001 and Homology. Berlin Heidelberg New York: Springer 1975, 526 pp., Zbl. 305.55001 III. Books on Elementary and PL-Topology of Manifolds, III. Books on Elementary and PL-Topology of Manifolds, Complex Manifold Theory, Fibration Geometry, Lie Groups

Complex Manifold Theory, Fibration Geometry, Lie Groups III. 1. Elementary Differential Topology, PL-Manifold Topology II!. 1. Elementary Differentiai Topology, PL-Manifold Topology *Dubrovin, B.A., Novikov, S.P., Fomenko, A.T.: Modern Geometry. Moscow: Nauka *Dubrovin, Novikov, S.P., Fomenko, 1980, 760 B.A., pp., Zbl. 433.53001. [English A.T.: transl.: Modern Grad. Geometry. Texts Math.Moscow: 93 and Nauka 1104. 1980,York 760 pp., 433.53001.Springer [English1984 transI.: New BerlinZbl. Heidelberg: and Grad. 19851 Texts Math. 93 and 1104. New YorkB.A., Berlin Heidelberg: 1985] Geometry. The Methods of *Dubrovin, Novikov, S.P., Springer Fomenko, 1984 A.T.: and Modern *Dubrovin, Novikov, S.P.,Nauka Fomenko, Modern Geometry. The Methods Homology B.A., Theory. Moscow: 1984, A.T.: 344 pp., Zbl. 582.55001. [English transl.:of Homology pp., ZbI. 582.55001. [English transI.: Grad. TextsTheory. Math. Moscow: 124. New Nauka York 1984, Berlin 344 Heidelberg: Springer 1990, 416 pp.] Grad. TextsL.S.: Math. 124. New York and Berlin 1990,416 pp.] Tr. *Pontryagin, Smooth manifolds theirHeidelberg: applications Springer in homotopy theory. *Pontryagin L.S.: Smooth manifolds their applications in homotopy theory. Tr. Mat. Inst. , Steklova 45 (1955) 1-136. and [English transl.: Transl., II. Ser., Ann. Math. Mat. Jnst. Steklova 1-136. [English trans!.: Trans!., II. Ser., Ann. Math. Sot. 11 (1959) l-1441.45 (1955) Zbl. 64,174 Soc. Il (1959) 1-144]. Zb!. 64,174

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*Pontryagin, L.S.: Smooth Smooth Manifolds Manifolds and and Their Their Applications Applications to Homotopy Homotopy Theory. Theory. to *Pontryagin, L.S.: 3rd edition. edition. Moscow: Moscow: Nauka Nauka 1985, 1985, 176 176 pp., pp., Zbl. Zbl. 566.57002 566.57002 3rd Postnikov, M.M.: M.M.: An An Introduction Introduction to Morse Morse Theory. Theory. Moscow: Moscow: Nauka Nauka 1971, 1971, 567 567 pp., pp., to Postnikov, Zbl. 215,249 215,249 Zbl. Hirsch, M.W.: M.W.: Differentiai Differential Topology. Topology. Grad. Grad. Texts Texts Math. Math. 33. 33. Berlin Berlin Heidelberg Heidelberg New New Hirsch, York: Springer Springer 1976, 1976, 221 221 pp., pp., Zbl. Zbl. 356.57001 356.57001 York: Milnor, J.: Topology Topology from from the the Differentiable Differentiable Viewpoint. Charl.: Univ. Univ. Press Press of of VirVirViewpoint. CharI.: Milnor, ginia 1965, 1965, 64 64 pp., pp., Zbl. Zbl. 136,204 136,204 ginia Munkres, J.R.: J.R.: Elementary Elementary differential topology. topology. Ann. Ann. Math. Math. Stud. Stud. 54 54 (1963) (1963) 107 107 Munkres, differential pp., Zbl. Zbl. 107,172 107,172 pp., Rourke, C.P., C.P., Sanderson, Sanderson, B.J.: B.J.: Introduction Introduction to Piecewise Piecewise Linear Linear Topology. Topology. Berlin Berlin to Rourke, Heidelberg New New York: York: Springer, Springer, Ergeb. Ergeb. Math. Math. 69, 69, 1972, 1972, 123 123 pp., pp., Zbl. Zbl. 254.57010 254.57010 Heidelberg *Seifert, H., H., Threlfall, Threlfall, W.: Variationsrechnung Variationsrechnung im Grossen Grossen (Theorie (Theorie von von Marston Marston *Seifert, W.: im Morse). Leipzig Berlin: Teubner Teubner 1938, 1938, 115 115 pp., pp., Zbl. Zbl. 21,141 21,141 Morse). Leipzig Berlin: Wallace, A.H.: A.H.: Differentiai Differential Topology. Topology. First First Steps. Steps. New New York York Amsterdam: Amsterdam: W.A. BenBenW.A. Wallace, jamin, Inc. Inc. 1968, 1968, 130 130 pp., pp., Zbl. Zbl. 164,238 164,238 jamin, 111.2. Forms, Sheaves, Complex and and Algebraic Algebraic Manifolds IlJ.2. Forms, Sheaves, Complex Manifolds *Dubrovin, B.A., Novikov, S.P., S.P., Fomenko, Fomenko, A.T.: A.T.: Modern Modern Geometry. Geometry. The The Methods Methods of of *Dubrovin, B.A., Novikov, Homology Theory. Moscow: Moscow: Nauka 1984, 344 Zbl. 582.55001. 582.55001. [English [English transI.: transl.: Homology Theory. Nauka 1984, 344 pp., pp., Zbl. Grad 124. New New York York Berlin Berlin Heidelberg: Heidelberg: Springer Springer 1990, 1990, 416 416 pp.] pp.] Grad Texts Texts in in Math. Math. 124. *Chern, manifolds. The Chicago, Autumn Autumn 1955-Winter 1955-Winter 1956, The Univ. Univ. Chicago, 1956, *Chern, S.S.: S.S.: Complex Complex manifolds. 181 pp., 74,303 181 pp., Zbl. Zbl. 74,303 Godement, R.: Topologie algebrique theorie des faisceaux. Actual. Actual. Sei. Sci. Ind. Ind. 1252. 1252. et théorie des faisceaux. Godement, R.: Topologie algébrique et Publ. Math. Univ. Univ. Strasbourg Strasbourg No. 13, 13, Paris: Paris: Hermann Hermann 1958, 1958, 283 283 pp., pp., Zbl. Zbl. Publ. Inst. lnst. Math. No. 80,162 80,162 Griffiths, J.: Principles Principles of Algebraic Geometry. New New York: York: John John WiWiGriffiths, P., P., Harris, Harris, J.: of Aigebraic Geometry. ley 1978, 813 408.14001 Sons 1978, 813 pp., pp., Zbl. ZbI. 408.14001 ley && Sons *Hirzebruch, F.: Neue Neue Topologische Methoden in Algebraischen Geometrie. *Hirzebruch, F.: Topologische Methoden in der der Aigebraischen Geometrie. Berlin Heidelberg New Springer, Ergeb. Math. g. 1956, 165 165 S., Zbl. 70,163 70,163 g. 1956, S., Zb!. Berlin Heidelberg New York: York: Springer, Ergeb. Math. *Hirzebruch, F.: Topological Methods in Algebraic Geometry. Grundl. Math. Wiss. *Hirzebruch, F.: Topological Methods in Aigebraic Geometry. GrundI. Math. Wiss. 131. New York Berlin Heidelberg: Springer 1966, 232 pp., Zbl. 138,420 131. New York Berlin Heidelberg: Springer 1966, 232 pp., Zbl. 138,420 *Springer, G.: Introduction to Riemann Surfaces. Reading, Massachusetts: Addison*Springer, G.: Introduction to Riemann Surfaces. Reading, Massachusetts: AddisonWesley Publ. Company, Inc. 1957, 230 pp., Zbl. 78,66 Wesley Pub!. Company, Ine. 1957, 230 pp., Zb!. 78,66 111.3. Foundations of Differential Geometry and Topology: JIl.3. Foundations of Bundles, Differential Geometry and Topology: Fibre Lie Groups Fibre Bundles, Lie Groups *Dubrovin, B., Novikov, S.P., Fomenko, A.T.: Modern Geometry. Moscow: Nauka * Dubrovin, B., Novikov, S.P., 1980, 760 pp., Zbl. 433.53001.Fomenko, [English A.T.: transl.:Modern Grad. Geometry. Texts Math.Moscow: 93 andNauka 104. Math. 93 and 104. 1980, 760 pp., Zbl. 433.53001. [English transI.: Grad. Texts New York Berlin Heidelberg: Springer 1984 and 19851 1985] New York Berlin Heidelberg: Groups. Springer Moscow 1984 andLeningrad: *Pontryagin, L.S.: Continuous ONTI 1983, 315 pp., *Pontryagin, L.S.: Continuous Groups. Moscow Leningrad: ONTI 1983, 315 pp., Zbl. 659.22001 Zbl. 659.22001 Pontryagin, L.S.: Topological Groups. 4th edition: Moscow: Nauka 1984, 520 pp., 4th edition: Pontryagin, Topological Groups.transl.: Zbl. 79,39 L.S.: and Zbl. 85,17. [English London Moscow: 19391 Nauka 1984, 520 pp., Zb!. 85,17. [English transI.: Zbl. 79,39 Bishop, R.L., and Crittenden, R.J.: Geometry of London Manifolds. 1939] New York London: Acad. Bishop, R.L., 273 Crittenden, Geometry of Manifolds. New York London: Acad. Press 1964, pp., Zbl. R.J.: 132,160 Press 1964, 273 pp., Zbl. W., 132,160 Gromoll, D., Klingenberg, Meyer, W.: Riemannsche Geometrie im Grossen. Gromoll, D., Klingenberg, W.,287 Meyer, W.: 155,307 Riemannsche Geometrie im Grossen. Lect. Notes Math. 55, 1968, pp., Zbl. Math. 55, 1968, 287 pp., 155,307Spaces. New York: Acad. Press Lect. Notes Helgason, S.: Differential Geometry and Zbl. Symmetric Helgason, Differentiai Geometry and Symmetric Spaces. New York: Acad. Press 1962, 486S.:pp., Zbl. 111,181 1962,486 D.: pp.,Fibre Zbl. 111,181 Husemoller, Bundles. New York: McGraw-Hill Book Company 1966, 300 Husemoller, D.: Fibre Bundles. New York: McGraw-Hill Book Company 1966, 300 pp., Zbl. 144,448 pp., Zbl.K.:144,448 *Nomizu, Lie Groups and Differential Geometry. Tokyo: The Mathematical So*Nomizu, K.: Lie 1956, Groups ciety of Japan 80 and pp., Differentiai Zbl. 71,154 Geometry. Tokyo: The Mathematical Society of Japan 1956, 80 pp., Zbl. 71,154

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Schwartz, J.T.: Differential Differentiai Geometry Geometry and and Topology. Topology. Notes Notes Math. Math. and and Appl. App!. New New Schwartz, J.T.: York: Gordon and and Breach Breach 1968, 1968, 170 170 pp., pp., Zbl. Zb!. 187,450 187,450 York: Gordon *Serre, *Serre, J.P.: J.P.: Lie Lie Algebras Algebras and and Lie Lie Groups. Groups. Lectures Lectures given given at at Harvard Harvard Univ. Univ. New New York, Amsterdam: W.A. W.A. Benjamin, Benjamin, Inc. Inc. 1965, 1965, 252 252 pp., pp., Zbl. Zbl. 132,278 132,278 York, Amsterdam: Spivak, Spivak, M.: M.: Calculus Calculus on on Manifolds. Manifolds. New New York York Amsterdam: Amsterdam: W.A. W.A. Benjamin, Benjamin, Inc. Inc. 1965, 1965, 144 144 pp., pp., Zbl. Zbl. 141,54 141,54 *Steenrod, *Steenrod, N.: N.: The The Topoloy Topoloy of of Fibre Fibre Bundles. Bundles. Princeton, Princeton, N.J.: N.J.: Princeton Princeton Univ. Univ. Press Press 1951, 224 pp., 1951,224 pp., Zbl. Zbl. 54,71 54,71 IV. IV. Surveys Surveys and and Text-Books Text-Books on on Particular Particular Aspects Aspects ofof Topology Topology and and Its Its Applications Applications IV.l. IV.1. Variational Var'iational Calculus Calculus “in "in the the Large” Large" Al’ber, S.J.: On the periodic problem of variational calculus AI'ber, S.J.: On the periodic problem of variational calculus in in the the large, large, Usp. Usp. Mat Mat Nauk Nauk 12, 12, No. No. 44 (1957) (1957) 57-124, 57-124, Zbl. Zbl. 80,87 80,87 Dubrovin, Dubrovin, B.A., B.A., Novikov, Novikov, S.P., S.P., Fomenko, Fomenko, A.T.: A.T.: Modern Modern Geometry. Geometry. The The Methods Methods of of Homology Homology Theory. Theory. Moscow: Moscow: Nauka Nauka 1984, 1984, 344 344 pp., pp., Zbl. Zbl. 582.55001. 582.55001. [English [English transl.: transI.: Grad. Grad. Texts Texts Math. Math. 124. 124. New New York York Berlin Berlin Heidelberg: Heidelberg: Springer Springer 1990, 1990, 416 416 pp.] pp.] Lyusternik, L.A., Shnirel’man, L.G.: Topological methods in variational Lyusternik, L.A., Shnirel'man, L.G.: Topological methods in variational problems problems and and their their applications applications inin the the differential differential geometry geometry of of surfaces. surfaces. Usp. Usp. Mat. Mat. Nauk Nauk 2, 2, No. 1 (1947) 166-217 No. 1 (1947) 166217 Bott, Bott, R.: R.: On On manifolds manifolds all an of of whose whose geodesics geodesics are are closed. closed. Ann. Ann. Math. Math. 60 60 (1954) (1954) 375-382, Zbl. 58,156 375-382, Zbl. 58,156 Klingenberg, Klingenberg, W.: W.: Lectures Lectures on on Closed Closed Geodesics. Geodesics. Grundlehren Grundlehren Math. Math. Wiss. Wiss. 230. 230. Berlin Heidelberg New York: Springer 1978, 227 pp., Zbl. 397.58018 Berlin Heidelberg New York: Springer 1978, 227 pp., Zbl. 397.58018 *Milnor, *Milnor, J.: J.: Morse Morse theory. theory. Ann. Ann. Math. Math. Stud. Stud. 51 51 (1963) (1963) 153 153 pp., pp., Zbl. ZbL 108,104 108,104 IV.2.

Knot

Theory

IV.2. Knot Theory Crowell, R.H., Fox, R.H.: Introduction to Knot Theory. Boston: Ginn and Company Crowell, RR, Fox, RH.: Introduction to Knot Theory. Boston: Ginn and Company 1963, 182 pp., Zbl. 126,391 1963, 182 pp., Zbl. 126,391 Bredon, G.: Introduction to Compact Tansformation Groups. New York: Acad. Press Bredon, G.: Introduction to Compact Tansformation Groups. New York: Acad. Press 1972, 459 pp., Zbl. 246.57017 1972, 459 pp., ZbI. 246.57017 Conner, P.E., Floyd, E.E.: Differentiable Periodic Maps. Ergeb. Math. 33. Berlin Conner, P.E., Floyd, E.E.: Differentiable Periodic Maps. Ergeb. Math. 33. Berlin Heidelberg New York: Springer 1964, 148 pp., Zbl. 125,401 Heidelberg New York: Springer 1964, 148 pp., Zbl. 125,401 IV.4.

Homotopy

Theory.

Cohomology

Operations.

Spectral

Sequences

IV.4. Homotopy Theory. Cohomology Operations. Spectral Sequences Boltyanskij, V.G.: Principal concepts of homology and obstruction theory. Usp. Mat Boltyanskij, V.G.: Principal concepts of homology Usp. 113Mat Nauk 21, No. 5 (1966) 117-139. [English transl.: and Russ.obstruction Math. Surv.theory. 21 (1966) Nauk Zbl. 21, No. 5 (1966) 117-139. [English transI.: Russ. Math. Sury. 21 (1966) 1131341, 152,403 134], ZbI. B.A., 152,403 *Dubrovin, Novikov, S.P., Fomenko, A.T.: Modern Geometry. The Methods of *Dubrovin, Novikov, S.P.,Nauka Fomenko, Modern Geometry. The Methods Homology B.A., Theory. Moscow: 1984, A.T.: 344 pp., Zbl. 582.55001. [English transl.:of Homology Theory. Moscow: Nauka 1984, 344 pp., Zbl. 582.55001. [English transi.: Grad. Texts Math. 124. New York Berlin Heidelberg: Springer 1990, 416 pp.] Grad. Texts 124. New York Berlin Springer pp.] Postnikov, M.M.:Math. Spectral sequences. Usp. Heidelberg: Mat. Nau 21, No. 51990, (1966)416141-148. sequences. Nau 13331401, 21, No. 5Zbl.(1966) 141--148. Postnikov, M.M.: Spectral [English transl.: Russ. Math. Surv. 21,Usp. No. Mat. 5 (1966) 171,440 [English Math. of Sury. 21, No. Usp. 5 (1966) Fuks, D.B.: transI.: Spectral Russ. sequences fibrations. Mat. 133-140], Nauk. 21,ZbI. No.171,440 5, 149-180, Fuks, Spectral sequences of fibrations. Nauk. 21, No. Zbl. 5, 149-180, 1966.D.B.: [English transl.: Russ. Math. Surv. 21,Usp. No. Mat. 5 (1966) 141-1711, 169,547 [English transI.: Russ. Math. Sury. 21, No. 5 (1966) 141-171], Zbl. 1966. Fuks, D.B.: Eilenberg-MacLane complexes. Usp. Math. Nauk 21, No. 5 (1966)169,547 213Fuks, Eilenberg-MacLane complexes. Usp. Nauk205-2071, 21, No. 5Zbl. (1966) 213215. D.B.: [English transl.: Russ. Math. Surv. 21, No.Math. 5 (1966) 168,210 215. D.B., [English transI.: Russ. Sury. 21, V.L.: No. 5Homotopic (1966) 205·-207], Zbl. Moscow: 168,210 *Fuks, Fomenko, A.T., Math. Gutenmakher, Topology. *Fuks, D.B., Fomenko, Gutenmakher, V.L.: Homotopie Topology. Moscow: University Press (MGU)A.T.,1969, 460 pp. [English transl: Budapest (1986)], Zbl. University Press (MGU) 1969, 460 pp. [English transi: Budapest (1986)], Zbl. 189,540 189,540 Adams, J.F.: Stable Homotopy Theory. Lect. Notes Math. 3, 1969, 74 pp., Zbl. Adams, 126,390J.F.: Stable Homotopy Theory. Lect. Notes Math. 3, 1969, 74 pp., Zbl. 126,390J.F.: Stable Homotopy and Generalized Adams, Homology. Chicago Lectures in Adams, Stable Homotopy and of Generalized Homology. Math., J.F.: Chicago London: The Univ. Chicago Press 1974,373 Chicago pp., Zbl.Lectures 309.55016in Math., Chieago London: The Univ. of Chicago Press 1974, 373 pp., Zbl. 309.55016

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*Hu, Homotopy Theory. New Acad. Press pp., ZbI. Zbl. 88,388 88,388 *Hu, ST.: S.T.: Homotopy Theory. New York: York: Acad. Press 1959, 1959, 347 347 pp., Mosher, M.C.: Cohomology and Applications Mosher, R.E., R.E., Tangora, Tangora, M.C.: Cohomology Operations Operations and Applications inin HomoHomotopy Theory. London: Harper&Row 1968, 214 214 topy Theory. New New York York Evanston Evanston London: Harper & Row Publishers Publishers 1968, pp., pp., Zbl. Zbl. 153,533 153,533 *Serre, singuliere des espaces espaces fib&s. Applications. Ann. Math. 54 *Serre, J.P.: J.P.: Homologie Homologie singulière des fibrés. Applications. Ann. Math. 54 (1951) 4255505, (1951) 425-505, Zbl. ZbI. 45,260 45,260 Serre, et classes Math. 58 Serre, J.P.: J.P.: Groupes Groupes d’homotopie d'homotopie et classes de de groupes groupes abeliens. abéliens. Ann. Ann. Math. 58 (1953) 258-294, 52,193 (1953) 258~294, Zbl. Zbl. 52,193 Steenrod, Epstein, D.B.A.: Cohomology operations. Ann. Math. Math. Stud. 50 (1962) (1962) Steenrod, N., N., Epstein, D.B.A.: Cohomology operations. Ann. Stud. 50 139 pp., 139 pp., Zbl. Zbl. 102,381 102,381 IV.5. Theory IV.5. Theory

of Cobordisms of Characteristic Chamcteristic Classes Classes and and Cobordisms

Anosov, fibrations and the K-functor. Usp. Mat. Nauk 21, Anosov, D.V., D.V., Golo, Golo, V.: V.: Certain Certain fibrations and the K-functor. Usp. Mat. Nauk 21, No. [English transl.: Russ. Math. Math. Surv. 21, No. 5, 5, (1967) (1967) 1733 No. 55 (1966) (1966) 181-212. 181~212. [English transI.: Russ. Sury. 21, No. 173 2031, 203], Zbl. Zbl. 168,211 168,211 Bukhshtaber, Novikov, S.P.: Formai Formal groups role Bukhshtaber, V.M., V.M., Mishchenko, Mishchenko, A.S., A.S., Novikov, S.P.: groups and and their their role in the algebraic topology apparatus. Usp. Mat. Nauk 26, No. 2 (1971) 131-154. in the algebraic topology apparatus. Usp. Mat. Nauk 26, No. 2 (1971) 131~154. [English transl.: Russ. Math. Math. Surveys 26, No. (1972) 63-901, Zbl. 224.57006 [English transI.: Russ. Surveys 26, No. 2,2, (1972) 63~90], Zb!. 224.57006 *Dubrovin, B.A., Novikov, S.P., Fomenko, A.T.: Modern Geometry. Methods of *Dubrovin, B.A., Novikov, S.P., Fomenko, A.T.: Modern Geometry. Methods of Homology Nauka 1984, 344 pp., Zbl. 582.55001. [English transl.: Homology Theory. Theory. Moscow: Moscow: Nauka 1984, 344 pp., Zbl. 582.55001. [English transI.: Grad. Math. 124. New York Berlin Heidelberg: Springer 1990, 416 pp.] Grad. Texts Texts Math. 124. New York Berlin Heidelberg: Springer 1990, 416 pp.] Novikov, The Cartan-Serre theorem and interior homology. Usp. Mat. Nauk Novikov, S.P.: S.P.: The Cartan-Serre theorem and interior homology. Usp. Mat. Nauk 21, No. 5 (1966) 217-232. [English tansl.: Russ. Math. Surv. 21, No. 5 (1966) 21, No. 5 (1966) 217~232. [English tans!.: Russ. Math. Sury. 21, No. 5 (1966) 209-2241, Zbl. 171,443 209~224], Zbl. 171,443 Rokhlin, V.A.: Theory of interior homology. Usp. Mat. Nauk 14, No. 4 (1959) 3-20, Rokhlin, V.A.: Theory of interior homology. Usp. Mat. Nauk 14, No. 4 (1959) 3~20, Zbl. 92,156 Zbl. 92,156 Borel, A.: Sur la cohomologie des espaces fib& principaux et des espaces homogenes Borel, A.: Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts. Ann. Math., II. Ser. 57 (1953) 1155207, Zbl. 52,400 de groupes de Lie compacts. Ann. Math., II. Ser. 57 (1953) 115~207, Zbl. 52,400 Borel, A.: La cohomologie mod 2 de certains espaces homogknes. Comment. Math. Borel, A.: La cohomologie mod 2 de certains espaces homogènes. Comment. Math. Helv. 27 (1953) 165-197, Zbl. 52,403 Helv. 27 (1953) 165~197, Zbl. 52,403 Borel, A., Serre, J.P.: Groups de Lie et puissances reduites de Steenrod. Am J. Math. Borel, A., Serre, J.P.: Groups de Lie et puissances réduites de Steenrod. Am J. Math. 75 (1953) 409-448, Zbl. 50,396 75 (1953) 409~448, Zbl. 50,396 Conner, P.E., Floyd, E.E.: Differentiable Periodic Maps. Ergeb. Math. 33. Berlin Conner, P.E., Floyd, E.E.: Differentiable Periodic Maps. Ergeb. Math. 33. Berlin Heidelberg New York: Springer 1964, 148 pp., Zbl. 125,401 148 pp., Zbl. 125,401 Heidelberg F.: NewNeue York:Topologische Springer 1964, *Hirzebruch, Methoden in der Algebraischen Geometrie. *Hirzebruch, F.: 9. Neue Topologische Methoden Springer in der Aigebraischen Geometrie. Ergeb. Math. Berlin Heidelberg New York: 1956, 165 S., Zbl. 70,163 Math. 9. Berlin Heidelberg New York: Springer 1956, 165 S., Zbl. Ergeb. Hirzebruch, F.: Topological Methods in Algebraic Geometry. Berlin Heidelberg 70,163 New Hirzebruch, F.: Topological Methods in Algebraic Geometry. Berlin Heidelberg New York: Springer 1966, 232 pp., Zbl. 138,420 York: Springer 1966, 232 pp., Zbl. 138,420 Milnor, J.W.: Survey of cobordism theory. Enseign. Math. 8, II. Ser., N. 1-2 (1962) Milnor, 16-23, J.W.: Zbl. Survey 121,399 of cobordism theory. Enseign. Math. 8, II. Ser., N. 1~2 (1962) 121,399 16-23, Zbl. *Milnor, J.W.: Lectures on Characteristic Classs. I. (Notes by James Stasheff). *Milnor, Princeton, ,LW.: N.J.:Lectures Princeton on Characteristic Univ. Press 1957Classs. 1. (Notes by James Stasheff). Princeton, Princeton Univ. Press 1957 *Milnor, J.W.: N.J.: Lectures on Characteristic Classes. II. (Notes by James Stasheff). *Milnor, J.W.: Lectures on Characteristic Classes. II. (Notes by James Stasheff). Princeton, N.J.: Princeton Univ. Press 1957 Princeton, Univ. Press 1957 Milnor, J.W., N.J.: Stasheff,Princeton J.D.: Characteristic classes. Ann. Math. Stud. 76 (1974) 330 Milnor, J.W., Stasheff, J.D.: Characteristic classes. Ann. Math. Stud. 76 (1974) 330 pp., Zbl. 298.57008 pp., Zbl. Snaith, V.P.: 298.57008 Algebraic cobordism and K-theory. Mem. Am. Math. Sot. 221 (1979) Snaith, V.P.: 152 pp., Zbl. Algebraic 413.55004 cobordism and K-theory. Mem. Am. Math. Soc. 221 (1979) 413.55004 152 pp., Strong, R.E.: Zbl.Notes on Cobordism Theory. Princeton, N.J.: Princeton Univ. Press Strong, R.E.: Notes on Cobordism N.J.: Princeton Univ. Press and Univ. Tokyo Press 1968, 387 Theory. pp., Zbl. Princeton, 181,266 and Univ. Tokyo Press 387 pp.,desZbl. 181,266 Thorn, R.: Quelques proprietes1968,globales varietes differentiables. Comment. Math. Thom, propriétes globales des variétés différentiables. Comment. Math. Helv. R.: 28 Quelques (1954) 17-86, Zbl. 57,155

Helv. 28 (1954) 17-86, Zb!. 57,155

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IV.6. 6. KK-Theory and the the Index Index of of Elliptic Elliptic Operators IV Theory and Operators Anosov, D.V., Golo, V.L.: Certain fibrations and the K-functor. Usp. Mat. Mat. Nauk Nauk Anosov, D.V., Golo, V.L.: Certain fibrations and the K-functor. Usp. 21, No. (1966) 181-212. 181-212. [English [English transI.: transl.: Russ. Russ. Math. Math. Sury. Surv. 21 21 No. No. 5, 5, (1967) (1967) 21, No. 55 (1966) 173-2031, ZbI. Zbl. 168,211 168,211 173-203], Dynin, A.S.: A.S.: Index Index of of an an elliptic elliptic operator operator on on aa compact compact manifold. manifold. Usp. Usp. Mat. Mat. Nauk Nauk Dynin, 21, No. No. 55 (1966) (1966) 233-248. 233-248. [English [English transI.: transl.: Russ. Russ. Math. Math. Sury. Surv. 21 21 No. No. 5, 5, (1960) (1960) 21, 225-2401, Zbl. Zbl. 149,411 149,411 225-240], Mishchenko, A.S.: Vector Vector Bundles Bundles and and Their Their Applications. Applications. Moscow: Nauka Nauka 1984, 1984, Mishchenko, A.S.: Moscow: 208 pp., Zbl. 569.55001 569.55001 208 pp., ZbI. *Atiyah, M.F.: M.F.: K-Theory. Mass.: Harvard Harvard University University Press Press 1965, 1965, 160 160 *Atiyah, K-Theory. Cambridge, Cambridge, Mass.: PP. pp. Atiyah, M.F.: M.F.: Power Power operation in K-t.heory. K-theory. Q.J. Q.J. Math. Math. 17 17 (1966) (1966) 165-193, 165-193, ZbI. Zbl. operation in Atiyah, 144,449 144,449 Atiyah, M.F., M.F., Singer, Singer, I.M.: The index index of of elliptic elliptic operators. I. Ann. Ann. Math., Math., II. II. Ser. Ser. Atiyah, LM.: The operators. I. 87 (1968) 484-530, Zbl. 164,240 87 (1968) 484-530, ZbI. 164,240 Atiyah, M.F., M.F., Segal, Segal, G.B.: The index index of of elliptic elliptic operators. operators. II. II. Ann. Ann. Math., Math., II. II. Ser. Ser. Atiyah, G.R.: The 87 (1968) 531-545, Zbl. 164,242 87 (1968) 531-545, ZbI. 164,242 Atiyah, M.F., M.F., Singer, Singer, LM.: I.M.: The The index index of of elliptic elliptic operators. operators. III. III. Ann. Ann. Math., Math., II. II. Ser. Ser. Atiyah, 87 (1968) 546-604, Zbl. 164,243 87 (1968) 546-604, ZbI. 164,243 Atiyah, M.F., Singer, operators. IV. IV. Ann. Math., II. II. Ser. Ser. Atiyah, M.F., Singer, I.M.: LM.: The The index index of of elliptic elliptic operators. Ann. Math., 93 (1971) 119-138, Zbl. 212,286 98 (1971) 119-138, ZbI. 212,286 Atiyah, M.F., Singer, I.M.: The index of elliptic elliptic operators. V. Ann. Ann. Math., Math., II. II. Ser. Ser. Atiyah, M.F., Singer, LM.: The index of operators. V. 93 (1971) 139-149, Zbl. 212,286 98 (1971) 139-149, Zbl. 212,286 Hirzebruch, F.: Elliptische Differentialoperatoren auf Mannigfaltigkeiten. Veroff. ArHirzebruch, F.: Elliptische Differentialoperatoren auf Mannigfaltigkeiten. Verëff. Arbeitsgemeinsch. Forsch. Land. Nordrhein-Westfalen, Natur und Ing. Gesellschaftsbeitsgemeinsch. Forsch. Land. Nordrhein-Westfalen, Natur und Ing. Gesellschaftswiss. 157 (1965) 33-60 wiss. 151(1965) 33-60 Karoubi, M.: K-theory. Grundlehren Grundlehren Math. Wiss. 226. New Berlin Heidelberg: Heidelberg: Karoubi, M.: K-theory. Math. Wiss. 226. New York York Berlin Springer 1978, 308 S., Zbl. 382.55002 Springer 1978, 308 S., ZbI. 382.55002 Palais, R.S.: Seminar on the Atiyah-Singer Index Theorem. Princeton, N.J.: PrincePalais, R.S.: Seminar on the Atiyah-Singer Index Theorem. Princeton, N.J.: Princeton Univ. Press 1965, 366 pp., Zbl. 137,170 t.on Univ. Press 1965, 366 pp., ZbI. 137,170 IV. 7. Algebraic K-Theory. Multiply Connected Manifolds IV 7. Algebraic K- Theory. Multiply Connected Manifolds Bass, H.: Algebraic K-Theory. New York Amsterdam: W.A. Benjamin, Inc. 1968, Bass, K-Theory. New York Amsterdam: W.A. Benjamin, Inc. 1968, 762 H.: pp., Aigebraic Zbl. 174,303 ZbI. 174,303 to algebraic K-theory. Ann. Math. Stud. 72 (1971) 184 pp., 762 pp., Milnor, J.: Introduction Milnor, J.: Introduction to algebraic K-theory. Ann. Math. Stud. 72 (1971) 184 pp., Zbl. 237.18005 ZbI. 237.18005 Milnor, J.: Whitehead torsion. Bull. Am. Math. Sot., New Ser. 72 (1966) 358-426, Milnor, J.: Whitehead torsion. Bull. Am. Math. Soc., New Ser. 72 (1966) 358-426, Zbl. 147,231 ZbI. 147,231 IV.8.

Categories

and

Functors.

Homological

Algebra.

IV.B. Categories and Functors. Homological Algebra. General Questions of Homotopy Theory General Questions of Homotopy Theory Bucur, I., Deleanu, A.: Introduction to the Theory of Categories and Functors. Bucur, Deleanu, Introduction to the Theory Categories LondonL, New York A.: Sydney: John Wiley & Sons LTD of 1968, 224 pp., and Zbl. Functors. 197,292 London York Sydney: John Wiley & Sons LTD 1968, pp., ZbI. 197,292 Cartan H., New Eilenberg, S.: Homological algebra. Princeton Math.224Series 19 (1956) 390 Cartan H., Eilenberg, S.: Homological algebra. Princeton Math. Series 19 (1956) 390 pp., Zbl. 75,243 pp., Zbl.S.:75,243 MacLane, Homology. Berlin Heidelberg New York: Springer 1963, 522 pp., Zbl. MacLane, 133,265 S.: Homology. Berlin Heidelberg New York: Springer 1963, 522 pp., Zbl. 133,265 IV.9. Four-Dimensional Manifolds. Manifolds of Few Dimensions IV9. Four-Dimensional Manifolds. Manifolds of Few Dimensions Volodin, I.A., Kuznetsov, V.E., Fomenko, A.T.: On the algorithmic recognition of Volodin, I.A., Kuznetsov, V.E., Fomenko, A.T.:Mat. On Nauk the algorithmic recognition the standard three-dimensional sphere. Usp. 29, No. 5 (1974) 71-168,of the Usp. Mat. 29, 5No. 5 (1974) 71-168, Zbl. standard 303.57002. three-dimensional [English transl.: sphere. Russ. Math. Surv. Nauk 29, No. (1974) 71-1721 ZbI. 303.57002. [English transI.: Russ. Math. Sury. 29, No. 5 (1974) 71-172]

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Volodin, I.A., Fomenko, A.T.: Manifolds. Semin, Voktorn, Voktorn, Volodin, I.A., Fomenko, A.T.: Manifolds. Knots. Knots. Algorithms. Aigorithms. Tr. Tr. Sernin, Tenzorn. Anal Prilozh. Prilozh. Geom. Geom. Mekh. Mekh. Fiz 18 (1978) [EnTenzorn. Anal Fiz 18 (1978) 94-128, 94-128, Zbl. Zbl. 443.57003. 443.57003. [English Math. Sov. Sov. 3 (1984) glish transl.: transI.: Sel. Sel. Math. (1984) 311-3411 311-341] Mandelbaum, R.: Math. Mandelbaum, R.: Four-dimensional Four-dimensional topology: topology: An An introduction. introduction. Bull. Bull. Am. Am. Math. Sot., 2, No. (1980) 1-159, 1-159, Zbl. Soc., New New Ser. Ser. 2, No. 11 (1980) Zbl. 476.57005 476.57005 Stallings, J.: J.: Group three-dimensional manifolds. New Haven London: Stallings, Group theory theory and and three-dimensional manifolds. New Haven London: Yale 1971, 65 65 pp., 241.5701 pp., Zbl. Zbl. 241.5701 Yale Univ. Univ. Press Press 1971, IV.10. Classification Problems of Manifolds Manifolds of the the Higher-Dimensional Higher-Dimensional Topology Topology of IV.lO. Classification Problems of *Dubrovin, Novikov, S.P., S.P., Fomenko, Fomenko, A.T.: A.T.: Modern Geometry. The Methods of of The Methods *Dubrovin, B.A., B.A., Novikov, Modern Geometry. Homology Nauka 1984, 1984, 344 582.55001. [English [English transl.: Homology Theory. Theory. Moscow: Moscow: Nauka 344 pp., pp., Zbl. Zbl. 582.55001. transI.: Grad 124. New York Berlin Berlin Heidelberg: Heidelberg: Springer Springer 1990, 1990, 614 614 pp.] pp.] Grad Texts Texts Math. Math. 124. New York Milnor, J.: On manifolds homeomorphic to the ‘i--sphere. Ann. Math., II. 64 Ser. 64 Milnor, J.: On manifolds homeomorphic to the 7-sphere. Ann. Math., II. Ser. (1956) 399-405, 72, 184 184 (1956) 399-405, Zbl. Zbl. 72, *Milnor, Math., vol. 2. New John New York: York: John *Milnor, J.: J.: Differential DifferentiaI Topology. Topology. Lect. Lect. Modern Modern Math., vol. 2. Wiley 1964, 165-183, 165-183, Zbl. Zbl. 123,162 Wiley and and Sons Sons 1964, 123,162 *Milnor, on the the h-Cobordism h-Cobordism Theorem. Theorem. Princeton, Princeton, N.J.: Univ. N.J.: Princeton Princeton Univ. *Milnor, J.: J.: Lectures Lectures on Press 1965, Zbl. 161,203 Press 1965, Zbl. 161,203 Smale, S.: S.: Generalized conjecture in greater than four. Ann. Ann. Smale, Generalized Poincare Poincaré conjecture in dimensions dimensions greater than four. Math. II. Ser. 74, No. 2 (1961) 391-406, Zbl. 99,392 Math. II. Ser. 74, No. 2 (1961) 391-406, Zbl. 99,392 Smale, the structure of manifolds. manifolds. Am. Am. J. 84, No. (1962) 387-399, On the structure of J. Math. Math. 84, No. 33 (1962) 387-399, Smale, S.: S.: On Zbl. 109,411 Zbl. 109,411 *Smale, S.: of sorne some recent differential topology. topology. Bull. Bull. Am. Am. *Smale, S.: AA survey survey of recent developments developments in in differential Math. Sot., New Ser. 69 (1963) 131-145, Zbl. 133,165 Math. Soc., New Ser. 69 (1963) 131-145, Zbl. 133,165 IV. 11. Singularities and Maps IV.1l. Singularities of of Smooth Smooth Functions Functions and Maps Arnol’d, V.I., Varchenko, A.N., Gusejn-Zade, E.M.: Singularities of Differentiable Differentiable Arnol'd, V.L, Varchenko, A.N., Gusejn-Zade, E.M.: Singularities of Maps. I. Moscow: Nauka 1982, 304 pp. [English transl: Boston: Birkhauser 19851, Maps. 1. Moscow: Nauka 1982, 304 pp. [English transi: Boston: Birkhiiuser 1985], Zbl. 513.58001 Zbl. 513.58001 Arnol’d, V.I., Varchenko, A.N., Gusejn-Zade, E.M.: Singularities of Differentiable Arnol'd, V.I., Varchenko, A.N., Gusejn-Zade, KM.: Singularities of Differentiable Maps. II. Moscow: Nauka 1984, 336 pp. [English transl.: Boston: Birkh”auser Maps. II. Moscow: Nauka 1984, 336 pp. [English transI.: Boston: Birkh"auser 19881, Zbl. 545.58001 1988], Zbl. 545.58001 *Milnor, J.: Singular points of complex hypersurfaces. Ann. Math. Stud. 66 (1968) *Milnor, Singular 122 pp.,J.:Zbl. 184,484points of complex hypersurfaces. Ann. Math. Stud. 66 (1968) 122 pp., Zbl. 184,484 IV.12. Foliations. Cohomology of Lie Algebras of Vector Fields IV . .12. Foliations. Cohomology of Lie Algebras of Vector Fields Fuks, D.B.: Cohomology of Infinite-Dimentional Lie Algebras. Moscow: Nauka 1984, Fuks, Cohomology Infinite-Dimentional Aigebras. Moscow: 1984, 272 D.B.: pp. [English transl.: of Contemp. Sov. Math., Lie New York (1986)], Zbl.Nauka 592.17011 272 pp. [English transI.: Contemp. Sov. Math., New York (1986)], Zbl. 592.17011 V. Selected

Research

Works

and

Monographs

on the

Same

Subjects

1-12

as in IV

V. Se!ected Research Works and Monographs on the Same Subjects 1-12 as in IV V.l. Variational Calculus “in the Large” V.l. Variational Calculus "in the Large" Al’ber, S.I.: The topology of functional manifolds and variational calculus “in the topology manifolds the Al'ber, large”. S.I.: Usp.TheMat. Nauk of 25,functional No. 4 (1970) 57-122, andZbl.variational 209,530. calculus [English "in transl. large".Math. Usp. Surv. Mat. Nauk Russ. 25, No.25,4 No. (1970)4 (1970) 51-1171 57-122, Zbl. 209,530. [English transI. 25, No. 4 L.G.: (1970)Topological 51-117] Russ. Math. Lyusternik, L.A.,Sury. Shnirel’man, Methods in Variational Problems. Lyusternik, Shnirel'man, L.G.: Moscow Topological Methods in Variational Proceedings L.A., of NII Mat. & Mech., University, Moscow: Gosizdat Problems. 1930, 68 Mech., Moscow Proceedings of NIl Mat. pp. [French transl.: Paris, & Hermann 19341, University, Zbl. 11,28 Moscow: Gosizdat 1930, 68 Hermann and 1934], 11,28 pp. [French Paris,formalism Novikov, S.P.: transI.: Hamiltonian theZbl. many-valued analogue of Morse thethe many-valued analogue MorseSurv. theNovikov, S.P.: ory. Usp. Mat.Hamiltonian Nauk 37, formalism No. 5 (1982)and3-49. [English transl.: Russ. ofMath. ory. Usp.5 (1982) Mat. Nauk 5 (1982) 3-49. [English transI.: Russ. Math. Sury. 37, No. l-561, 37,Zbl.No.571.58011 37, No.S.P.: 5 (1982) 1-56],generalized Zbl. 571.58011 Novikov, Analytic Hopf invariant. Many-valued functionals. Usp. Novikov, S.P.: 39, Analytic invariant. Many-valued Mat. Nauk No. 5 generalized (1984) 97-106.Hopf [English transl.: Russ. Math.functionals. Surv. 39, Usp. No. Nauk 39, No. Zbl. 5 (1984) 97-106. [English transI.: Russ. Math. Sury. 39, No. 5Mat. (1984) 113-1241, 619.58002 5 (1984) 113-124], Zbl. 619.58002

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Fomenko, A.T.: A.T.: Variational Methods in in Topology. Topology. Moscow: Moscow: Nauka Nauka 1982, 1982, 344 344 pp., pp., Fomenko, Variational Methods Zbl. 526.58012 526.58012 Zbl. Fomenko, A.T.: Topological Variational Problems. Moscow: Moscow: Moscow Moscow University University Fomenko, A.T.: Topological Variational Problems. Press 1984, 1984, 216 216 pp. pp. Press Fomenko, A.T.: A.T.: Multidimensional variational methods methods in in the the topology topology of extremals. extremals. Fomenko, Mult.idimensional variational Usp. Mat. Mat. Nauk Nauk 36, No. No. 6 (1981) (1981) 105-135. 105-135. [English [English transI.: transl.: Russ. Russ. Math. Math. Sury. Surv. 36, 36, Usp. No. 6 (1981) (1981) 127-165], 127-1651, Zbl. Zbl. 576.49027 576.49027 No. Simons, J.: Minimal Minimal varieties in Riemannian manifolds. Ann. Ann. Math., Math., II. II. Ser. Ser. 88 88 Simons, varieties in Riemannian manifolds. (1968) 62-105, 181,497 (1968) 62-105, Zbl. Zbl. 181,497 V.2.2. K Knot Theory V. not Theory Farber, M.Sh.: M.Sh.: Classification of simple simple knots. Usp. Mat. Mat. Nauk Nauk 38, No. No. 5 (1983) (1983) 5959Farber, Classification of knots. Usp. 106. [English [English transi.: transl.: Russ. Russ. Math. Math. Sury. Surv. 38, 38, No. No. 5,63-117 5, 63-117 (1983)], (1983)], Zbl. Zbl. 546.57006 546.57006 106. V.3. Theory Theory of of Finite Finite and and Compact Compact Transformation Transformation Groups V.3. Groups Montgomery, D.: Compact groups of transformations. In: Differential analysis. Montgomery, D.: Compact groups of transformations. In: Differentiai analysis. Oxford Univ. Press 1964, 43-56, Zbl. 147,423 147,423 Oxford Univ. Press 1964, 43-56, Zbl. V.4. Homotopy Theory. Cohomology Cohomology Operations. Operations. Spectral Spectral Sequences Sequences V.4. Homotopy Theory. Postnikov, M.M.: Localization of topological spaces. Usp. Mat. Nauk 32, 32, No. No. 66 Postnikov, M.M.: Localization of topological spaces. Usp. Mat. Nauk (1977) 117-181. 117-181. [English [English transl.: Math. Surv. 32, No. (1977) 121-184], 121-1841, (1977) transI.: Russ. Russ. Math. Sury. 32, No. 66 (1977) Zbl. Zbl. 386.55014 386.55014 Rokhlin, V.A.: groups. Usp. Mat. Nauk 1, No. No. 5-6 5-6 (1946) (1946) 175-223, 175-223, Zbi. Zbl. Rokhlin, V.A.: Homotopy Homotopy groups. Usp. Mat. Nauk 1, 61,410 61,410 Adams, J.F.: On the the non-existence of elements of Hopf invariant one. one. Ann. Ann. Math., Math., Adams, J.F.: On non-existence of elements of Hopf invariant II. Ser. Ser. 72, IL 72, No. No. 11 (1960) (1960) 20-104, 20-104, Zbl. Zbl. 96,174 96,174 Adams, loop spaces. spaces. Ann. Math Stud Stud. . 90 (1978) 214 214 pp., 398.55008 J.F.: Infinite Infinite loop Ann. Math .90 (1978) pp., Zbl. Zbl. 398.55008 Adams, J.F.: Boardman, J.M., Vogt, R.M.: Homotopy Homotopy invariant invariant algebraic structures on on topologitopologiBoardman, J.M., Vogt, KM.: algebraic structures cal spaces. Lect. 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Kervaire, A manifolds which does structure. Kervaire, M.A.: M.A.: A manifolds whieh does not not admit admit any any differentiable differentiable structure. Comment. (1960) 257-270, 257-270, Zbl. Zbl. 145,203 145,203 Comment. Math. Math. Helv. Helv. 34, 34, No. No. 4 (1960) Kervaire, M.A., Milnor, Groups of spheres. 1. I. Ann. II. Ser. Kervaire, M.A., Milnor, J.: Groups of homotopy homotopy spheres. Ann. Math., Math., II. Ser. 77, No. 115,405 No. 3 (1963) (1963) 5044537, 504-537, Zbl. Zbl. 115,405 Kirby, R., Siebenmann, on topological topological manifolds, smoothKirby, R., Siebenmann, L.: L.: Foundational Foundational essays on manifolds, smoothings triangulations. Ann. Math. Stud. Stud. 88 (1977) 355 355 pp., 361.57004 ings and and triangulations. Ann. Math. 88 (1977) pp., Zbl. Zbl. 361.57004 Madsen, R.J.: The The classifying spaces for for surgery surgery and cobordism of Madsen, I., L, Milgram, Milgram, R.J.: classifying spaces and cobordism of manifolds. 92 (1979) pp., Zbl. Zbl. 446.57002 446.57002 manifolds. Ann. Ann. Math. Math. Stud. Stud. 92 (1979) 279 279 pp., Sullivan, I. Localization, and Galois SymGeometrie Topology, Topology, Part Part 1. Localization, Periodicity Periodicity and Galois SymSullivan, D.: D.: Geometric metry. Mass.: Massachusetts Inst. Techn. 1970, Zbl. 366.57003 metry. Cambridge, Cambridge, Mass.: Massachusetts Inst. Techn. 1970, Zbl. 366.57003 Wall, C.T.C.: Surgery on on compact compact manifolds. Lond. Math. Sot. Monogr. Monogr. 1. London, London, Wall, C.T.C.: Surgery manifolds. Lond. Math. Soc. N.Y.: 1970, 280 280 pp., N.Y.: Acad. Acad. Press Press 1970, pp., Zbl. Zbl. 219.57024 219.57024 V.ll. Singularities Singularities of Smooth Smooth Functions and M Maps V.ll. of Functions and aps Arnol’d, V.I.: Singularities of smooth maps. Usp. Mat. Nauk 23, No. 11 (1968) (1968) 3-44. 3-44. Arnol'd, V.L: Singularities of smooth maps. Usp. Mat. Nauk 23, No. [English Russ. Math. Surv. 23, 23, No. No. 1 (1968) l-431, Zbl. 159,536 [English transl.: transI.: Russ. Math. Sury. (1968) 1-43], Zbl. 159,536 Arnol’d, of functions in the the neighborhood neighborhood of degenerate critical critical Arnol'd, V.I.: V.l.: Normal Normal forms forms of functions in of degenerate points. Nauk 29, No. 2 (1974) transl.: Russ. Russ. Math. points. Usp. Usp. Mat. Mat. Nauk 29, No. (1974) 11-49. 11-49. [English [English transI.: Math. Surv. 29, (1974) 10-50], 10-501, Zbl. Sury. 29, No. No. 22 (1974) Zbl. 298.57022 298.57022 Arnol’d, of smooth smooth functions functions and their normal normal forms. Arnol'd, V.I.: V.l.: Critical Critical points points of and their forms. Usp. Usp. Mat. Mat. Nauk 30, No.5 3-65. [English [English transI.: transl.: Russ. Math. Surv. No. 55 (1975) (1975) Nauk 30, No.5 (1975) (1975) 3-65. Russ. Math. Sury. 30, 30, No. l-751, 1-75], Zbl. Zbl. 338.58004 338.58004 Arnol’d, Critical points of functions functions on manifold with Simple Lie Lie Arnol'd, V.I.: V.l.: Critical points of on aa manifold with boundary. boundary. Simple Groups BkCkF4 and singularities of evolutes. evolutes. Usp. Nauk 33, 33, No. (1978) Groups B k CkF4 and singularities of Usp. Mat. Mat. Nauk No. 55 (1978) 91-105. Russ Math. Surv. 33 (1978) 99-116], 99-1161, Zbl. 91-105. [English [English transl.: transi.: Russ Math. Surv. 33 (1978) Zbl. 408.58009 408.58009 Arnol’d, V.I.: Indices of singular singular points points of l-forms on on aa manifold boundary; conArnol'd, V.l.: Indices of of 1-forms manifold with with boundary; convolution of group invariants induced induced by maps and singular projections projections of of smooth smooth volution of group invariants by maps and singular surfaces. No. 22 (1979) [English transI.: transl.: Russ. Russ. Math. surfaces. Usp. Usp. Mat. Mat. Nauk. Nauk. 34, 34, No. (1979) 3-38. 3-38. [English Math. Surv. 34 l-421, Zbl. Sury. 34 No. No. 22 (1979) (1979) 1-42], Zbl. 405.58019 405.58019 Mather, J.: Stratifications mappings. Preprint. Harvard 1971, 1971, 68 appeared Mather, J.: Stratifications and and mappings. Preprint. Harvard 68 pp. pp. appeared in: Usp. Mat. No. 55 (167) 85-118, Zbl. 253.58005 in: Usp. Mat. Nauk Nauk 29, 29, No. (167) (1972) (1972) 85-118, Zbl. 253.58005 Thorn, R.: Lect. Notes Notes Math. Math. 197,1971, 197, 1971, Thom, R.: The The bifurcation bifurcation subset subset of of aa space space of of maps. maps. Lect. 202-208, 202-208, Zbl. Zbl. 216,459 216,459 Whitney, H.: On ideals ideals of of differentiable functions. Am. Am. J. 70, No. No. 33 (1948) (1948) Whitney, H.: On differentiable functions. J. Math. Math. 70, 635-658, 635-658, Zbl. Zbl. 37,355 37,355 Whitney, H.: Internac. Whitney, H.: Singularities Singularities of of mappings mappings of of Euclidean Euclidean spaces. spaces. Sympos. Sympos. Internac. Topologia Mexico (1958) (1958) 285-301, 2855301, Zbl. 92,284 Topologia Algebraica Algebraica México Zbl. 92,284 V.12. Foliations. Cohomology ofof Lie Vector Fields V.12. Foliations. Cohomology Lie Algebras Algebms of of Vector Fields Bernshtejn, Rozenfel’d, B.I.: spaces of infinite-dimensional Lie Bernshtejn, I.N. LN. Rozenfel'd, B.l.: Homogeneous Homogeneous spaces of infinite-dimensional Lie algebras classes of of foliations. foliations. Usp. Usp. Mat. Mat. Nauk No. 44 (1973) (1973) algebras and and characteristic characteristic classes Nauk 28, 28, No. 103-138. [English Surv. 28, (1973) 107-142], 107-1421, Zbl. Zbl. 103-138. [English transl.: transI.: Russ. Russ. Math. Math. Sury. 28, No. No. 44 (1973) 285.57014 285.57014 Novikov, of Foliations. 14 (1965) Novikov, S.R.: S.R.: Topology Topology of Foliations. Tr. Tr. Mosk. Mosk. Mat. Mat. 0-va a-va 14 (1965) 2488278. 248-278. [English. Most. Math. Sot. 14 14 (1967) Zbl. 247.57006 247.57006 Trans. Mosc. Math. Soc. (1967) 268-3041, 268-304], Zbl. [English. transl.: transI.: Trans. Fuks, Characteristic classes of No. 22 (1973) (1973) Fuks, D.B.: D.B.: Characteristic classes of foliations. foliations. Usp. Usp. Mat. Mat. Nauk Nauk 28, 28, No. 3-17. [English Math. Surv. 28, No. (1973) 1-16], l-161, Zbl. Zbl. 272.57012 3-17. [English transl.: transI.: Russ. Russ. Math. Sury. 28, No. 22 (1973) 272.57012 Fuks, Cohomology of Infinite-Dimensional Algebras and and Characteristic Characteristic of Infinite-Dimensional Lie Lie Algebras Fuks, D.B.: D.B.: Cohomology Classes of Ser. Sovrem. Sovrem. Probl. Probl. Mat. 10, 1978, 1978, 179179Classes of Foliations. Foliations. Itogi Itogi Nauki Nauki Tekh., Tekh., Ser. Mat. la, 285. [English transl.: J. Sov. Math. 11, 922-980 (1979)], Zbl. 499.57001 285. [English transI.: J. Sov. Math. Il,922-980 (1979)], Zbl. 499.57001 Godbillon, C.: Cohomologies d’algebres de Lie de de champs champs de de vecteurs formels. Lect. Lect. Godbillon, C.: Cohomologies d'algèbres de Lie vecteurs formels. Notes Math. 383, 1974, 69-87, Zbl. 296.17010 Notes Math. 383,1974,69-87, Zbl. 296.17010

Index Index

311 311

Index Index Abel, 10 10 Abel, action functional, 196 action functional, 196 Adams, 95, 95, 97, 97, 110, 110, 118, 118, 119, 119, 191, 191, 194, 194, Adams, 210, 223 223 210, - conjecture, conjecture, 192 192 -- differentials, differentials, 235 235 - operations, operations, 116-121, 116-121, 137 137 - resolution, 110-l 11 - resolution, 110-111 - spectral spectral sequence, sequence, 97-103, 97-103, 110-112, 110-112, 129, 223 129, 223 ~~ for for spheres, spheres, 100-103, 100-103, 111-112 111-112 Adams-Novikov spectral sequence, sequence, 129, 129, Adams-Novikov spectral 1344136 134-136 Adyan, 246 246 Adyan, Alania, 279 Alania, 279 Alexander, 44, 53, 54 Alexander, 44, 53, 54 - duality, duality, 54, 54, 59 59 Alexander polynomial, see knot Alexander polynomial, see knot Alexandrov, 67 Alexandrov, 67 Alexandrov, P. S., 273 Alexandrov, P. S., 273 algebraic algebraic - K-theory, see K-theory K-theory, see K-theory - L-theory, 265 - L-theory, 265 ~ variety (projective), 165 variety (projective), 165 Anderson, 124, 225, 234 Anderson, 124, 225, 234 Annulus conjecture, 267-270 Annulus conjecture, 267-270 Arf ~Arf function, 209 fun ct ion ,209 209 - identity, -~ invariant, identity, 209 209, 234, 262 invariant, problem, 209, 234, 101, 262 234-235 ~ invariant 101,234-235 -Arnol’d, invariant problem, 273 Arnol'd, principle, 273 Arzela’s 188, 201 Arzelà's principle, 188, 201 associated fiber bundle, see fibration a.'3sociated fiber bundle, 296 see fibration associativity equations, associativity equations, 296 Atiyah, 14, 113, 118, 120-122, 124, 141, Atiyah, 14, 113, 194, 220, 241, 118, 264 120-122, 124, 141, 194, 220, 241, 264 - -Bott formulae, 123, 141 -Bott formulae, 123, 141 - -Hirzebruch spectral sequence, see - spectral -Hirzebruch spectral sequence, see sequence spectralindex sequence - -Singer theorem, 14, 122 - -Singer292 index theorem, 14, 122 Axelrod, Axelrod, 292 Barden, 258 Barden, 101, 258 235 Barrat, Barrat, base, see101,235 fibration base, see fibration Bass -Bass projector, 258 projector,259 258 -- theorem, - theorem, Beck, 265 259 Beck,265

Becker, 139 139 Becker, Belavin, 124, 124, 274 274 Belavin, -Polyakov equation, equation, 124 124 -- -Polyakov Bernoulli numbers, numbers, 210, 210, 222 222 Bernoulli Bernshtein, 273 273 Bernshtein, Betti numbers, numbers, 50 50 Betti Birman, 247, 247, 289 289 Birman, - -Wenzl -Wenzl algebra, algebra, 286 286 blackboard frame, frame, 293 293 blackboard Boardman, 273 273 Boardman, Bockstein Bockstein - homomorphism, homomorphism, 70, 75, 75, 90, 90, 93, 93, 156, 156, 70, 169 169 operator, 93 93 -- operator, Boltyanskii, 84 Boltyanskii, 84 bordism, 113, 139 139 bordism, 113, - classes classes of, 128 128 --- - intersection intersection of, - groups, 210, 216 216 groups, 210, ~ normal, normal normal, see see normal - relative, 125 - relative, 125 bordism theory, 79, 2088226 bordism theory, 79, 208-226 -SU, 129, 225 --- -SU, 129, 225 - complex, 125, 129-140, 2255226 - complex, 125, 129-140,225-226 - framed, 208-210 framed, 208-210 - oriented, 125, 140, 225 - oriented, 125, 140, 225 - spin, 125, 225 125, 225 - spin, symplectic, 125, 139, 225 125, 139, 225 - symplectic, unoriented, 125 unoriented, 125 129 - with singularities, with singularities, 129 Borel, 106 Borel, 106 ~ theorem, 1066107 106107 theorem, Bott, 151, 175, 189, 273 151, 175,114, 189,273 -Bott, periodicity, 191 -Botvinnik, periodicity, 114, 191 129 Botvinnik, 129 boundary -boundary of simplex, see simplex of simplex,47see simplex - operator, -bouquet, operator, 16 47 bouquet, 16 braid -braid closed, 284 closed, 281 284 -- group, - group, 281257 Brakhman, Brakhman, Brezin, 274 257 Brezin, 274273 Brieskorn, Brieskorn,168 273 Brouwer, Brouwer, 44, 168 192, 196, 235, 237-244, Browder, Browder, 259, 26544, 192, 196, 235, 237-244, 259, 265 - -Levine-Livesay theorem, 256 -Levine-Livesay 256 -- -Novikov theorem, theorem, 119 -Novikov theorem, 119

312 312

Index Index

Brown, 138, 138, 144, 144, 225, 225, 234, 234, 270 270 Brown, Brown-Mazur theorem, 267 267 Brown-Mazur theorem, Brown-Peterson theory, 138 138 Brown-Peterson theory, Buchstaber, 115, 121, 121, 136, 136, 138 138 Buchstaber, 115, bundle bundle ~ complex, complex, 148 148 - cotangent, cotangent, 148 148 - normal normal spherical, spherical, 237 237 ~ - principal, principal, 150 150 - real, 148 -" real, 148 - tangent, 148 ~ tangent, 148 - universal, 150 ~ universal, 150 Burau representation, representation, 286 Burau 286 Cairns, 166 166 C'3.irns, Calaby-Yau manifolds, 274 274 Calaby-Yau manifolds, Candelas, 274 274 Candelas, cap product, product, 54, 54, 127 127 cap Cartan, 40, 40, 53, 53, 93, 93, 98, 98, 107, 107, 155, 155, 178, 178, Cartan, 193, 223, 239, 254 193, 223, 239, 254 - -Killing -Killing theorem, 148 theorem, 148 - -Serre theorem, 109 ~ -Serre theorem, 109 - -Serre-Adams -Serre-Adams method, 223 223 rnethod, - type of Steenrod algebra, see Steenrod type of Steenrod algebra, see Steenrod Casson, 249, 249, 266, 266, 270, 271 Casson, 270, 271 category of triples, 29 category of triples, 29 Cauchy, 10 Cauchy, 10 Tech, 67, 273 èech, 235 67, 273 Cerf, Cerf, 235 chain -chain cellular, 60 cellular, 60 - complex, 47 --~ complex, pacyclic, 4773 p-acyclic, -~ boundaries, 7347 boundaries, -- cycles, 47 47 cycles, 47 -- distinguished basis of, 73 distinguished basis G, of, 48 73 -- over abelian group over abelian group C, 48 -- relative, 57 relative,4757 - integral, integral, 57 47 -~ relative, relative, 57 - singular, 65 singular, 44, 65 76 Chapman, Chapman, 44,classes, 76 characteristic 155 classes, -characteristic Chern, 129-133, 153155 Chern, 129~133, - of manifold, 158 153 of manifold, 44, 158154, 157 - Pontryagin, 154, 157of, 270 --~ Pontryagin, topological 44, invariance topological invariance ~~ ~Stiefel-Whitney, 116, 129of, 270 Stiefel-Whitney, 116, 217 129 characteristic numbers, characteristic Cheeger, 183 numbers, 217 Cheeger, Chen, 202 183 Chen, 202 Chern, 88 -Chern,88 -Dold character, 138 -Dold character, -~ -Simons integral, 138 291 -Simons integral, 291

~

character, 117,120, 117, 120, 138 138 character, characteristic classes, classes, see see characteristic characteristic classes characteristic classes Chernavskii, 269 269 Chernavskii, chiral field, field, see see field field chiral Chogoshvili, 273 273 Chogoshvili, circle action, action, 140141 140-141 circle closed set, 15 15 closed closure, 15 15 closure, cobordism, 113 113 cobordism, - groups, groups, 210 210 ~ cochain cochain - cellular, 60 cellular, 60 - complex, 48 cornplex, 48 -- ~ coboundaries, coboundaries, 48 ~ 48 -- cocycles, cocycles, 48 48 - groups groups -over abelian abelian group group C, G, 48 48 ~~ over Cohn, 255 255 Cohn, cohomology -cohornology cellular, 60 60 ~ cellular, -~ groups, groups, 41, 41, 48, 48, 49 49 ~- of of the the covering, covering, 67 67 -- relative, relative, 57 57 -with coefficients in sheaf, 49, 81 ~ ~ with coefficients in sheaf, 49, 81 - operation, 89-101 operation, 89~ 101 - product product structure, structure, 52 52 - ring, ring, 53 53 -~ spectral spectral or or èech, Tech, 68 68 - with respect to representation, representation, 72 with respect to 72 compact-open topology, see topology compact~open topology, see topology complex, 5, 6 complex, 5, 6 ~ CW-, 59, 61 ~ CW-, 59, 61 -- n-skeleton, 61 n-skeleton, 61 -- cells, 61 ~ ~ cells, 61 - cubic, 42 42 -~ cubic, simplicial, see simplicial complex simplicial, see simplicial complex cone, 46 cone,46 270 Connell, Connell, 129, 270 139, 140, 226 Conner, Conner, 129, 139, 140, 226 connexion, 151 -connexion, G-, 151 151 C-, 151 - differential-geometric, 23 differential-geometric, 23 -~ homotopy, 154 homotopy, 154 - horizontal subspace, 151 horizontal subspace, 151 contractible, see homotopically trivial contractible, see hornotopically trivial convergence, 17 convergence, 17 convex hull, 41 convex hull, 41 convolution, 28 convolution, Conway, 276 28 Conway, 276 cotangent bundle, see bundle cotangent functor, bundle, see see functor bundle covariant covariant functor, see functor covering -covering map, 21 ~ map, 21 ~

Index Index - regular, regular, 33 33 space, 21 21 -- space, -- universal, universal, 33 33 covering-homotopy property, see see covering-homotopy property, homotopy connexion connexion homotopy cross-section, see see fibration fibration cross-section, curvature curvature - form, form, 152 152 Gaussian, 7-8 7-8 -- Gaussian, - positive positive scalar, scalar, 255 255 de Rham, Rham, 74, 74, 76, 76, 180 180 de - theorem, theorem, 181,203 181, 203 declinations in liquid liquid crystals, crystals, 24 24 declinations in deformation, 18 deformation, 18 retract, 19 19 -- retract, degree of of map, 204 degree map, 204 Dehn, 245 245 Dehn, Deligne, 110, 110, 178 178 Deligne, Devinatz, 97 97 Devinatz, Dieck, 139 139 Dieck, differential 146 differential form, form, 146 146 -- exterior exterior product, product, 146 - harmonic, 183 harmonie, 183 -- primitive, 185 primitive, 185 - quantized, 196 quantized, 196 differential-geometric connexion, differential-geometric connexion, see connexion see connexion Dijgraaf, 296 Dijgraaf, 296 Dirac monopole, 194 Dirac monopole, direct product, 16194 direct product, 16 divisor, 186 divisor, 186 Dolbilin, 42 Dolbilin, Dold, 118,42249 Dold, 118, 249 Donaldson, 124, 273 Donaldson, Douglas, 255124, 273 Douglas, 255 Drinfeld, 287 287 Drinfeld, Dror Bar-Natan, 247, 289 Dror Bar-Natan, 247,297289 Dubrovin, 279, 296, Dubrovin, 279, 296, 297 Edwards, 44, 76, 272 -Edwards, theorem, 44, 45 76, 272 theorem, 45 Ehresmann, 88 Ehresmann,88,8898, 135, 254 Eilenberg, 98, 135, 254 899109 -Eilenberg, -MacLane 88,space, 39940, -MacLane space, 3940, 89-109 - -Steenrod axioms, 58 - -Steenrodautomorphisms, axioms, 58 elementary 245 elementary automorphisms, 245 Eliashberg, 274 Eliashberg, elliptic genera,274274 elliptic operator, genera, 274 elliptic 122 operator, 122 122 -elliptic difference element, 122 difference - symbol of, element, 122 symbolof, 122 Engel, 165 Engel, 7-8 165 Euler, Euler, 7--8

313 313

- class, class, 164 164 theorem, 66 -- theorem, - theorem theorem on on convex convex polyhedra, polyhedra, 7 Euler characteristic, characteristic, see Euler-Poincaré Euler-Poincare Euler see characteristic characteristie Euler-Poincark characteristic, 42, 44, Euler-Poincaré characteristic, 42, 50, 74, 74, 141, 141, 206, 206, 211 211 50, exact homotopy homotopy sequence sequence of of aa pair, pair, 30 30 exact exact sequence sequence exact - of fibration, 31 of fibration, 31 of pair, pair, 57 57 --- of of triple, triple, 58 58 -- of exactness conditions, conditions, 30 30 exactness excision excision - axiom, axiom, 59 59 theorem, 58 58 -- theorem, Faddeev D_ D. K., K., 254, 254, 287 287 Faddeev Farber, 199 199 Farber, Farrell, 196, 196, 254, 254, 259 Farrell, 259 Fermat, 187 187 Fermat, Fet, 189 189 Fet, Feynman amplitude, amplitude, 198, 292 292 Feynman 198, fiber, see fibration fiber, see fibration fiber bundle, see fibration fibration 79 79 fiber bundle, see -- map, 80 map, 80 -- smooth, smooth, 81, 148 - universal, 81,8 1148 universal, 81 - with singularities, 178 with singularities, 178 fibration, 19, 79-86 fibration, 19, 79-86 - associated fiber bundle, 81 associated fiber bundle, 81 - base, 20 - base, 20 - cross-section, 80 80 ~ cross-section, fiber, 20 - fiber, 20 - locally trivial, 22, 105 locally trivial, 22, 105 - projection, 20 projection, 20 - Serre, 32, 105 Serre, 32, group, 105 -- structure 79-86 structure - total space,group, 20 79-86 total space,function, 20 -- transition 79 fieldtransition function, 79 -field chiral, 197 chiral, 197 - electromagnetic, 196 electromagnetic, 196 196 ~ generalized magnetic, 196 - generalized Finsler length, magnetie, 187 Finsler length, 187 Floer -Floer homology, 274 homology, - theory, 201 274 - theory, Floyd, 129,201 139, 140, 226 Floyd, 129, 139, 140, 226 foliation, 195-200 foliation, 195-200 Fomenko, 248, 279, 297 248, 137-141 279, 297 Fomenko, formal group, formaI group, 137-141 FrCchet, 15 Fréchet,bordism, 15 framed see bordism theory framed bordism, see bordism theory

314 314

Index Index

Frank-Kamenetskii, Frank-KamenetskiI, 247 247 Fredholm Fredholm -- Alternative, Alternative, 14 -- representation, representation, 254, 254, 255 255 free acyclic acyclic resolution, 135 free resolution, 135 Freedman, Freedman, 272, 272, 273 273 Freudenthal, Freudenthal, 208 208 F’robenius, Frobenius, 97 97 -- algebra, 296 algebra, 296 Fuks, Fuks, 273 273 functional functional -- local, local, 197 197 -- multi-valued, 194 multi-valued, 194 functor, 30 functor,30 fundamental class, 55, 127 fundamental class, 55, 127 fundamental group, 2444267 fundamental group, 12, 12, 27, 28, 244-267 -- abelianized, abelianized, 13 13 ~- of surface, 38 38 of a non-oriented non-oriented closed closed surface, ~- of an oriented closed surface, of an oriented closed surface, 38 38 Gabrielov, Gabrielov, 250 250 Gamkrelidze, 129 Gamkrelidze, 129 gauge gauge -- field, 155 field, 155 153 - transformation, transformation, 153 Gauss Gauss ~- -Bonnet -Bonnet formula, formula, 7-8 7-8 -- formula, formula, 7, 7, 99 -- map, map, 150 150 Gaussian curvature, see curvature Gaussian curvature, see curvature Gel’fand, I.1. M., M., 250, Gel'fand, 250, 264, 264, 273 273 general 143, 166 general position, position, 143, 166 genus, 37 genus,37 Geometric conjecture, Geometrie conjecture, 248 248 Gersten, 258 258 Gersten, Ginzburg, Ginzburg, 200 200 Givental, 296 Givental, 296 Gliick, 270 Glück,270 glueing function, see see transition transition function glueing function, function G-manifold, manifold C-manifold, see see manifold Golo, 257 Golo, 257 Gottlieb, Gottlieb, 139 139 graph, graph,66 Grassmannian Grassmannian manifold, manifold, see see manifold manifold Griffiths, Griffiths, 110 110 Grinevich, Grinevich, 200 200 Gromoll, 189 Gromoll, 189 Gromov, 255, 273, Gromov, 255, 273, 274 274 Gross, Gross, 274 274 Grothendieck, 120, 136, 136, 186, 186, 254 Grothendieck, 120, 254 group group characters, 53 53 - of of characters, -~ of of homotopy homotopy spheres, spheres, 231-236 231-236 group ring, 28 group ring, 28

Gudkov, 273 Gudkov,273 Gusein-Zade, 140 Gusein-Zade, 140 Gysin homomorphism, homomorphism, 120 120 Gysin Haefliger, 244, 244, 273 273 Haefiiger, Haken, 245 Haken,245 Hauptvermutung, 269, 271 271 Hauptvermutung, 43345, 43-45, 269, Hausdorff topologieal topological space, space Hausdorff space, see space h-cobordism, 176, 232 232 h-cobordism, 176, Hecke algebra, 287 287 Hecke algebra, Heegaard Heegaard diagram, 247 -~ diagram, 247 177 splitting, 177 -~ splitting, Hessian, 188 Hessian, 188 higher signatures, 253 higher signatures, 253 Hilbert theorem theorem on on the Hilbert the existence existence of aa geodesic, 188 of geodesie, 188 Hilden, Hilden, 247 247 Hironaka, 186 186 Hironaka, Hirsch, 228, 228, 243 Hirsch, 243 Hirzebruch, 120, 137, 141, 141, 220 220 Hirzebruch, 120, 137, ~- formula, formula, 119 119 -- series, series, 141 141 Hodge Hodge ~- decomposition, decomposition, 183 183 lemma, 184 184 -~ lemma, Hodgkin, 121 Hodgkin, 121 121 -- theorem, theorem, 121 Hoffer, 201, 201, 274 Hoffer, 274 homeomorphism, 17 homeomorphism, 17 homology homology cellular, 60 -- cellular, 60 41, 47 -- groups, groups, 41, 47 ~- integral, integral, 13, 13, 47 47 -- relative, relative, 57 57 ~- with from abelian abelian with coefficients coefficients from group group G, C, 48 48 -- with respect to 72 with respect to representation, representation, 72 Tech, 68 68 -- spectral spectral or or Cech, homology sphere, 78 78 homology sphere, homology homology theory theory generalized, 59 59 -- generalized, homotopic homotopie maps, 18 18 -- maps, homotopically 19 homotopieally trivial, trivial, 19 homotopy, 18 homotopy, 6, 6, 18 ~ class, class, 12, 12, 18 18 -- connexion, connexion, 20, 20, 32, 85 32, 85 equivalence, 19 19 -~ equivalence, -- groups, groups, 23 23 - - n-dimensional, 24 n-dimensional, 24 -- bilinear bilinear pairing pairing of, of, 28 28 -- of of 0, U,, 190 On and and Un, 190 -- of of spheres, 112 spheres, 112

Index Index relative, 29 29 --- - relative, groups (stable), (stable), 151,208 151, 208 -- groups -- of of M(Om) A&(0,) and M M(SO,), 214 and (SOm) , 214 -- of of spheres, spheres, 208-210 208-210 invariance, 51 51 -~ invariance, - invariants, invariants, 44 44 - inverse, inverse, 19 19 - simplicial, 51, see simplicial - simplicial, 51, scc simplicial tori, 266 266 -- tori, - type, type, 19 19 Hopf, 95, 95, 193, 193, 206, 206, 253 253 Hop~ -- algebra, algebra, 96-97, 96-97, 192, 192, 287 287 -- diagonal diagonal homomorphism, homomorphism, 97 97 - - Frobenius-Adams Frobenius-Adams operator, 97 operator, 97 ~- symmetric, symmetric, 97 97 fibration, 39 39 -- fibration, - invariant, invariant, 95, 95, 101, 101, 191, 191, 194, 194, 207 207 - problem,88 problem, 88 - theorem, theorem, 40, 40, 63, 63, 109, 109, 206 206 Hopkins, 97 Hopkins, 97 horizontal subspace, see connexion see connexion horizontal subspace, Hsiang, 196, 249, 254, 259, 270, 271 Hsiang, 196, 249, 254, 259, 270, 271 H-space, 191 H-space, 191 Hurewicz, 28, 32 Hurewicz, 28, 32 -... theorem, 64, 70 70 theorem, 64, hyperbolic groups, 255 hyperbolic groups, 255 immersion, 144, 244 immersion, 144, 244 - classification, 227-229 classification, 227229 incidence number, 60-61 incidence number, 60-61 index index ~ of operator, 14, 122 of singular operator,point, 14, 122 -- of 205 of singular122 point, 205 - problem, -intersection problem, 122 index, 56, 166 index, -intersection of cycles, 55, 56 56, 166 of cycles, 55, 56 inverse -inverse image, 15 image, 15 -- system, 67 - system, 67 Ivanovskii, 139 IvanovskiI, 139 Jaeger, 278 Jaeger, 287 278 Jimbo, Jimbo,287 Jones algebra, 286 Jones J. algebra, Jones D. S., 286 101, 235 J. S., 275-294 101, 2.35 Jones Jones V., D. 247, Jones V., 247, 275-294 Kahler manifold, 110 Kahler manifold, 110 Karoubi, 258 Karoubi, 258 Kasparov, 140, 254, 255 Kasparov, 140, 255 Kauffman, 247, 254, 2766279 -Kauffman, polynomial, 247,278276-279 polynomial, Kazakov, 274 278 Kazakov, 274

315 315

Kervaire, 136, 136, 176, 176, 194, 194, 210, 210, 222, 222, 229, 229, Kervaire, 234, 236, 236, 244 244 234, Kharlamov, 273 273 Kharlamov, Kirby, 44, 44, 249, 249, 270, 270, 271 271 Kirby, ~ operations, operations, 292 292 Kirilov, 273 273 Kirilov, Klein bottle, bottle, 230 230 Klein Kniznik-Zamolodchikov monodromy, Kniznik-Zamolodchikov monodromy, 291 291 knot, 12, 12, 40 40 knot, - Alexander Alexander polynomial polynomial of, of, 246, 246, 275-282 275-282 - diagram, diagram, 275 275 -- oriented, oriented, 275 275 ~- unoriented, unoriented, 275 275 -~ genus genus of, of, 246 246 ~ group, group, 245 245 - HOMFLY HOMFLY polynomial poIynomia1 of, of, 276-282 276-282 isotopy of, of, 13 13 -~ isotopy Jones polynomial polynomial of, of, 247, 247, 275-283 275-283 -- Jones ~ Kauffman Kauffman polynomial polynomial of, 276-279 2766279 of, - ribbon, 288 ribboll, 288 Kochman, 129 Kochman, 129 Kolmogorov, 53, 273 273 Kolmogorov, 53, Kontzevich, 288, 288, 291, 291, 296 296 Kontzevich, -- formula, formula, 290 290 Kreck, 255 Kreck,255 Krichever, 140, 140, 296 296 Krichever, K-theory, 79, 79, 113-118 113-118 K-theory, -- algebraic, algebraic, 258-259 258-259 - Hermitian, Hermitian, 263 263 KG-theory, 124 KG-theory, 124 Kulish, 287 Kulish,287 Kummer 216 Kummer surface, surface, 216 Lagrangian, 187 Lagrangian, 187 Landweber, 274 Landweber, 274 - -Novikov -Novikov ~- algebra, 131 algebra, 131 -- operations, 131 - - operations, 131 Laplace operator, 183 Laplace operator, 183 Lashof, 249, 271 Lasbof, 249, Lawrence, 291271 Lawrence, 291273 Lawson, 255, Lawson, Lees, 271 255, 273 Lees, 271 123, 178 Lefschetz, 123, 178 -Lefschetz, duality, 59 -- formula, duality, 59 123, 168, 186 formula, 123, 168, 18653 ~- ring of intersections, ring of intersections, 53 lens space, 74-79, 139, 177 space, 74-79, 139, 92 177 -lens infinite-dimensional, - infinite-dimensional, 92 Leray, 68, 104, 193 68, 104, 193 see spectral -Leray, spectral sequence, - sequence spectral sequence, sec spectral sequence

316 316 theorem, 104 104 -- theorem, Levine, 196, 196, 244, 244, 259 259 Levine, Lickorish, 292 292 Lickorish, Lie algebra, algebra, 147 147 Lie Lie group, group, 147 147 Lie action, 139 139 -~ action, maximal torus, torus, 156 156 -- maximal Weyl group, group, 156 156 -- Weyl Lie-Whitehead superalgebra, 193 Lie-Whitehead superalgebra, 193 Lin X-S., X-S., 289 289 Lin linking coefficient, coefficient, 9, 9, 207 207 linking links, 12, 12, 275-282 275-282 links, Liulevicius, 97 Liulevicius, 97 Livesay, 259 259 Livesay, Lobachevskian plane, 37 37 Lobachevskian plane, loop space, 22, 22, 32 32 loop space, Losik, 250 Losik,250 Louis H., 247 Louis H., 247 Lusztig, 254 Lusztig, 254 Lyusternik, 189, 193 193 Lyusternik, 189, 174-189 -- -Shnirelman -Shnirelman category, category, 174-189 Miiller, 183 Müller, 183 Mobius, 37 Mobius,37 -- band, 162 band, 162 MacLane, 135, 254 MacLane, 135, 254 Madsen, Madsen, 44, 44, 243 243 Mahowald, Mahowald, 101, 101, 235 235 Makarov, Makarov, 248 248 manifold, manifold, 55 -- G-, G-, 125, 125, 215 215 -- Grassmannian, Grassmannian, 149, 149, 190 190 -- Hodge, 165 Hodge, 165 -- Kahler, Kahler, 165 165 -- piecewise piecewise linear, linear, 45 45 ~~ - - orientable, orientable, 45 45 -- real-analytic, real-analytic, 142 142 -- smooth, smooth, 142 142 -- topological, topological, 17 17 Manin, Manin, 296 296 map, 15 map,15 ~- continuous, continuous, 15 15 -- covered, covered, 19 19 mapping mapping cylinder, cylinder, 17 17 Markov, 246, Markov, 246, 281 281 -- moves, moves, 247, 247, 281 281 -- trace, 284 trace, 284 Massey Massey bracket, bracket, 94 94 Mather, 273 Mather,273 Matveev, Matveev, 248 248 Maupertuis Maupertuis -- or Fermat-Jacobi-Maupertuis DT Fermat-Jacobi-Maupertuis principle, 187 principle, 187 - functional, 196 - functional, 196

Index Index maximal torus, torus, see see Lie Lie group group maximal Maxwell, 7-8, 7-8, 288 288 Maxwell, -Yang-Mills -- -Yang-Mills -- equations, equations, 124 124 -fields, 23 23 --- - fields, Mazur, 243, 243, 258, 258, 267 267 Mazur, McDuff, 274, 274, 296 McDuff, metric metric Hermitian, 165 -- Hermitian, 165 pseudo-Riemannian, 147 -- pseudo-Riemannian, 147 Riemannian, 147 -- Riemannian, 147 Meyer, 189 189 Meyer, microbundles, 242 microbundles, 242 Migdal, 274 Migdal,274 Milgram, 44, 243 243 Milgram, Miller, 136, 136, 138 138 Miller, Mills, 155 155 Mills, Milnor, 44, 44, 102, 102, 128, 128, 136, 136, 176, 176, 194, 194, 210, 210, Milnor, 216, 222, 222, 223, 223, 225, 225, 229, 229, 234-236, 234-236, 244, 244, 216, 258, 268, 268, 269, 269, 272, 272, 273 273 258, operations, 93 93 -- operations, Mishchenko, 121, 137, 137, 140, 140, 255, 255, 263 263 Mishchenko, 121, Mislin, 192 192 Mislin, Moise, 43, 43, 76, 76, 267 267 Moise, monodromy, 11, 11, 33, 178 monodromy, 33, 178 group, 195 195 -- group, Morava K-theories, Morava K-theories, 138 138 Morgan, 110 Morgan, 110 Morrey, 144 Morrey,144 Morse, Morse, 193 193 -- -Novikov 199 -Novikov theory, theory, 199 -- function, function, 169, 169, 177 177 - index, 170 - index, 170 ~- inequalities, inequalities, 172 172 -- surgery, surgery, 171, 171, 233 233 -- theorem, theorem, 171 171 -- theory, theory, 199 199 ~~ of functionals, functionals, 188 188 - - of -- unequalities, unequalities, 199 199 Munkres, 243 Munkres,243 Murakami Murakami H., H., 281 281 Muskhelishvili, Muskhelishvili, 13 13 Nadiradze, Nadiradze, 139 139 nerve nerve of of covering, covering, 67 67 Newman, Newman, 243 243 Nielsen, Nielsen, 177, 177, 244 24".1 Nishida, 97 Nishida,97 Noether, Noether, 13, 13, 50 50 Noether-Muskhelishvili Noether-Muskhelishvili formula, formula, 122 122 normal normal -- bordism, bordism, 238 238 -- homotopy, homotopy, 240 240 -- map, map, 238 238

Index Index normal normal spherical spherical bundle, bundle, see bundle bundle normalization axiom, see Eilenbergnormalization axiom, EilenbergSteenrod Steenrod Novikov, 45, 97, 97, 102, 102, 128, 129, Novikov, 42, 42, 44, 44, 45, 128, 129, 133, 177, 194, 194, 200, 213, 133, 138, 138, 140, ] 40, 177, 200, 201, 201, 213, 216, 247, 253, 216, 225, 225, 226, 226, 234, 234, 236-244, 236-244, 247, 253, 255, 273, 297 255, 257, 257, 261, 261, 273, 297 - -Siebenmann 257 -Siebenmann obstruction, obstruction, 257 - -Wall -Wall groups, groups, 262 262 - conjecture on the higher signatures, - conjecture on the higher signatures, 264 264 Novikov Novikov P. P. S., S., 246 246 number 139 number of of fixed fixed points, points, 139 obstruction, obstruction, 82-86 82-86 - cochain, 83 cochain, 83 ~- to cross-section, 82-86 82-86 to extension extension of of cross-section, - to extension of homotopy, 82-86 to extension of homotopy, 82-86 -- to map, 82-86 82-86 to extension extension of of map, open open set, set, 15 15 operator operator -- differential differential -- cokernel, cokernel, 13 13 -- compact, 14 compact, 14 -- elliptic linear, elliptic linear, 13 ]3 -- kernel, kernel, 13 13 -- Noetherian, 13, 14 14 Noetherian, 13, ~ - pseudo-differential, 13 pseudo-differential, 13 ~ Schrodinger, 288 - Schrodinger, 288 Oshanine, 274 Oshanine, 274 Papakyriakopoulos theorem, 40, 246 Papakyriakopoulos theorem, 40, 246 partition of unity, 142 partition path, 16 of unit y, 142 path, 16 path-component, 16 path-component, 16 Pazhitnov, 199 Pazhitnov,255, 199 265 Pedersen, Pedersen, 138, 255, 225, 265 234 Peterson, Peterson, 138, 225, 234 178 Picard-Lefschetz theory, Picard-Lefschetz piecewise linear theory, 178 -piecewise map, 45 linear map, 45 17 - structure, structure, 17 Piunikhin, 297 Piunikhin,28,29739, 40, 50-52, 72, 179, 206 Poincare, 39, 40, -Poincaré, -Atiyah 28, duality, 12850-52, 72, 179, 206 -Atiyah duality, 128 - complex, 237 complex, 237 55, 68, 72 -- dual complex, dual complex, -- duality, 54, 55, 55, 59 68, 72 -- duality, 54, 55, 59 lemma, 179 -- lemma, problem 179 on three geodesics, 189 problem on - theorem, 50 three geodesics, 189 theorem,50 Polyakov, 124, 201 Polyakov, 124,201 Pontryagin, 53, 84, 88, 93, 207, 209, Pontryagin, 53, 84, 211, 215, 218, 273,88, 27993, 207, 209, 211, 215, 218, 273, 279

317 317

- characteristic classes, see characterischaracterischaracteristic classes, tic classes tic classes - characteristic numbers, 211 211 characteristic numbers, -- class, 124 class, 124 - duality, duality, 53, 53, 54 - squares squares and and powers, 90 powers, 90 Postnikov, 84 Postnikov, - invariants, invariants, 109 109 -- tower, tower, 109 109 presheaf, 65570 presheaf, 65--70 - restriction restriction homomorphism, homomorphism, 65 65 principal fiber boundle, principal fiber boundle, 81 principle of time, 187 187 principle of least least time, problem the three three pipelines three problem of of the pipelines and and three wells, wells, 66 projection, fibration see fibration projection, see projective space projective space - complex, complex, 39-40, 39-40, 62 62 -- real, real, 34-35, 62 34-35, 62 quantization of the the group, group, 287 287 quantization of Quillen, 119, 119, 138, 138, 192, 192, 258 Quillen, 258 Rabin, Rabin, 246 246 Rabinovich, 201 Rabinovich, 201 Ranicki, 199, 265 265 Ranicki, 199, Ravenel, 101, 136, 138 138 Ravenel, lOI, 136, Ray elements, 129 129 Rayelements, Ray-Singer 183 Ray-Singer torsion, torsion, 183 real division algebras real division algebras ~ classification, 95 - classification, 95 Reeb, 273 Reeb,273 Reeb fibration, 292 Reeb fibration, 292 Reidemeister Reidemeister ~ -Ray-Singer invariant, 291 -Ray-Singer invariant, 291 ~ -Whitehead torsion, 44 - -Whitehead torsion, 44 - moves, 278 - moves, 278 ~ torsion, 44, 74-79, 183 74-79, 183 torsion, 44,279, Reshetikhin, 287, 291 Reshetikhin,19279, 287, 291 retraction, retraction, 1977 Richardson, Richardson, Riemann, 10,77165 Riemann, 165 - C-function,10, 186 (-function, 186 120 - -Roth theorem, -- -Roth-Hirzebruch -Roch theorem, 120 Hirzebruch -- -Rochformula, 220 --- - formula, theorem, 220 123, 220 - -surface, theorem, 12 123, 220 surface, Rohlin, 44, 12128, 209, 211, 213, 222, 226, Rohlin, 44, 128, 249, 254, 273 209, 211, 213, 222, 226, 249, 254, 210 273 - theorem, -Rosenberg theorem, J., 210255 Rosenberg J.,249, 255 271 Rothenberg, Rothenberg, 249, 271 Ruan, 296 Ruan, 296

318 318

Index

Rubinstein H., H., 247 247 Rubinstein Salomon, 201 201 Salomon, Sard, 144 144 Sard, scalar product product scalar of n-cochain n-cochain and and n-chain, n-chain, 49 49 -- of Schwarz, 44, 44, 124, 124, 184, 184, 241, 241, 247, 247, 249, 249, Schwarz, 274 274 S-duality, 241 241 S-duality, second variation, 188 second variation, 188 Segal, 124 124 Segal, Serre, 22, 22, 90, 90, 93, 93, 102, 102, 106, 106, 107, 107, 109, 109, Serre, 192, 193, 193, 223, 223, 239, 239, 254 254 192, -Rohlin theorem, theorem, 249 249 -- -Rohlin fibration, see see fibration fibration -- fibration, Serre-Hochschild spectral sequence, see Serre-Hochschild spectral sequence, see spectral sequence sequence spectral Shaneson, 249, 249, 265, 266, 270, 270, 271 271 Shaneson, 265, 266, Sharpe, 265 Sharpe, 265 sheaf, 81 64-70, 81 sheaf, 64-70, -- germs 70 germs of of sections, sections, 70 -- of of non-abelian 70 non-abelian groups, groups, 70 quotient sheaf, -- quotient sheaf, 69 69 - subsheaf, 69 - subsheaf, 69 Shenker, Shenker, 274 274 Shnirelman, 189 Shnirelman, 189 Shtogrin M. I., Shtogrin M. L, 42 42 Siebenmann, Siebenmann, 44, 44, 249, 249, 252, 252, 256, 256, 266, 266, 270, 271 270, 271 signature signature of of manifold, manifold, 217 217 simplex simplex - n-dimentional, - n-dimentional, 41 41 -- algebraic algebraic boundary, boundary, 47 47 -- combinatorial combinatorial neighbourhood, neighbourhood, 45 45 -- face, 41 face, 41 -- orientable, orientable, 45 45 -- singular, singular, 64 64 simplicial simplicial -- approximation, approximation, 51 51 -- approximation approximation theorem, theorem, 45 45 -- complex, 41 41 complex, -~ m-skeleton, 43 m-skeleton, 43 - - n-dimensional, n-dimensional, 43 43 - ~ combinatorial combinatorial equivalence, equivalence, 43 43 -~ cone, 46 cone,46 - -- subdivided subdivided into into cells, cells, 60 60 -- subdivision, 43 subdivision, 43 -~ suspension, suspension, 46 46 -- homotopy, 51 homotopy,51 -- map, map, 45 45 simply-connected space, see space simply-connected space, see space Singer, 14, 122, 183, 292 Singer, 14, 122, 183, 292 Sirokav, 199 Sirokav, 199 Sitnikov, 273 Sitnikov, 273

Sklyanin, 287 287 Sklyanin, Smale, 173, 173, 176, 176, 232, 232, 269, 269, 292 292 Smale, function, 171 171 - function, - theorem, theorem, 228, 228, 258 258 smash product product of of spaces, spaces, 27 27 smash Smith, 77 Smith,77 exact sequences, sequences, 78 78 -- exact Smith J., 97 Smith smooth fiber fiber bundle, bundle, see see fiber fiber bundle bundle smooth smooth structures structures smooth spheres, see see group group of homotopy homotopy -- on spheres, spheres spheres smooth triangulation, triangulation, 165 smooth 165 Solov’ev, 255 255 Solov'ev, space space - compact, compact, 15 Hausdorff, 15 15 -- Hausdorff, metric, 15 - metric, path-connected, - path-connected, 16 16 -- pointed, pointed, 16 -- simply-connected, simply-connected, 33 33 spectral sequence, 103-112 spectral sequence, 103-112 see Adams -- Adams, Adams, see Adams -- Adams-Novikov, Adams-Novikov, see see Adams-Novikov Adams-Novikov spectral spectral sequence sequence -- Atiyah-Hirzebruch, Atiyah-Hirzebruch, 115-118, 115-118, 128 128 -- Leray, Leray, 104-106 104-106 -- Serre-Hochschild, Serre-Hochschild, 103 103 -- transgression, transgression, 106 106 stable stable -- homeomorphism, homeomorphism, 270 270 -- manifolds, manifolds, 270 270 Stallings, Stallings, 244, 244, 258, 258, 269, 269, 273 273 Stank0 Stanko M. M. A., A., 42 42 Steenrod, Steenrod, 84, 84, 273 273 -- algebra, 91-98 algebra, 91-98 --- - Cartan Cartan type type of, of, 93, 93, 98 98 --- - Milnor Milnor operations, operations, 93 93 -- powers, 91-93 powers, 91-93 -- problem, problem, 169 169 -- squares, squares, 90-93 90-93 Steinberg, Steinberg, 258 258 Stiefel-Whitney Stiefel-Whitney - characteristic - characteristic numbers, numbers, 211 211 Stiefel-Whitney Stiefel-Whitney - characteristic - characteristic classes, classes, see see characterischaracteristic classes tic classes Stolz, Stolz, 255 255 subdivision, subdivision, see see simplicial simplicial complex complex submanifold submanifold -- critical, critical, 175 175 submersion, submersion, 144 144 Sullivan, Sullivan, 44-45, 44-45, 109, 109, 119, 119, 192, 192, 202, 202, 241, 241, 243 243

Index Index surface surface closed, non-oriented, 38 -~ closed, non-oriented, 38 - closed, orientable, 37 closed, orientable, 37 surgery theory, 237-244 surgery theory, 237-244 suspension, 46, 58 58 suspension, 46, - isomorphism, isomorphism, 58 58 symbol of operator, operator, 122 symbol of 122 Taimanov, 189, 200, 200, 201 201 Taimanov, 189, Tait, 287 287 Tait, Takhtadjan, 287 Takhtadjan, 287 Tamura, 273 Tamura,273 tangent tangent bundle, 145 -~ bundle, 145 ~ space, space, 145 145 ~ vector, vector, 145 145 Taubes, 274 Taubes,274 tensor tensor fields, 146 -~ fields, 146 - product, 48 product, 48 Thorn, 44, 169, 211, 213, 214, 249, 273, Thom, 44, 169, 211, 213, 214, 249, 2n, 279 279 -- -Pontryagin construction, 225 -Pontryagin construction, 225 -- -Rohlin-Schwarz theorem, 44 -- -Rohlin-Schwarz theorem, 44 ~ class, 212 class, 212 - isomorphism, 212 - isomorphism, 212 - space, 211 space, 211 - theorem, 126, 128, 225 theorem, (Lord 126, 128, 225 288 Thompson Kelvin), Thompson (Lord Kelvin), 288 Thurston, 248, 273 Thurston, 248, 273 Tian, 296 Tian, 296 Toda, 97, 223 Toda, genus, 97, 223120, 219 Todd Todd genns, 120, 219 topological -topological group, 17 group, 17 5, 44 - invariants, invariants,field5, 44 ~ quantum theories, 291-298 - ring, quantum field theories, 291-298 17 ring, 1715 ~ space, --- space, pointed, 15 18 pointed, ~ vector space, 18 17 -topology, vector space, 5, 15 17 ~topology, compact-open,5, 15 16 - induced, compact-open, 16 17 - induced, 17 torus, 36 torus,36 total space, see fibration total space, seeseefibration transgression, spectral sequence transgression, transition function,see spectral 23, see sequence fibration transition function, 23,215sec fibration transversal-regularity, transversal-regularity, 215 transverse knot, 282 transverse knot, 282 tree, 35 tree, 3.5 t-regularity, see transversal-regularity t-regularity, Turaev, 291 see transversal-regularity Turaev, 291

319 319

- -Reshetikhin -Reshetikhin invariant, 279, 294 294 invariant, 279, two-valued formal group, 138 two-valued formaI group, 138 Tyupkin, 124, 274 274 Tyupkin, 124, universal uni versai - covering covering space, see see covering covering space, fiber bundle, bundle, see fiber fiber bundle bundle -~ fiber see Vafa, 274 Vafa,274 Van Kampen, Kampen, 53, 88 88 Van 53, variational variational derivative, 188 -- derivative, 188 - periodic periodic problem, 186 problem, 186 Vassiliev invariants, 289-298 Vassiliev invariants, 289-298 vector bundle, see bundle bundle see vector bundle, Verlinde brothers, 296 Verlinde brothers, 296 Vershinin, 129, 139 139 Vershinin, 129, Vey, 273 273 Vey, Vietoris, 67 Vietoris, 67 Vigman, 201 Vigman, 201 Villamayor, 258 Villamayor, 258 Viro, 273, 292 Viro, 273, 292 Volodin, 258 Volodin, 258 Voronovich, Voronovich, 287 287 Wagoner, 258 VVagoner, 243, 243, 258 Waldhausen, 276 VValdhausen, 276 Wall, 128, 226, 249, 261, 266, 270, 271 VVall, 128, 226, 249, 261, 266, 270, 271 Wallace, 269, 292 VVallace, 269, 292 Weyl group, see Lie group VVeyl group, see Lie group Whitehead, 28, 76, 88, 166, 241, 243 Whitehead, 28, 76, 88, 166, 241, 243 - formula, 201 - formula, 201 ~ group, 77-79, 257 - group, 77-79, 257 - homomorphism, 210 homomorphism, 210 Whitney, 142, 144, 244, 273, 279 VVhitney, 142, 144, 244, 273, 279 ~ formula, 130 formula, 130 Wilson, 136, 138 VVilson, 247, 136, 274, 138 291, 296 Witten, VVitten, 247, 274, 291, 296 wu -VVu formulae, 169, 212 - formulae, 169, 212 ~ generators, 160

- generators, 160

Yang, 155, 284 284 ~Yang, -Baxter 155,equations, 247, 282, 284 -Baxter equations, 247, 282, 284 - -Mills --- -Mills equations, 123 - - fields, equations, ~~ 123 123 123 Yau, fields, 296

Yau,296

Zamolodchikov, 283 Zamolodchikov, 283 Zariski topology, 186 Zariski topology, 186 Zieschang, 292

Zieschang, 292

Novikov (Ed.)

Topology 1 General Survey This book constitutes nothing less than an up-to-date survey of the whole field of topology (with the exception of "general (set-theoretic) topology"), or, in the words of Novikov himself, of what was termed at the end of the19th century "Analysis Situs", and subsequently diversified into the various subfields of combinatorial , algebraic, differential , homotopie, and geometric topology. The book gives an overview of these subfields, beginning with the elements and proceeding right up to the present frontiers of research. Thus one finds here the whole range of topological concepts from fiber spaces (Chapter 2) , CW-complexes, homology and homotopy, through bordism theory and K-theory to the Adams-Novikov spectral sequence (Chapter 3), and in Chapter 4 an exhaustive (but necessarily concentrated) survey of the theory of manifolds. An appendix sketching the recent impressive developments in the theory of knots and links and low-dimensional topology generally, brings the survey right up to the present. This work represents the flagship, as it were, in whose wake follow more detailed surveys of the various subfields, by various authors.

ISBN 3-540-17007-3 ISBN 0-387- 17007-3