Homotopy Theory
PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks
Edited by EILENBERG PAULA. SMITHand SAMUEL
Columbia University, New York
I: ARNOLDSOMMERFELD. Partial Differential Equations in Physics. 1949 (Lectures on Theoretical Physics, Volume VI) 11: REINHOLD BAER.Linear Algebra and Projective Geometry. 1952 111: HERBERT BUSEMANN AND PAULKELLY.Projective Geometry and Projective Metrics. 1953 IV: STEFAN BERGMAN AND M. SCHIFFER. Kernel Functions and Elliptic Differential Equations in Mathematical Physics. 1953 V: RALPHPHILIPBOAS, JR. Entire Functions. 1954 VI: HERBERT BUSEMANN. The Geometry of Geodesics. 1955 VII: CLAUDE CHEVALLEY. Fundamental Concepts of Algebra. 1956 VIII: SZE-TSENHu. Homotopy Theory. 1959 IX: A. OSTROWSKI. Solution of Equations and Systems of Equations. 1960 X: J. DIEUDONNE. Foundations of Modern Analysis. 1960 Curvature and Homology. 1962 XI: S. I. GOLDBERG. XII: SIGURDURHELGASON. Differential Geometry and Symmetric Spaces. 1962 XIII. T. H. HILDEBRANDT. Introduction to the Theory of Integration. 1963 XIV: SHREERAM ABHYANKAR. Local Analytic Geometry. 1964 XV: RICHARD L. BISHOP AND RICHARD J. CRITTENDEN. Geometry of Manifolds. 1964 XVI: STEVEN A. GAAL.Point Set Topology. 1964 I n prepnratioii JOSELUISM A s s E R A A N D J U A N JORGE SCHAFFER. Linear Differential Equations and Function Spaces.
Homotopy Theory
SZE-TSEN HU Wayne State University, Detroit, Michigan
1959
@
ACADEMIC PRESS New York and London
COPYRIGHfl 1959 BY
ACADEMIC PRESS INC.
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PRINTED IN THE UNITED STATES OF AMERICA
Preface
The recognition of the branch of mathematics now called homotopy theory took place in the few years after the introduction of homotopy groups by Witold Hurewicz in 1935. Since then, with numerous advances made by various workers, it has been playing an increasingly important role in the expanding field of algebraic topology. However, there exists no textbook on the subject at any level except the extremely condensed Cambridge tract of P. J. Hilton entitled “An Introduction to Homotopy Theory.” The present book is designed to guide a reader, who might be a beginning student or a newcomer to this branch of mathematics and who has a little knowledge of elementary algebraic topology, through the basic principles of homotopy theory. The author has aimed to provide the reader with sufficient detail for him to understand the fundamental ideas and master the elementary techniques so that he may be able to study the more advanced and more complicated results directly from the original papers. The main problem in homotopy theory is the extension problem as formulated in Chapter I and illustrated in Chapter 11.The fiber spaces, which are of fundamental importance, are defined and studied in Chapter 111. Homotopy groups are constructed and axiomatized in Chapter IV while theelementary techniquesof computation are given in ChapterV. ChapterVI gives an introduction to the obstruction theory of continuous maps, and Chapter VII contains an account of the cohomotopy groups. In the next three chapters, one will find an exposition of the spectacular results obtained mostly by the French school after Leray’s discovery of the spectral sequence. The techniques developed in these chapters are applied to compute the first few homotopy groups of spheres in the final chapter. As indicated in the second paragraph above, this book is by no means designed to be an exhaustive treatment of its subject; for example, the recent celebrated contribution of M. M. Postnikov is not included. Besides, homotopy theory is advancing so rapidly that any treatment of this subject becomes obsolete within a few years. At the end of each chapter is a list of exercises.These cover material which might well have been incorporated in the text but was omitted as not essential to the main line of thought. The inexperienced reader should not be discouraged if he cannot work out these exercises. In fact, if he is interested in one of the exercises, he is expected to read the papers indicated there. The bibliography at the end of this book has been reduced to the minimum essential to the text and the exercises. References to this bibliography are included for the convenience of the reader so that he can find more details V
vi
PREFACE
concerning the material; the references are not intended to be a historical record of mathematical discovery. (Thesereferences are cited in the text by numbers enclosed in square brackets). Frequently, expository articles are preferred to the earlier original papers. Whenever no reference is given concerning some subject or in some exercise, it means only that the reader does not have to look for further details in order to understand the material or to work out the problem. Cross references are given in the form (11; 7.1), where I1 stands for Chapter I1 and 7.1 for the numbering of the statement in the chapter. A list of special symbols and abbreviations used in this book is given immediately after the Table of Contents. Certain deviations from standard set-theoretic notations have been adopted in the text; namely, is used to denote the empty set and A\B the set-theoretic difference usually denoted by A-B. On the other hand, the symbol I indicates the end of a proof and the abbreviation iff stands for the phrase “if and only if.” Finally, for the algebraic terminology used in this book, the reader may refer to Claude Chevalley’s “Fundamental Concepts of Algebra,” published in this series. The author acknowledges with great pleasure his gratitude to Professor Norman Steenrod who has read several versions of the manuscript and whose numerous suggestions and criticisms resulted in substantial improvements. The author also wishes to express his appreciation of the friendly care with which Dr. John S. Griffin, Jr., and Professor C. T. Yang have read the final manuscript, of the many improvements they suggested, and of their help in the proofreading. It is a pleasure to acknowledge the invaluable assistance the author received in the form of partial financial support from the Office of Naval Research when he was a t Tulane University and from the Air Force Office of Scientific Research while at Wayne State University. SZE-TSENHu
Wayne State University,Detroit, Michigan
Contents
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PREFACE
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LIST OF SPECIAL SYMBOLS AND ABBREVIATIONS
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V
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CHAPTER I MAIN PROBLEM AND PRELIMINARY NOTIONS 1. Introduction . . . . . . . . . . . . . . . . . . . 2 The extension problem . . . . . . . 3 . The method of algebraic topology 4. The retraction problem . . . . . . . . . 5 Combined maps . . . . . . . . . . 6 . Topological identification . . . . . . . . . . . . . . . . 7 The adjunction space . 8. Homotopy problem and classification problem . . . . . . . . . . 9 . The homotopy extension property 10. Relative homotopy . . . . . . . . . . 1 1 Homotopy equivalences . . . . . . . . . 12 The mapping cylinder . . . . . . . . . 13 A generalization of the extension problem . . . . . . . . . . . . 14 The partial mapping cylinder . . . . . . . 15 The deformation problem . . . . . . . . . . 16. The lifting problem . . . . . . . . 17 . The most general problem . Exercises . . . . . . . . . . . . .
3 5 7 8 9 11 13 15 17 18 20 21 22 24 25 25
CHAPTER I1. SOME SPECIAL CASES OF THE MAIN PROBLEMS 1. Introduction . . . . . . . . . . . . . . . . . 2 . The exponential map p I R -+ S1 . . . . . . 3. Classification of the maps S' -+ S1 . 4. The fundamental group . . . . . . . . . . . . . . . . . 5 . Simply connected spaces . . . . . 6 . Relation between n , ( X . xo) and H,( X ) 7. The Bruschlinsky group . . . . . . . . . 8. The Hopf theorems . . . . . . . . . . . . . . . . . . 9 The Hurewicz theorem . Exercises . . . . . . . . . . . .
35 35 35 37 39 42 44 47 52 56 57
CHAPTER I11. FIBER SPACES . 1. Introduction . . . . 2 . Covering homotopy property
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3. Definition of fiber space . . . . . 4 . Bundle spaces . . . . . . . 5 . Hopf fiberings of spheres . . . . 6 Algebraically trivial maps X -+ S2 . . 7. Liftings and cross-sections . . . . 8. Fiber maps and induced fiber spaces . . 9 . Mapping spaces . . . . . . 10. The spaces of paths . . . . . 11. The space of loops . . . . . . 12. The path lifting property . . . . 13. The fibering theorem for mapping spaces . 14. The induced maps in mapping spaces . . 15. Fiberings with discrete fibers . . . 16. Covering spaces . . . . . . 17. Construction of covering spaces . . . Exercises . . . . . . . . .
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CHAPTER IV . HOMOTOPY GROUPS . 1. Introduction . . . . . 2 . Absolute homotopy groups . . 3. Relative homotopy groups . . 4 The boundary operator . . . 5. Induced transformations . . 6. The algebraic properties . . . 7 The exactness property . . . 8 . The homotopy property . . . 9. The fibering property . . . 10. The triviality property . . . 11. Homotopy systems . . . . 12. The uniqueness theorem . . 13. The group structures . . . 14 The role of the .basic point . . 15. Local system of groups . . . 16 %-Simplespaces . . . . Exercises . . . . . . .
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62 65 66 68 69 71 73 78 79 82 83 85 86 89 93 97
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CHAPTER V . T H E CALCULATION O F HOMOTOPY GROUPS . 1. Introduction . . . . . . . . . . . 2 . Homotopy groups of the product of two spaces . . . 3. The one-point union of two spaces . . . . . . 4 . The natural homomorphisms from homotopy groups to homology groups . . . . . . . . . . . 5. Direct sum theorems . . . . . . . . . 6. Homotopy groups of fiber spaces . . . . . . .
113 114 115 117 118 119 119 121 123 125 129 131 135 143 143 143 145 146 150 152
ix
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7. Homotopy groups of covering spaces 8. The n-connective fiberings . . 9 . The homotopy sequence of a triple 10 The homotopy groups of a triad . . . 11 Freudenthal's suspension Exercises . . . . . . .
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CHAPTER VI OBSTRUCTION THEORY 1. Introduction . . . . . 2 The extension index . . . 3. The obstruction c n f l (g) . . . . . 4. The difference cochain . 5 Eilenberg's extension theorem . 6 . T h e obstruction sets for extension 7. The homotopy problem . . . 8. The obstruction an(/.g; ht) . . 9 The group Rn(K.L ; f) . . . 10.The obstruction sets for homotopy 11. The general homotopy theorem . 12. The classification problem . . 13. The primary obstructions . . 14 Primary extension theorems . . 15. Primary homotopy theorems . 16. Primary classification theorems . 17. T h e characteristic element of Y . Exercises . . . . . . .
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175 175 175 176 178 180 181 182 183 184 185 186 187 188 190 191 191 193 193
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CHAPTER VII . COHOMOTOPY GROUPS . 1. Introduction . . . . . . 2 . The cohomotopy set n m ( X . A ) . . 3. The induced transformations . . 4 . The coboundary operator . . . 5. The group operation in n m ( X . A ) . . 6 . The cohomotopy sequence of a triple . . . . 7. An important lemma . 8 . The statement (6) . . . . . 9 . The statement (5) . . . . . 10. Higher cohomotopy groups . . . 11 Relations with cohomology groups . 12. Relations with homotopy groups . . . . . . . . . Exercises .
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CHAPTER VIII . EXACT COUPLES AND SPECTRAL SEQUENCES . . . . . . . . 1. Introduction . . . . . . . .
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2 Differential groups . . . . . 3 Graded and bigraded groups . . . 4 Exact couples . . . . . . 5. Bigraded exact couples . . . . 6. Regular couples . . . . . . 7 . The graded groups R(%)and S(V) 8. The fundamental exact sequence . . . . 9. Mappings of exact couples . . . 10 Filtered differential groups . 11. Filtered graded differential groups . 12. Mappings of filtered graded d-groups . Exercises . . . . . . . .
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CHAPTER I X . THE SPECTRALSEQUENCE OFA FIBER SPACE 1 Introduction . . . . . . . . . . . 2 . Cubical singular homology theory . . . . . . 3. A filtration in the group of singular chains in a fiber space . 4 The associated exact couple . . . . . . . . 5 The derived couple . . . . . . . . . . 6. Homology with arbitrary coefficients . . . . . . 7. The spectral homology sequence . . . . . . . 8 Proof of Lemma A . . . . . . . . . . 9. Proof of Lemma B . . . . . . . . . . 10. Proof of Lemmas C and D . . . . . . . . . . . . . . . 11. The PoincarC polynomials . 12. Gysin’s exact sequences . . . . . . . . . 13. Wang’s exact sequences . . . . . . . . . . . . . . . . 14. Truncated exact sequences . 15 The spectral sequence of a regular covering space . . . . . . . . . . 16. A theorem of P. A . Smith . 17 Influence of the fundamental group on homology and cohomology groups . . . . . . . . . . . 18 Finite groups operating freely on S r . . . . . . Exercises . . . . . . . . . . . . .
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CHAPTER X . CLASSES OF ABELIAN GROUPS 1. Introduction . . . . . . . 2 . The definition of *classes. . . . . 3. The primary components of abelian groups 4 The %?-notions on abelian groups . . . 5 . Perfectness and completeness . . . 6. Applications of classes to fiber spaces . . 7 . Applications to n-connective fiber spaces . . . 8 The generalized Hurewicz theorem
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259 259 259 262 263 266 269 271 272 274 275 277 280 282 284 285 287 288 290 292
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9. The relative Hurewicz theorem . 10. The Whitehead theorem Exercises . . . . . .
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CHAPTER X I HOMOTOPY GROUPS O F SPHERES . . . 1. Introduction . . . . . . . . . . . . . . . . . . . 2 . The suspension theorem 3. The canonical map . . . . . . . . . . 4 . Wang's isomorphism p,, . . . . . . . . . 5. Relation between p., and i# . . . . . . . . 6. The triad homotopy groups . . . . . . . . 7. Finiteness of higher homotopy groups of odd-dimensional spheres . . . . . . . . . . . . 8. The iterated suspension . . . . . . . . . . . . . . 9 The p-primary components of 7rm(S3) . 10. Pseudo-projective spaces . . . . . . . . 11. Stiefel manifolds . . . . . . . . . . 12. Finiteness of higher homotopy groups of even-dimensional spheres . . . . . . . . . . . . 13. The p-primary components of homotopy groups of evendimensional spheres . . . . . . . . . 14. The Hopf invariant . . . . . . . . . . 15. The groups n n + l ( S n )and nn+2(Sn). . . . . . . . . . . . . . . . 16 The groups nn+3(Sn) . . . . . . . . . 17. The groups nn+4(Sn) 18. The groupsnn,,.(Sn), 5 < Y < 15 . . . . . . . Exercises . . . . . . . . . . . . .
31 1 311 311 313 314 315 316
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BIBLIOGRAPHY INDEX
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317 318 319 321 323 325 325 326 328 329 330 332 333
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343
List of Special Symbols and Abbreviations The symbols and abbreviations listed below are followed by a brief statement of their meaning and by the number of the page on which they first appear. Symbols and abbreviations which are universall'y used in most branches of mathematics, such as E, U, n, w , >, sup, inf, etc., are not listed. End of proof, 3 Set-theoretic difference, 5 Is homotopic to, 1I Empty set, 17 Tensor product, 261 Boundary of, 47 Coboundary of, 50 Closed unit interval [0,1], 2 n-cube, 3 Euclidean n-space, 4 Unit n-sphere, 4 Unit n-simplex, 7 n-Dimensional homology group of, 3 %-Dimensionalcohomology group of, 5 Fundamental group of, 40 n-th homotopy group of, 109 n-th cohomotopy group of, 205 Group of integers, 109 Group of integers mod p, 281 The set X consists of a single element, 13 Restriction of the map f on A, 1 Space of all paths f:I --f Y such that f ( 0 ) ~ and A f ( l ) ~ B78 , Space of all maps X - t Y,12 One-point union of X and Y,145 ACHEP ACHP AHEP ANR AR BP CHEP CHP
Absolute covering homotopy extension property, 62 Absolute covering homotopy property, 62 Absolute homotopy extension property, 13 Absolute neighborhood retract, 26 Absolute retract, 26 Bundle property, 65 Covering homotopy extension property, 62 Covering homotopy property, 24 xii
LIST O F S P E C I A L S Y M B O L S A N D A B B R E V I A T I O N S
Coker deg dim HEP
iff
Im Int Ker LPLP NHEP PCHEP PCHP PLP re1 SSP Tor
Cokernel of, 298 Degree of, 37 Dimension of, 49 Homotopy extension property, 13 If and only if, 2 Image of, 215 Interior of, 8 Kernel of, 215 Local path lifting property, 98 Neighborhood homotopy extension property, 30 Polyhedral covering homotopy extension property, 62 Polyhedral covering homotopy property, 62 Path lifting property, 82 Relative to, 17 Slicing structure property, 97 Torsion product of, 270
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CHAPTER I M A I N PROBLEM A N D P R E L I M I N A R Y NOTIONS
1. Introduction There is a general type of topological problem which will be called the extension problem. One of the principal objectives of the book is to show that this problem is fundamental in topology. It will be shown that many theorems of topology and most of its applications in other fields of mathematics are solutions of special cases of the extension problem. The objective of the first chapter is to formulate the extension problem precisely, and to study the problem in its most general terms. It will be shown that various other problems are fully equivalent to extension problems. We shall begin with a restricted form of the extension problem, and one of its special cases, namely, the retraction problem. The solution is shown to depend only on the homotopy class of the map involved, and then only on the homotopy types of the spaces. These considerations lead naturally to a moregeneral type of problem. The latter is then shown to be reducible to the simplest type of problem, namely, the retraction problem. Finally, dual problems of deformation and lifting (finding a cross-section) are discussed in an analogous fashion together with their interrelations with the extension problem. Thus it will appear that underlying these various questions is one central question, namely, the extension problem.
2. The extension problem By a map, or mapping, f :X .+ Y of a space X into a space Y , we mean a single-valued continuous function from X to Y . The space X is called the domain off or the anti-image of f ; and the space Y is called the range of f . We shall not recall the definition and the elementary properties of continuous functions, since these can be found in any textbook on general topology, for example, [K; pp. 84-88]. On the other hand, we assume that the reader is familiar with the popular notions and notations concerning maps such as given in [E-S] . Let f : X .+ Y be a map and A a subspace of X.Then f defines a unique map g : A + Y such that g(x) = f(x)for each x E A . This map g is called the restriction of f to A or the partial map of f on A and is denoted by
g = f IA;
f will be called an extension of g over X . 1
2
I. M A I N P R O B L E M A N D P R E L I M I N A R Y N O T I O N S
If h : A c X denotes the inclusion m a p defined by h ( a ) = a E X for each a E A , then the relation g = f I A is equivalent to the commutativity relation / h = g in the diag-ram: A L + 1’
x The (restricted) extension problem is concerned with whether or not a given map g : A + Y defined on a given subspace A of a space X has an extension over X . When X , A , Y and g are given in some reasonably effective manner, the problem is to find an effective procedure for deciding whether g has an extension over X,and finding one when it exists. As will be seen, solutions have been given in numerous special cases; these are quite varied in nature, and the situations in which they apply are very restricted. As yet there is no reasonably complete theory. Actually it is convenient to concentrate on a broader problem which will be stated in 5 9, but several important ideas arise in connection with this problem. Let us begin by considering a few simple examples. 1. Let X be a given space and A be a subspace of X which consists of two points x, and x l . Let Y be a 0-sphere, say the boundary sphere of the closed unit interval I.Consider the map g: A -+ Y defined by g(xo) = 0 and g(xl) = 1. Then g has an extension over X iff x,, x1 lie in different quasi-components of X,[E-S; p. 2541. Hereafter, the symbol “iff” will stand for “if and only if”. 2. Let X = I and A be the boundary sphere of I . Let Y be any given space, and g : A -+ Y a given map. Then g has an extension over X iff g(O), g( 1) lie in a compact, connected and locally connected subspace of Y satisfying the second countability axiom. 3. Let A be the union of two disjoint closed subspaces B, C of a normal space X , let Y = I , and let g : A + Y denote the map defined by
g(B) = 0, g(C)
=
1.
Then, by Urysohn’s lemma [L,; p. 271, g has an extension over X . 4. Let A be a closed subspace of a normal space X , let Y = I, and let g : A + Y denote any map. Then, by Tietze’s extension theorem [L,; p. 281, g has an extension over X . See Ex.D a t the end of the chapter. In the last example given above, the space Y = I has the property that the extension problem is always trivial regardless of the domain ( X ,A ) provided that X is normal and A is closed. The class of spaces having this property are the solid spaces. Precisely, a space Y is said to be solid if every map g :A -+ Y of any closed subspace A of an arbitrary normal space X has an extension over X . Proposition 2.1.
Any topological product of solid spaces is solid.
3. T H E M E T H O D
OF ALGEBRAIC TOPOLOGY
3
Proof. Let { Y , 1 p E M } be a collection of solid spaces and Y = P,Y, denote the topological product of this collection, [L,; p. 101. We are going to prove that Y is solid. Let A be a closed subspace of a normal space X and g : A + Y any given map. Denote by p,,: Y --+ Y,, u , E M , the natural projection of Y onto Y,, and set g, = p,,g:A + Y,,.
Since Y,, is solid, g,, has an extension f,: X + Y p ,Define a map f :X+ Y by taking P p f W = f,(4, ( x E X).
It is obvious that f is an extension of g over X.I Since the closed unit interval Z is solid as noted above, it follows from (2.1) that any compact parallelotope, [L,; p. 191, is solid; in particular, the n-cube I n and the Hilbert cube I" are solid. Their homeomorphs are likewise solid, hence the n-cell and the n-simplex are solid. 3. The method of algebraic topology In the preceding section, we formulated the extension problem and gave examples in which the extension existed. It is natural to look for examples where the extension does not exist. The primary method of proving nonexistense is to apply homology theory and derive an algebraic problem from the geometric one and, finally, show that the algebraic problem has no solution. For this purpose, let us consider the triangle A
L
Y
x of maps as described in the preceding section. In any homology theory satistying the Eilenberg-Steenrod axioms [E-S; pp. 10-121, the maps f, g, h induce for each m the homomorphisms f,, g,, h, indicated in the following diagram :
According to Axiom 2, [E-S; p. 111, the relation fh mutativity relation f* h, = g*
=
g implies the com-
in the triangle of homomorphisms given above. Hence, the existence of an extension f :X + Y of the map g : A -+ Y gives a solution of a derived algebraic problem, namely, to find a homomorphism
+ : H m ( X )+ H m ( Y )
4
I. M A I N P R O B L E M A N D P R E L I M I N A R Y N O T I O N S
such that the commutativity relation $h, = g, holds. Thus, the existence of the homomorphism $ is a necessary (though not generally sufficient) condition for the existence of an extension of the given map g over X . On many occasions, this necessary condition provides us a method to show that a particular given map g : A --f Y fails to have an extension over X . For example, let us take X to be the unit n-cell En of the n-dimensional euclidean space R n , A = Y to be the boundary (n-1)-sphere Sn-l of En, and g : A + Y to be the identity map on 9 - l . We shall prove the following Proposition 3.1.
For each n 2 1, the identity map g:Sn-l --fSn-l has no
extension over En. Proof. Assume that there is some extension f :En + 9 - l of g. We shall deduce a contradiction as follows. Assume n > 1 and consider the homology theory with the group Z of integers as the coefficient group. Take m = n - 1, then we have
Since g is the identity map on Sn-l, it follows from Axiom 1 that g, is the identity automorphism of Hm(Sn-l). Since Hm(Sn-l) # 0, this implies that g, # 0. On the other hand, since H,(En) = 0, the inclusion map h:Sn-l c En induces h, = 0. Thus, we obtain f,h, = 0 and g, # 0. This contradicts the relation f,h, = g., In fact, the derived algebraic problem has no solution. It remains to dispose of the case n = 1. In this case, Sn-l consists of two points and hence is disconnected. On the other hand, En is connected and so is its continuous image f (En).Since f is an extension of the identity map g, we have f (En) = 9 - l . This is a contradiction. I Note. The case n
1 of (3.1) can also be proved from the derived algebraic problem provided that one uses the reduced homology groups, [E-S; pp. 18-19]. As an important application of (3.1), we shall give the following =
Theorem 3.2. (The Brouwer Fixed-Point Theorem). Every map f :En -+En has a fixed point, that is to say, there exists a point x of En such that f(x) = x . Proof. Assume that f : En + En is free of fixed points. Then, we may define a map r :En 9 - l as follows. Let x E En. Since f has no fixed points, we have f ( x ) # x . Draw the line from f ( x ) to x and produce until it intersects 9 - 1 a t a point r ( x ) . One verifies that the assignment x + r(x) defines a continuous function Y:En --f 9 - 1 . If x E Sn-1, it is obvious from the construction that r ( x ) = x . Hence r is an extension of the identity map on 9 - l . This contradicts (3.1). I This application of (3.1)shows that the negative nature of a “non-existence” --f
4. T H E R E T R A C T I O N P R O B L E M
5
theorem may not diminish its interest. A reformulation sometimes gives it a positive aspect. In the derived algebraic problem formulated above, one may of course use cohomology theory instead of homology theory.
4. The retraction problem If Y = A and g = i is the identity map on A , then we obtain an important special case of the extension problem which will be referred to as the retraction problem. If i has an extension r : X + A , then A is called a retract of X , r is called a retraction of X onto A , and we will write
r : X 3A. According to (3.1), the boundary (n-1)-sphere 9 - l of En is not a retract of E n . On the other hand, if X denotes the space obtained by deleting from En an interior point which may be assumed to be the origin 0 without loss of generality, then 9 - 1 is a retract of X.In fact, a retraction Y : X Sn-1 ~ is given by Y(X) =
(5,. . . 5) ,
1x1
for every point x = ( x l , . . . , xn) of X = En \ 0, where I x 1 denotes the distance between 0 and x . The same formula also gives a retraction Y of Rn \ 0 onto 9 - l . For another example of retracts, let us consider the topological product X = A x B. Pick a point b, from B. Then A can be considered as a subspace of X by means of the homeomorphism h :A -+ X defined by h ( a ) = (a, b,) for each a E A . This having been observed, it becomes clear that the natural projection X = A x B + A gives a retraction of X onto A . I n particular, a meridian of the torus T , = 9 x S1is a retract of T,. Observe that if A is a retract of X then the extension problem becomes trivial regardless of the range Y . Indeed, we have the following proposition 4.1. A is a retract of X has a n extension over X .
io,for a n y space Y ,every m a p g : A -+ Y
Proof. If A is a retract of X with a retraction r : X 3 A , then g r : X + Y is an extension of g. Conversely, assume that the condition holds and take Y = A . Then the identity map i on A should have an extension Y : X + A . I The retraction problem gives rise to a derived algebraic problem as follows. In any homology theory or cohomology theory satisfying the EilenbergSteenrod axioms, the inclusion map i : A c X induces for each m the homomorphisms i * : H , ( A ) + H,(X), i * : H m ( X )+ H m ( A ) .
6
I. M A I N P R O B L E M A N D P R E L I M I X A R T N O T I O N S
The derived algebraic problem is to determine whether or not there exist homomorphisms
4 : H m ( X )-+ H m ( A ) , y : H m [ A )+ Hm(X) such that +i* and i*y are the identities on H m ( A )and Hm(A) respectively. The existence of the homomorphisms 4 and y is a necessary condition for A to be a retract of X . In fact, if r : X 2 A is a retraction, then ri is the identity map on A and hence4 = r* and y = r* are solutions of the derived algebraic problem. Furthermore, since r,i, and i*r* are the identities, it follows that i,, r* are monomorphosms, that r*, i* are epimorphisms, and that H m ( X ) , H m ( X ) decompose into the following direct sums:
H,(X) = Image i, H m ( X ) = Kernel i*
+ Kernelr,, + Imager*.
If the coefficient group of the cohomology theory is a ring, then the cohomology groups H m ( X ) , m = 0, 1, * * , constitute a ring H * ( X ) with the cup product as multiplication. The inclusion i:A c X and the retraction r : X 3 A induce the ring homomorphisms
-
i * : H * ( X )+ H * ( A ) , r * : H * ( A )-+ H * ( X ) . Since ri the identity map on A , it follows that i*r* is the identity automorphism of the ring H * ( A ) . Hence r* is a monomorphism, i* is an epimorphism, and H * ( X ) decomposes into the direct sum
H * ( X ) = Kerneli*
+ Imager*
where Kernel i* is an ideal and Image r* is a subring isomorphic to H * ( A ) under r*. These necessary conditions can be used to prove that a particular given subspace A of a certain space X fails to be a retract of X . For example, let X denote the complex projective space of complex dimension n > 1 and A a linear subspace of X of complex dimension r with 0 < r < n. Then A is not a retract of X . To prove this fact, let us assume that there is a retraction r : X 2 A . Then we obtain a ring monomorphism r * : H * ( A )-+ H * ( X ) of the cohomology rings with integral coefficients. Let a and E denote generators of the free cyclic groups H 2 ( A )and H2(X) respectively. Since r* is a monomorphism, there is a non-zero integer k such that r*(a) = k t . Since n > r , we have a n = 0. Since r* preserves multiplication, we obtain kntn
This contradicts the fact that
H" (S) .
=
En
.*(an)
=
0.
is a generator of the free cyclic group
Finally, let us give an important example of retract in the form of the following
5. COMBINED
MAPS
7
Proposition 4.2. I f ( X ,A ) is a (finitely) triangulable pair, [E-S; p. 601, then the closed subspace L = ( X x 0) U ( A x I )
of the product space M
=
X x I is a retract of M .
Proof. First, let us prove the special case where X is the unit n-simplex An of the euclidean (n + 1)-space, [E-S; p. 551, and A is the boundary (n-1)-sphere of A , which is empty if n = 0. Then a retraction r : M 3 L can be constructed geometrically as follows. Since I R, it follows that M is a subspace of An x R. Then we define r to be the central projection of M onto L from the point (c, 2) of A n x R, where c denotes the centroid of A,.
This proves the special case. For a finitely triangulable pair ( X ,A ) , we may assume that X is a finite simplicia1 polyhedron and A is a subpolyhedron. Since a retract of a retract is also a retract, one can easily prove the proposition by induction on the number of simplexes in X but not in A and by the aid of the special case proved above. I A strengthened form of (4.2) will be given in 9 10. Besides, (4.2) can also be generalized to some non-triangulable pairs; see Ex.0 a t the end of the chapter. 5. Combined maps Frequently, a function is constructed by prescribing it on pieces of its domain. The purpose of this section is to give sufficient conditions, for the continuity of functions so constructed. Let { X , I p E M } be a given system of subspaces of a space X , indexed by the elements of a set M , such that the union of all subspaces X,, p E M , is the whole space X . Let D,, = X,, il X , for each pair of indices p, v in M . For any given map g : X --f Y of X into a space Y , the partial maps g, = g I X , are well-defined and satisfy the relation g, I D,, = g , I DPv for every pair of indices p,v in M . Hence, our problem in this section is to study the inverse of this process described as follows. Let us assume that, for each index p E M , there is given a map f,:X, + Y such that I Dpv = f v I D,v
f,
for each pair of indices p and v in M . Then we may define a function f :X + Y by taking We are concerned with the problem whether or not f is continuous. Proposition 5.1. If M is finite and all the subspaces X,, p E M , are closed in X , then the combined function f is continuous.
Proof. Let F be any closed set in Y . Then it follows from the continuity of f,, p E M , that f;'(F) is a closed set of X,. Since X, is closed in X, this
8
1. M A I N P R O B L E M A N D P R E L I M I N A R Y N O T I O N S
implies that f;l(F) is a closed set of X . Since M is finite,
is a closed set of X . Hence f is continuous. I A frequent application of this proposition is as follows. Let f,g:X x I --f Y be two given maps such that f (x,l) = g(x,O) for every x E X . Then we may define a function h : X x I -+ Y by taking:
( X E X , O < t <+): ( X E X , 4 < t < 1).
h(x,t) = ),l{:t,;
An application of (5.1) shows that h is continuous. This uniquely defined map h will be called the sum of the maps f and g denoted by f + g. This operation is obviously not commutative. Proposition 5.2. I f x is an interior point of some subspace X,, then the combined function f is continuous at x . Proof. Let U be an open neighborhood of the point f ‘ Y ) = f,(x) in Y. It follows from the continuity of f , that there is an open neighborhood V of x in X , such that f,(V) c U . Call
W
= I/
nInt(x,).
Then W is an open neighborhood of x in X and f ( W ) = f,(W) c U. Hence f is continuous a t x . I Corollary5.3. If all the subspaces X,, p~ M , are open in X , then the combined function f is continuous.
Finally, as a generalization of (5.1), we have the following Proposition 5.4. If the system { X , I p E M } forms a locally-finite closed covering of X , then the combined function f is continuous. Proof. Let x be an arbitrary point of X . It suffices to prove that f is continuous at x . Since the system { X , I p E M } is locally finite, there exists a neighborhood X , of x in X which meets only a finite number of the subspaces { X , I p E M }. Since X, is closed in X , it follows that X , n X , is closed in X,. An application of (5.1) proves that the restriction = f I X , is continous. So, we may enlarge the system { X , I p E M } by adjoining X,. The new system usually fails to be a closed covering of X . However, since x is a n interior point of X,, (5.2) implies that f is continuous at x. I
f,
6. Topological identification Suppose that X is a space in which an equivalence relation is given. Then the points of X are divided into disjoint equivalence classes. Let us denote
7. T H E A D J U N C T I O N S P A C E
9
by 2 the set of all these equivalence classes and by [XI the equivalence class which contains the point x in X . The assignment x + [XI defines a function
p:x-+z of X onto Z called the natural proiection. The set Z can be topologized as follows: a set W in 2 is called open iff the inverse image P-l(W) is an open set in X . This topology of 2 is called the identification topology determined by p or the quotient topology defined by the given equivalence relation in X . Having been topologized in this way, Z is said to be the quotient space obtained from X by topological identification. This method of constructing new spaces from old is of extreme importance in combinatory topology. Most well-known spaces can be obtained from simpler spaces by topological identification. For example, the n-sphere is obtained from the n-cell by identifying the boundary to a single point and the real projective n-space is obtained from the n-sphere by identifying the antipodal points. As another example of topological identification, let us consider a given groupn of homeomorphisms of a space X . The groupn defines an equivalence relation in X as follows: two points a, b in X are equivalent iff b = t ( a ) for some 6~ n The . quotient space thus obtained is called the orbit space and is denoted by X / n . More examples of topological identification will be found in the exercises at the end of the chapter. Proposition 6.1. Let f : Z + Y be a given function of the quotient space Z ilzto a space Y . If the composed function g = f p : X -+ Y i s continuous, then so i s f.
V be any open set in Y . I t suffices t o show that the inverse f -l(V)is an open set in 2. Since g = f p , we have
Proof. Let
image W
=
g-'(V)
=
P-l[f-l(V)]= P-l(W).
The continuity of g implies that g - l ( V ) is an open set in X.Hence, by definition of the identification topology, W is open. I For further information about quotient spaces, see [I<; pp. 94-1001.
7. The adjunction space As an important application of the process of topological identification, let us consider the construction of the adjunction space of a diagram
where g is a map defined on a closed subspace A of a space X into another space Y .
I0
I. M A I N P R O B L E M .4ND P R E L I M I N A R Y N O T I O N S
For this purpose, consider the disjoint union
W=XUY of the spaces X and Y . W is aspace with its topology defined as follows: a set V c W is open iff V II X and V n Y are open sets of the spaces.X and Y respectively. This space W will be called the topological sum of the spaces X and Y . If we identify each x E A with g ( x ) E Y , W becomes a space Z which will be called the adjunction space obtained by adjoining X to Y by means of the map g : A +. Y . In more detail, Z is the quotient space obtained from W by means of topological identification as follows. Two points x E X and y E Y are said to be equivalent iff x E A and g ( x ) = y ; two points x and of A are equivalent iff g ( x ) = g(x’). Letting, of course, each point be equivalent to itself, we have an equivalence relation in W ;we take Z to be the quotient space of W with respect to this equivalence relation and p : W + 2 the natural projection. One can easily verify that the natural projection p maps Y homeomorphically onto a subspace p ( Y ) of Z . By means of this imbedding, Y will be considered as a closed subspace of Z. Obviously, Y = Z iff A = X . Furthermore, we have the following X I
Proposition 7.1. Y is 1 retract of 2 if the
map g : A -+ Y has an extension
over X . Proof.
Necessity. Let r : Z 3 Y be a retraction. Using the natural projection : X -,Y by taking
p : W +.Z, we define a map f f(x)
=
YP(X),
( X E X C
W).
If x E A , then we have f ( x ) = rp(x) = rg(x) = g ( x ) . Hence f is an extension of g over X . Suficiency. Let f : X +. Y be a map with f I A = g. Define a function r : Z +. Y as follows. Let z E 2. If z E Y , we define r(z) = z. If z 4 Y , then there is a unique point x E X \A such that p ( x ) = z. In this case, we define ~ ( z= ) f ( x ) . Let h = re. Then h I X = f and h I Y is the identity map. According to (5.1),h is continuous. By (6.1), this implies that r is continuous. Since r 1 Y is the identity map, 7 is a retraction. I Thus the extension problem for the map g : A +. Y over X is equivalent to a retraction problem which is a special case of the extension problem. Next, let us note another interesting property of the adjunction space. The natural projection p : W +.Z maps X into 2 and A into Y . Hence it defines a map p : ( X ,A ) +. (2, Y ) . This map p is evidently a relntive homeomorphism, that is to say, it carries X \ A homeomorphically onto 2 \ Y .
8. H O M O T O P Y
PROBLEM AND CLASSIFICATION PROBLEM
I1
For examples of adjunction spaces, let us take X = En and A = Sn-1. Consider a map g:Sn-1 +. Y and the adjunction space Z obtained by adjoining E n to Y by means of g. Then the pair (2,Y) is called a relative n-cell. In particular, if Y = Pn-1 is the real projective (fz-l)-space and if g:Sn-1+ P - 1 is the map obtained by identifying antipodal points, then the adjunction space 2 can be identified with the real projective n-space P n . Hence, the pair ( P n , P*-1) is a relative n-cell. Similarly, (CPn, CPn-1) is a relative 2n-cel1, where CPn denotes the complex projective n-space which is of real dimension 2n.
8. Homotopy problem and classification problem A family of maps h t : X +. Y (0 Q t Q l ) , indexed by the real numbers t E I , is called a homotopy if the function H : X x I + Y,defined by
H ( x , t)
=
ht(x), ( % E X t, E I )
is continuous. h,, and h , are called the initial map and the terminal map of the homotopy ht. In the sequel, the homotopy ht and the map H will be considered essentially as the same thing and we shall use whichever appears more convenient for the particular purpose at hand. Two maps f :X+. Y and g : X + Y are said to be homotopic (notation: f 32 g), if there exists a homotopy, h t : X -+ Y , (0 Q t Q l ) , such that h, = f and h, = g. In this case, ht is called a homotopy connecting f and g, and is denoted by h t :f
N
g.
Intuitively, f and g are homotopic iff each can be changed continuously into the other. Given two maps f , g : X +. Y , it is not always true that f and g are homotopic. For example, if X = Y is the n-sphere Sn, f the identity map on Sn, and g a constant map, then it follows from the Homotopy Axiom of homology theory that / and g are not homotopic, [E-S; p. 111. On the other hand, if f, g : X +. Sn are two maps such that for any x E X the points f ( x ) and g ( x ) are never antipodal, then f and g are homotopic. In fact, there is always defined uniquely the minor arc of the great circle joining f ( x ) and g ( x ) and we may define ht(x)as the point dividing this arc in the ratio t : 14. As a consequence, if a map f : X + S* leaves a point free from the image f ( X ) ,then f is homotopic to a constant map. The homotopy problem is to determine whether or not two given maps f, g : X -+ Yare homotopic. This problem is actually a special case of the extension problem of $ 2 . In fact, consider the product space X x I and its subspace M = ( X x 0 ) u ( X x 1) that is to say, M consists of the bottom and the top of X x I. Define a map + Y by setting +(% 0) = +(x,l) = g ( 4
+ :M
f
(
4
8
I2
I. M A I N P R O B L E M A N D P R E L I M I N A R Y N O T I O N S
for each x E X . Then every extension H : X x I + Y of 4 is a homotopy connecting f and g and vice versa. Thus f and g are homotopic iff 4 has an extension over X x I . The homotopy axiom of the homology and cohomology theories gives necessary conditions for two maps to be homotopic. More precisely, if f, g : X --f Y are homotopic maps, then they induce the same homomorphisms on the homology groups and the cohomology groups; that is, f* = g, and f* = g*. These necessary conditions can be used to prove that two particular given maps f , g : X + Y fail to be homotopic. For example, let X and Y be two oriented closed n-dimensional manifolds. In the homology theory with integral coefficients, let a E H n ( X ) and j3 E Hn(Y ) denote the generators determined by the orientations of X and Y . The degree of any given map f : X +. Y is defined to be the integer deg(f) such that f,(cr) = deg(f).j3. Then two maps f , g :X -+ Y are not homotopic if deg(f ) # deg(g). Some special cases of homofopies are of importance. Let us suppose that X is a subspace of Y . Then a homotopy h t : X + Y , (0 Q t Q I ) , is said to be a deformation of X in Y if h, is the inclusion map i : X c Y . Furthermore, if h, is a constant map, then the deformation ht is called a contraction of X in Y . They are simply called deformations and contractions of X if Y = X. If a contraction of X (in Y ) exists, X is said to be contractible (in Y ) . If x, E X and a contraction ht : X + Y , (0 Q t Q l), exists such that It&,) = x, for each t E I , then X is said to be contractible to the point x, in Y . For example, every proper subspace of the n-sphere Sn is contractible to a point in Sn. Now, for any two given spaces X and Y , let us denote the totality of maps of X into Y by 52 = YX. Proposition 8.1. The relation
N
between the maps 52 is an equivalence
relation. Proof. We have to show that the relation N is reflexive, symmetric, and transitive. (1) To prove that it is reflexive, let us define for any f E Q a homotopy f t : X + Y (0 < t < 1) by taking f t = f for every t E I . Then f t : f N f. (2) To prove the symmetry, let ht :f N g be a homotopy connecting two given maps f E 52 and g E 52. Define a homotopy Kt:X + Y (0 Q t Q 1) by taking kt = hl-t for each t E I . Then k t : g N f. (3) To prove the transitivity assume4t: f N g and yt :g 1: h. Define a system of maps Xt :X + Y (0Q t Q 1) by taking
=+,
By (5.1), Xt is a homotopy. Since X, = f and X , = y1 = h, we have N h. I As a consequence of (8.1) the maps D are divided into disjoint equivalence
Xt:f
9.
T H E HOMOTOPY E X T E N S I O N P R O P E R T Y
I3
classes, called the homotopy classes of these maps. Let us denote by n ( X ;Y ) the totality of these homotopy classes and denote by [ f ] the homotopy class of f , that is to say, the homotopy class which contains the map f EQ. If g : X -+ X is a map and if the maps f o , f , : X ’ + Y are homotopic, then obviously we have fog N flg. This means that n ( X ; Y ) is a contravariant functor in X , [E-S; p. 1111. On the other hand, if g: Y’ + Y and if the maps f o , f l : X + Y’ are homotopic, then g f , N gf,. This means that n ( X ; Y ) is a covariant functor in Y . Consequently, n(x;Y ) i s a functor of two variables contravariant in X and covariant in Y . In a great many cases, it is possible to enumerate the homotopy classes n ( X ; Y ) whereas the set Y xhas the power of continuum. Given two spaces X and Y , the classification problem is to enumerate the homotopy classes n ( X ; Y ) of the maps Y xand to exhibit a representative map in each homotopy class. For example, if X is a paracompact Hausdorff space and Y is a solid space, then it is easily seen that any two maps are homotopic and hence n ( X ; Y ) consists of a single homotopy class; in symbols,
n (X ;Y ) = 0. The same is true if X is any space and Y is contractible. Some interesting non-trivial examples will be given in the next chapter.
9. The homotopy extension property In the present section, we propose to relate the concept of homotopy to the extension problem. The main result can be stated as follows: I n a great many cases, the extension problem for a given m a p g : A + Y over X 3 A depends only 0% the homotopy class of g. Let f : X + Y be a given map and A a given subspace of X . A homotopy ht:A -+ Y ; (0 < t < l), is called a partial homotopy of f if f 1 A = ho;it will be simply called a homotopy of f in case A = X . Restrictions and extensions of homotopies are defined in the same manner as those for maps, Definition 9.1. A subspace A of a space X is said to have the Itomotopy extension property (abbreviated HEP) in X with respect to a space Y , if every partial homotopy ht:A + Y , (0 < t < l) ,
of an arbitrary map f :X
-+
Y has an extension
g t : X + Y , (0 < t
< l),
such that go = f . A is said to have the absolute homotopy extension property (abbreviated AHEP) in X , if it has the HEP in X with respect to every space Y .
14
I. M A I N P R O B L E M A N D P R E L I M I N A R Y N O T I O N S
The homotopy extension property is basic to various constructions in homotopy theory. Fortunately, this important property exists in reasonably smooth spaces ; in particular, we have the following two propositions. Proposition 9.2.
A H E P in X .
If ( X ,A ) i s a (finitely) triangulable pair, then A has the
Proof. Let f : X + Y be a given map and ht:A -+ Y , (0 f t Q I), a given partial homotopy of f . Consider the product space M = X x I and its closed subspace L = ( X x 0 ) U ( A x I ) . Define a map H :L -+ Y by setting
H ( x l t,
=
1
f(x), ifxEX,t = 0 , It&), if x E A , t E I .
According to (4.2), there is a retraction r : M D L. Define a homotopy g t : X + Y , (0 Q t < I), by taking
gt(x) = H r ( x , t ) , ( x E X , t E I ) . Then gt is obviously an extension of ht such that go = f . Since Y is arbitrary, this proves that A has the AHEP in X . I A space X is said to be binormal if X x I is normal. Paracompact Hausdorff spaces (and hence metrizable spaces) are binormal. Proposition 9.3. If Y i s a (finitely) triangulable space, then every closed subspace A of any binormal space X has the H E P in X with respect to Y . Proof. Without loss of generality, we may assume that Y is a subcomplex of the unit n-simplex An, where n depends on Y . According to a classical theorem, [E-S; p. 721, there exists an open neighborhood U of Y in A , such that Y is a retract of U . Let r : U 3 Y be a retraction. Let A be any closed subspace of a given binormal space X , f :X -+ Y a given map, and h t : A +. Y , (0 Q t Q l ) , a given partial homotopy of f . Consider the spaces M = X x I and L = (Xx 0 ) U ( A x I ) , and define a map H : L -+ Y as in the proof of (9.2). By tj 2, A n is solid. Since L is closed in the normal space M , the map H : L + Y c An has an extension G : M + An. Let V = G-l(U). Then V is an open neighborhood of L in M . Since the unit interval I is compact, one can easily prove that there exists an open neighborhood W of A in X such that W x I is contained in V . Since X is normal, it follows fromurysohn’s lemma, [L,; p. 271, that there exists a continuous real function X:X -+ I such that X(X \ W ) = 0 and X(A) = 1. Define a map F : M + Y by taking
F(x,t) = r [ G ( x ,x ( x ) t ) l ,
( x E X , t €I ) .
Obviously F is an extension of H over hl. Hence the partial homotopy ht of f has an extension g t : X - + Y , (0 Q t Q I ) , defined by gt(x) = F(x,t) for each x E X and t E I . Clearly go = f . I
10. R E L A T I V E H O M O T O P Y
15
Generalizations of (9.2) and (9.3) will be given in Ex.N and Ex.0 a t the end of the chapter. Now let A be a given subspace of a space X and g : A + Y a given map. If A has the HEP in X with respect to Y , then the restricted extension problem in 4 2 obviously depends only on the homotopy class of g. In this case, the restricted extension problem is equivalent to a broadened form, namely to ask whether there is a map f : X -, Y such that f h N g, where h : A c X denotes the inclusion map. This new broadened form is apparently weaker than the original restricted form. However, once we reflect upon the concept of homotopy, we realize that the original form of the problem was formulated in too narrow a fashion. Hereafter, when we refer to the extension problem, we shall mean its broad ened form unless otherwise stated.
10. Relative homotopy Let M be an abstract set and X , Y be two given spaces. Let { X , }, { Y, } be two systems of sets indexed by ,u E M such that X , c X and Y , c Y for each index p E M . For the sake of brevity, we shall denote by { M } the detailed notation { X,, Y , l,u E M } whenever there is no danger of ambiguity. Consider the maps f : X -+ Y such that f ( X , ) c Y, for each ,u E M . The totality of these maps will be denoted by
52
=
YX{ M } = Y " { x,, Y , l,uEM}.
For example, if M consists of a single element ,u and if X , = A , Y , = B , then 8 is the set of all maps f :( X ,A ) -+ ( Y ,B ) , see [E-S; p. 31. For a second example, let M contain two elements ,u and v . If X , = A , Y, = B are as above and X , = { xo }, Y,= { yo } are singletons with xo E A , yo E B , then 52 is the set of all maps f : ( X ,A , xo) + ( Y ,B , yo). I t is primarily these cases that the notation is intended to cover. Two maps f and g in 52 are said to be homotopic relative to the system { M } if there exists a homotopy h t : X -+ Y , (0 Q t Q l ) , such that h, = f , h, = g and ht E Q for every t E I. In notation, f egrel(M}.
In this case, hl is called a homotopy connecting f and g in Q and is denoted by
ht:f - g r e l { M } . If X,, is a single point and Y , =f ( X , ) , we may identifyp with X , . If this holds for all p E M , then M is identified with a subset of X . In this important case, we shall use the usual notation: f -grelM,
ht:f -g relM.
16
I. M A I N P R O B L E M A N D P R E L I M I N A R Y N O T I O N S
In particular, if M is the empty set, the notation will be simply f g and ht:f N g as in 5 8. As in (8.1) one can easily prove that the relation of homotopy relative to { M } of the maps SZ is an equivalence relation. Hence, 9 is divided into disjoint classes, called the homotopy classes relative to { M } or the homotopy classes i n 9. The classification problem of the maps 9 is to enumerate these classes in terms of known topological invariants. Let B be a subspace of Y which intersects Y p for each EM. A map f E 52 is said to be deformable into B relative to { M }, if there exists a map g : X + B such that f N i g re1 { M } p1
where i denotes the inclusion map i :B c Y . In particular, B may consist of a single point yo in the intersection of the subspaces Y p .Let 0 denote the constant map O ( X ) = yo. Then 0 is in SZ. A map f is said to be null-homotopic or inessential relative to { M } if f N 0 re1 { M }, otherwise it is said to be essential relative to { M }. Now, let us define the important notion of deformation retract. A subspace A of a space X is said to be a deformation retract of X if there exists a deformation ht:X -+ X , (0 < t < l), such that h, is a retraction of X onto A . If the deformation ht satisfies a further condition that ht(a) = a for each a E A and each t E I , then A is said to be a strong deformation retract of X . In this case, lat will be called a retracting deformation of X onto A . In Ex.T a t the end of this chapter, we shall see that, in a great many cases, these two notions are equivalent. If X denotes the space obtained by deleting from En an interior point which may be assumed to be the origin 0 without loss of generality, then Sn-1 is a strong deformation retract of X . In fact, a retracting deformation ht: X + X , (0 < t < l), is given by ht(x) =
(1 - t
+" )x 1x11r
for each x E X and t E I , where I x I denotes the distance between 0 and x . On the other hand, a point of the n-sphere Sn is a retract of Sn but not a deformation retract of Sn. For another example of strong deformation retract, we strengthen (4.2) by the following proposition which can be proved by an easy modification & the arguments used in the proof of (4.2). Proposition 10.1 I f ( X ,A ) is a (finitely) triangulable pair, then the closed subspace L = ( X x 0) U ( A x I ) is a strong deformation retract of the product space M = X x I .
More examples of strong deformation retracts are given in Ex.S a t the end of the chapter.
11. H O M O T O P Y E Q U I V A L E N C E S
I7
11. Homotopy equivalences Under the general notations of the preceding section, let u s consider two given maps f:X+Y, g:Y+X such that f (X,,) c Y,, and g( Y,,)c X,, for each ,u E M . g is called a left homotopy inverse of f relative to { M } and f is called a right homotopy inverse of g relative to { M }, if there exists a homotopy h l : X + X (0 < t < 1) such that h, = g f , h, is the identity map on X, and ht(X,,)c X,, for each ,u E M and each t E I . g is called a two-sided homotopy inverse of f relative to { M } if it is both left and right homotopy inverse of f relative to { M }. The map f : X + Y is said to be a homotopy equivalence relative to { M }, if it has a two-sided homotopy inverse relative to { M }. In notation,
f : X = Yrel{M}. The spaces X and Y are said to be homotopically equivalent relative to { M }, if there exists a homotopy equivalence f : X II Y re1 { M }. In this case, we also say that X and Y are of the same homotopy type relative to { M } and we shall use the notation
X
N
Y re1 { M }.
The following two special cases are of frequent occurrence in the sequel. Firstly, if M = 0 , that is to say, if there is no system of relativity, then we simply omit the phrase “relative to { M } ” in the terminology given above. For example, two spaces X and Y are said to be hotnotopically equivalent, X II Y, if there are maps f :X + Y and g: Y + X such that the compositions gf and fg are homotopic to the identity maps on X and Y respectively. Secondly, assume that M consists of a single element p and thaVX,, = A , Y,,= B. Then a map f : (X, A ) + (Y, B) is said to be a homotopy equivalence and is denoted by f : ( X , A ) = ( Y , B )if f : X - Y r e l { A , B } . The pairs (X, A ) and (Y, B) are said to be homotopically equivalent, ( X ,A ) N (Y, B), if there exists a homotopy equivalence f : (x,A ) N (Y, B). The notion of homotopy equivalence introduced above is justified by the fact that the extension problem as broadened in 5 9 depends essentially only on the homotopy type of the pair ( X ,A ) and that of the space Y.This result can be precisely formulated as follows. Assume that (X, A ) e (X‘,A’) and Y N Y’and let g : A + Y be a given map. Let h : A c X and h’:A‘c X’ denote the inclusion maps. By definition, there exist maps + : ( X , A ) +(X‘,A1),+’:(Xt,A’)+ ( X , A ) , y : Y + Y ’ , y I : Y ’ + Y such that composed maps 4‘4,
#I,
y‘y and yy’ are homotopic to the corre-
18
I. M A I N P R O B L E M A N D P R E L I M I N A R Y N O T I O N S
sponding identity maps. Let X:A + A ’ and X‘:A’ + A denote the maps defined by the maps 9 and 9’ respectively, and let g’ = ygX‘:A’-+ Y’.
If the extension problem for g has a solution over X,i.e., if these exists a map f :X-+ Y such that f h N g, then the extension problem for g’ also has a solution over X ’ . In fact, f ’ = yf+’:X’ + Y’ satisfies the relation f ‘h‘ = yf+’h‘
= yf
hX’
N
ygx’
= g’.
Conversely, if there exists a map e’:X’ -+ Y’ such that e‘lz’ N g’, then the extension problem for g has a solution over X . In fact, e = y’e’+:X -+ Y satisfies the relation
eh
=
y’e’+h
= y‘e’h’X N
y’g’X
=
y’ygX’X N g.
Examples of homotopy equivalences: (1) If f : X + Y is a homeomorphism of X onto Y , then f - l is obviously a two-sided homotopy inverse of f . Hence homeomorphs are of the same homotopy type. ( 2 ) If X is a deformation retract of Y , then the inclusion map f : X c Y is a homotopy equivalence. According to the homotopy axiom and the algebraic axioms, a homotopy equivalence induces isomorphisms on the homology groups and the cohomology groups. Hence, homotopically equivalent spaces have isomorphic homology groups and cohomology groups. A property of spaces is said to be a homotopy invariant if it is preserved by homotopy equivalences. Almost all invariants studied in algebraic topology are homotopy invariants. The role of homotopy equivalences having been clarified, it is natural to search for two-sided homotopy inverses, if any, of a given map f : X + Y relative to a given system { M } . For this purpose, it is sometimes helpful to observe the following fact. If a given map f : X -+ Y has both left and right homotopy inverses relative to a given system { M }, then it has a two-sided one. In fact, if g’: Y -+ X is a left homotopy inverse of f relative to { M } and if g”: Y + X is a right homotopy inverse of f relative to { M }, then it is easy to verify that the composed map g = g’fg”: Y + X is a two-sided homotopy inverse off relative to { M }.
12. The mapping cylinder Let f :X+ Y be a given map. By the process of topological identification, we shall construct a space M f which is called the mapping cylinder of f . For this purpose, let us consider the topological sum
w
=
(X x I)
u
Y
of the spaces X x I and Y . If we identify ( x , 1) E X x I with f ( x ) E Y for
I9
12. T H E M A P P I N G C Y L I N D E R
every x E X , we obtain a quotient space M f , the mapping cylinder of f , and a natural projection p:W-+Mf. One can easily verify that p maps Y homeomorphically onto a closed subspace p(Y)of M f . By means of this imbedding, Y will be considered as a closed subspace of MI. On the other hand, the map
g : X -+M f given by g(x) = p ( x , 0 ) for each x E X carries X homeomorphically onto the closed subspace p ( X x 0) of M f . By means of this topological map g, X is imbedded as a closed subspace of MI. Thus X and Y are considered as disjoint closed subspaces of M f and will be called the domain and the range of the mapping cylinder M f .
range Y of the mapping cylinder MI Y is a strong deformation retract of M f .
Proposition 12.1. The
f :X
+.
Proof. Define a system of functions h t : M f +. M f , (0
hdP(x, 41 = P(% s MY)
+ t - st )
of
a map
by taking
=Y
for every x E X , y E Y , s E I and t E I. By (6.1), it is easily verified that ht is a homotopy. It is obvious that h, is the identity map on M I , h, is a retraction of MI onto Y , and ht I Y is the identity map on Y for each t E I. Hence Y is a strong deformation retract of M f . I Proposition 11.2.
For any given map f : X +. Y , the maps g : X
Proof. Let h t : X
M f , (0 < t
p f : X --* M f are homotopic. +.
+. MI
and
< l), denote the homotopy defined by
ht(4
=
P(% t )
for every x E X and t E I . Then we have h, = g and h, = p f . Hence g N pf. I TWOmaps f : X -+ Y and f ' : X ' +. Y' are said to be homotopically equivalent if there exist homotopy equivalences 4 : X N X' and y : Y N Y' such that yf N f'4. Let +':XI N X and y ' : Y' N Y denote two-sided homotopy inverses of 4 and y respectively. Then the three conditions Yf
= f'4* f = y ' f ' h yf4' = 1'
are mutually equivalent. As an immediate consequence of (12.1) and (12.2), we have the following theorem which justifies the introduction of the notion of mapping cylinders. Theorem 12.3. Evevy mil$ f :X clusioii map, namely, g : X c M f .
-,Y
is homotopically equivalent to an in-
I. M A I N P R O B L E M .4ND P R E L I M I N A R Y N O T I O N S
20
Finally, the following special case of mapping cylinder is of importance. If Y consists of a single point v , then the map f :X-+ Y has to be the constant map f ( X ) = v. In this case, the mapping cylinder M f is called the join of X to v or the cone over X and will be denoted by k ( X ) .The point v is called the vertex of k ( X ) .
13. A generalization of the extension problem The extension problem as broadened in 3 9 suggests a generalized problem which can be displayed diagrammatically by
where f and g are given maps and h must be found such that hg N f. However, this generalized problem is equivalent to the extension problem of the map f :X + Y over the mapping cylinder M , 2 X of the map g :X + Z . More precisely, we have the following Proposition 13.1. For any two maps f :X-+ Y and g : X -+Z, the following two statements aye equivalent : (i)There is a map h : Z -+ Y such that hg N f. (ii)There i s a map F : M , -+ Y such that Fq N f , where q : X c M , denotes the inclusion map. Proof.
(i)-+ (ii).According to (12.1), there is a retraction r : M , 3 .Z
defined by Hence the map k : Z -+ Y has an extension F
(ii)+ (i).Let h
=F
=
h r : M , + Y . By (12.2),
12. Then, by (12.2), we have
hg=FPg-Fq-/.I The derived algebraic problem of this general problem is as follows. In any homology theory or cohomology theory satisfying the EilenbergSteenrod axions, the given maps f : X -+ Y and g : X -+ Z induce the homomorphisms /*, g,, /*, g* indicated in the following diagrams: H , (X)-'-+ /
gh\
H%a(Z)
H, (Y )
H y X ) + L H m ( Y)
a;, /4 Hm(Z)
Then the derived algebraic problem is to determine whether or not there exist homomorphisms 4 and y such that #Jg* = f* and g*y = f *.
14. T H E P A R T I A L
MAPPING CYLINDER
21
The existence of the homomorphisms 4 and y is a necessary condition for the existence of a map h : X + Y such that hg = f . In fact, if such an h exists, then 4 = h, and y = h* are solutions of the derived algebraic problem. In many cases, this necessary condition can be used to prove the nonexistence of h. For example, let us assume X = Sm = Y , Z = S" with m # 0 and m # n. If f : X -+ Y is of degree different from zero and if g : X -+Z is any given map, then there exists no map h : Z -+ Y such that hg N f . 14. The partial mapping cylinder The objective of the present section is to show that the generalized problem in 9 13, which is so far the broadest form of the extension problem, is equivalent to a retraction problem, which is apparently the narrowest form of the problem. For this purpose, let us introduce the partial mapping cylinder M g ( X )of a given map g : A --f Y defined on a subspace A of a space X . Consider the
W=XU(A x I)UY
topological sum
of the spaces X , A x I and Y . For each a E A c X , identify a with (a, 0 ) E A x I , and ( a , 1) with g(a) E Y . In this way, we obtain a quotient space M g ( X ) which is called the partial mapping cylinder of g over X and a natural projection p : W-+M,(X). Then one can easily see that 9 maps X and Y homeomorphically onto disjoint closed subspaces p ( X ) and p ( Y )of M g ( X )respectively. Thus, X and Y will be considered as disjoint closed subspaces of M g ( X )and will be called the domain and the range of the partial mapping cylinder M g ( X ) . Lemma 14.1. If the range Y of M g ( X )is a retract of M g ( X ) then , there exists
a map f : X
Y such that f I A
g. Proof. Let r : M g ( X )3 Y be a retraction and define f : X -+ Y by f Consider the homotopy he : A + Y , (0 < t < l), defined by +
N
h(4 = r ( P ( a , t ) ) , I t is obvious that h,
=
f I A and h,
= g.
of
r 1X.
(aEA,tEI).
I
Lemma 14.2. If there exist a ma+ f : X
range Y
=
-+
Y such that f I A
N
g, then the
M g ( X )is a retract of M g ( X ) .
1A
g, there is a homotopy ht : A + Y (0 S t S 1) such = g. Let r : M g ( X )-+ Y be defined by (if z = p ( x ) ,x E X ) , ~ ( 2= ) ht(x) (if z = p ( x , t ) . ( x , t ) E A x I ) , ( i f 2 = P ( Y ) ,Y E Y ) . Then i t is obvious that r is a retraction. I Proof. Since f
that h,,
=
N
f 1 A and Iz,
{I("
22
I. M A I N P R O B L E M A N D P R E L I M I N A R Y N O T I O N S
Now, consider two given maps f : X + Y and g : X +Z. Since X is a subspace of the mapping cylinder M8 of g, the partial mapping cylinder Mf(M,) of f over k ? g is defined. Then we have the following Theorem 14.3. There exists a map h : 2 + Y such that hg N f if the range Y of Mf(M,) is a retract of Mf(kfg). Proof. Suficiency. Assume that Y is a retract of Mf(M,). Then, by (14.1) there exists a map F : k f g --f Y such that F I X N f . According to (13.1), this implies the existence of a map h : 2 + Y such that hg N f . Necessity. Assume that there is a map h :2 + Y such that hg N f . Then, by (13.1), there is a map F : M g + Y such that F I X N f . According to (14.2),this implies that Y is a retract of M f ( M g ) .I The construction of the partial mapping cylinder is closely related to that of the adjunction space given in 5 7. In fact, for reasonable spaces X and A , (see Ex.P), the adjunction space obtained by adjoining X to Y by means of a map g : A + Y is homotopically equivalent to Mg(X). Finally, if Y consists of a single point v , then the map g : A + Y has to be the constant map g ( A ) = v. In this case, the partial mapping cylinder M,(X) is called the partial cone over the subspace A of the space X . Obviously, Mr(x) = x u k(A) k(X).
15. The deformation problem Let us consider a given map f : ( X ,A ) + ( Y ,B ) and a given subspace Y oof Y . According to our general definition in 5 10, f is said to be deformable into Y o if there exists a homotopy f t : ( X ,A ) -+ ( Y ,B ) , (0 Q t < l), such that f, = f and f l ( X )c Yo. The deformation problem of f : ( X ,A ) + ( Y ,B ) into Yo is to determine whether or not f is deformable into Yo.Let B, = B n Yo.Then this problem can be described diagrammatically by
where h : (Yo,B,) c ( Y ,B ) denotes the inclusion map and g : ( X ,A ) + (Yo,B,) must be found such that hg N f . For the important special case that A = 0 , we may always take B = a and hence B, = a.In this case, we just omit A , B and B, from the notation given above. Hence, the deformation problem is actually the dual problem of the extension problem given in 5 9. The derived algebraic problem and the necessary conditions of the deformability are dual to those of the extension problem in an obvious way. They can be formulated and applied similarly and hence are omitted. Now we are in a position to prove the interesting result that the broadest
15.T H E D E F O R M A T I O N P R O B L E M
23
extension problem in 3 13 is equivalent to a deformation problem. For this purpose, let f : X + Y and g : X + Z be given maps. Let
W = Mf(Mg), V = k ( Y ) , then Y c W and Y c V . Define a map : ( W , Y ) + (V, Y ) by taking
+
(if w E Y ) , (if w E M,), ( q ( f ( x ) .1 - t ) , (if w = p ( x , t ) , x E x , t E I ) , w,
+(w) = v ,
where v denotes the vertex of the cone k(Y)and p : M , U ( X x I ) U Y + W and q : Y x I -+ V denote the natural projections. Theorem 15.1. There exists a map h : 2 + Y such that hg
+ : (W,Y )
+ (V, Y
)is deformable into Y .
N
f i# the map
Proof. Necessity. According to (14.3) there is a retraction Y : W 3 Y . Define two homotopies &, qt : ( W , Y ) -+ ( V , Y ) ,(0 < t Q I ) , by taking
lt(4
=
lw' 1))
q[rp(x,1 - t
w,
+ s t ) , 1 -s],
q H 4 , 1 -t l ,
V t ( 4 =
(if w E Y ) , (if w E M,), (if w = p ( x , s), EX, S E I ) , (if w E Y ) , (if w E M,), (ifw = p ( x , s ) , x E X , s E I ) .
( q I r p ( x , s ) ,(1 - s ) (1 - t ) j , Then one can easily verify that lo = l1 = qo and ql = r. Hence is deformable into Y . Szcficiency. There is a homotopy #Jt : ( W , Y ) + ( V , Y ) ,(0 Q t Q I ) , such that #Jo = and bl(W) c Y . Then we may define a retraction r : W 3 Y by taking (if w E Y ) , (if w E M,), r(w) = (if w = p ( x , t ) , x E X , 0 Q t Q i), ;:P(% 2 4 , (if w = p ( x , t ) , x E X , 4 Q t Q 1). $2-21 f (4,
+,
+
+
[
Hence, by (14.3), there exists a map h : 2 + Y such that hg f. I On the other hand, the deformation problem is equivalent to an extension problem with side conditions. To see this, let f : ( X ,A ) + ( Y ,B ) be a given map and Y o a given subspace of Y . Consider the subspace X x 0 of X x I and the map F : X x 0 -+ Y defined by F ( x , 0) = f ( x ) for each x E X . Then the following proposition is obvious. Proposition 15.2. f : ( X ,A ) -+ ( Y ,B) is deformable into Y o i# F has an extension G : X x I -+ Y such that G ( A x I ) c B and G ( X x 1) c Yo.
The special case of the deformation problem where Y o = B is of importance. In this case, we have nice necessary conditions as given in the following proposition the proof of which is immediate and hence omitted.
24
I. M A I N P R O B L E M A N D P R E L I M I N A R Y N O T I O N S
Proposition 15.3. I f a m a p f : ( X ,A ) + ( Y ,B) i s deformable into B, then it induces trivial homomorphisms on homology and cohomology groups, that i s to say, f* = 0 and f * = 0.
I n particular, if ( X ,A ) = ( Y ,B ) and if f is the identity map, we have the following definition. The pair ( X ,A ) is said to be deformable into A if the identity map on ( X ,A ) is deformable into A . If this is the case, then the inclusion map i : A c X is a homotopy equivalence. Furthermore, in case A has the H E P in X with respect to A , then the pair ( X ,A ) is deformable into A iff A is a deformation retract of X .
16. The lifting problem Consider a given map
p:E-+B of a space E onto a space B. Let X be a given space and f : X +- B a given map. Then the lifting problem in the restricted form is to look for a map g : X + E such that p g = f .The map g is called a lifting of f (relative to p). This problem is dual to the restricted extension problem of Q 2.
Dual to the homotopy extension property of Q 9, we have the following important notion of covering homotopy property.
Definition 16.1. A map p : E + B is said to have the covering homotopy property (abbreviated CHP) for a space X,if, for every map g : X -+ E and every homotopy f t : X -+ B , (0 < t < l), of the map f = Pg, there exists a homotopy gt : X + E , (0 < t < I), of g which covers f t , that is to say, figt = f t for every t E I . The covering homotopy property is basic to many constructions in homotopy theory. Fortunately, this important property holds in a great many of cases, namely, the various fiber spaces. See Chapter 111. Now, if p : E + B has the CHP for X , then the restricted lifting problem for f : X + B obviously depends only on the homotopy class of f . In this case, it is equivalent to the broadened form, namely to ask for a m a p g : X + E such that fig N f . Hereafter, when we refer to the lifting problem, we shall mean this broadened form unless otherwise stated. Once the lifting problem is broadened as above, the original condition that p is onto is no longer important. In fact, we have a generalized problem displayed diagrammatically by X - L
Y
where f and h are given maps and g must be found such that hg N f . This general problem is dual to the generalized extension problem of 3 13. If h is onto, then i t is the lifting problem for f ; if h is an inclusion map, then it is the deformation problem for f into the subspace 2 of Y . On the other
17. T H E M O S T G E N E R A L
PROBLEM
25
hand, this general problem is equivalent to a deformation problem. More precisely, we have the following Theorem 16.2. For any two maps f : X + Y and h : Z + Y , the following two statements are equivalent: (i) There exists a map g : X +. 2 such that hg N f . (ii) T h e m a p pf : + Mh is deformable into 2, where p : Y C Mh denotes the inclusion map.
x
Proof. (i) +. (ii). According to (12.2), we have ph N q, where q : Z c Mh denotes the inclusion map. Then, (i) implies that pf N phg 3i qg. Hence, f i f is deformable into 2. (ii) +. (i).There is a homotopy gt : X + Mh, (0 < t < l), such that go = pf and g,(X) c 2. Let g : X + Z denote the map defined by g,. By (12.1), there is a retraction 7 : M h z Y such that r I Z = h. Let f t : +. Y, (0 < t < l ) , denote,the homotopy given by f t = rgt for each t E I . Then we have f o = rpf = f and f , = rqg, = hg. Hence hg N f . I
x
17. The most general problem In this final section of the chapter, we shall formulate the most general problem which contains essentially all problems described in this chapter as special cases. Consider three given spaces X,Y ,Z together with three given systems of subspaces { X, }, { Y , }, { 2, } indexed by p E M . Following our general ideas in 4 10, we are concerned with only those maps and homotopies X +. Y,X +.2,Z +. Y which are relative to { M } in the obvious sense. Then our most general problem can be displayed diagrammatically by X - L
Y
z where the maps are understood to be relative to { M }, f and one of g, h are given, and the other is to be found such that hg N f re1 { M }. Nearly everything done in this chapter for the simplest extension, deformation and lifting problems admits generalizations some of which will be given in the exercises. Although the problem is formulated in the most general form, the only cases considered in remainder of the book have 0, 1 or 2 elements as the domain M of the index p.
EXERC I S E S A. Elementary properties of retracts
1. Every retract of a Hausdorff space is closed. 2. Every subspace which consists of a single point is a retract of the containing space.
I. M A I N P R O B L E M A N D P R E L I M I N A R Y N O T I O N S
26
3. If A is a retract (deformation retract) of X and if B is a retract (deformation retract) of Y , then A x B is a retract (deformation retract) of x Y. 4.If X is a retract (deformation retract) of Y and if Y is a retract (deformation retract) of 2, then X is a retract (deformation retract) of Z.
x
B. More properties of solid spaces
1. A retract of a solid space is solid. 2. Any two maps of a binormal space into a solid space are homotopic. 3. A binormal solid space is contractible to a point. C. AR’s and ANR’s
I n classical algebraic topology, attention is confined to very well-behaved spaees, namely polyhedra. One class of spaces whose members retain many of the desirable properties of polyhedra is the absolute neighborhood retracts. A subset A of a space X is said to be a neighborhood retract of X, if it is a retract of some open subspace U of X . A metrizable space Y is said to be an absolute retract (abbreviated AR), whenever a topological image of Y as a closed subset 2, of any metrizable space Z is necessarily a retract of Z. A metrizable space Y is said to be an absolute neighborhood retract (abbreviated ANR), whenever a topological image of Y as a closed subset 2, of any metrizable space 2 is necessarily a neighborhood retract of Z. Prove that every compact AR is solid. D. Dugundji’s extension of Tietze’s theorem
Let X be a metrizable space and A be a closed subspace of X . Prove the following assertions. 1. Existence of canonical coverings. There exists of canonical covering of X \A ; by this we mean an open covering { U } of X \A satisfying the following conditions : (CC1) { U } is locally finite, [K; p. 1261. (CC2) Every neighborhood of a boundary point of A in X contains infinitely many sets of { U }. (CC3) For each neighborhood V of a E A in X , there exists a neighborhood WofainX,WcV,suchthat U c v i f U ~ U ( }andUmeetsW. 2. Replacement by polyhedra. By the aid of a canonical covering of X \A, prove that there exists a space Y and a map p : X -P Y with the following properties : (RPl) The partial map p I A is a homeomorphism of A onto a closed subset p ( A ) of Y . (RP2) Y \p(A) is an infinite simplicial polyhedron with Whitehead’s weak topology and ,u(X \A) c Y \ p ( A ) .
EXERCISES
27
(RP3) Every neighborhood of a boundary point of p ( A ) in Y contains infinitely many simplexes of the simplicia1polyhedron Y \ p ( A ) . 3 . The extended Tietze’s theorem. If f : A -+ S is a map of A into a locally convex linear space S , then there exists an extension F : X + S of f such that F ( X )is contained in the convex hull off ( A ) in S. Proofs are given in [Dugundji I].
E. Kuratowski’s imbedding Let X be a given metrizable space and d a distance function which makes X a metric space. Replacing d by d / (1 + d ) if necessary, we may assume that d is bounded. A metric space X with a bounded distance function d is usually called a bounded metric space. 1. Let S denote the totality of bounded continuous real functions defined on X . Prove that S forms a Banach algebra with its norm defined by
If I
=
SUPZ€XI f
(4 I
2. For an arbitrary point a E X,consider the bounded continuous real function f a E S defined by fa(x) =
d(a, X )
for each x E X.Prove that the correspondence a -+fa defines an isometric map X : X + S. X is called Kuratowski’s imbedding of the bounded metric space X,[Kuratowski 11. 3 . Wojdyslawski’s theorem. The image X ( X ) of Kuratowski’s imbedding X : X -+ S is a closed subset of the convex hull 2 of X(X)in S. If X is separable, then so is 2, [Wojdyslawski I]. F. Spaces obtained by topological identification
1. Mobius strip. Take a rectangle ABCD and identify the side A B with the side C D so that A goes into C and B into D as shown in the following figure:
Fig. I . On the left, there is a rectangle A B C D with M a n d ” as mid-points of two sides. The figure on the right is obtained by pasting t h e two sides of the rectangle together so t h a t A goes into C, B goes into D, and M goes into N.
The quotient space X thus obtained is called the Mobius strip. Let the construction be so carried out that the mid-points M , N of A B and CD are identified. As a consequence the line M N will go into a Jordan curve J
28
I. M A I N P R O B L E M A N D P R E L I M I N A R Y N O T I O N S
on the strip X . Prove that J is a deformation retract of X and that X \ J is pathwise connected. 2. Torus and Klein bottle. Take the 1-sphere S1represented as the complex numbers z with I z I = 1 and consider the product space W = S1 x I. If we identify (z, 0) and (z, 1) for each z E S1,we obtain from W a quotient space T , called the torus. On the other hand, if we identify (z, 0) and (2-1, 1) for each z E S1,we obtain from W a quotient space U , called the Klein bottle. In each case, the line 1 x I will go into a Jordan curve J on the quotient space. Prove that J is a retract of the quotient space but not a deformation retract of the quotient space. 3. Closed orientable surfaces. Consider a plane convex region W bounded by a 4g-sided regular polygon P.Let the sides of P be denoted by
- - a,bgcgdg in a certain positive sense. For each i = 1, - - -,g, let us identify a$ with ct-1 P
=
alb,cldl
and br with dr-1, where ct-1 denotes ci with reverse direction. The quotient space Zg thus obtained is called the closed orientable surface of genus g. Prove that the Betti numbers of Zgare
RO= 1, R1 =2g,
R 2 = 1.
4. Closed non-orientable surfaces. Consider a plane convex region W bounded by a 2n-sided regular polygon P. Let the sides of P be denoted by
P
=
albla,b,
* * *
anbn
-
in a certain positive sense. Identify at with bt for each i = 1,2, * * , n. Thus we obtain a quotient space f i n , which is called the closed non-orientable surface of characteristic 2 - n. Prove that the Betti-numbers of s2, are
RO= 1, R 1 = n - I ,
R2=0.
G. An application of the cone k(X)
Let us use the notation of 9 10. Prove that a map f E a is null-homotopic relative to { M } iff there exists a map F : k ( X ) + Y of the cone k ( X ) into Y such that F 1 X = f , F ( v ) = yo, and F [ k ( X , ) ] c Y , for each p E M . An important special case is as follows: a map f : S n + Y is homotopic to a constant map iff f has an extension F : En+l -+ Y . Using this, prove that the identify map on S n is not null-homotopic. H. Extension properties
A subset A of a space X is said to have the extension property in X with respect to a space Y , if every map f : A + Y can be extended over X . The subset A is said to have the neighborhood extension property in X with respect to Y , if every map f : A -+ Y can be extended over some open set U of X which contains A .
EXERCISES
29
Thus, a space Y is solid iff every closed subset A of any normal space X has the extension property in X with respect to Y . The subset A is said to have the absolute extension property in X , if it has the extension property in X with respect to every space Y . The subset A is said t o have the absolute neighborhood extension property in X , if it has the neighborhood extension property in X with respect to every space Y . Prove the following two assertions: 1. A subset A of a space X is a retract of X iff it has the absolute extension property in X . 2. A subset A of a space X is a neighborhood retract of X iff it has the absolute neighborhood extension property in X . 1. Borsuk's extension theorems
By means of the results in the exercises D and E, prove the following two extension theorems: 1. A metrizable space Y is an AR iff every closed subset A of an arbitrary metrizable space X has the extension property in X with respect to Y . 2. A metrizable space Y is an ANR iff every closed subset A of an arbitrary metrizable space X has the neighborhood extension property in X with respect to Y . As a consequence of the theorem 1, show that every convex subset of a locally convex linear space is an AR; in particular, every simplex is an AR and every euclidean space Rn is an AR. J. Separable ANR's and compact ANR's
Prove the following two theorems: 1. For a separable metrizable space Y , the following statements are equivalent : (i) Y is an ANR. (ii) A topological image of Y as a closed subset 2, of any separable metrizable space 2 is necessarily a neighborhood retract of 2. (iii) Every closed subset A of an arbitrary separable metrizable space X has the neighborhood extension property in X with respect to Y . 2. For a compact metrizable space Y , the following statements are equivalent : (i) Y is an ANR. (ii) Y is homeomorphic with a neighborhood retract of the Hilbert parallelotope I". (iii) Every closed subset A of an arbitrary normal space X has the neighborhood extension property in X with respect to Y . State and prove the analogous theorems for the AR's. It follows from the theorem 1 that, for a separable metrizable space Y ,
30
I. M A I N P R O B L E M A N D P R E L I M I N A R Y N O T I O N S
the definitions of AR’s and ANR’s given in Ex.C are equivalent to Kuratowski’s modifications of Borsuk’s original definitions. In fact, the statement (ii) in theorem 1 is used as the definition of ANR’s by Kuratowski as well as by Lefschetz, [L,; p. 581. K. Operations on ANR’s
Prove the following properties of ANR’s: 1. Every open subspace of an ANR is an ANR. 2. Let Y be an ANR and B a closed subspace of Y . B is an ANR iff B is a neighborhood retract of Y . 3 . Let Y and 2 be metrizable spaces. The product space Y x 2 is an ANR iff both Y and 2 are ANR’s. 4. Let Y , and Y , be two closed subspaces of a metrizable space Y with Y = Y , U Y , . If Y , , Y , , Y , f l Y , are ANR’s, then so is Y . [Borsuk 21. 5. Let a metrizable space Y be covered by a collection { Y,, I p E M } of disjoint open subspaces. If Y,, is an ANR for every p E M , then so is Y . 6. Let a metrizable space Y be covered by a countable collection { Y , I n = 1,2, * * } of open subspaces. If Y , is an ANR for each n = 1,2, * , then so is Y . [Hanner 11.
-
- -
L. Locally finite simplicial polyhedra
By induction on the number of simplexes and with the aid of the property 4 in Ex.K, prove that every finite simplicial polyhedron is an ANR and hence a compact ANR. [Borsuk 21. Then, by means of the properties 1, 5 and 6 in Ex.K, prove that every locally finite simplicia1 polyhedron is an ANR. [Liao 1 ; Hanner 11. M. The general homotopy extension theorem
The notion of the HEP given in $ 8 can be localized and apparently generalized as follows. A subspace A of a space X is said to have‘the neighborhood homotopy extension property (abbreviated NHEP) in X with respect to a space Y , if every partial homotopy
ht:A-+Y,(O
g t : V + Y , (0 < t G l ) , over some open neighborhood V of A in X such that go = f I V . However, if A is a closed subspace of a normal space X , then the NHEP is equivalent to the HEP. In other words, A has the H E P in X with respect to Y iff A has the NHEP in X with respect to Y . By the aid of a Urysohn’s
EXE RCI SES
31
characteristic function [L,; p. 271, prove this general homotopy extension theorem. N. Borsuk’s homotopy extension theorems
By means of Borsuk’s extension theorem in Ex.1 and the general homotopy extension theorem in Ex.M, prove the following theorems : 1. If Y is an ANR, then every closed subspace A of an arbitrary metrizable space X has the H E P in X with respect to Y . [H-W; p. 861. 2. If Y is a compact ANR, then every closed subspace A of an arbitrary binormal space X has the H E P in X with respect to Y . As an application of these theorems, prove that every contractible ANR is an AR. 0. Closed A N R subspaces in an A N R
Let X be an ANR, A a closed subspace of X , and T denote the subspace ( X x 0) U ( A x I ) of the product space X x I . Prove that the following statements are equivalent, [Hu 31 : (1) A has the AHEP in X . (2) T is a retract of X x I . (3) T is an ANR. (4) A is an ANR. As a well-known special case, every subpolyhedron A of a locally finite simplicia1 polyhedron X has the AHEP in X . P. Relation between partial mapping cylinder and adjunction space
Let g : A -+ Y be a given map defined on a given subspace A of a space X into a space Y . Consider the partial mapping cylinder M g ( X )and the adjunction space Z obtained by adjoining X to Y by means of g. There is a natural map t:M,(X)-+Z defined by
I
q ( w ) , (if u t ( u ) = qg(a), (if u
= =
p(w),w E X U Y ) , +(a,t ) , a € A ,t E I ) ,
where p : X U (A x I ) U Y -+ M g ( X ) and q : X U Y + Z denote the natural projections. By means of these natural projections, Y can be considered as a subspace of both M,(X) and Z. Hence, t is actually a map (M,(X), Y ) into (Z, Y ) such that t I Y is the identity map on Y . Using the equivalence of (2) and (4) in the preceding exercise, prove that the map t : ( M g ( X ) ,Y ) + ( Z , Y ) is a homotopy equivalence if X is an ANR and A is a closed ANR subspace of x.
32
I. M A I N P R O B L E M A N D P R E L I M I N A R Y N O T I O N S
Q. Local contractibility of ANR’s
A space X is said to be locally contractible if, for each x E X and every open neighborhood U of x , there is an open neighborhood V c U of x which is contractible to the point x in U. By means of Kuratowski’s imbedding, prove that every ANR is locally contractible. As a partial converse of this result, [Borsuk 21 proved that every locally contractible compact metrizable space of finite dimension is an ANR. R. Dominating polyhedra of ANR’s
A space X is said to be dominatedby another space D if there are two maps f : X +-D, g : D
-P
X
such that the composed map gf is homotopic to the identity map on X. In this case, D is called a dominating spGce of X . Prove the following assertions : 1. Every ANR has a dominating simplicial polyhedron. 2. Every separable ANR has a dominating locally finite simplicial polyhedron. 3. Every compact ANR has a dominating finite simplicial polyhedron. S. Some strong deformation retracts
Prove the following assertions: 1. If A is a strong deformation retract of a compact space X,Y is a Hausdorff space, and f : ( X ,A ) + ( Y ,B ) is a relative homeomorphism, then B is a strong deformation retract of Y . [Spanier 1; p. 2081. 2. In the relative n-cell ( Y ,B) obtained by adjoining En to B by means of a map g :Sn-l+ B defined on the boundary (n - 1)-sphere Sn-l of En, the subspace B of Y is a strong deformation retract of the space obtained from Y by deleting an interior point of En. 3. In the product space S m x Sn, the subspace Sm V S n = (Sm x to) U (so x Sn), where so E Sm and to E Sn are given points, is a strong deformation retract of the space obtained from Sm x Sn by deleting one point which is not in Sm v Sn. 4. In the product space Sl x S m x Sn, the subspace T = (Sl x Sm x to) U (Sl x so x Sn) U (ro x S m x Sn), where yo E Sl, so E Sm, to E Sn, is a strong deformation retract of the space obtained from S1 x Sm x Sn by deleting one point which is not in T . 5. In the product space ( S l v S m ) x Sn, the subspace S l V S m V S n is a strong deformation retract of the space obtained from (Sl \/ Sm) x Sn by deleting a point of Si x Sn not in Sl\/ Sn and a point of Sm x Sn not in
33
EXERCISES
Generalize this result to the one-point union of a finite number of spheres. 6 . If A is a closed subspace of a metrizable space X and if both A and X are AR’s, then A is a strong deformation retract of X . 7. If A is a strong deformation retract of a metrizable space X , then
Sm V S n .
T = ( X x O ) U ( A x I ) U ( X X 1) is a strong deformation retract of the product space X
x I. [Hu 21.
1.Relations between different notions of deformation retracts Assume that X and A are ANR’s and that A is a closed subspace of X . Prove that the following statements are equivalent, [Fox I] : 1. The subspace A is a strong deformation retract of X . 2. The subspace A is a deformation retract of X . 3. There exists a homotopy Izt : ( X ,A ) + ( X ,A ) , (0 Q t < I), such that Ido is the identity map on X and h,(X) c A .
U. Expansion of the relativity of a
null homotopy
Let X , Y be given spaces, A c X , B c Y given subspaces, and xo E A , yo E B given points. Consider the maps
f2 = Yx{ A , B ; x,, y o } . According to 8 10, SZ consists of the totality of the maps f : X + Y such that f ( A )= B and f b o ) = Yo. Let f E Q and let 0 denote the constant map O ( X ) = yo. Prove that f = 0 re1 { A , B } implies f N 0 re1 { A , B ;xo, yo } provided that the following side conditions are satisfied: (1) The subspace A U k(xo) of the cone k ( A ) has the HEP in K(A) with respect to B. (2) The subspace X U K(A) of the cone K(X) has the H E P in K(X) with respect to Y . These side conditions are satisfied, for example, if X is a finite simplicia1 polyhedron and A is a closed subpolyhedron of X . For more general cases see N and 0. V. A general expansion theorem
Let X be a metrizable space and
Nc M c A c X be closed subspaces. Let Y be a space and B a subspace of Y not necessarily closed. Establish the following Theorem. Let f , g : X -+ Y be two given maps such that f ( A ) = B, f I M
=
= B.
g I M , g(A)
34
I. M A I N P R O B L E M A N D P R E L I M I N A R Y N O T I O N S
Then f N g re1 { A , B ; .N } implies f N g re1 { A , B ; M } provided that the following conditions are satisfied. [Hu 41 : (1) N is a deformation retract of M . ( 2 ) The subspace P = ( A x 0 ) U ( M x I ) U ( A x 1) has the HEP in A x I with respect to B. ( 3 ) The subspace Q = ( X x 0) U ( A x I) U ( X x 1) has the HEP in X x I with respect to Y . The conditions (2) and (3) are satisfied, for example, if X is a finite simplicia1 polyhedron and A , M are closed subpolyhedra of X .
CHAPTER II S O M E SPECIAL CASES O F T H E M A I N PROBLEMS
1. Introduction In the first chapter, we have described the main problems in homotopy theory. However, the examples and their solutions given there are essentially trivial cases. The present chapter takes up the first nontrivial cases. Answers will be obtained by using homology and cohomology groups. By means of the elementary properties of the exponential map # : R --t S1 in 3 2, we begin with the classification problem of the maps S1+S1 in 3 3. Then, the classification problem of the maps of S1 into an arbitrarily given space X is treated in 3 4 and 3 5 by studying the fundamental group. In this connection, the classical relation between the fundamental group and the first homology group is given in 3 6. Dually, the classification problem of the maps X + S1 is studied in 3 7 by means of the Bruschlinsky group. Finally, we take up the higher dimensional sphere Sn. The Hopf theorems concerning the maps Sn + Sn and Kn + Sn are proved in 3 8. Dual to the Hopf theorem is the Hurewicz theorem, which is stated in fj 9; the proof of this important proposition is deferred until Chapter V, since it seems to depend in an essential way on the elementary properties of the homotopy groups.
2. The exponential map p: R + S' Represent the 1-sphere S1as the unit circle in the space K of all complex numbers, that is to say S ' = ( Z E K I l Z l = 1). Therefore, S' is a compact abelian topological group with the usual multiplication of complex numbers as group operation. Consider the map # : R + S' defined on the space R of real numbers by the formula p ( x ) = exp (2nxi) = eBnXi, ( x E R ) , where e denotes the base of natural logarithms and i the unit of imaginary numbers. This map p will be called the exponential map of R onto S'. The continuity of p and the following proposition are obvious. Proposition 1.1. The ex#onentiaZ map
P(x
p
is a homomorphism, that is, #-l( 1) is the subgrow# Z of
+ y ) = p ( x ) p ( y )for any x and y in R . The kernel
all integers.
Since p ( x ) = cos ( 2x4 + i sin (274, it is easy to see that p is one-to-one on
36
11. S O M E S P E C I A L C A S E S O F T H E M A I N P R O B L E M S
the open interval (- +, 4)of R. By a standard theorem in general topology, p maps every closed subinterval of (- $, 4) homeomorphically and hence p is a homeomorphism on the open interval itself. Since p is a homomorphism, this implies that p is a homeomorphism on each Baaslate of (- 4, 4). Therefore, for every open interval ( a , b) with b - a < 1, fl carries ( a , b) homeomorphically onto an open subspace of S1.This and (2.1) imply the following proposition 2.2. For every proper connected subspace U of S1, every component of f 1 (U ) homeomorphically onto U . By a path in a space X,we mean a map u: I
p carries
+x
of the unit interval I into X . The points u(0) and a( 1) are called respectively the initial point and the terminal point of the path u. They are said to be connected by u. If u(0) = u(l), then u is called a loop in X with u(0) = a(1) as its basic point. Proposition 2.3. (The covering path property). For every path u : I + S1 and every point x, E R such that p ( x 0 ) = a(O), there exists a unique path t : I --f R such that t(0)= x, and p z = u. Proof. By the continuity of u and the compactness of
partition
O=to
<***
I, there exists a
1
of I such that the image of every closed subinterval It = [ti, 0 < i < n, under u is a proper connected subspace Ug of S1. We shall establish the proposition by proving inductively the following assertion A t : There exists a unique map ti defined on the closed interval [0, ti] into R such that d o ) = x , , + t r = 0 I [O,ti]. A , is obvious. Assume 0 < i < n. It remains to prove that At implies For this purpose, let us assume A t . By the choice of the partition, Ut = u(It)is a proper connected subspace of S1. Let Vr denote the component of p-l(Ug) which contains the point tg(tg).By (2.2), the partial map p i = p 1 Vi is a homeomorphism of Vd onto Ui and hence the inverse P 6 - l : Ug + Vi is + R by taking defined. Define a map zt+] : [0, (if 0 < t < t i ) , (if tt < t < tt+J This proves the existence of ti+l.Since Vi is a component of p - l ( U i ) and pg is a homeomorphism, it follows easily that ti+,is also unique. I If the reader is familiar with the logarithmic function and its analytic continuations, he can easily see that the unique covering path t : I + R in (2.3) is given by 1 t(0)= xo, z ( t ) = - log, u(t), t E I . 2na
ta+1(t) =
tt(t), [ flg-ldt),
3.
CLASSIFICATION O F THE MAPS
s' +sl
37
Proposition 2.4. (The covering homotopy property). For every m a p f : X + R of a space X into R and every homotopy ht :X +S1, (0 < t < l), of the m a p h = p f , there exists a unique homotopy f t : X + R, (0 Q t < I ) , o f f such that p / t = ht for every t E I . Proof. For each point x E X , the partial homotopy ht 1 x gives a path in S1. Define f t : X + R, (0 < t < l ) , by taking f t I x to be the unique path in R' which starts from f ( x )and covers ht I x. Let
H : X x I+S,
F : X x I+R
be defined by H ( x , t ) = ht(x) and F ( x , t ) = f t ( x ) . For the existence part of the proof, it remains to establish the continuity of F . For this purpose, let xo E X . By the continuity of H and the compactness of I , there exist a neighborhood M of xo in X and a partition
0
=
to
< t, <
* * *
< tn-l < tn
=
1, It
=
[ti, ti+l]
of I such that the image of M x It under H is contained in a proper connected subspace of S1. By an inductive construction similar to that used in the proof of (2.3), we can define a map G : M x I + R such that
G ( x ,0)
=
f ( x ) , PG(x, t ) = H ( x , t )
for every x E h i ' and 1 E I . By the uniqueness part of (2.3),this implies that G = F 1 M x I . Since M is a neighborhood of xo in X , it follows that F is continuous at (xo, t ) for every t E I . Since xo is arbitrary, this completes the proof. I These properties of the exponential map p generalize to a very important class of spaces, namely, the covering spaces. See Chapter 111.
3. Classification of the maps S' -t S1 Let A denote the set of all maps of S1 into itself. According to ( I ; 5 8), these maps are divided into disjoint homotopy classes and the set of these homotopy classes is denoted byn(S1; 9). In this section we will see that the members ofn(S1; Sl) correspond in a natural way to the integers. Use will be made of the homology group Hl(S1),but the reader will realize that this may be avoided by adopting some other definition of degree. Before going further, let us first note that the classification problem of the maps f : S1+S1is equivalent to that of the maps4 : I +SLwitli+(O) = 4(1). In fact, the exponential map p : I +S1 of 9 2 is an identification map, ( I ; 5 6), and hence the assignment f + 4 = f p gives a one-to-one correspondence preserving homotopy. Now, let us consider the homology group H,(S1) which is isomorphic to the additive group 2 of integers. According to (I ; 4 8), the degree of a given map f : S1 -+ S1 is the integer deg(f) uniquely determined by the relation
f*W
=
d%(f)'C
38
11.
S O M E SPECIAL C A S E S O F THE MAIN PROBLEMS
for every c E H,(S1). Obviously, deg(f)depends only on the homotopy class [f] of f. Hence we may define the degree of a homotopy class a ~ n ( S 1Sl) ; by taking deg(a) = deg(f) where f E a. Thus we obtain a function 3 2. deg :n(S1;S1)
We are going to show that deg mapsn(S1;S1)ontoZis a one-to-one fashion. The exponential map p : R -+S1maps the open interval (0,l) homeomorphically onto the open subspace W = S1\ 1. Let q : W -+ (0,l) denote the unique homeomorphism such that pq is the inclusion map W c S1. To prove that deg is onto, let n be any integer and let : s1 +.S'
denote the map defined by+%(z)= zn for every z E S'. One can easily see that is also given by the following formula
+n
where t n : I +. R denotes the map defined by t n ( t ) = nt for every t E I. The following lemma is obvious and implies that deg is onto. Lemma 3.1. deg (&)
= n. To prove that deg is one-to-one, it suffices to establish the following
Lemma 3.2. If the degree of a given map f : S1-tS1 is n, t k n f Proof.
(0 Q t
Pick 8 E R such that p ( 8 ) = f (1). Define a homotopy
< l), by taking
-
ft(z) = f(z)ip(et),
(2 E
-+,,. ft
: S1 +.S1,
s1, t E I).
Set g = fl. Then we have f g and g(1) = 1. Let u = g p : I +S1.Since a(0) = 1 = p ( O ) , it follows from (2.3) that there exists a unique path t : I +. R such that t(0)= 0 and p t = u. Since pt(1) = u(1) = 1, t ( 1 ) is an integer, say t ( 1 ) = m. Define a homotopy qt : I -+ R, (0 < t <.I), by taking qt(s) = t ( s )
+ it&) - t t ( s ) ,
(s E I , t E I).
Then qo = t,q1 = tm and qt(0) = 0, qt( 1) = m for every t E I . Finally, define a homotopy gt :S1+.S', (0 < t Q I), by taking gt(z) =
{ Pqtdz), 1,
(ifZES'\ l), (if z = 1 ~ 9 ) .
Then go = g and g, = & This implies f N g = +m. I t follows that deg ( f ) = deg (q&). Since deg ( f ) = n and deg (&J = m, we obtain m = n and hence f I Thus we have proved the following II#J,,.
41 T H E
39
FUNDAMENTAL GROUP
Theorem 3.3. (Classification theorem). The homotopy classes n (S 1 ;Sl) are in a one-to-one correspondence with the integers Z under the function deg: n(S1; S1)+Z. The homotopy class of degreenisrepresentedby the map+n:Sl+.Sl.
To strengthen (3.3), let us make use of the fact that S1is a topological abelian group under the usual multiplication of complex numbers. The maps A form a ring with addition and multiplication defined as follows: If f , g : S1--f S1 are any maps, then f g and fg are the maps in A defined by
(f
+ g)(z)
+
= f(z)g(z),
(fd(z) = f[g(z)l
for every z E S1.Since the homotopy classes [ f + g] and [fg] obviously depend only on the classes [ f ] and [g], this ring structure of A induces a ring structure in n(S1; 9).Now, (3.3) can be strengthened by the following Corollary 3.4. The fwzction deg is an isomorphism of the ring n(S1; Sl)onto the ring Z of integers.
Proof. Let f , g ELI. It suffices to prove that
+
g) = deg(f) + deg(g), deg(fg) = deg(f)deg(g). dedf Let deg(f) = m and deg(g) = n. Then f = + m and g = + n . It is obvious from the definitions that
+ +n
+m
= +m+n,
+m+n = +mn.
This implies the corollary. I
4. The fundamental group Let X be a space and let x, E X ; let S2 denote the set of all maps of S1 into X such that the point 1 of S1is mapped at the point x, of X . For any map f : S1 +. X in S2, the composition f p : I +. X of f with the exponential map p of 3 2 is a loop in X with its basic point at x,. This correspondence is obviously one-to-one. Hence, by identifying f with f p , we may consider S2 as the set of all loops in X with given basic point x,. In symbols, we have
S2 = { f : I x I f (0)= x, = f ( 1) }. We may define a multipltication in S2 as follows. For any two loops f , g ES2, their product fag is the loop defined by --f
(f*g)
(')
=
f (2t)>
g(2t - 11,
(if 0 Q t Q &), (if 4 < t < 1).
Intuitively speaking, f .g is the loop traced by moving along the loops f and g in succession. Two loops f , g E S2 are said to be equivalent, (in symbols: f g), if there exists a homotopy ht : I +. X , (0 < t < l), such that h, = f , h, = g and ht(0) = x, = ht(1) for every t . This relation is reflexive, symmetric and
-
-
40
11. S O M E S P E C I A L C A S E S O F T H E M A I N P R O B L E M S
transitive; hence the loops S2 are divided into disjoint equivalence classes. Let n , ( X , x,) denote the set of these classes and [ f ] denote the class containing f E S ~ The . loop f is said to be a representative for the class [ / ] . Let d denote the degenerate loop d ( I ) = x,,. For any loop f E S2, let f -l denote the reverse of f defined by f - l ( t ) = f (1 - t) for each t E I . Then clearly ( f - l ) - l = f . The following elementary properties can be easily verified : (1) If f f' and g g', then f - g f'eg'. (2) ( f * g ) . h- f * ( g . 4 . (3) f a d - f N d . f . (4) f * f -1 a f -1.f. According to property ( l ) , the class [ f . g ] depends only on the classes [ f ] and [g]. Hence we may define a multiplication in n , ( X ,x,) by taking
- --
N
By (2), (3) and (4), this multiplication makes n , ( X , x,) a group which is called the Poincart group or the fundamental group of X at x,. The class e = [ d ] is the neutral element of n,(X, x,), and the class [ f -l] is the inverse element of [ f ] . For example, take X to be the S1of 5 2 and x, the point 1 of S1. Then S2 becomes a subset of the set A of all maps of S1into itself. The inclusion i :S2 c A induces a function
i, :n1(9,1) +n(S' ; 9). By (3.2), i, is onto. On the other hand, i, is also one-to-one. To prove this, let f , g E S2 be any two maps such that f N g. Then there is a homotopy ht : S1+S1,(0 Q t < l ) , such that h, = f and h, = g. Define a homotopy kt : S1 +S1, (0 < t Q l), by taking kt(z) = ht(z)/ht(l) for each ZES' and tEI.Thenk,=f,k,=gandKt(l) = lforeveryt~I.Hence,f-grell. This proves that i, is one-to-one. Finally, i t is not difficult to see that +mp.+mfi is equivalent to +m+n+. Therefore, n,(S1, 1) i s a free cyclic group generated by the class [ p ] of the exponential map p : I + S1. The definition of the fundamental group n,(X,x,) is by no means constructive and the problem of computing the group effectively is quite difficult. In a finite simplicial complex, an effective rule can be given for writing down generators and relations (finite in number) for the group (see Ex. A at the end of the chapter). This reduces the problem for finite simplicia1 complexes to solving the word problem of group theory. For many of the simpler complexes, this has been done. Now let x1 be another point of X and let u : I + X be a path connecting x, to xl. For each i = 0, 1, let f2i denote the set of all loops in X with basic point at x i . The path u defines a transformation u#:sZ, +Q, as follows. Let f E Q , and t : I + X denote the reverse of u defined by t ( t ) = u(1 - t) for each t E I . Then u#( f ) is the loop u * f * t defined by
4. T H E F U N D A M E N T A L ( a - f - t (t) )
=
GROUP
(if 0 < t (if4 Q t (if 8 Q t
f(3t-1). a(3t). t(3t -2).
< f). < 1).
<6).
It is straightforward to verify the following three properties: (5) Iff - g , then a - f - t - a . g . t . (6) a . ( f . g ) - t - ( a . f * t ) . ( a . g . t ) . (7) t . ( a - f . t ) * a -f. According to (5), the class [ a e f - t ] depends only on the class [ f ] . Hence a# induces a transformation xl) -+izl(X, xo). a* :n,(X,
The property (6) implies that a* is a homomorphism. Similarly, the reverse t of a determines a homomorphism t* :n , ( X , xo)
+n,(X, X I ) . It follows from (7) that the composition z*a* is the identity automorphism on n d l ( Xxl). , Since the roles of a and t are interchangeable, this implies that a* is an isomorphism and z., = (a*)-,. Thus we have proved the following Proposition 4.1. Every path a : I + X induces an isomorphism a* : iz, (X,x l ) w n , ( X , xo), where xo = a(0) and xl = a(1).
A space X is said t o be pathwise connected if every pair of points of X can be connected by a path in X.If X is pathwise connected, then it follows from (4.1) that the fundamental groups n,(X, xo) a t various basic points xo E X are all isomorphic. As abstract groups, these may be considered as ths same group which will be denoted by n l ( X ) and called the (abstract) fundamental group of the pathwise connected space X. Any two paths a, t : I X are said to be equivalent, (in symbols, a t), if a(0) = t(O),a(1) = z ( l ) , and a E t relative to the extremities of I . Then one can easily verify the following
-
--f
Proposition 4.2.
If two paths a,
t : I +X
are equivalent, then a*
= t*.
If the path u : I -+ X is a loop with basic point xo E X , then a represents an element [ in n,(X, xo). By the definition of o*,one can easily see that a*(w) =
[WE-'
for every w in n,(X, xo). Hence a* is the inner automorphism of nl(X, xo) determined by the element [ = [a]. Finally, let 4 : (X, xo) ( Y ,yo) be a given map. For every loop f in X with xo as basic point, the composition 4f is a loop in Y with yo as basic point. The following properties are obvious : (8) If f g, then 4f 4g. (9) If h = feg, then4h = t$f$g. -+
-
-
11. S O M E S P E C I A L C A S E S O F T H E M A I N P R O B L E M S
42
+, +
(10) If two maps y~ : (X, xo) -+ ( Y ,yo) are homotopic, then +f By (8),the map : (X, xo) -+ ( Y ,yo) induces a transformation
-lyf.
+* :n1 ( X ,xo) -+n1( Y ,Yo). The property (9) implies that +., is a hamoniorphism which will be called the induced homomorphism of (on the fundamental groups). Thus the operations (X, xo) + n , ( X , xo) and form a covariant functor for the categories involved, [E-S : p. 1111.
+ + -++,,,
5. Simply connected spaces A pathwise connected space X is said to be simply connected provided that every pair of paths a, t : I -+ X with a ( 0 ) = t(0)and a(1) = t ( 1 ) are homotopic with end points held fixed. Proposition 5.1. A pathwise connected nonvacuous space X is simply
connected
i8 n l ( X ) = 0.
Proof. Necessity. Let xo E X and let f : I + X be any loop with xo as basic point. By the definition of simple connectedness, f is equivalent to the degenerate loop d at xo. This implies that n l ( X , xo) = 0 and hencen,(X) = 0. Suficiency. Assume that nl(X) = 0 and let a, t : I -+ X be two paths such that a ( 0 ) = xo = t(0)and a ( 1) = x 1 = T(1). Then we obtain a loop f : I -+ X with basic point xo defined by j = sat-1, that is to say,
424 t ( 2-2t), I
f(t) =
(if 0 < t (if 4 < t
< &), < 1).
Since n I ( X )= 0, f is equivalent to the degenerate loop at x,,. It follows that there exists a map F : I2 -+ X such that
F(0, t )
= x0,
F ( 1 , t ) = x0, F ( t ,0 ) = f ( t ) , F ( t , 1)
=
x0
for every t E I. Now, i t is not difficult to construct a homotopy & : I + X, (0 < t < l), such that lo= a, tl = t, and &(O) = xo, &(I) = x1 for every t E I. For this purpose, let us first consider the following geometric situation in the unit square I 2 in the euclidean plane R2. Let q denote the point (4,O) of 12.Then the line Lt through q with angle of inclination (1 - t)nmeets the boundary of lain q and another point #t if 0 < t < 1. Let #, = (0,O) and fil = (1,O). Define a homotopy ht : I -+ P , (0 < t Q l), by taking ht(s) to be the point which divides the segment # l q in the ratio s : ( 1 - s). Now, let & = Fht for each t E I. Then one verifies immediately lo = a, ljl = t and &(O) = x,, &( 1) = F(q) = x1 for every t E I. Hence a t. I I t follows immediately from (5.1) that every contractible space is simply connected. For examples of non-contractible simply connected spaces, we have the following
-
5 . SIMPLY C O N N E C T E D
SPACES
Proposition 5.2. T h e n-sphere Sn i s simply connected
43
i8 n > 1.
Proof. The O-sphere SO is not simply connected since it is not pathwise
connected. The 1-sphere S1 is not simply connected since nl(S1) m 2. It remains to prove that Sn, n > 1, is simply connected. For this purpose, we shall prove that every loop f : I +. S* at a basic point xo E Sn is equivalent to the degenerate loop d at xo. By the method of simplicia1 approximation [E-S; p. 641, f is equivalent to a loop g : I + S n which is a simplicia1 map on some triangulations of I and Sn. Since n > 1, g ( I )is a proper subspace of Sn and hence is contractible in S n to the point xo. This implies g w d and hence f ~ dI . A space X is said to be locally Pathwise connected at a point xo E X if, for every open neighborhood U of xo in X,there exists an open neighborhood V c U of x,, such that every pair of points in V can be connected by a path in U.If X is locally pathwise connected at every point of X , then it is said to be locally pathwise connected. For example, triangulable spaces are locally pathwise connected. An important property of spaces which are both simply connected and locally pathwise connected is that the restricted lifting problem with respect to the exponential map p : R +. S1in 9 2 has a unique solution. Mote generally, we have the following Proposition 5.3. (Covering map property). Let X be a connected and locally pathwise connected space, and let xo be a given point in X . For every m a p f : X -+ S1and every point ro E R such that p(ro) = f(xo)and f * n l ( X , xo) = 0 , there exists a unique m a p g : X +. R such that g(xo) = ro and p g = f . Proof. We shall first construct g : X +. R as follows. Let x be an arbitrary point in X , then there exists a path n : I +. X such that n(0) = xo and n(1) = x . Let a : I S1denote the path defined by a = fn.Then a(0) = f fxo) and a(1) = f f x ) .According to (2.3), there is a unique path t : I -+ R such that t(0)= ro and p t = u. Since f * n l ( X , xo) = 0 , it is easy to see that different choices of the path n in X give rise to equivalent paths a = fn in S1. Hence, by (2.4), one can readily prove that t(1) does not depend on the choice of n and we may define g ( x ) = t(1). Clearly we have pg = f and g(x,) = r,. For the existence proof, it remains to prove that g is continuous. Let x1 be any point in X . To show the continuity of g, it suffices to prove that g is continuous at x l . To this end, we shall prove that g coincides with a map in some neighborhood of x , in X . Let z1 = f ( X I ) and W = S1\ z1en'. Let J denote the component of # - I ( W ) which contains rI = g(x,). Then J is an open interval of length 1 and p maps J homeomorphically onto W . Let q : W +. J denote the homeomorphism such that pq(z) = z for every z E W. Let U = f-l(W) and define a map h : U +. R by taking h(x) = qf(x) for every X E U . Obviously we have h(x,) = rl and ph = f I U.We are going to prove that g coincides with h in an open neighborhood of x1 in X . --f
44
11. S O M E S P E C I A L C A S E S O F T H E M A I N P R O B L E M S
Since X is locally pathwise connected and U is an open neighborhood of x1 in X , there exists an open neighborhood V c U of x1 in X such that every point of V can be connected to x, by a path in U . We shall prove that g ( x ) = h(x)for every x E V . For this purpose, let x GV be given. Then there is a path q : I + U c X such that y(0) = x I and y(1) = x. Since X is pathwise connected, there is a path 6:I + X such that [ ( O ) = xo and [( 1) = x,. Let n = [.q : I + X be the path defined in the obvious way. Set p = f 5 and u = fn. Denote by 8 : I + R the unique path such that O(0) = 7, and PO = p. By the very construction of g, we have O(1) = g(x,) = yl. On
the other hand, we also have h(xl) = rl. Hence, we may define a path t : I + R by taking e(qt ( i f 0 < t <#, t(4 = (if 4 < t < 1). y(2t - 11,
{
Then t ( 1 ) = h ( x ) . On the other hand, since PO = p = f5 and phy = f y , we obtain p t = u. Further, we also have t(0)= O(0) = yo. Hence, i t follows from the construction of g that t ( 1 ) = g(x). This proves that g ( x ) = h(x) whenever x is in V . Since V is an open neighborhood of x1 in X and h is continuous, it follows that g is continuous at xl. This completes the existence proof. To prove the uniqueness, let g and g’ be any two maps of X into R such that
Ps = f
=
Pg’,
dxo)
= Yo = g ’ ( x 0 ) .
We are going to prove that g = g’. Let x be an arbitrary point in X. It suffices to prove that g(x) = g‘(x). There is a path n : I + X connecting xo to x. Let u = fn, t = gn and t‘ = g‘n.Then we have
pt
=
pt‘,
t(0)= Yo
t = t’.In
particular,
u
According to (2.3), we get
=
=
t’(0).
g(x) = t ( 1 ) = t’(1) = g’(x). I
6. Relation between m(X,xo) and Hl(X) In the present section, we shall give a detailed exposition of the classical result on the relation between the fundamental groupn,(X, xo) of a pathwise connected space X at any given basic point xo and the I-dimensional integral singular homology group H , ( X ) of X. We shall assume that the reader is familiar with the singular homology theory as given in [E-S ; Chapter VII] and [Eilenberg 21. First of all, let us construct a natural homomorphism
h* :zi(X,xo)
+
Hi(X)
described as follows. Let u be any element of n , ( X , xo). Choose a representative loop f : 9 + x , f ( 1 ) = xo,
6. R E L A T I O N B E T W E E N n l ( X , xo) A N D H , ( X )
45
for a. This map f induces a homomorphism f, : H,(S1) +. H , ( X ) . Let L denote the generator of the free cyclic group H,(S1) corresponding to the counter-clockwise orientation of S1.By the homotopy axiom, the element f,(L) in H , ( X ) depends only on the class u andi therefore, we define h, by taking h,(a) = f,(c). By an elementary property of induced homomorphisms in homology theory, [E-S ; p. 361, it is easy to see that h, is a homomorphism. Let us consider the unit n-simplex An in the ( n 1)-space Rn+l which consists of the points (to,t,, * -,In) such that to t , * * * tn = 1 and ti 0 for each i = 0,1, +.,n. Let j : I + A l denote the homeomorphism defined by j(t) = (1 - t, t) for each t E I and let k : A , +.S1denote the map defined by k(to,t,) = pft,) for every point (to,t,) of A , , where p stands for the exponential map in 3 2. Then we have kj = p . As a singular 1-simplex in X , T = f k : A , + X is in the group C,(X) of integral singular 1-chains. Since dT = 0, T is a singular 1-cycle. It is not difficult to verify that T represents the homology class h,(a). Since H , ( X ) is abelian, the commutator subgroup Comm { n , ( X , xo) } of 3c1( X , xo) must be contained in the kernel of h,. In fact, we have the following classical theorem.
+ + +
-
+
Theorem 6.1. If X is pathwise connected, then the natwal homomorphism h, maps n , ( X , xo) onto H,(X) with the commutator subgroup of n,(X, xo) as its kernel. Hence H , ( X ) is isomorphic with the group n , ( X , xo) made abelian.
To prove this theorem, let us first carry out a preliminary reduction. Consider the singular complex S ( X ) of X , [E-S; p. 1861, and denote by S , ( X ) the subcomplex of S ( X ) defined as follows. A singular simplex T : A , 3 X is in S , ( X ) iff T sends the vertices of A, into the point xo. We shall call S , ( X ) the first Eilenberg subcomplex of S ( X ) . Since X is pathwise connected, it follows from Eilenberg’s reduction theorem, [Eilenberg 2 ; p. 4401,that the inclusion cellular map induces an isomorphism
1:S,
= S,(X)+S(X)
1, : HIISl) m H , ( X ) . On the other hand, let u be any element of n,( X , xo) and choose a representative loop f : S1 + X for a. Then the singular simplex T = f k : A , +. X is a 1-cycle in S, and hence determines a homology class p ( a )E Hl(S,) which does not depend on the choice of f . The operation a -+ p(u) defines a homomorphism /J : n i ( X ,xo) +. H i ( S i ) . Obviously we have the relation h, the following
=
1,p. Hence the theorem reduces to
Lemma 6.2. The homomorphism p maps n , ( X , xo) onto H , ( S , ) with the commutator subgroup of n , ( X , xo) as its kernel.
46
11. S O M E S P E C I A L C A S E S O F T H E M A I N P R O B L E M S
Proof. To prove that p is onto, let B be an arbitrary element of H,(S,). Choose a 1-cycle z of S , which represents B. Since Z E C , ( S , ) , it can be written in the form n
z
arTt
= i-1
-
where T,; * * , Tn are 1-simplexes in S, and a,, , an are integers. Since Tt maps the vertices of d, into xo, it is a 1-cycle in S , and hence represents an element Br of H,(S,). Then we have U
On the other hand, the loop Tfj : I -+ X represents an e!ement at ofn,(X,x,). According to the definition of p, we have ,u(ar) = Pi. Let
a
=
a"1a"a. * .a:" € n , ( X , xo). i a
Since p is a homomorphism, it follows that n
n
p(a) = Z atp(ar) = 2 atpi i=1
I--1
=
B.
This implies that p mapsnl(X, xo) onto H,(S,). To study the kernel of p, let us first note that, because of the commutativity of H,(S,), the kernel of p contains the commutator subgroup Comm { n , ( X , xo) } of n,(X, xo). The quotient group
n , * ( X , xo)
=n,(X,
xo)/Comm{ n,(X, xo) }
is commutative and is known as the group x,(X, xo) made abelian. We shall use the additive notation in n,*(X, xo) and consider the natural projection Y
:n,(X, xo) +n,*(X, xo).
Since the kernel of p contains Comm { n , ( X , xo) }, p induces a unique homomorphism such that p = p * v . To prove that the kernel of p is Comm { n,(X, xo) }, it suffices to show that p* is a monomorphism. Indeed, we will produce a left inverse x* for p*. Sincen,*(X, xo) is abelian and C,(S,) is the free abelian group generated by the 1-simplexes of S,, we may define a homomorphism x : C,(S,)
-+n,*(X, xo)
described as follows. Let T be any 1-simplex of S,. Then the loop T j :I -+ X represents an element [Tj]of zl(X, xo) and x is defined by taking
x(T)= 4 [Ti]). We assert that x maps B,(S,) into the zero element ofn,*(X, xo). To prove this, let T be an arbitrary 2-simplex in S,. Then T is a map T : d -+ X
7.
THEBRUSCHLINSKY GROUP
47
which sends the vertices of A , into xo. Let TW, T N , T(3 denote the 1-dimensional faces of T ; then we have
x(aT) = x(T(0)- T(1) + TP))= x(T(')+ T(")- TP)) = %( T(2)) %( T ( s )- %( T(')) = Y( [ W ) j ] ) Y( [T(o)j]) -Y( [TNj]) = Y( [TP)j] [TWj] [TNjl-l) = 0
+
+
because of the relation [TWj] = [ T ( 3 j ][T(o)j]since T is defined throughout
A,. Since B,(S,) is generated by all aT with T running over the 2-simplexes of S,, this proves our assertion. Thus x induces a homomorphism x* : H,(S,) +n1*(X, x0).
Finally, x*p* is the identity. To prove this, let a* be any element of n l * ( X ,xo). Choose a E ~ , ( Xxo), with v ( a ) = a*. Let f :S1 + X be a representative loop for a. Then the singular simplex T = f k : A , + X represents the class p(a) and hence we have x*p*(a*) = x*p*v(a) = y
= x*,u(a) = Y(
( [ f h j l ) = Y(
[Tj]) = a*,
UP])=4 .)
where p denotes the exponential map. This implies that ,u* is a monomorphism. I An important consequence of (6.1) is the fact that, for any pathwise connected space X , the fundamental group nl(X ) completely determines the integral singular homology group H , (X) .
7. The Bruschlinsky group Let ( X , A ) be an arbitrary pJr consisting of a space X and a subspace A of X which may be empty. Let us consider the set
w
=
{ f : ( X , A ) +(S1,1)}
of all maps f of the pair ( X , A ) into the pair (Sl, l) , where S1 denotes the unit circle in the complex plane as in 2. Since S1is an abelian group under multiplication of complex numbers, we may define an addition in W as follows. If f , g are any maps in W , then f + g is the map in W defined by
(f
+ g) (4 = f (4 g ( x )
for every x E X . In this way, W becomes an abelian group. The homotopy class (relative to A ) of the map f + g depends only on that of f and that of g. Hence the set n l ( X ,A ) of all homotopy classes of the maps of W forms an abelian group under the addition defined as follows: For any a, /Iin d ( X ,A ) , we have a
+ B = [f + gl,
f
gEB*
48
11. S O M E S P E C I A L C A S E S O F T H E M A I N P R O B L E M S
This abelian group d ( X ,A ) will be called the Bruschlinsky group of the pair ( X , A ) . If A is empty, then i t will be denoted by n l ( X ) and called the Bruschlinsky groufi of the space X . See [Bruschlinsky I]. For example, we have
nl(S1) M n'(S1, 1)
M
n,(S1, 1) M 2.
To determine the structure of n l ( X , A ) , let us assume throughout the remainder of the section that the pair ( X ,A ) is triangulable. Thus, we assume that X is a finite simplicial complex, [E-S; p. 561, and that A is a subcomplex of X . Under these assumptions, let us consider the integral cohomology group P ( X ,A ) of the pair ( X ,A ) and construct a natural homomorphism h* :d ( X ,A ) + H'(X, A ) as follows. Let L denote the generator of the free cyclic group H l ( 9 , 1) determined by the counter-clockwise orientation of S1.For an arbitrary element a ~ n l ( XA,) , pick a map f : ( X ,A ) + (9,1) which represents a. f induces a homomorphism
f * : H'(S1, 1) -+ H y X , A ) which depends only on a according to the homotopy axiom of cohomology theory. Hence we may define h* by taking h*(a) = f * ( L ) . It remains to verify that h* is a homomorphism. Let a ~ n l ( XA,) . By means of the homotopy extension property ( I ; 5 9), one can easily see that a contains a map f : ( X ,A ) + ( 9 , 1) which sends all vertices of X into the point 1 of S1.Such a map f determines an integral cochain cl( f ) E P ( X ,A ) as follows. Let u = vovl be any 1-simplex of X and let 1 :I +u denote the linear map which sends 0 to vo and 1 to vl. The composed map f 1 : I + S1is a loop in S1and hence the degree of f 1 is a welldefined integer deg ( f 1)which depends on f and u. Then the cochain c1( f ) is defined by [ c ' ( f ) 1 (4= deg(f 4. This cochain cl( f ) is a cocycle. To prove this assertion, let an arbitrary 2-simplex. Then we have
aZ = - u1 + uz,
Let At : I + uf,i follows that
[ W )fI ( d
=
=
be
v0v2, uz = vovl.
0,1,2, denote the linear maps as described above. I t
)I(W = [ C Y f )I(uo) - [ C Y f )I(%) + deg(f 1,) - deg(f 1,) deg(f 1,) = 0
= [CYf =
oo = vIv2,
t = vovlvz
+
[CYf
)I(%)
since f is defined throughout t.This proves that cl( f ) is a cocycle. By the definition of the induced homomorphism f *, it is not difficult to see that this cocycle cl( f ) € Z 1 ( X A , ) represents the element f * ( L ) E H 1 ( X ,A ) . Now let /Ibe another element of n l ( X ,A ) and pick a representative map
7. T H E B R U S C H L I N S K Y G R O U P
49
g : ( X ,A ) -+ (9,1) of b which sends all vertices of X into 1. Then the map f + g represents cc + B and sends all vertices of X into 1. For every 1-simplex u = vovl, we have
+ g ) 4 = deg(f 2) + deg(g4. It follows that cl(f + g) = c l ( f ) + cl(g) and hence deg ( ( f
h*(a
+ /I) it*(.) + h*(/3). =
This proves that h* is a homomorphism and completes its construction. Theorem 7.1. The natural homomorphism h* i s an isomorphism of d ( X ,A )
onto H1(X,A). Proof. Let z E Z1(X, A ) be arbitrarily given. To prove that h* is an epimorphism, i t suffices to construct a map f : (X, A ) -+ (9,1) which sends all vertices of X into 1 and such that c1( f ) = z. For each n = 0, 1; . , dim X, let Kn denote the subcomplex of X which consists of all simplexes in A together with the simplexes in X \A of dimension not exceeding n. We are going to construct a sequence of maps
-
fn:Kn+S1, n
=
1,2;*.,dimX,
as follows. To construct f,, let us take for each 1-simplex u a loop g, : I -+ S1such that g,(O)
=
1
=
=
vovl in X \A
gu(l), deg (go) = zb).
Let 1 , : I -+ u denote the linear map which sends 0 t o vo and 1 to v , . Then we define f , : K , -+ S1by taking (if X E K O ) , fl(4= (if X E ~ KE , \A). Next, let us construct the map f , : K , + S1. Let z = vOvlvabe an arbitrary 2-simplex in X \A and uo = vlv,, = vov,, u, = vovl. Then f , I at is a loop in S1of degree
(I,
z(uo)-z(ul)
+ z(uJ = z ( d t ) = (dz)(z)= 0
since z is a cocycle. By (3.3),f l I d t is homotopic to the constant map. Hence it follows from the homotopy extension property that f l I dz has an extension h1 : z + S*. Then we define f by the following formula:
,
fzW
fl(x)J
=
h,(X)>
(if x E K , ) , (if x E t E K , \A).
Now we shall complete the construction of the sequence { fn } by induction. Let m > 2 and assume that fm-1 : Km-1 -+S1has already been constructed. Let 8 be an arbitrary m-simplex in X \A, and vo be a vertex
50
11. S O M E S P E C I A L C A S E S O F T H E MA I N P R O B L E M S
of 0. Since m > 2, the ( m - 1)-sphere80 is simply connected by (5.2).Then, according to (5.3), there is a unique map j e : a0 -+ R such that ie(vo)
=
0,
Pie = f m - 1 180
where p : R +S1 denotes the exponential map in 3 2. Since R is solid, the map iehas an extension ke : 0 -+ R. Then we define fm by setting (if x E Km-l), (if X E 0 E Km \A).
im-i(x),
f m f x )=
pke(x),
This completes the construction of the sequence { fn }. Since fn I Kn-1 = fn-,, we obtain a map f : X +S1 such that f I Kn = fn. In particular, /(KO) = 1 and f I K , = f l . By the construction of f l , this implies that cl( f ) = z. Hence, k* is an epimorphism. To prove that k* is a monomorphism, let GC be any element of n l ( X ,A ) with h*(a) = 0. Pick a representative map f : (X, A ) -+ (Sl, 1) of a which sends all vertices of X to 1. Then the cocycle c1( f ) is a coboundary of X modulo A and hence there is a cochain c0 E CO(X,A ) such that cl( f ) = 6co. For each n = 1,2;. dim X + 1, let Jn denote the subspace of X x I defined by Jn = (X x 0) U (Kn-1 x I ) U (X x 1). a ,
To prove that f is homotopic to the constant map O(X) = 1 relative to A , let us construct a sequence of maps
F n : Jn-+S1, n
=
1,2;*.,dimX
+1
as follows. To construct F,, let us take for each vertex v in X \A 5, : I -+S1such that
L(0) = 1
=
tAl), deg (E,)
=
a loop
c0(~).
Then we define F , : J 1 -+S1 by taking
F,(x,4
=
1
1,
W), f (4,
(if x E A or if t (if x = v ) , (if t = 1).
=
0),
Next, let us construct the map F , : J , -+ S1. Let a = v,,v,, be any 1-simplex in X\A. Then the partial map F , I d(a x I ) on the boundary 8(a x I ) of a x I is a loop in S1 of degree
- co(vo) = (dco) (a)- [c'(f)l (a)= 0. Hence F , I d(a x I) has an extension q,, : a x I +S1.We define F , by the co(vl) - [c'(f)I
(0)
Now we shall complete the construction of the sequence { Fn } by induction. Let m > 2 and assume that Fm-, : Jm-l + S1 has already been
7.
51
T H E BRUSCHLINSKY GROUP
constructed. Let t be any (m - 1)-simplex in X\ A . Since m > 2, the (m- 1)-sphere a(t x I ) is simply connected. Hence, it follows just as before, that F,-#(t x I ) has an extension 5,: t x I -,S1. Then we define F,: J , S1by setting
-.
This completes the construction of the sequence { Fn }. Since Fn I Jn-1 = Fn-l, we obtain a map F : X x I -+S1 such that F I Jn = Fn. In particular, we have F ( x , 0) = 1, F ( x , 1) = f ( x ) , F ( a ,t ) = 1 for every x E X , a E A and t E I . This implies that f is homotopic to the constant map O(X) = 1 relative to A . Hence a = 0 and h* is a monomorphism. I Since H l ( X , A ) is effectively computable, the theorem (7.1) solves the classification problem for the maps ( X ,A ) -+ (9,1). In particular, if we take A = 0 , then it gives a solution of the classification problem for the maps X +S1. On the other hand, it also solves the homotopy problem and the extension problem in the form of the following corollaries. Corollary 7.2. Two maps f , g : ( X ,A )
ifl I*(&) = g*(1).
-+
(9,1) are homotopic (relative to A )
This corollary is an immediate consequence of (7.1). In particular, let us take A = a. The inclusion map j : S1c (9,1) induces an isomorphism j* : Hl(S1, 1) M H1(S1)and x = j * ( i ) is the generator of the free cyclic group H1(S1)determined by the counter-clockwise orientation of S1. Then (7.2) gives the following corollary as a special case. Corollary 7.3. Two maps
f, g : X +S1 are homotopic it f
* ( x ) = g*(x).
Corollary 7.4. A map f : A -+S1 can be extended over X iff the element f * ( x ) of H 1 ( A )is contained in the image of the homomorphism i* : H 1 ( X )-+ H 1 ( A )induced by the inclusion map i : A c X . In fact, if a is an element of H 1 ( X )such that i*(a)= f * ( x ) , then f has an extensiong : X -+ S1wilhg*(x) = a. Proof. The necessity of the condition is obvious. For the sufficiency, it suffices to establish the second assertion. By (7.1), there exists a map k : X + S1 such that (jk)* ( L ) = a.Then'we have
( j k i ) * ( i ) = i*(jk)*(t) = i*(a)= f * ( x ) = f * j * ( i ) =
(jf)*(i).
By (7.2), this implies that jki N j f and hence ki N f . According to the homotopy extension property, there exists an extension g : X + S1 of f such that g = k . Then we have g*(x) = k * ( x ) = k*j*(i) = ( j h ) * ( i ) = a. I Following the definition of triangulable pairs in [E-S; p.601, we have assumed above that X is a finite simplicia1 complex. However, the proof of
52
11. S O M E S P E C I A L C A S E S O F T H E M A I N P R O B L E M S
(7.1) is so arranged that it extends to the case that X is an infinite simplicia1 complex and A is any subcomplex of X provided that Whitehead's weak topology is used in X . Hence all results in this section are true for the infinitely triangulable pairs ( X , A ) defined in the obvious way. On the other hand, these results can be extended to more general pairs ( X ,A ) by using various Cech cohomology theories. Some of these generalizations will be given as exercises a t the end of the chapter.
8. The Hopf theorems With some dimensional restrictions on the triangulable pair ( X ,A ) , the nice results of the preceding section for S' can be generalized to higher spheres. For this purpose, let us first consider some preliminaries. For every n > 1, let Sn denote the boundary n-sphere of the unit (n + 1)simplex A = An+,. For each m = 0, l;.., n 1, let A ( m ) denote the m-th (n-dimensional) face of A . Then Sn is the union of the n-simplexes d ('4, A W , -,A@+'). The n-dimensional cocycle $ of Sn defined by$ (A(O))= 1 and + ( A ( m ) ) = 0, m = 1; .,n + 1, represents a generator x of the free cyclic cohomology group Hn(Sn). For any map f : X +S", the element f*(x)of P ( X ) depends only on the homotopy class [ f ] off and will be called the degree of the map f or of the class [ f ] . In particular, if X = 9, then the element f * ( x ) of Hn(Sn) determines a unique integer deglf) such that f * ( x ) = deg(f)x.In this case, it is this integer deg(f)which is traditionally known as the degree of the map f : Sn +Sn. If n = 1, this definition of deg(f) is obviously equivalent to the one given in 5 3. Let vo denote the leading vertex of Sn. For each q = 0, 1; n + 1, let IW denote the union of all A @ ) with m # q. Then the inclusion maps
+
--
-
-
5 : Sn c (9, ?lo), ?jQ : (Sn, vo) induce isomorphisms sions. Let
t*,
a ,
(9, F(Q))
on the cohomology groups of positive dimen-
& = f*-l
(4
J
AQ = ?jg*-'(
I).
Then L and & are generators of the free cyclic groups Hn(Sn, vo) and Hn(Sn, I'M)) respectively. Next, let us consider the unit n-simplex dn and its boundary (n- 1)sphere ad,. The n-cocycle y of An given by y(dn) = 1 represents a generator ,u of the free cyclic group Hn(& ad,). For any map f : ( A n ,ad,) -+ (Sn, vo), the element f * ( ~determines ) a unique integer deg(f)such that /*(L) = deg(f)* p This integer deg(f) will be called the degree of the map f. For each q = 0, 1; * *, n + 1, let cQ: (An,ad,) + (Sn, D O ) ) denote the map defined by the order-preserving one-to-one assignment of vertices of An into A W . Then it is not difficult to verify that (8.1)
&* (3.9) = (-
1Pp
8. T H E
53
HOPF THEOREMS
Now, let f : S n - + S n be a map which sends the ( n - 1)-dimensional skeleton of S n into vo. Then, by (8.1) and a theorem in cohomology theory, [E-S; p. 37, Theorem 14.6~1,one can easily deduce the following relation n+i
deg(f) =
(8.2)
(q=o
1)g
deg(ftA.
Finally, let us prove the following Lemma 8.3. For every integer m, there exists a m a p fm : (An, adn)+(Sn, vo) with deg(fm) = m. Proof. If m = 0, then the constant map fo(An) = vo is obviously of degree 0. Next, assume m > 0. Take a sufficiently fine simplicial subdivision K of An so that we may pick m mutually disjoint closed n-simplexes u1; * -,a m contained in the interior of An. Let the vertices uij of at be ordered in such a way that the orientation of at =
(utu ~
2 , *' * ,
utn+J
agrees with that of An. We now define fm to be the unique simplicial map of K into Sn which sends utj into the vertex vj of S n for each i = 1,. * , m and each j = 1, -,n + 1 and sends every other vertex of K into vo. Then i t is easily seen that fm(aAn) = vo and deg(f,) = m.Finally, let 2 denote a linear homeomorphism of An which interchanges a pair of vertices of An and leaves other vertices fixed. Then deg(fmR) = - m. I An organization of the Hopf theorems is as follows.
-
-
Theorem H". (Homotopy). If a m a p f : S n -+Sn has degree deg(f) = 0 , then f i s homotopic to the constant map O(Sn) = vo. Theorem En. (Extension). Let (X,A) be atriangulablepairwith dim(X\A) -+ S n be a given map. If there exists a n element a E Hn(X) such that i*(u)= f*(x), where i : A c X , then f admits a n extension g : X +. Sn such that g*(x) = u.
< n + 1 and f : A
Theorem C". (Classification).If X is a triangulable space with dim X < n, then the assignment f f*(x) sets up a one-to-one correspondence between the homotopy classes of the maps f : X + S n and the elements of the cohomology group Hn(X). --f
The converse of En is trivial: if f admits an extension g : X + S n , then there exists an element u E Hn(X) such that i*(u)= f*(x). In fact, we have u
=
g*(x).
Since H' is a special case of (3.2), these theorems can be established inductively by proving H" + En, En + H"+l, En * C" for every n > I . Also, it worth noting that Hnis a special case of C", C1is a special case of (7. l), and El is a special case of (7.4).
54
11. SOME S P E C I A L C A S E S O F T H E M A I N
PROBLEMS
Proof of Hn En. Let X be a simplicia1 complex and A a subcomplex of X. We may assume that f sends the (n - 1)-dimensionalskeleton of A into the point wo of Sn, for otherwise we may replace f by a homotopic map which satisfies this condition by applying the method of simplicial approximation and observing that every proper subspace of S n is contractible in Sn. Under these ass,umptions,we may define an n-cochain cn(f) of A as follows. Let u be any n-simplex of A and 1, :An + u denote the linear homeomorphism which preserves the order of vertices. Then f& is a map of (An, ad,) into ( 9 , zio) and hence we may define the cochain cn(f) by taking = deg(fL7) “)l(u) for every n-simplex (I of A . By (8.2), one can easily prove that cn(f) is a cocycle. By the definition of the induced homomorphism f*, it is not difficult to see that cn(f) represents the cohomology class f*(x) E Hn(A). Assume that a is an element of Hn(X) such that ;*(a)= f*(x). Then it follows that there exists an n-cocycle zn of X which represents ci and satisfies the relation zn (4 = [cn(Pl(4 for every n-simplex u of A. Let K denote the union of A and the n-dimensional skeleton of X . We shall define a map k : K + S n as follows. Let u be any n-simplex of K which is not in A. By (8.3),there is a map /u
: (An,
a&)
+
(Sn,vo)
with deg(f,) = zn(u). Let A, : An + u denote the linear homeomorphism which preserves the order of vertices. Then we define k by setting
[
k ( x ) = f(X)J
(4,
(if x € A ) , (if ~ E U E K \ A ) .
Finally, we are going to construct an extension g : X + S n of k as follows. Let t be any (n 1)-simplex of X \A and 1, :An+, + t the linear homeomorphism which preserves the order of vertices. Denote k, = k& 19.By (8.2), we have n deg (k,) = X (- l)(zn(t(O) = zn(at) = dzn(t) = 0.
+
i-0
According to Theorem H”,this implies that k, is homotopic to the constant map 0. Then it follows from the homotopy extension property that k, has an extension g, :An+, + 9.Then we define g : X +S n by taking
Since g is obviously an extension off, it remains to verify that g*(x) = a. According to the construction of g, it can be seen that g maps the (n - 1)dimensional skeleton of X into vo and that cn(g) = 2”. Since cn(g) represents g*(x) and 2%represents a, this implies g*(x) = a. I
8. T H E HOPF
THEOREMS
55
Proof of E n * Hn+'. Let f : Sn+l+ Sn+l be a given map with deg(f) = 0. We are going to prove that f is homotopic to a constant map. Using the method of simplicial approximation, we may assume that f is a simplicial map of a triangulation,K of Sn+l into 1 triangulation J of Sn+l. Pick an (n 1)-simplex (T = (u,, u l ; * , Hn+l) of J such that the cocycle defined by +(u) = 1 and+(t) = 0 for every (n 1)-simplext of J other than (I represents the generator x of Hfi+l(Sn+l). Let M denote the subcomplex of K consisting of the closed ( n + 1)simplexes of K which are mapped into (T by the simplicial map f. We may assume that no two of these (n + 1)-simplexes have a common vertex, for otherwise we could replace K and J by their second barycentric subdivisions K" and J",and take for a an (n + I)-simplex of J" none of whose vertices is also a vertex of J . Under these assumptions, we have H9(M) = 0 for every q > 0. The simplicial map f : K -+ J induces an (n 1)-cocycle y = I#(+) of K which represents the element
-
+
+
+
+
f * ( x ) = deg(f).x = 0 of Hn+l(K).Hence there exists an n-cochain c of K such that y = dc. The simplicial inclusion map 9 : M c K induces the following two cochains of M c1 = Y1 = P#(w) = P#f#(+) = (fP)#(+).
m),
The relation y = 6c in K implies that y 1 = dc, in M . Next, consider the leading n-face a(") = (ul,* * , u n + l ) of the (n 1)-simplex(T and let 4, denote the n-cochain of J defined by+,(d0)) = 1 and+,(t) = 0 for other n-simplexes t of J . Then the simplicial map fp : M + J induces an n-cochain c, = = (fp)#(+,) of M . Since+ is the coboundary of+, on the subcomplex u of J and fp maps M into (T, it follows that y 1 = dc,. Then d(c, - c,) = y1 -y1 =O and hence c1 - c, is an n-cocycle of M . Since H n ( M ) = 0, this implies that c1 - c, is a coboundary of M . Let M n denote the n-dimensional skeleton of the complex M . Then f defines a simplicial map X of Mn into the n-sphere au. By the construction given above, it is clear that c, is a cocycle of Mn and represents the degree of the map X . Since c1-c, is a coboundary, the degree of X is also represented by the cocycle c1 of Mn. Let N = (K \ M ) U M n and consider the inclusion map q : N c K . Since dc(t) = y ( t ) = 0 for any (n + 1)-simplex t which is not in M , it follows that cz = q#(c) is a cocycle of the complex N . Since c2 is an extension of c,, we may apply Theorem En to conclude that X has an extension p : N +da. Define a map g : Sn+l+ Sn+l by taking
-
+
56
11. S O M E S P E C I A L C A S E S O F T H E M A I N P R O B L E M S
Since f maps N into the space W
=
(J\ o) U do which is solid, the maps
f I N and p are homotopic in W modulo Mn. This implies f II g. Next, since the image of g is contained in u, it follows that g is homotopic to a constant
map. Since Sn+l is pathwise connected, this implies that f
II
g
II
0. I
X be a simplicia1complex with dim X < n. First, let -f S n such that f*(x)= u. Let A denote the (n- 1)-dimensional skeleton of X and i : A c X the inclusion map. Then we have P ( A ) = 0 and i*(a)= 0. Hence, by E", the constant map k(A) = oo has an extension f : X +Sn such that f * ( x ) = u. Next, assume that f, g : X +Sn are two maps such that f*(x) = g * ( x ) . We are going to prove f =g. For this purpose, let us study some elementary properties of the product space X x I . Obviously, X x I is triangulable and dim(X x I\ < r z + 1. Let p : X x I + X and q r : X - f X x I, [i =0,1), denote the maps defined by Proof of En 3 C". Let
a be any element of P ( X ); we shall find a map f : X
P ( x , t) = x , q o ( 4 = ( x , O), q1(4 = ( x , 1). Since $9, is the identity map on X , we obtain qe*fi*(u) = u for every u E Hn(X). Next, consider the subspaces x , = x x o , x , = x x 1, A = X , U X , of X x I . Then, according to an elementary theorem [E-S; p. 331, Hn(A) is the direct sum of Hn(X,) and Hn(X,). Hence the elements of Hn(A) can be represented by the pairs (p, y ) of elements of Hn(X). Let r : A C X x I denote the inclusion map. Then we have r*p*(a) = (u,a). Define a map k : A -+Sn by taking h(x,t ) =
f (x)J
dx),
(if x E X and t = 0), ( i f x E X a n d t = 1).
Then, h*(x) = (u,a) E Hn(A). According to En, there exists an extension H : X x I +Sn such that H * ( x ) = qo*(u).This proves f N g. I Thus we have proved the celebrated Hopf theorems H",En,C" for every n = 1,2; * In particular, if we take X = Sn in Theorem C", we obtain the following a .
Corollary 8.4. The homotoPy classes of the maps f : Sn + S n are in a oneto-one correspondence with the integers. The correspondence i s given by the assignment f + deg ( f ) . Finally, the remarks given at the end of fj 7 are also true in the present circumstances.
9. The Hurewicz theorem The dual of the Hopf classification theorem is the Hurewicz theorem, the statement of which is the main purpose of the present section. To this aim, let us define the important notion of n-connected spaces.
9.
57
THE HUREWICZ THEOREM
Let n > 0 be a given integer. A space X is said to be n-connected if, for every triangulable space T of dimension < n, any two maps f , g : T + X are homotopic. So, X is 0-connected iff it is pathwise connected, and X is 1-connected iff it is simply connected. Using the extension property, one can easily prove that an ( n - 1)-connected space X is n-connected iff every map f : Sn + X is homotopic to a constant map. Now, let X be a given space and consider the maps f :Sn+X,
+
where Sn denotes the boundary n-sphere of the unit (n 1)-simplex = A,,,. Since we have fully studied the case n = 1 in 3 4 and 3 6 , we will assume that n > 2. The fundamental n-cycle
d
n+1
2 =
(-
l)tA(O
i=o
of Sn represents a generator i of the free cyclic group Hn(S”). For each map f : Sn + X , the element f , ( i ) of H n ( X ) depends only on the homotopy class of f and will be called the degree of f . If X = Sn, then one can easily verify that f , ( i ) = deg ( f )- 1 , where deg(f ) denotes the integer defined in 3 8. Theorem 9.1. (Hurewicz Theorem). If X i s a n ( n - 1)-connected space with n > 2 , then the assignment f + f , ( i ) sets up a one-to-one correspondence between the homotopy classes of the maps f : Sn + X and the elements of the singular homology group H n ( X ) .
As an immediate consequence, we have the following important Corollary 9.2.
for every m
=
2;
A simply connected space X i s n-connected ifl H m ( X ) = 0 * , n.
In all existing proofs of the Hurewicz theorem, one must use the group structure and a few elementary properties of the homotopy group. In the sequel, we shall prove a much more gencral theorem. See (V; $4) and (X ;$ 8 ) .
EXERCISES A. The fundamental group of a connected simplicial complex
Let X be a connected simplicial complex. Denote the vertices of X by vo, v,; . * , vm. A broken line joining a vertex to another is a path which consists of a finite number of 1-simplexes. Since X is connected, we can join vo to of by a broken line 11. Suppose that these broken lines &, ( i = 0,l; * m),have been chosen such that 1, consists of only a single point vo. Let Acl denote the reverse of At. To each oriented 1-simplex vrvj of X , consider the loop a ,
58
11. S O M E S P E C I A L C A S E S O F T H E
M A I N PROBLEMS
which represents an element au E ~ , ( Xuo). , Prove that these elements ail form a system of generators of n,(X, uo) with the set of relations described as follows. First, for the 1-simplex v j q , we have trivial relation
ajr
=
(afj1-l.
Second, for each 1-simplex ucvj, the loop f i j is a product of the I-simplexes up,, of X, say
Then we have the relation
ftl = pu
ail
("€Vq).
= Pij (a€,,),
where Pi, (at,,) is the product obtained by replacing Z'CV,, with the corresponding element a h in n,(X, uo). Third, for each 2-simplex u = u t u p k of X, we have the relation atjawkr = 1. This gives an effective method for computing the fundamental group nl ( X ) of X. For examples, prove the following assertions: 1. The fundamental group n l ( X ) depends only on the 2-dimensional skeleton X2 of X . More precisely, the inclusion map i : X2 c X induces an isomorphism i, :n,(X2, uo) M n,(X, vo). 2. The fundamental group of a closed orientable surface of genus g is the abstract group generated by 2g elements a$,/$, (a' = 1; -,g), with a single relation
-
Hence the fundamental group of the (2-dimensional) torus is the free abelian group with two generators. If g > 1, then the fundamental group is nonabelian. 3. The fundamental group of a closed non-orientable surface of genus g is the abstract group generated by g elements a,, * * , agwith a single relation
-
alala2a2** .a@,
=
1.
Hence the fundamental group of the projective plane is the cyclic group of order two. If g > 1, then the group is non-abelian. 4. If a connected simplicial complex X is the union of two connected subcomplexes A and B with connected intersection D = A rl B, then nl(X) is the quotient group of the free product n,(A) On,(B),[S-T; p, 301, obtained by identifying, for each element 6 € n , ( D ) , the element t,(d) E n , ( A ) with the element q*(6)E ~ , ( Bwhere ), t : D c A and q : D c B denote the inclusion maps. In particular, if D is simply connected, then we have
n d X ) = n , ( A ) On,(B). 5. If X is a connected 1-dimensional simplicial complex, then n l ( X ) is a free group. In particular, if X is the space which consists of two circles intersecting at a single point, thenn,(X) is the free group on two generators.
EXERCISES
59
B. The bridge theorems
Assume that (1) X is a normal space, (2) Y is a connected ANR, and (3) either X or Y is compact. Let f : X -+ Y be a given map and a be a finite open covering of X . A map g, : N u -+ Y of the geometric nerve N , of a into Y will be called a bridge map for f if g,h, N f for every canonical map ha : X -+ N,, [L,; p. 401. If such a bridge map g, exists, a is said to be a bridge for the given map f . Prove the following assertions, [Hu 51 : 1. Every map f : X -+ Y has a bridge. 2. Every refinement of a bridge is also a bridge. 3. If a, ,!Iare two bridges for a given map f : X -+ Y , where X is compact, and if g, : N, -+ Y ,gp : Np -+ Y are bridge maps, then there exists a common refinement y of a and tf? such that gapya N gppyp, where pya : N , + N u and pyp : N , --f N p are arbitrary simplicia1 projections. Study the analogous assertions for a map f : X -+ Y of a paracompact Hausdorff space X into a connected ANR Y by considering the locally finite open coverings of X . C. Maps of compact spaces into spheres
The bridge theorems in Ex. B furnish us with a link between a map on a compact Hausdorf space and maps of simplicia1 complexes. Therefore, if one uses the tech cohomology groups and the notion of dimension both defined by means of finite open coverings, one can extend the results in fj 7 and fj 8 to compact Hausdorff spaces. Establish the following theorems: 1. Hopf extension theorem. Let ( X ,A ) be a compact Hausdorff pair with dim X Q n + 1, u a generator of the free cyclic group Hn(Sn),and f : A -+ Sn a given map. If there exists an element a E H n ( X ) such that i*(a)= f * ( u ) , where i : A c X , then there exists an extension g : X + Sn such that g*(u) = a. 2. Hopf classification theorem. If X is a compact Hausdorff space with dim X < n and if u is a generator of Hn(Sn),then the assignment f -+ f *(u) sets up a one-to-one correspondence between the homotopy classes of the maps f : X -+Sn and the elements of the Cech cohomology group H n ( X ) . In the case h = 1, the conditions on the dimension of X may be removed. Furthermore, the assignment f -+ f * ( u ) ,in this case, is an isomorphism h* :nyx) M H'(X).
For further generalizations to non-compact spaces, see [Dowker 11. D. The degree of a suspended maps
Consider the n-sphere Sn as the equator of an ( n + 1)-sphere Sn+I with north hemisphere E";tl and south hemisphere E?+l. By Tietze's extension
60
11. S O M E S P E C I A L C A S E S O F T H E M A I N P R O B L E M S
theorem, every map f : Sn + Sn has an extension f * : Sn+l+ Sn+l such that f *(E:+l) c E:+l and f *(En_+')c EE+l, which is called a suspended m a p of f. Prove 1. deg(f*) = d e g ( f ) . 2. (8.3) can be obtained inductively by the construction of suspended maps of a map f : S1-f S1of degree m.
CHAPTER I l l
F I B E R SPACES 1. Introduction The concept of fiber space is crucial in homotopy theory: it usually appears in the application of homotopy theory to geometric problems; it is a powerful weapon in the computation of the homotopy groups of various spaces; and it plays a key role in the axiomatization of homotopy theory. Thus, before turning to the discussion of the homotopy groups themselves, it is appropriate to develop certain properties of fiber spaces in some detail. Already we have seen one important example, namely the exponential map p : R +S1 of Chapter 11: in the language of the present chapter, we should say that i? is a fiber space (indeed a covering space) over S1with projection p . As to the usefulness of this particular example, recall that upon it was based the classification of maps of a space X into S1 of $ 7 of Chapter 11. Historically, there were a number of examples, and definitions of fiber spaces were abstracted from these in a variety of ways; but in each case it was possible to prove a so-called covering homotopy theorem. It remained for J.-P. Serre in 1950 to single this theorem out as the crucial property, and to base on it his study of the singular homology of fiber spaces. The influence of this study on homotopy theory has been profound, and it now seems quite clear that his is the proper definition; hence we adopt it and consider some of its immediate consequences in $$ 2-3. On the other hand, the classical examples of fiber spaces belong to a much more narrow class, in that all of them have a local product strzcctzcre. This concept, together with an important example, the Hopf fiberings of spheres, is considered in $5 4-6. After considering certain mappings of fiber spaces ($6 7-8) we develop (for later use, especially in Chapter IV) certain properties of spaces of paths ( $ 5 9-14). In particular, it will appear that they are fiber spaces in the sense of Serre mentioned above, but they do not have a local product structure. Finally ($9 15-17) we study the case of discrete fibers and in particular the classical covering spaces.
2. Covering homotopy property Throughout the present section, we shall consider a given map P:E+B of a space E called the total space into a space B called the base space. 61
62
111. F I B E R S P A C E S
Let X beagivenspace,f : X +. B agivenmap, andft : X + B , (0 Q t Q I ) , a given homotopy of f . A map f* : X +. E is said to cover f [relative to p ) if $f* = f . Similarly, a homotopy ft* : X -+ E , (0 Q t Q l),off* is said to cover the homotopy f t (relative to p ) if pft* = f t for each t e 1 ; ft* is called a cuvering homotopy of f t . The map p : E +. B is said to have the covering homotopy property (abbreviated CHP) for the space X if, for every map f* : X -f E and every homotopy f t : X +. B, (0 < t Q I ) , of the map f = pf* : X -f B, there exists a homotopy ft* : X -+ E , (0 Q t < l ) , of f* which covers the homotopy f t . The map p : E + B is said to have the absolute.covering homotopy property (abbreviated ACHP) if it has the CHP for every space X . The map p : E +. B is said to have the polyhedral covering homotopy property (abbreviated PCHP) if it has the CHP for every triangulable space X . The map p : E -+B is said to have the covering homotopy extension property (abbreviated CHEP) for the space X relative to a given subspace A of X if, for every map f* : X +. E and every homotopy f t : X +. B, (0 Q t Q l ) , of the mapf=pf*:X+B,everyhomotopygt*:A+E,(O Q t
3. Definition of fiber space A map p : E +. B is said to be a fibering if it has the PCHP. In this case, the space E will be called a fiber space over the base space B with projection p : E B. For each point b E B, the subspace p - l ( b ) of E is called the fiber over b. --f
Remark. We do not assume that p ( E ) = B and hence the fiber p-'(b) may be vacuous for some point b of B . However, applying CHP for a point, one can easily prove that p ( E )is a union of path-components of B. Therefore, if B is pathwise connected, we must have p ( E ) = B unless E itself is empty.
From time to time, it will be useful to know that, for a map p : E + B, the PCHP is equivalent to certain other covering properties. Precisely, we have the following
3.
DEFINITION OF FIBER SPACE
63
Theorem 3.1. For an arbitrary map p : E + B, the following statements are equivalent : (i) p : E +. B is a fibering, i.e. has the PCHP. (ii) For each m 2 0 , p : E -+ B has the CHP for the m-simplex Am. (iii) For each m 2 0 , p : E + B has the CHEP for the m-simplex A m relative to its boundary (m - 1)-sphereSm-l. (iv) p : E + B laas the PCHEP. (v) If ( X ,A ) is a triangulable pair such that A is a strong deformation retract of X and i f f : X + B and g* A + E are maps such that pg* = f I A , then g* has an extension f* : X + E such that pf * = f . Proof. (i) -+ (ii) is obvious.
(ii) + (iii). If m = 0, then A , is a single point and its boundary sphere S m - l is empty. Hence, the implication (ii) + (iii) is obvious in this case. For the general case m > 0, we construct a homeomorphism h of A , x I onto itself which carries ( A , x 0) U (Sm-l x I) homeomorphically onto d m x 0. For this purpose, let us first construct a homeomorphism h, of ( A , x 0) U (Sm-l x I ) onto A , x 0 as follows: For an arbitrary point w = ( x o ; * . , x, ; t) of ( A , x 0) U ( P - l x I ) , we define ho(w)to be the point (yo; y , ; 0 ) of A , x 0, where
-
a ,
I t f o l l o w s t h a t h , ( ~ , ; ~ ~Xm; , 1) = (x,;.-,x,;O)foreach(x,, x,) ~ 9 - l Similarly, wecandefine a homeomorphism h, of (Am x 1) U (Sm-l x I ) onto A , x 1 such that h, ( x , ; * * , x , ; O ) = ( x , ; . . , ~ ~ 1) ; for each ( x , ; * * , Xm) E Sm-l. Next, we define a homeomorphism h, of the boundary a(& x I ) of A , x I onto itself by taking
Mw)
=
ho(4, 1 h-l( 4, 1
if w E ( A , x 0) U i f w e d m x 1.
(9-l
x I),
Finally, the homeomorphism h, of a ( A , x I ) may be extended to a homeomorphism h of A , x I by radial extension from the center (c, 4) of A , x I , where c denotes the center of A,. Since h is an extension of h,, it carries ( A , x 0) U (Sm-l x I ) onto A , x 0. The following picture
4
Illustrates the homeomorphism h for the case m
=
1.
.
64
111. F I B E R S P A C E S
Now, let f* :Am + E be a map, ft :Am + B , (0 Q t Q l), a homotopy of the map f = Pf*, and gt* :Sm-l-+ E , (0 Q t Q l ) , a homotopy of g* = f* I Sm-l which covers f t I Sm-l. The homotopy f t and the partial homotopy gt* give rise to maps
F:dm x I +B,
G*:(Am x 0 ) U ( S m - 1 x I ) + E
in the obvious way. Let
@=Fh-l:AmX I + B ,
Y*=G*ho':AmxO+E.
Then we have PY* = @ I Am x 0 . By (ii), Y* has an extension @*:Am x I + E such that $@* = @. Define a homotopy ft* : Am + E , (0 f t < l), by taking ( X E Am, t E I ) . ft*(x) = @*h(x,t ) , Then it is easy to verify that fo* = f*, ft* I Sm-l = gt* and pft* = ft for every t E I. (iii) -,(iv). Let ( X ,A ) be a triangulable pair, f* : X + E a map, ft : X B , (0 Q t < l), ahomotopyof themap f = pf*, andgt* : A + E , (0 4 t Q 11, a homotopy of g* = f* I A which covers ft I A . We are going to extend gt* to a homotopy ft* : X -+ E , (0 < t < l), off* covering ft. We may assume that X is a simplicia1 complex and A is a subcomplex of X . For each m > 0, let Xm denote the m-dimensional skeleton of X . Let Km denote A U Xm. By successive application of (iii), one can construct for each integer m = 0, 1,2; a homotopy htm : Km + E , (0 Q t < l), such that --f
--
hr
= f*
hy I A
=
I Km, gt*,
= ft
hY+' I Km
I Km, =
hr.
Then the required homotopy ft* is defined by taking ft* I Km = h r for each m = 0, 1,2; * * (iv) + (v). Since A is a strong deformation retract of X , there exists a homotopy ht : X + X , (0 Q t < I), such that ho is a retraction of X onto A , h, is the identity map on X , and ht (a) = a for every a E A and t E I. Define a map f # :X +E and a homotopy ft : X + B , (0 Q t < l), by taking f # = g*ho and ft = fht for each t E I. Then fo = pf#. Since ft(a) = f ( a ) for each a E A and t E I , we may define a partial covering homotopygt* : A +E , (0 Q t < l), of f # by setting gt* = g* for every t E I. According to (iv), gt* has an extension ft* : X + E , (0 Q t Q I), such that Pft* = f t for every t €1. Let f* = f l * . Then we have f* 1 A = g* and pf* = f l = fh, = f. (v) + fi). This implication follows inmmediately from the fact that X x 0 is a strong deformation retract of X x I. I As noted a t the end of 3 2, the product space E = B x D is obviously a fiber space over B relative to the natural projection p : E --f B .
4. B U N D L E S P A C E S
65
4. Bundle spaces A map p : E +. B is said to have the bundle property (abbreviated BP) if there exists a space D such that, for each b E B, there is an open neighborhood U of b in B together with a homeomorphism +,y:
U x D+p-l(U)
of U x D onto p-l(U) satisfying the condition
(DF)
p + , y ( ~ d, ) = U , (U E
U , d E D).
In this case, the space E is called a bundle space over the base space B relative to the projection p : E +. B. The space D will be called a director space. The open sets U and the homeomorphisms +U will be called the decomposing neighborhoods and the decomposing functions respectively. As an immediate consequence of the definition, we have + ( E ) = B except in the trivial case where E is empty. The main idea of this definition is that a bundle space is a space with a local product structure over every point of the base space. In particular, the product space E = B x D is a bundle space over B relative to the natural projection fi : E + B with D as director space. In this definition of bundle spaces, we have essentially followed that of Ehresmann and Feldbau, [S; p. 181. For relations with coordinate bundles in the sense of Steenrod, see [S ; pp. 18-20]. Many examples can be found in
[SI. Theorem 4.1. Every bundle space E over B relative to p E -+ B i s a fiber space over B relative to p . Since CHP is trivial for product spaces, the idea of the following proof is to reduce the construction to local ones where the local product structure is available. Proof. Let X be a (compact) triangulable space, f * : X -+ E a given map, and f l : X -+B, (0 < t < l), any homotopy of the map f = pf * : X +. B. I t suffices to construct a homotopy f t * : X + E , (0 < t < l ) , of f * which covers ft. Pick a collection o = { U } of decomposing neighborhoods U which covers the base space B and define a map F : X x I +. B by taking
F(x,t)
= ft(x),
(x E
x,t E I ) .
The collection { F - l ( U ) I U E O } of open sets of X x I forms an open covering of X x I . Since X x I is compact, { F - I ( U ) I U E o } has a refinement of the form { W Ax I , }, where { W A} is a finite open covering of X and { I , } = { I , , * * , I f } is a finite sequence of open subintervals of I which covers I . We may assume that I , meets only I,,-l and I,+l for each p = 2; * .,r - 1. Choose numbers 0 = to < t , < . . * < tr = 1
66
111. F I B E R S P A C E S
so that t, is in the intersection I , n I,+,. We shall assume, inductively, that the covering homotopy ft* has already been defined for all t Q t, where p > O is a given integer less than 7. We proceed to extend ft* over the closed subinterval [t,, t p + l ] of I . Taking a sufficiently fine triangulation of X,we may assume that X is a simplicial complex such that every closed simplex of X is contained in some W Aof the finite open covering { W A }constructed above. Hence, for each closed simplex n, we may choose a decomposing neighborhood U, E W such that f t ( x ) E ua, ( X E 0 s t, Q 1 Q tp+l). Let yua : U, x D + D denote the natural projection. Let a be a vertex of X.Then ft* can be defined by taking
ftp*(a)l, ( f , Q t t,+J. Thus ft* is defined on the zero-dimensional skeleton X o of X for each t E [t,, t,+l]. We shall assume that ft* has already beendefinedon the ( n- 1)dimensional skeleton Xn-l for each t E [t,, t,+J, where n > 0 is a given integer. We proceed to extend ft* over the n-dimensional skeleton X n of X . Let a be any closed n-simplex of X . Then ft* has been defined on the boundary aa for every t E [t,, t,+l]. Let M = a x [t,, t,+l] and consider the subspace N = (a x t,) u (80x [t,, t,+,l) ft*(a) = +ua [ft(a),WU~+U,'
of M . According to (I;4.2), these exists a retraction p : M 2 N . Let O : N +E denote the map defined by O(x, t ) = f t * ( x ) . Then we may extend ft* over a by setting f t * W = +u, [ f t b ) , yua+u,' O p b , 01 for every x E a and t E [t,, t,+J. This completes the construction of ft*. I The preceding theorem is a corollary of the following general theorem which is known as the covering homotopy theorem of bundle spaces. Theorem 4.2. If a map p : E + B has the BP, then it has the C H P for every paracompact Hausdorf space. For a proof of (4.2), see [Huebsch 11 and also [S; p. 501.
5. Hopf fiberings of spheres Among the early examples of bundle spaces were the three fiberings of spheres p : S2n-l + S n , n = 2,4, 8 discovered by Hopf [2] in 1935. We shall examine the first of these (the case n = 2) in detail here, and show in 6 that it may be applied to the classification problem for the maps f : X +S2, where X denotes a triangulable space of dimension not more than 3. To construct the fibering for n = 2, let us represent S3 as the unit sphere in the space C2 of two complex variables, that is to say, S3 consists of the points (zl,z 2 ) in C2 such that Z 1 Z l + z2z2 = 1.
5. H O P F F I B E R I N G S O F S P H E R E S
67
-
Let S2 be represented as the complex projective line, that is to say, as pairs [z,, z2] of complex numbers, not both zero, with equivalence relation [z,, z2] [hl, h , ] where 1 # 0. Then the Hopf map p : S3+S2 is defined by p ( z l ,z 2 ) = [z,, z,] for each ( z l , z 2 ) E S ~The . continuity of p is obvious. Since any pair [z,, z z ] can be normalized by dividing by (zlZ1 + z,Z2)*, p maps S3 onto S2. To prove that S 3 is a bundle space over S2 relative to p , let us represent S1 as the set of all complex numbers 1 with I 1 I = 1. Consider the points a = [l, 01 and b = [0, 11 of S2 and the open sets
U
V = S2\b. Then U and V cover S2. Every point in U can be represented by a pair [z, 11. Hence we may define a map +v of U x S1 into S3 by taking = S2\a,
for each [z, 11 E U and each 1 E S1. One can easily verify that +v maps U x S1 homeomorphically onto p - l ( U ) and that p + ~ ( ud,) = zc for each u E U and d E D. Hence +u is a decomposing function. Similarly, we can construct a decomposing function This completes the proof that S 3 is a bundle space over S2 relative to the Hopf map p . If ( z l , z 2 ) E S3, then one verifies immediately that the fiber p - l [ z l , z2] consists of all the points (h,, h2) with 1 ~ 9 Hence . the fibers are just great circles of S3. In this way the 3-sphere is decomposed into a family of great circles with the 2-sphere as a quotient space. The Hopf fiberings : S7 +S4 and p : S15 + SS are constructed in an analogous fashion from the quaternions and the Cayley numbers respectively; a concise and clear description may be found in [S; pp. 105-1 101. In these fiberings, the fibers are 3-spheres and 7-spheres respectively. The Hopf maps are all essential; in fact, this is a consequenceof the following
+".
Proposition 5.1. I f a sphere Sn is a fiber space over a base space B which contains more than one point, then the projection p : Sn -+ B is an essential map. Proof. Assume that p were inessential. Then there exists a homotopy ht : Sn -+ B (0 Q t Q 1) such that h, = p and Izl(S")is a single point b, of B . Let i denote the identity map on Sn; then we have pi = p . According to the CHP, there exists a homotopy kt : Sn -+ Sn, (0 < t Q 1),such that k , = i and pkt = ht for every t E I . Now, k , maps Sn into the fiber p-l(b,) which is a proper subset of Sn since B contains more than one point. Hence, according to ( I ; 3 8), k is inessential. This implies that the identity map i is homotopic to a constant map. This is impossible by ( I ; Q 8). I I t is interesting to observe that the Hopf maps are algebraically trivial, that is to say, their induced homomorphisms on the homology groups and the cohomology groups are all trivial. Historically, these Hopf maps were the first examples of essential maps which were algebraically trivial. The
68
111. F I B E R S P A C E S
existence of these maps shows that induced homomorphisms are not by themselves sufficient to classify all maps.
6. Algebraically trivial maps X --f S2 In the present section, we shall show that, by the aid of the Hopf map p : S3 -+ S2, one can solve the classification problem of the algebraically trivial maps f : X -+ S2of a 3-dimensional triangulable space X into S2. Let X be any given triangulable space. For an arbitrary map F : X -+ S3, i t is obvious that the composed map f = pF : X +S2 is algebraically trivial and that the homotopy class o ff depends only on that of F. Proposition 6.1. For any given triangulable space X , the assignment F + f = pF sets up a one-to-one correspondence between the homotopy classes of the maps F : X + S3 and those of the algebraically trivial maps f : X -+ S2. As an immediate consequence of (6.1) and the Hopf classification theorem C3 in (I1 ; fj 8), we have the following Theorem 6.2. T h e homotopy classes of the algebraically trivial maps f : X -+ S2 of a 3-dimensional triangulable space X into the 2-sphere S2 aye in a one-to-one co~respondencewith the elements of the integral cohomology group H3(X).For any u E H 3 ( X ) ,the homotopy class which corresponds to u contains the m a p f = PF : X +S2, where F : X -+S3 i s a m a p with a as its degree. In particular, if X = S3, then every map f : X +S2 is algebraically trivial. Hence, we have the following Corollary 6.3. T h e homotopy classes of the maps f : S3 -+ S2 are in a one-toone corresfiondence with the integers. For a n y integer n, the homotopy class which corresponds to n i s represented by the composition f = pF : S3-+ S2 of the Hopf m a p p : S3 -+ S2 and a m a p F : S3 -+ S3 with deg ( F ) = n. The proof of (6.1) consists of the following two lemmas. Lemma 6.4. If a m a p f : X +S2 i s algebraically trivial, then there exists a m a p F : X + S 3 such that PF = f . Proof. Since S3 is a bundle space over S2 relative to the Hopf map p ;S3-+ S2, we may choose a collection w = { U } of decomposing neighborhoods U which covers 9. Taking a sufficiently fine triangulation of X , we may assume that X is a simplicia1 complex such that the given map f carries every closed simplex u of X into some decomposing neighborhood U , E w . Let Xm denote the m-dimensional skeleton of X , and let f m = f I Xm. For each m = 2,3; * * , we shall construct a map Fm : Xm + S3 such that pFm = f m , Fm+i I Xm = Fm. First, let us construct F,. Since the given map f is algebraically trivial, so is f 2 . According to the Hopf classification theorem C2 of (11; fj 8), f , is homotopic to a constant map which can be lifted. Hence it follows from the CHP that there is a map F , : X 2 -+ S3 such that p F , = f 2. Next, assume that n > 2 and that Fm has been constructed for every
7. L I F T I N G S
A N D CROSS-SECTIONS
69
m < n. Let u be an arbitrary closed n-simplex of X and choose a decomposing neighborhood U , E w which contains f (a).Let 4, : u, x s1-+ p - l ( U,), yo : u, x s1-+ S' denote the decomposing function and the natural projection respectively. Define a map [, : du -+ S1 by taking 5,(x ) = yo&lFn -1 ( x ) for every x E da. By (11; fj 7), [, has an extension 7, : u -+ S1. Then we define a map Fn : X n + S3by taking (if x E Xn-l), Fn -I(%), Fn(x) = (if x E u E X n ) . 4 J f n ( 4 ,~ d x1 ) Obviously we have PFn = f n and Fn I Xn-l = F n -l. This completes the inductive construction of the maps Fm, m = 2,3; -. Finally, define a map F : X + S3by taking F I X m = Fm for every rn > 2. Then we have PF = f . I Lemma 6.5. If F , G : X +- S3aye two maps such that pF N PG, then F N G. Proof. Since pF N PG, it follows from the PCHP that F is homotopic to a map F' such that PF' = PG. Hence we may simply assume that PF = PG. Consider S3 as the group of quaternions q with qq = 1. Then the fiber which contains the quaternion 1 is a subgroup S1 of S3 and the other fibers are cosets of S1 in S3.In fact, in the usual representation q = x1 x2i + x 3 j x4ij = x1 x2i ( x , x.,i)j = z1 zzj where z , = x1 + x 2 i and z2 = x 3 + x4i, the multiplication is based on the rules j 2 = - 1 and z j = $5. Then S1is the subgroup defined by z2 = 0 and the right cosets of S1 are the fibers of the Hopf map p : S3+- S2. Define a map H : X -+S3by taking H ( x ) = F ( x )* [G(x)]-' for every x E X . Since pF = pG, F ( x ) and G ( x ) are contained in a coset of S1 and hence H carries X into a proper subspace S1 of S3.Then it follows that there exists a homotopy Ht : X +S3, (0 < t < l ) , such that H , = H and H , ( X ) = 1. Define a homotopy J t : X +S3, (0 < t < l ) , by taking It(%)= H t ( 4 .G(x) for each x E X and t E I . Then we have J , = F and J 1 = G. Hence F N G. I Since the proof of (6.4) and (6.5) is based on special properties of S1 andS3, there are no analogous results for the maps X -+ S4 and X -+ SS.
{
-
+
+ +
+
+ +
7. Liftings and cross-sections Let E be a fiber space over a base space B with projection p : E -+ B and let f : X -+ B be a map of a space X into B. By a lifting of f in E , we mean
70
111. F I B E R S P A C E S
a map g : X + E such that p g = f . As a special case of (I; 4 16), the lifting problem for the given map f is to determine whether or not f has a lifting in E . For example, if p : S3 + S2is the Hopf map and X is a triangulable, space, then (6.4) and (6.5) solve the lifting problem for any map f : X + S2, namely, f has a lifting in S3 iff f is algebraically trivial. If p : E + B has the CHP for X , then the lifting problem for a map f :X -+ B is equivalent to a broadened lifting problem, namely, to find a map g : X + E such that p g = f . According to the definition of fiber spaces, this is always the case if X is triangulable. By (4.2), this is also true if E is a bundle space over B relative to p and X is a paracompact Hausdorff space. If X is a subspace of B and f : X B is the inclusion map, then the notion of a lifting for f reduces to that of a cross-section over X . A cross-section in E over a subspace X of B is a map x : X E such that
-.
$x(x)
=
x,
-.
(XEX).
Thus, a map x : X + E is a cross-section iff the image x ( x ) of an arbitrary point x E X is contained in the fiber over x. I t is easy to verify that every cross-section x : X + E maps X homeomorphically onto x ( X ) with p I x ( X ) as its inverse. Therefore, a cross-section x : X + E is considered intuitively as lifting the subspace X of the base space B up into E . If E is a bundle space over B relative to p : E + B and U c B is a decomposing neighborhood with decomposing function and natural projection q5u: U x D+P-'(U), y u : U x D + D , then, for any point e in P-l(U), there is a cross-section xe : U + E given by
xe(u) = $u(u, 4, d = YU#JU-'(~) €or each U E U . If u = p ( e ) , then xe(u)= e. Thus, in bundle spaces, local cross-sections always exist. However, the existence of a global cross-section, i.e. a cross-section over the whole base space B, is a rather strong condition on the structure of the fiber space. In fact, if a global cross-section x : B + E exists, then, in any homology theory satisfying the Eilenberg-Steenrod axioms, the projection p : E + B and the cross-section x : B + E induce for each m the homomorphisms 9,: Hm(E)+ H m ( B ) , x * : Hm(B)+ H m ( E ) . Since px is the identity on B , p,x, must be the identity on Hm(B).I t follows that x + is a monomorphism, that p , is an epimorphism, and that Hm(E) decomposes into the direct sum Hm(E) = Kernel p , + Image x * . An immediate consequence of this necessary condition is that the Hopf fiberings in 4 5 do not have global cross-sections. Dually, one can deduce necessary conditions for the existence of a global cross-section in terms of cohomology : Hm(E) = Imagep* + Kernel%*.
8. F I B E R
MAPS A N D I N D U C E D F I B E R SPACES
7'
These conditions (both those on the groups Hm(E) and those on the groups Hm(E))resemble those for retracts. This is no accident : if x is a cross-section, then the image of x is a homeomorph of B and is a retract of E . Let x : X + E be a given cross-section. Corresponding to the extension problem of maps in (I ; 5 2), we have the extension Problem of cross-sections to determine if x can be extended over the whole base space, that is to say, whether OT not there exists a cross-section x* : B + E such that x* 1 X = x . This extension problem of cross-sections is a generalization of that of maps in (I; 3 2). Indeed, if E = B x D where D is a given space, then E is a bundle space over B with projection fi : E + B defined by p ( b , d ) = b. For a given map f : X -+ D on a subspace X of B, we have a cross-section x f : X +E defined by xf (4= ( x , f (4) for each x E X . Then it is obvious that f has an extension over B iff the crosssection xf can be extended throughout B. Similar to the classification problem for maps in (I; tj 8) is the classification problem of cross-sections. Let K denote the set of all cross-sections x : B + E . Introduce an equivalence relation in K as follows: For any two crosssections f , g E K , f g iff there exists a homotopy ht : B + E , (0 < t < l ) , such that h, = f, h , = g, and ht E K for every t E I . Then the classification problem of cross-sections is to enumerate the classes of K divided by this equivalence relation -. An argument similar to that used above for the extension problem shows that this classification problem of cross-sections is a generalization of that of maps in ( I ; 5 8).
-
-
8. Fiber maps and induced fiber spaces Let p : E +B and p' : E' + B' be any two fiberings. A map F : E
+ E' is said to be a fiber map if it carries fibers into fibers. Precisely, F is a fiber map iff, for every point b in B, there exists a point b' in B' such that F carries p-'(b) into p'-l(b'). Now let F : E + E' be a given fiber map. Then F induces a function f : B + B' defined by f ( 4 = P'FP-W
for every b E B. For any arbitrary set U in B', we have
f-'(U)
=
PF-'P'-l(V).
Hence f is continuous if p is either open or closed. If this is the case, f is called the induced map of the fiber map F . In particular, if E is a bundle space over B relative to p , then p is obviously open and so f is continuous. The following rectangle is commutative: F E-+E'
72
111. F I B E R S P A C E S
Some special cases of fiber maps are important. First, let us take p' : E' +.B' to be the trivial fibering over B, that is to say, B' = B, E' = B, and p' is the identity map. In this case, the projection : E +. B is a fiber map and its induced map is the identity map on B. Second, let us take p : E +. B to be the trivial fibering over a space X , that is to say, B = X , E = X , and p is the identity. In this case, every map F : X +. E' is a fiber map with f = p'F : X -+ B' as induced map. This suggests the following extension of the lifting problem in 3 7. Let : E +. B and 9' : E' -+ B' be two fiberings. By a lifting of a given map f : B +. B', we mean a fiber map F : E -+ E' which induces f . Hence, a map F : E -+ E' is a lifting of f : B +. B' iff P'F = fp. The lifting problem for f : B -+ B' is to determine whether or not f has a lifting F : E +. E'. As we have seen in the special case of cross-sections, the answer to this problem is not always affirmative. However, for a given fibering p' : E' B' and a given map f : B -+ B' of a given space B into B', we can construct a fibering : E +. B together with a lifting F : E +. E' of f as follows. Let E denote the subspace of B x E' given by E = { (b, e') E B x E' I f ( b ) = p'(e') } and let p : E +. B denote the natural projection defined by p ( b , e') = b. Let F : E + E denote the map defined by F ( b , el) = e'. We are going to prove that p : E -+ B is a fibering and that F is a lifting of f . By the preceding construction, we have f p = p'F. Hence it remains to prove that p : E -+ B is a fibering. Let : X +. E be a given map of a triangulable space X into E and ht : X +. B , (0 < t Q l ) , be a homotopy of the map y = +p. Let 6 = F+ : X +. E', kt = fht : X +. B', (0 Q t Q 1). Then kt is a homotopy of the map k , = f h , = fp+ = pfF+ = p ' l Since p' : E' -+ B' is a fibering, there exists a homotopy kt* : X +. E', (0 < t < l ) , of 6which covers kt. Define a homotopy ht* : X -+ E , (0 Q t < I ) , by taking ht*(x) = (ht(x),k t * ( x ) ) for every x E X and t E I . This definition of ht* is justified by the relation f k t * = kt = fht. Since
+
--f
+
ko*(x) = ( h O ( X ) > ko*(x)) = F + ( x ) ) = 9(x) for every x E X , ht* is a homotopy of +. Since ht* obviously covers ht, this completes the proof that p : E -+ B is a fibering and F is a lifting off. The fibering p : E -+ B constructed above is said to be induced by f ; the lifting F : E +. E' off will also be said to be induced by f . Note that, for each b E B, F maps the fiber p - l ( b ) homeomorphically onto the fiber @'-l(b'), where b' = f ( b ) . Finally, it is straightforward to verify that, if E' is a bundle space over B', then so is E over B. (
P
W
8
9. M A P P I N G
SPACES
73
Finally, if x' : B' + E' is a cross-section, then the map x : B + E defined by x(b) = (b, x ' f ( b ) ) is also a cross-section. We shall call x the induced cross-section of x' by f . The following special case will be used in the sequel. If B is a subspace of B' and f : B + B' is the inclusion map, then E can be identified with p'-l(B) and p with p' I B in an obvious way. In this case, the induced fibering p : E -+ B will be called the restriction of p' : E' + B' on B. Hence we have the following
p
Proposition 8.1. If E i s a fiber space over a base space B with projection : E + B and if A i s a n y subspace of B, then p-l(A) i s a fiber space over A
with p 1 p-I(A) as projection.
9. Mapping spaces Let
X and Y be arbitrarily given spaces and denote by
a= Y X
the totality of maps of X into Y . There are various ways of topologizing $2, but we will be concerned only with the compact-open topology, [Fox 31. I t is also called the k-topology, [Arens 21, and the topology of compact convergence, [B; I111 and [K]. For any two sets K c X and W c Y , let M ( K , W ) denote the subset of Q . defined by M ( K ,W ) = { I f ( K )c W 1. M ( K , W ) will be called a subbasic set of $2 if K is compact and W is open. The compact-open topology of D is defined by selecting as a subbasis for the open sets of IR the totality of the subbasic sets M ( K , W )of 52. According to the usual definition of a subbasis, every subbasic set is open in D and every open set of f2 is the union of a collection of the finite intersections of subbasic sets. Throughout the present book, mapping spaces are understood to be topologized by their compact-open topologies, unless otherwise stated. Lemma 9.1. I f X i s a Hausdor8 space and { U } i s a subbasis for the open sets of Y , then the totality of the sets M ( K , U ) , for K a compact subset of X and U E { U }, constitutes a subbasis for the compact-open topology of a [Jackson 21. Proof. It suffices to show that if K is a compact subset of X and W an open subset of Y , and if f E M ( K , W ) , then there exist compact subsets K,; . * , Km of X and members U,; . * , Um in { U } such that
Let X say U f ;
E
n M(Km, Urn)c M ( K , W ) . / E M ( K , , V,) n K . Since f ( x ) E W , there are a finite number of sets in { U }, * , U& such that f(X)EU:n
n U$,C
w.
I l l . FIBER SPACES
74
Since f is continuous, there is a neighborhood G, of x in X such that As a compact Hausdorff space, K is regular. So there is an open neighborhood H , of x in K such that the closure K , = H , is contained in G,. The collection { H , I x E K } is an open covering of the compact space K , and hence there are a finite number of points in K , say x1; x,, such that K = Hzl U * * * U HZ,. a ,
In the subscripts and superscripts involved above, we shall simply replace x j b y j , j = l;*.,q. Now the sets K,; ., K, are compact. Moreover,
-
Hence we have
a
ni
Suppose that g e l 2 is contained in the set on the right-hand side of the preceding formula. If x E K , then xis in some H f and hence is in Kj. Therefore Thus g E M (K, W ) ,and so a
ni
Next, there is a natural function w:QxX+Y defined by w (f , x ) = f ( x ) for each f E Q and each x E X . w will be called the evaluation of the mapping space Q. Proposition 9.2. If w of Q i s continuous.
A’ i s a locally compact regular s+ace, then the evaluation
Proof. Let f E 9,x E X, and an open set W of Y which contains f ( x ) be arbitrarily given. Since f is continuous, the inverse image f -l( W ) is an open set containing x . Since X is regular and locally compact, there exists an open neighborhood V of x such that the closure is compact and is contained in f -1(W).Then U = M ( 7 , W ) is a subbasic open set of i2 which contains f . U = M ( V , W ) implies that w maps U x V into W . This proves the continuity of w . I For the necessity of the local compactness of X in (9.2), see Ex.J at the end of the chapter. For each point y E Y, let j ( y ) denote the constant map in f2 which maps X into the single point y . The assignment y + j ( y ) defines a function
j:Y+Q
9. M A P P I N G
75
SPACES
called the natural injection of Y into B.It can be easily verified that j maps Y homeomorphically onto a subspace j ( Y )of B.Furthermore, if Y is a Hausdorff space, then j ( Y ) is closed in a. For any given point a E X , let pa:.R+
Y
denote the function defined by p,( f ) = f (a) for each f E B.Clearly, p, is a map and sends the subspace j ( Y ) of B onto Y . We shall call p, the projection of Sa onto Y determined by the given point a E X . From the definitions given above, we observe that p,j is the identity map on Y and i p , is a retraction of 9 onto its subspace j ( Y ) .Hence we have the following Proposition 9.3. The natural injection i maps Y homeomorphically onto a retract i(Y ) of B. Next, let us consider three given spaces T , X , Y and the mapping spaces Q = yx, @ = y x x * , y = Q T .
For each map taking
+ :X
x T
+
[O(+)(t)l(x)
Y in @, define a function 8(#) : T +B by = +(x,
O(+) is said to be associated with 9.
4,
Proposition 9.4. The associated function
( J E Tx~ , X).
8(+) of
+
E
@ is continuous.
Proof. Let y = 8(+) and let U = M ( K , W )be any subbasic open set in B. I t suffices to prove that y-l( U ) is an open set of T. Let to be any given point in y - l ( U ) . By definition, we have
K x t0c+-1(W) The continuity of implies that +-'(W) is an open set of X x T . Hence #-l(W) is the union of a collection of open sets of the form G, x H,, where C, and H , are open sets of X and T respectively. Since K is compact, K x to i. contained in the union of a finite number of these open sets, say
+
GI x HI, Gz x H z , . . . , Gn x Hn with to E Ht for each i
=
1,2;
* *,
n. Then
H=H,nH,n *-*nH, is an open set of T containing to and is contained in y - l ( V ) . Therefore, y-l( V )is an open set of T. I As a consequence of (9.4) the assignment -+ 8(+) defines a function
e:o+y; when continuous, this function will be called the association map; that this is usually the case is shown by the following Proposition 9.5.
If T i s a Hausdor8 space, then 8 :@ + Y is continuous.
76
111. F I B E R S P A C E S
Proof. Since T is a Hausdorff space and the totality of the subbasic sets M ( K , W) constitutes a subbasis of Q, it follows from (9.1) that the subsets
W L , M ( K , W)l
=
{y
EY
I y ( L )= M ( K , W) 1
form a subbasis for Y, where L runs through the compact subsets of T , K runs through the compact subsets of X, and W runs through the open subsets of Y . It follows clearly from the definition of 8 that
8-1 { M [ L , M ( K , w) 3 1 = M ( K x L, w). Since K x L is compact, M ( K x L, W )is open
[email protected] { M [ L , M ( K , W ) ]} is a subbasis of Y, it follows that 8 is continuous. I It is obvious from the definition that 8 carries @ into Y in a one-to-one fashion. In general, 8 is not onto. However, we have the following Proposition 9.6. T h e evaluation o of Q i s continuous i f , for every space T , the function 8 : @ + Y i s onto.
Proof. Necessity. Let y E Y and consider the map X : X x T + Q x defined by X ( X , t ) = f y ( t ) ,x ) , (xE t E T).
X
x,
Let 4
= W X E @.
Since
[W(t)l(x) = 4(%4 = W X ( % t ) = o [ y ( t )XI * = [y(4l(x) for every 1 E T and x E X,we have 8(+) = y. Hence 8 is onto.
Suficiency. Assume that the condition holds. In particular, select T = 52 and take y E Y to be the identity map on Q. Then there is a map 4 E @ with O(4) = y . Since
4(% f ) = [ Y ( f ) I ( X = ) f(x) =4f4 , for every x E X and f E Q, the continuity of 4 implies that of o.I As an immediate consequence of (9.2) and (9.6), we have the following Corollary 9.7. If X is a locally compact regular space, then the function 8 : @ + Y sends @ onto Y in a one-to-one fashion. Proposition 9.8. If X and T are Hausdorf spaces, then the association m a p
0 : @ + Y i s a homeomorphism of @ onto a subspace of Y [Jackson 21.
Proof. Since 8 is one-to-one and continuous by (9.5), i t remains to prove that 8-1 is continuous on 0(@)c Y. For this purpose, it suffices to prove that, if J is a compact subspace of X x T and W is an open set in Y , then the image O [ M ( J ,W)] is an open set of 8 ( @ ) . Let y E 8 [ M (J , W)] be arbitrarily given. Choose a #IJ E M (J , W) with 8(4) = y . Let J X and J T denote the projections of J in X and T respectively. For each point z = ( x , t ) E J , choose an open neighborhood U, of x in J X and an open neighborhood V zof t in J T , such that
+(UZx Vz)=
w.
9.
MAPPING S P A C E S
77
As compact Hausdorff spaces, J X and JT are regular. Hence we may shrink U, and V, a little bit so that
L)=w,
x
+(&
where Kz denotes the closure of Uz in J X and L, that of V, in J T . The collection { (U, x V,) n J 1 z E J } is an open covering of the compact space J. Hence there is a finite number of points in J , say z1; ., z,,, such that J C ( U Zx~ VzJ U U (Uz, x Vz,).
-
For the subscripts in the notations of the various sets involved above, we shall simply replace zt by i, i = 1, -,n. Now the sets Kg and Lr, i = 1, * n, are compact. Moreover,
-a ,
for each i
=
[y(Ldl (4= +(Kix Lt) = W l;.., n. Hence Y E
e(@)n {.;
1=1
W L t , M(&,
Wl 1.
Suppose that X E Y is contained in the set on the right-hand side of the preceding formula. Since X E O(@), there is a 5 E @ with X = 8 ( 5 ) . If z = ( x , t ) E J , then z is contained in some Ui x Vt and hence in Kt x Li. Since X is in M [ L i , M ( K t , W ) ]and since x E Kr, t E Lt, we have 5(z) = t ( x ,t )
=
[X(t)l ( x ) E w.
This proves that t ( Jc) W . Therefore, 8 [ M ( J ,W ) ] Thus, . we obtain n
Y E ~ P )n {.n M L , S=l
t E M ( J , W ) and hence X
MW, w)i I =
=
8 ( [ )E
e w ( J ,wi.
This proves that 8 [ M ( J ,W ) ]is an open set of 8 ( @ ) .I As an immediate consequence of (9.7) and (9.8), we obtain the following Theorem 9.9. Let X and T be Hausdor8 spaces and Y any space. If X i s locally compact, then the association map 8 i s a homeomorphism of the mapping space @ = Y x x onto the mapping space Y = ( Y X ) T .
Hereafter, when the assumptions of (9.9) are satisfied, the two mapping spaces will be identified by means of the homeomorphism 8 . In symbols, we have y x x T = (Y")T. This will be called the exponential law of mapping spaces. Now let us go back to (9.3) and look for a sufficient condition that j ( Y ) be a strong deformation retract of Q. Such a condition is given by the following Proposition 9.10. If X i s a locally compact regular space and contractible to a point a E X , then j ( Y ) i s a strong deformation retract of Q.
78
111. F I B E R S P A C E S
Proof. By (9.2) the evaluation o :9 x X + Y is continuous. Since X is contractible to the point a E X , there exists a map h : X xI+X such that h(x,0 ) = x , h(x, 1) = a, ( x E X ) ; h(a, t ) = a, (t E I ) . Define a map C# : X x 9 x I + Y by taking C#(x,f, t ) = o [ fh(x, , t ) l , ( X E X , f €9, 4. According to (9.4),the associated function =:
e(+) :JZ x
is continuous. Define a homotopy
Xt
I +Q
: 9 +J2, (0
< t < l ) , by setting
X d f ) =p(f,t) (fEQ,tEI). Then it is easily verified that X, is the identity map, X , fcr every f~ j(Y)and t E I . I Since it is easily verified that Xt[P-YY)I = P 3 Y L we have proved the following
= jPa,
and &(f)
=f
(YEy , t E 4 ,
Proposition 9.1 1. If X is a locally compact regular space and contractible to a point a E X , then, for each y E Y , the subspace p;'(y) is contractible to the point j ( y ) . 10. The spaces of paths Let Y denote a given space. By a path in Y we mean a map a:I+Y of the unit interval I = [0, I ] into Y . The points a(0) and a(1) are called respectively the initial point and the terminal point of the path a, and are said to be connected by the path CT. The relation that two points a and b can be connected by a path in Y is obviously symmetric, reflexive and transitive; hence the points of Y are divided into disjoint classes, called the path-components of Y . We shall denote by Cy the path-component of Y which contains the point y E Y . Y is said to be pathwise connected if it consists of a single path-component and hence every pair of points in Y can be connected by a path. The totality of paths in Y forms a space 9 = Y' with the compact-open topology defined in the previous section. This space 9 together with certain of its subspaces is of fundamental importance in homotopy theory as well as in the functional topology of Morse, [ M , and M J . By a generalized triad ( Y ;A , B ) , we mean a space Y together with two subspaces A and B. ( Y ;A , B ) is said to be a triad if the intersection A fl B is non-empty. For a given generalized triad ( Y ;A , B ) , let us denote by [ Y ;A , Bl
11. T H E S P A C E OF LOOPS
79
the subspace of l? which consists of the paths u in Y such that u(0)E A and B. The following particular cases are of importance. If A = Y and B consists of a single point y E Y , then we shall denote the subspace [Y ; Y , y ] of 8 simply by 8,.It is called the space of paths in Y with a given terminal point y. A loop in Y is a path u : I + Y such that a(0) = a(1). The point o(0) = a( 1) will be called the basic point of the loop u. The set of all loops in Y forms a subspace A of B which will be called the space of loops in Y . If both A and B of a given generalized triad ( Y ;A , B) consist of the same single point y E Y , then we shall denote the subspace [Y ;y, y ] of B simply by A,. It is obviously a subspace of A and will be called the sface of loops in Y with a given basic point y. Clearly we have
a(1) E
A,
=A
n 8,.
The projections Po, p , : 8 -+ Y determined by the points 0, 1 as in the preceding section shall be called the initial projection and the terminal projection respectively. If Y is a Hausdorff space, then the continuity of Po and p , implies that the subspaces A, A,, 8,of 8 are all closed. Let 6, denote the degenerate loop &(I) = y. By (9.11), we have the following
6,. Intuitively, a contraction of 8, to the point 6, is obtained by pushing all paths of Q, along themselves simultaneously to the terminal point y. Proposition 10.1.8, is contractible to the point
11. The space of loops Let Y denote a given space and y E Y a given point. The space A , of loops in Y with y as basic point has a central role in the study of homotopy groups; therefore, we shall study its important properties in the present section. There is a natural multiplication defined in A, as follows. For any pair of loops f , g E A,, the product f g E A , is the loop in Y defined by
f (24, ( f g ) ( t ) = g(*t - I),
I
(0 Q t < 41, (3 Q t Q 1).
Intuitively speaking, fg is the loop in Y which travels along the loop f with double speed while t is in the first half of I; then f g travels along the loop g with double speed while t is in the second half of I . The correspondence (1, g) +. ,~i(f, g) = fg defines a function /L
: A , x A , +Ag
called the multiplication function in A,. Proposition 11.1. The natural multiplication in A , is continuous; by this we mean that the multiplication function ,u is continuous.
80
111. FIBER SPACES
Proof. Let U = M ( K , W ) n A, be an arbitrary non-empty subbasic open set of A,. It suffices to show that the inverse image p-l( U)is an open set of A, x A,. Let 4, y : I + I denote the maps defined by
+
+(t) = i t , y w = i ( t 1) for each t E I . Let A = +-l(K) and B = y-l(K) ; then A and B are compact subspaces of I. Consider the subbasic open sets F = M(A, n A,, G = M ( B , n A ,
w)
w)
of A,. For any pair of loops f , g E A,, it is easy to see that fg E M ( K , W )iff f E M ( A , W ) and g E M ( B , W ) . Hence p-l(U) = F x G. This proves that p-l(U) is open. I Corollary 11.2. Every loop f
EA,
determines two maps L f , Rf :A , + A y defined by Lf(g)= f g and R f ( g ) = gf for every g E A,. Proposition 11.3. I f SEA, denotes the degenerate loop & I ) = y , then L.3 and R6 are deformations of A,. I n fact, there exists a homotopy ht : A , +A,, (0 t l), such that h, = L a ,h, i s the identity m a p , and h,(d) = S for each t E I ; and analogously for Rd. Proof. According to (9.2), the evaluation (o : A, x I + Y is continuous. Hence we may define a map 4 :I x A, x I + Y
< <
By (9.4), the associated function y = O(+) is continuous. It is easily seen that maps A , x I into the subspace A , of Q and, by definition, y is given by t E 1,s E 1). [ y ( f t ) l ( s ) = +(s, f , t ) , ( f Hence we may define a homotopy kt : A , +A,, (0 Q t < l ) , by taking k t ( f ) = ~ ( ft ),, ( f 0 < t Q 1). Then it is easy to verify that k , = Ra, k , is the identity map, and kt(S) = 6 for each t E I . This implies that R6 is a deformation of Ay. Similarly, we can prove that L6 is a deformation of A,. I These properties of A , show that it belongs to the class of H-spaces, which are defined as follows. By a continuous multiplication in a space X , we mean a map p : X x X + X . Let ,U be a given continuous multiplication in X . For any pair of points a and b of X , p(a, b) is customarily denoted by ab and is called the product of a and b. For any given point a E X , the correspondences x + a x and x + xu determine respectively the maps L a : X + X , Ra: X + X
81
11. T H E S P A C E O F L O O P S
called the left and the right translation of X by a. A point a E X is said to be a left homotopy unit if a is an idempotent, that is to say, aa = a, and if La is homotopic with the identity map relative to a. Similarly, an idempotent a E X is said to be a right homotopy unit if Ra is homotopic with the identity map relative to a. An idempotent a E X is called a homotopy unit if it is both left and right homotopy unit. By an H-space, we mean a space X with a given continuous multiplication p which has a homotopy unit. According to the propositions (1 1.1) and (1 1.3),A, is a n H-space under its natural multiplication with the degenerate loop 6 as a homotopy unit. As another important example, any topological group is obviously an H-space under its group multiplication with the neutral element as a homotopy unit. An important property of H-spaces which will be used in the sequel is expressed by the following Proposition 11.4. If X i s a n H-space with xo as a homotopy unit, then the fundamental group n , ( X , xo) is abelian. Proof. Let u,j3 be any two elements of n,(X, xo) and let f , g : I X be representative loops of u,B respectively. Since xo is an idempotent, we may define a loop h : I .+ X a t xo by taking --f
h(4 = f (t)g(t) for each t E I . This loop h represents an element y of n , ( X , xo) which obviously depends only on u and j3. I t suffices to prove that y = up and y = flu. To prove y = UP,we may assume that the representive loops f , g have been so chosen that f ( t ) = xo for each t 2 4 and g ( t ) = xo for each t Q 3. It
Since xo is a homotopy unit of X , there exist two homotopies #s, ys : -+ X , (0 < s < l), suchthat#&) = xox,+,(x) = y0( x ) = x x 0 9 W I ( X ) = x for each x E X and +,(x0) = xo = ys(xo)for each s E I . So, we may define a homotopy h, : I -+ X , (0 < s < l), by taking
4,
X
=
Then we have h,
=
a f (4, { WSdt) +
(if0 < t (if 4 < t
< +), < 1).
h and hs(0) = xo = &(I) for each s E I . Since h, satisfies
i t obviously represents the element uj3. This implies that y = uj3. Similarly, we can prove y = j3u by choosing f and g such that f ( t ) = xo for each t < 4 and g(t) = xo for each t > 4. I In the preceding proof, we have also proved the following
82
111. F I B E R S P A C E S
Corollary 11.5. Under the hypothesis of (11.4),iftwo elements a,/? €zl ( X ,xo) are represented by the loops f , g : I -+ X at x,,, then the element a/?is represented by the functional product h : I -+ X of f , g defined by
h(t) = f (4 g(t)n tt E 4 . As we shall see later, this assertion also holds for the higher homotopy groups. Now let us come back to the space A , of loops and consider its pathcomponents. By (9.9), one can easily see that the path-components of A , are exactly the equivalence classes of the loops of AV as defined in (11, 9 4). Hence we have the following Proposition 11.6. Under the natural multiplication of A,, the path-components of A, furm a group which is essentially the ficndamental grouPz,(Y, y ) .
12. The path lifting property In the present section, we are going to define, for any given map p :E -+ B the path lifting property (abbreviated PLP) which was recently introduced by Hurewicz and Curtis, and to prove that it is equivalent to the absolute covering homotopy property (abbreviated ACHP) defined in 9 2. Roughly speaking, a map p : E -+ B is said to have the PLP if, for each e E E and each path f : I --* B with f (0) = p ( e ) , there exists a path g : I -+ E such that g(0) = e, p g = f , and that g depends continuously on e and f . For a precise definition, let Z denote the subspace of the product space E x B' defined by z = (e, f ) E E x B ' I p ( e ) = /(o) 1. Define a map q:E'+Z by taking q(g) = (g(O),p g ) for each g :I -f E in E'. Then p : E -+ B is said to have the PLP if there exists a map r:Z-+E' such that qr is the identity map on Z.In this case, Y is clearly a homeomorphism of Z onto a retract of E'.
A map p : E -+ B has the PLP in it has the ACHP. Proof. PLP j. ACHP. Let g : X -+ E be any given map of a space X into E and f t : X - + B , (0 < t Q l ) , a given homotopy of the map f = p g . According to (9.4), the homotopy ft gives a map h : X + B' defined by [h(X)l(4 = f&), ( X E X , t E 4. Let k : X -+Zdenote the map defined by Proposition 12.1.
k(x) = (g(x), h(x)). ( % E X ) . Then the composition rk is a map of X into E'. According to (9.9),we may define a homotopy gt : X -+ E , (0 < t < l), by taking g t ( 4 = [ ~ k ( x ) l ( t ) (, X E X , t E 1).
13. T H E F I B E R I N C
THEOREM FOR M A P P I N G SPACES
-
83
One can easily verify that go = g and pgt = f t for each t E I . This implies ACHP. ACHP PLP. Let q :2 + E denote the natural projection defined by q(e,f ) = e for each (e, f ) €2.Define a homotopy & :2 .+B , (0 < t < l), by taking Me,/)= f(t), ( ( ~ , ~ ) E Z , ~ E I ) .
Then we have to= h.According t o ACHP, there exists a homotopy qt :2 + E , (0 < t < l), such that qo = q and pqt = 66 for each t E I . According to (9.4), qt gives a map Y : Z + E' defined by
[r(z)l(t)= q&), ( z E 2,t E 4. Then it is easily verified that qr(z) = z for each Z E Z .Hence, p : E + B has the PLP. I Corollary 1 2 3 If a map over B relative to p .
p
:E
+B
has the PLP, then E is a fiber space
13. The fibering theorem for mapping spaces Let X be a locally compact ANR and A be a closed subspace of X which is also an ANR. On the other hand, let Y be an arbitrary space. Let us consider the mapping spaces E = Yx, B = Y A . There is a natural map
p:E-+B defined by p ( f ) = f I A for every map f : X + Y . We are going to prove that E is a fiber space over B with p as projection; in fact, we have the following stronger result.
p : E + B has the PLP. Let us consider the subspace Z of E x B'
Theorem 13.1. The map
Proof. and the map q : Ex + 2 described in the preceding section and try to construct a map r :2 -+ E' such that qr is the identity map. Consider the closed subspace T = ( X x 0) U ( A x I ) of X x I . By (I; Ex. 0),T is a retract of X x I . Let p : X x I 1 T denote a retraction. Consider the mapping space YT.For each g : T -+ Y in YT,define two maps e : X --f Y and f : I -+ Y" as follows:
e ( x ) = g(x, 0). ( x E X ) ;
[ f ( t ) l ( a= ) g(a, 4, ( ~ E ~ I, € 4 . The continuity of f follows from (9.4). Hence, the assignment g + 8(g) = (e, f ) determines a function 8 yT -+z. By the methods used in establishing the exponential law of mapping spaces in 5 9, one can prove that 8 is a homeomorphism of YTonto Z. Using the inverse of 8, we can define a map r :Z + E' by taking { [r(z)I(t)1 (4= [8-'(z)lp(x, t )
84
111. F I B E R S P A C E S
for every t E I and x E X . Since p is a retraction, it follows immediately that qr(z) = z for each z in Z. I Note. The condition that X is locally compact is inessential. In fact, the theorem holds without this condition; for (9.7) may be replaced by Ex. L at the end of the chapter.
Now let { A , j p E M } and { Y , I p E M } be given families of subspaces of A and Y respectively, both indexed by the elements of a set M . Consider the subspace B' of B which consists of the maps g E B such that g(A,) c Y , for each p E M . Let E' = P-l(B'). Then E' is the subspace of E consisting of the maps f : X + Y such that f (A,) c Y , for every p E M . By (8.1) and (12.2), we have the following
p' : E' -+ B' i s a fibering. In fact, it is also obvious that p' : E' -+ B' has the PLP. So, this generalizes (13.1). For important examples, let us take X to be the unit interval I . First, take A to be the single point 1 of I . In this case, the space E is the space SZ of all paths in Y , and the space B can be identified with Y in an obvious way. Furthermore, the natural projection is essentially the terminal projection p , :SZ+ Y . Hence P I : 52+Y has the P L P by (13.1). Let 2 be a subspace of Y , then P,-l(Z) = [ Y ;Y , Z ] . Corollary 13.1. T h e m a p
Therefore, by (13.2), we have the following Corollary 13.3. For any subspace Z of a given space Y , the space [ Y ;Y , Z ] i s a fiber space over Z relative to the terminal projection. Similarly, the space [ Y ;Z , Y ]i s a fiber space over Z relative to the initial projection.
Next, let X = I and take A to be the subspace of I which consists the boundary points 0 and 1 of I . Let M be an index set containing a single element p. Define A , and Y , to be the single points 1 E A and y E Y , where y is a given point. In this case, the space E' in (13.2) becomes the space 52, of all paths in Y with y as terminal point, and the space B' can be identified with the path-component C, of Y in an obvious way. Furthermore, the natural projection p' is essentially the initial projection p,. Hence, we have the following important Corollary 13.4. The space i s a fiber space over Y relative to the initial projection. T h e fiber over the given point y E Y i s the space A , of all loops in Y with y as basic point.
Let Z be a subspace of Y . Then the following corollary is an immediate consequence of (13.4) and (8.1). Corollary 13.5. For every subspace 2 of a space Y and any given point y in Y , the space [ Y ; Z , y ]i s a fiber space over Z relative to the initial projection.
14.T H E I N D U C E D M A P S
I N MAPPING SPACES
85
14. The induced maps in mapping spaces Let X , Y ,Z be arbitrarily given spaces and : Y + Z a given map. For
+
each f E Yx,the composition +f is in Zx.The assignment f ++f defines a + x : Y X +zx function which is obviously continuous and will be called the induced map of on Y x into Zx.Let C denote the category composed of all spaces and all maps, [E-S; p. 1 101. The operation Y + Y xand+ ++x, where X is a given space, defines a covariant functor of C into itself, [E-S; p. 11 11. Furthermore, if we consider the natural injection and the projection p a determined by a given point a E X , the commutativity relatioiis
+
j+ = +xj, +pa = pa+x obviously hold in the following rectangles of maps: Y-+Z4
4 Y-+z
Throughout the remainder of the present section, we are concerned with the important special case that X = I . Consider a given map : Y + 2. Let y be a given point of Y and denote z = +(y) E 2. As above, we shall use the following notation:
+
YI:
Q, = [ Y ;Y , YI, A , = [Y; Y, Q, = [ Z ; Z,z], A , = [ Z ;z, ZI. Then the induced map obviously sends Q, into Q, and Av into A,. More
+'
generally, 4' maps [Y ; A , B] into [Z ; + ( A ) ,+(B)] for any given subspaces A and B of Y . According to 3 11, there are natural multiplications defined both in A, and in Az. I t follows from the definition that (14.1) +'(fg) = +'(f)+'(g) for any f E A , and g E A,. Since 4' is a map, it sends the path-components of Av into those of A,. Hence, (14.1) shows that 4' induces a homomorphism of the group of path-components of A , into that of A, which is the induced homomorphism +* of n,(Y , y) into nt,(Z,z). Now let Y , Z be given spaces, A C Y , B C Z given subspaces, and yo E A , z,, E B given points. Let u = [Y; Y , Yo], c = [ Y A;, yo], V = [Z ;2, z0], D = [Z ; B, z0]. and denote by uoE C and vo E D the degenerate loops u,(Z) = yo and vo(Z) = zo respectively. Let P : U - + Y , q:v+z denote the initial projections. We are going to establish a covering map theorem which will be important in the next chapter.
86
111. F I B E R S P A C E S
Theorem 14.2. If y : U -+ Z is a map swch that y(C) C B and d u o )= z,, then there exists a map y* : U+V such that qy* = yl y*(C) c D , and
v*(uo) = Vo. Proof. According to (9.2). the evaluation o:U x I + Y is continuous. Define a map a : I x I x U -,Y by setting a(s,t,f) = o ( f , s + t - s t ) , (SEI,tEI,fEU).
By (9.3),the associated function X : I x U
+U
defined by
W ,f ) I ( s ) = a(s,t , f ) , (t E I,f E u,s E 4, is continuous. Let 6 = yX : I x U +Z. By (9.3), the associated function y* : U + V defined by
w,
[y*(f)I(t) = f L ( t E I,f E U), is continuous. It remains to verify the relations qy* = p, y*(C) c D and Y*(”O) = .V, From the definition of X , one can easily see that X(0,f) = f for every fcs U. Hence we have qy*(f) = [Y*(f)l(O)= f ) =w(f) for every f E U. This proves that qy* = y. y*(C) c D is an immediate consequence of qy* = y and y(C)c D.To check y*(uo),= u, we first note that X ( t , u,) = u, for every 1 E I.Then we have l u X ( O 9
[y*(u,)I(t) = 6V, uo) = @(t, u,) = y ( 4 = 2,. for every t E I.This implies y*(uo)= u,. I If we removed the condition y*(uo) = uo from the conclusion, then (14.2) would be easier to prove. By (13.1), the map q : V + Z has the PLP and hence also the ACHP. Since U is contractible, an application of the CHP for U gives a lifting y* : U -+ V of the map y : U + 2.That y*(C) c D is obvious but the condition y*(uo)= u, does not necessarily hold. The lifting y* constructed in the proof of (14.2) may be called the canonical lifting of y .
15. Fiberings with discrete fibers Motivated by the results concerning the exponential map p : R +S1 in Chapter 11, we are going to establish similar results for fiber spaces with discrete fibers, i.e. in which all fibers are discrete. Let E be a given fiber space over B relative to p : E + B with discrete fibers. Lemma 15.1. Fm each path u : I + B joining b, to b1 and for each e, E p-l(b0), there exists one and only one (covering) path a* : I + E such that o*(O) = e, and pa* = a. Proof. The path u may be considered as a homotopy of the partial map a I 0; hence, the existence of a* is an immediate consequence of the covering homotopy property. To prove the uniqueness, let u*, a# :I + E be any two paths in E such
15. F I B E R I N G S W I T H D I S C R E T E F I B E R S
87
that fiu* = u = #a# and u*(O) = e, = u#(O). Let s E I be arbitrarily given. It remains to show that u*(s) = a#(s). For this purpose, let us define a map g : I + E by taking u*(s - 2st), (if 0 Q t Q i), g(t) = u#(2st - s ) , (if 4 < t < 1). Then the map f = fig : I + B has a homotopy fr : I + B , (0 < I Q l), defined by u(s -2st Brst), (if 0 < t < i), fr(4 = u(2rs 2st - s - 2rst), (if 4 < t < 1). Since fr(0)= u(s) = fr( 1) for every r E I , it follows from (iv) of (3.1) that g has a homotopy gr : I --f E , (0 < I < l ) , such that
(
[
+
+
gr(0) = u*(s), gr(1) = a#(s) for every Y E I . Since / , ( I ) = a(s), p g , = f l implies that the connected set g,(I) is contained in the fiber over u(s) which is discrete. Therefore, g,(I) must be a single point. This implies that a*(s) = u#(s). I We recall that two paths u , t : I B joining b, to b, are equivalent, u N t,if they are homotopic with the end points held fixed. The paths in B joining b, to b, are thus divided into disjoint equivalence classes which are actually the path-components of the space [ B ; b,, b , ] . Pgr
= fr,
--f
Lemma 12.2. The terminal point u*(l)of the covering path a* : I + E i n (15.1) depends only on e,E#-l(b,) and the equivalence class of the path u:I-+B. Proof. Assume that u, t : I + B are equivalent paths joining b, tb b, and that u*, t*:I - + E are the covering paths with common initial point e o E f i - l ( b o ) . It suffices to prove that u*(l) = t*(1). Since u N t,there exists a homotopy ht : I + B , (0 Q t < l), such that h, = u, h, = t and ht(0) = b,, ht(1) = b, for each t E I. According to (iv) of (3.1), there exists a homotopy ht* : I + E , (0 < t < l), such that ho* = u*, fiht* = ht, ht*(O) = e,, and ht*(1) = u*(l) for every t E I . Since h,*(O) = e, and fib,* = h, = t, it follows from the uniqueness part of (15.1) that h,* = T*. Hence we have t*(1) = h,*(l) = ~ * ( 1 )I.
Lemma 15.3. For each b, E B and each e, E fi-l(bo),the projection p : ( E , e,) +
(B, b,) induces a monomorfihism
p,
: n , ( E ,e,) + n , ( B , bob Proof. Assume that a E ~ , ( Ee,), and $*(a) = 1. Let g : I --f E be a representative loop of a. Since the loop f = p g : I + B represents the element fi+(a)= 1, there exists a homotopy ft : I B , (0 < t < l), such that f,= f, / , ( I ) = b, and ft(0) = b, = ft(1) for each t~ I . According to (iv) of (3.1), there exists a homotopy gt : I + E , (0 < t < l), such that go = g, #gt = ft --f
88
111. F I B E R S P A C E S
and g,(O) = eo = g,(l) for each t E I . Since f l ( I ) = b, and Pg, = f l , the connected set gl(l) is contained in the fiber p-l(b0) which is discrete. Hence, g!(I) must be a single point. This implies that g , ( l ) = eo and a = 1. I Therefore, n,(E, e,) is isomorphic to a subgroup p,[nI(E, e,)] of n l ( B , b,) which obviously depends on the choice of e,. In general, there are no relations among these subgroups p , [ n l ( E ,e,)] for various choices of e, E p-l(bo) unless they can be joined by paths in E. Hence, let us assume that E is pathwise connected. It follows, of course, that B must also be pathwise connected. Now let el be another point in p-l(bo). Since E is pathwise connected, there exists a path a : I -+ E joining e, to el. According to (11; 4.1), u determines an isomorphism
u, :n l ( E ,el) M n l ( E , 8,). On the other hand, pa : I + B is a loop a t b, and hence represents an element w of n,(B, b,). By the definitions of p , and a,, one can easily see that +,a*(.) = w*p,(a)*w-l for each a E n l ( E , el). This implies that p,[n,(E, e l ) ] is the transform ~ - l . P * [ n I ( Ee0)l.w , of P*[n,(E,e0)I. Conversely, let w be an arbitrary element of n,(B, b,). Pick a representative loop t : I + B for w . By (15.1) there exists a path a : I -+ E such that a(0) = e, and pa = t. By (15.2), the point e, = a(1) in fi-l(b,) depends only on the element w . One can also easily see that el = e, iff w is in the subgroup fi,[nl(E, e,)]. Hence, we have proved the following Theorem 15.4. If E i s a pathwise connected fiber space over B relative to
p :E
B with discrete fibers, then, for each b, E B, the images p,[nc,(E,e,)] of the induced monomorphisms -+
P, :n,(E, 8,) +n,(B, b,) for all e, E p-l(b,) constitute a class of conjugate subgroups of n,(B, b,). Fwthermore, for a fixed e, E #-1(b0), there i s a natural one-to-one correspondence between the poivrts of p-l(b,) and the right cosets. of fi, [nl(E,e,)] in ntl(B,b,) with e, corresponding to the subgvorrfi p,[n,(E, eo)]. This class X(E : b,) of conjugate subgroups { P,[nl(E, e,)] I e, E p-l(b,) } of n , ( B , b,) will be called the characteristic class of E a t b,. Each group in X(E : b,) is isomorphic to the fundamental group n,(E) of E. X(E : b,) will consist of a single group iff p , [nl(E,e,)] is an invariant subgroup of n l ( B ,b,) for some and hence every e, E #-l(b,). If X(E, b,) consists of a single group, then so does X(E, b,) for any other b, E B. In this case, we say that the fibering ( E ,p , B ) is regular. For a fixed e, E p-l(bo), the natural one-to-one correspondence is defined as follows: The right coset of p,[nl(E, e,)] in n,(B, b,) corresponding to el E p-l(b,) is the one which contains the element w E ~ , ( Bb,), represented by the loop pa : I --f B, where a : I -+ E is any path joining e, to el. Note. All the assertions as well as their proofs obviously hold for fiber
16.C O V E R I N G S P A C E S
89
spaces with totally pathwise disconnected fibers. Here, a space is said to be totally pathwise disconnected if it has no pathwise connected subspaces except single points. 16. Covering spaces Throughout the present section, let B be a given connected and locally pathwise connected space. A connected space E is said to be a covering space over B relative to a map p : E -+ B if the following conditions are satisfied: (CSl) p maps E onto B. (CS2) For each b E B, there is a connected open neighborhood V of b in B such that each component of p - l ( V ) is open in E and is mapped homeomorphically onto V by p. As immediate consequences of (CS2), the following two conditions are also satisfied: (CS3) E is locally pathwise connected. (CS4) p : E + B is an open map. Examples. According to (11; 2.2), the real line R is a covering space over S1relative to the exponential map p : R + S1.Next, let us consider the unit n-sphere S n in the euclidean ( n + 1)-space Rn+l. If we identify the antipodal points of Sn, we obtain the real projective n-space Pn with natural projection p : S n -+ Pn. It is easy to verify that S n is a covering space over Pn relative to p and the fibers are the pairs of antipodal points in S”. Covering spaces are the earliest examples of fiber spaces. In fact, we have the following theorem which is an easy consequence of (CS2) and the connectedness of B. Theorem 16.1. Every covering space E over B relative to p : E -+ B is a bundle space over B relative to p with discrete fibers. For generalizations and converses of (16.1), see Ex. P a t the end of the chapter. Now, let us consider the lifting problem for a map f : X + B of a connected and locally pathwise connected space X into the base space B of a covering space E relative to a projection p : E + B. Pick x, E X and denote b, = f (x,). Let e, be a point of E with p(e,) = b,. The following theorem is a generalization of (11; 5.3). Theorem 16.2. (The lifting theorem). There exists a unique map g : X + E such that g(xo) = e, and p g = f 28 the image of f* : n , ( X , x,) + n l ( B ,b,) is contained i n that of p , : n l ( E ,e,) +7c1 ( B , b,). Proof. The necessity of the condition is obvious. In fact, if a map g : ( X , x,) + ( E ,e,)
exists such that p g = f, then we have p*g, = f* and hence the image off* is contained in that of p , . I t remains to establish the sufficiency of the condition. The remainder of the proof is motivated by that of the special case (11; 5.3).
111. F I B E R S P A C E S
90
To construct the map g , let x be an arbitrary point in X. Then there exists a path n : I -, X with n(0) = x, andn(1) = x. The composed map u = f n : I + B i s a p a t h i n Bwithu(0) = f(x,)andu(l) =f(x).Accordingto(15.1), there exists a unique path t : I -+ E such that t(0)= e, and P t = u. We assert that the point ~ ( 1 of ) E does not depend on the choice of the path n : I --f X. To prove this, let n’: I -+ X be another path in X joining x, to x and let t’: I -+ E denote the unique path such that ~ ‘ ( 0= ) e, and +t’ = fn’. We are going to prove that t(1) = ~ ’ ( 1 )The . loop A = n ~ . ’ -: ~ I + X represents an element a € n l ( X ,x,) and so the loop f I : I + B represents the element f*(a) of nl(B, b,). According to our condition, this element is contained in the subgroup p , g 1 ( E , e,). Hence there exists a loop p : I + E at e, such that p p = f I . Since I = n d - 1 , it follows from the uniqueness of the path in (15.1) that t ( t ) = p(#) and t ‘ ( t ) = p(1 - i t ) for each t E 1. In particular, t(1) = p(4) = t’(1). This proves our assertion. Because of the preceding assertion, we may define a function g : X + E by setting g(x) = t(1) for each x E X. By the construction of the point t(l), it is obvious that g(xo) = e, and p g = f . By the same method as used in the proof of (11;5.3), one can establish the continuity and the uniqueness of the function g : X -+ E. I If we omit the condition g(xJ = e, in (16.2), we obtain the result that there exists a map g : X + E with p g = f iff f* maps n,(X, x,) into a group of the characteristic class x (E, b”). Now, let E be a covering space over B relative to p : E --f B, E’ a covering space over B’ relative to p’ : E‘ -+ B’, and f : B -+ B’ a given map. Let e, E E, b, E B, e’, E E‘, and b,,’ E B‘ be given points such that
p(eo) = b,, The maps diagram:
p , p’
p’(e’,)
=
b’,,
f(b,) = b’,.
k*- I*
and f induce homomorphisms indicated in the following n,(E,4
g*
n,(E’, c‘o,
f*
n,(B,4)
nl(B’, b’o)
Theorem 16.3. (The fiber map theorem). There exists a unique map
-+ E‘ such that g(eo)= e‘, and p’g that of PI*.
g :E
=
f p i# f* carries the image of p , into
Since X can be considered as the trivial covering space over itself, (16.2) is a special case of (16.3). However, (16.2) also implies (16.3) by considering the map f p : E -+ B’. This fiber map theorem has quite a few important consequences to which we devote the remainder of this section. First, let us assume that B = B‘ and that f is the identity map on B. Then we obtain the following
16. C O V E R I N G
SPACES
91
Theorem 16.4. (The covering theorem). Let E and E' be two covering spaces over the same space B relative to p : E -+ B and p' : E' + B respectively, and let b, E B. I/ e, E E and e', E E' are such that
#(eo) = bo = fl'(e'o)* p*[n,(Ele0)l = #J'*rZl(E'* e'J1, then there exists a unique map g : E -+ E' such that = e'o, P'g = P. Furthermore, E is a covering space over E' relative to g .
Proof. The first assertion is a special case of (16.3). Hence, it remains to prove that E is a covering space over E' relative to g . For this purpose, let us first verify (CSI). Let e E E , b = p ( e ) and e' = g(e). Choose a connected open neighborhood V of b in B such that (CS2) holds for both covering spaces E and E' over B . Let W denote the component of p-l(V) containing e and W' that of p'-l( V )containing e'. Then the restrictions q=pJw, q'=p')w'
are homeomorphisms onto V . Since p'g = p , we have g I W = q'-lq. Hence g maps W homeomorphically onto W'. This implies that g(E)is both open and closed in the connected space E'; so g maps E onto E'. Next, let us verify (CS2). Let e' E E' and b = $'(el). Choose a connected open neighborhoodv of b in B such that (CS2) holds for both covering spaces E and E' over B . Let W' denote the component of p'-l(V) which contains e'. Then W' is a connected open neighborhood of e' in E' and g-l(W) is the union of a set of components of p-l(V). It follows that every component of g-l(W') is open in E and is mapped homeomorphically onto W' by g. I In particular, if E is simply connected, then (16.4) implies that E is a universal covering space over B relative to fi : E -+ B. Here, a covering space E over B relative to p : E -+ B is said to be ulziversal if, for every covering space E' over B relative to p' : E' + B, there exists a map g : E + E' such that p'g = p and that E is a covering space over E' relative to g. Next, let us define the notion of equivalent covering spaces as follows. Two covering spaces E and E' over a same base space B relative to projections p : E -+ B and p' : E' -+ B are said to be equivalent if there exists a homeomorphism g :E -+ E' of E onto E' such that p'g = p . By (16.2) and (15.4), the characteristic classes X(E, b,) and X(E', b,) are defined for every b, E B. Theorem 16.5. (The equivalence theorem). For any given point 6 , E B , two covering spaces E and E' over B relative to projections p : E -+ B and p' : E' -+ B are equivalent ifl X(E, b,) = X(E', b,). Proof. Necessity. Let g : E -+ E' be a homeomorphism of E onto E' such that p'g = p. Let e, E p-l(b,) and e', = g(eo).Then g induces an isomorphism g, of n,(E, e,) onto n,(E', e l , ) . On the other hand, p'g = p gives p', g, = p,. This implies p,[n,(E, e,)] = p',[n,(E', e',)] and hence X(E, b,) = X(E', b,).
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111. F I B E R S P A C E S
Suficiency. Assume that X(E, b,) and e’, E E‘ such that
=
X(E‘,b,). Then there are points e,
E
E
#(e,) = bo = p‘(e’o)> $ * b , ( E #e0)I = p’*rnl@‘>e’0)l. According to (16.3), there exists a map g : E + E‘ such that g(e,) = e’, and p’g = p . Similarly, there exists a map h : E‘ +. E such that h(e’,) = e, and ph = p’. Now, consider the composed map hg : E +. E . Since hg(e,) = e, and phg = p’g = p , it follows from the uniqueness part of (16.4) that hg is the identity map on E. Similarly, gh is the identity map on E‘. Hence g is a homeomorphism and the covering spaces E , E‘ are equivalent. I In particular, any two simply connected covering spaces over the same base space are equivalent. Next, let us consider the regular covering spaces. A covering space E over a base space B relative to p : E +. B is said to be regular if the fibering p : E +. B is regular in the sense of 5 15. Hence, E is a regular covering space over B iff for every b, E B, X(E, 6,) consists of a single invariant subgroup of n,(B, b,). Now, let E be a given regular covering space over B relative to p : E +. B and let b, be a given point in B. Since X(E, b,) is an invariant subgroup of nl(B,b,), the quotient group
W = ni(B, bo) / X(E, b,) is well-defined. One can easily verify that, as an abstract group, W does not depend on the choice of the basic point b, E B. Then, we have the following Theorem 16.6. T h e group W operates on the right of the regular covering space E. More precisely, to each element w of the group W and each point e of the space E , there corresponds a unique point ew of E such that
(ewJw2 = 4 . F 2 ) and, for each w E W , the correspondence e -f ew defines a homeomorphism w* o f E onto itself. Furthermore, the operation has the following two properties : (i) p(ew) = p ( e ) for every e E E and w E W . (ii) For a given e E E , ew = e implies w = 1. In words, the condition (ii) is equivalent to saying that W operates freely on E. Proof. Pick a point e, E $-1(b,). According to (15.4), there is a natural oneto-one correspondence v : W + p-l(b,) of W onto p-l(b,) with v ( 1) = e,. For an arbitrary element w E W , let el = v ( w ) . Since E is a regular covering space over B, we have
P*[Zl(E>e0)l = $*[n,(E, e l ) ] . Hence, just as in the proof of (16.5),there is a unique homeomorphism w* of E onto itself such that w*(e,) = el and pw* = p . By the construction of v and w * , one can easily verify that (w1w2)* = w2*w1*,
(w,, W 2 E W ) .
17.
CONSTRUCTION O F C O V E R I N G SPACES
93
If the homeomorphism w* of E admits a fixed point e E E , then it follows from the uniqueness part of (16.3) that w* must be the identity map on E. This implies that el = w*(e0) = e, and hence we have w = v-l(e0) = 1. Then, the theorem follows immediately if we set ew = w*(e) for every e E E and W E W . I In particular, if E is a simply connected covering space over a space B, then the fundamental group n,(B) operates freely on E. In the classical theory, the homeomorphisms w* in (16.6) are referred to as the covering transformations (Deckbewegungen) of the regular covering space E. 17. Construction of covering spaces A space B is said to be locally simply connected if, for every point b E B and every open neighborhood V of b in B, there exists an open neighborhood U c V of b in B such that, for any two points u, and ul in U ,every pair of paths in U joining uo to u1 are homotopic in V with end points held fixed. Obviously, every locally contractible space is locally simply connected and hence so is every simplicia1complex. For our purpose in the present section, a slightly weaker condition is enough: we can replace the open neighborhood V by the space B itself. In this case, the space B is said to be semi-locally simply connected. Observe that every locally simply connected space is semilocally simply connected and so is every simply connected space. Throughout the present section, we assume that B is a given space which is connected, locally pathwise connected, and semi-locally simply connected. Let b, be a given point in B. Theorem 17.1. (The existence theorem). For every subgroup G of the fundamental group nl(B,b,), there exist a covering space E over B relative to a projection p : E -+ B and a point e, E p-l(b,) such that G i s exactly the image of the induced homomo7Phism p., : n l ( E ,e,) +nl(B,b,). Proof. Let us consider the space of paths 52 = [ B ;b,, B ] . As in $ 10, we denote by p , : 52 -+ B the terminal projection defined by p , (a) = u(1) for each u E 52. We shall define a new equivalence relation in 52 as follows. Two paths u, t E 52 are said to be equivalent modulo G, (in symbols: CJ t mod G), if $,(a) = p,(t) and the element of n,(B, b,) represented by the loop o * T - ~is in G. Let E denote the quotient space of 9 defined by this equivalence relation. Then the points of E are equivalence classes (mod G) of the paths 9.We shall denote by [u] the class which contains the path u E 52. Define a map p:E+B by taking P[u] = $,(a) for every [a]E E. The continuity of p follows from (I; 12.1). Since B is pathwise connected, it follows that p maps E onto B. We shall construct a convenient basis for the open sets of E as follows. For a given e E E and a given open neighborhood U of p(e) in B, choose a path N
94
111. F I B E R S P A C E S
a E e and denote by N(e, U)the subset of E which consists of the classes each containing a path of the form t : I -+ B such that t ( t ) = a(2t) for every t < 4 and t ( t ) E U for every t 2 4. It is obvious that N(e, U)does not depend on the choice of the representative path a from the class e G E. Since B is locally pathwise connected and semi-locally simply connected, it is straightforward to prove that N(e, U)is open in E. Then it follows easily that the collection { N ( e , U ) }for all e E E and all open neighborhoods U of p ( e ) in B constitutes a basis for the open sets of E. An immediate consequence of this result is that p : E -+ B is open. To prove that E is a covering space over B relative to p , let us establish (CSl) and (CS2). Since p maps E onto B, (CS1) is satisfied. To prove (CS2), let b be an arbitrarily given point of B. According to our assumption on B, there exists a pathwise connected open neighborhood U of b in B such that, for any two points uoand u1in U,every pair of paths in U joining uoto u1 are homotopic with end points held fixed. Then it suffices to prove that p-l( U )is the disjoint union of the open sets { N(e, U)I e E $-'(a) } in E each of which is mapped homeomorphically onto U by p. For this purpose, we shall first prove that p maps N(e, U)homeomorphically onto U for each e in P-l(b). By the definition of N(e, U), it is obvious that p maps N(e, U )into U.Since U is pathwise connected, there is a path 8 : I -+ U joining b to u. Choose a path aEl2 with [a] = e and consider the path z = a.8. Clearly [t]E N(e, V) and p [ z ] = u. This proves that p maps N(e, U)onto U.Let us assume that el and eBare any two points in N(e, U) such that #(el) = p(ea). There are two paths t(EB, (i = 1,2), such that [tt] = et, t t ( t ) = a(2t)whenever 1 Q 4, and t t ( t ) E U whenever t 2 4. Since +(e,) = P(ea),we have tl(1) = t g1). ( Call this common terminal point v E U. Denote by 6s :I -+ U the path defined for each i = 1,2 by ( t E I).
Then t , and [g are two paths in U joining b to v. According to the choice of U, El and t Bare homotopic in B with end points held fixed. This implies that tl and t gare homotopic with end points held fixed and hence e, = eB. This proves that p maps N(e, U)in a one-to-one fashion. Since p is both open and continuous, we conclude that p maps N(e, U)homeomorphically onto U. Next, we are going to prove that the open sets N(e, U),e E p-l(b), are disjoint. Assume that N(e,, U)and N(e,, U)have a common point x. Choose paths ut E B,(i = 1 , 2 ) , with [at] = et. Then there are paths ti E 8,such that [tt]= x , t t ( t ) = a424 if t Q IJ, and q ( t )E U if 1 2 4. Let 51 : I + U , ( i = 1,2), denote the path defined by (i).Then El and Ea are two paths in U joining b to p ( x ) . By the choice of U,t1and tB are homotopic in B with end points held fixed. This implies that
t,-t,l-al.t,-t~l-o;;l-ul.a$l.
17.C O N S T R U C T I O N
O F COVERING SPACES
95
Since [t,]= x = [ta],t l - t s l represents an element of G and so does a,.a,l. Hence el = e,. This proves that the open sets N(e, U),e E $-'(a), are disjoint. Now, we shall prove that p-l(U) is the union of the collection { N(e, U)I e E # - l ( b ) }. Since # maps N(e, U ) into U,N(e, U ) is contained in p-l(U). Let x E p-l(U) be arbitrarily given. We have to prove that there is some e E p-l(b) such that N(e, U) contains x. For this purpose, chQose a path z € 8with [t]= x and let 2c = p ( x ) = t(1). Since U is pathwise connected, there exists a path 8 : I -+ U joining b to u. Let a = t . 8 - 1 and t = a.8. Set e = [a]. Since a(1) = O(0) = b, we have e E p - l ( b ) . By an easy homotopy, one can prove that t and t are homotopic with end points held fixed. It follows that x = [t]= [t]E N(e, U).Hence, p-l(U) is the union of the collection { N ( e , U )I e E +-,(/I) }. This completes the proof of (CS2). As a continuous image of a pathwise connected space 8, E is pathwise connected. This completes the proof that E is a covering space over B relative to p . Let 6 E 8 denote the degenerate path at b,, that is to say, 6(I)= b,. Denote e,, = [6]E E. Then p(e,) = b, and p induces a monomorphism 9, :n,(E, e,) +nl(B, bob It remains to prove that the image of p , is esactly the given subgroup G of n,(B, b,). For this purpose, let us first prove an assertion as follows. Let a : I -+ B be any path with a(0) = b,. According to (15.1), a has a unique covering path a* : I -+ E with a*(O) = e,. We assert that, for each t E I , a*(t) is the class which contains the path at : I -+ B defined by at(s) = cr(st) for each s E I . This follows immediately from the fact that the assignment t -,at defines a path t:I -,52 according to (9.10)and that E is a quotient space of 52. Now, let us prove that the image of p , is exactly G. Let a be any element in n,(E, e,). Choose a representative loop u* : I -+ E for a, then the loop u = pa* represents the element #,(a) of n,(B, b,). According to the foregoing [a] = a*(1) = e, = [d]. assertion, we have This implies that #,(a) E G. Conversely, let p be any element of G. Choose a representative loop a : I -+ B for p. By (15.11, a has a unique covering path a* : I -+ E with a*(O) = e,,. According to the foregoing assertion, we have a*(1) = [a]. Since a represents p E G, this implies that a*(1) = e,. Hence a* represents an element a of n,(E, e,). This implies that p = p,(a). Thus, we have proved that G is exactly the image of 9., I In particular, if G is the trivial subgroup of n,(B, b,) consisting of the neutral element 1 of nl(B,b,), then E is a simply connected covering space over B. According to (16.4) this implies that E is a universal covering space over B. By (15.3)it follows that every universal covering space of B is simply connected. Finally, by (16.5) we conclude that every pair of universal covering spaces over B are equivalent. Hence, E is essentially the only
96
111. F I B E R S P A C E S
universal covering space over B. Hereafter, E will be referred to as the universal covering space of B. Combining (16.5) and (17.1), we obtain the following classification of covering spaces. Theorem 17.2. (The classification theorem). For any connected, locally pathwise connected, and semi-locally simply connected space B, the equivalence classes of the covering spaces over B are in a one-to-one correspondence with the conjugate classes of subgroups of the fundamental group n,(B).
The following simple examples are given t o illustrate the preceding results. (i) Covering spaces of S1. The real line R is the universal covering space over S1 relative to the exponential map p : R -f 9. Since nl(S1)is abelian, it follows that every covering space over S1is regular. Since nl(S1)is free cyclic, the non-trivial subgroups of nl(S1) are the free cyclic subgroups Gn,(n = 1 , 2 , * * where Gn is of index n in n,(S1).The covering space which corresponds to Gn is S1itself relative to the projection p , : S1--f S1defined by p,(z) = 2% for each z E S1.These are essentially all the covering spaces over S1. This example reveals the fact that homeomorphic covering spaces over the same base space are not necessarily equivalent. (ii)Covering spaces of S n and P" with n > 1. Since S n is simply connected, every covering space over Sn is equivalent to the trivial covering space by which we mean the covering space S n over itself relative to the identity map. On the other hand, S n is the universal covering space over Pn relative to the natural projection p : Sn -f Pn. Since the fibers of this covering space are the pairs of antipodal points in 9, it follows from (15.4) thatn,(Pn) is the cyclic group of order 2. Hence, by (17.2), the universal covering space is essentially the only non-trivial covering space over Pn. (iii) Covering spaces of closed surfaces. Let M denote a closed surface other than S2and Pz.Then, by (11; Ex. A), n l ( M ) is an infinite group. It follows from (15.4) that the universal covering space E of M is not compact. Hence, E is a simply connected infinite 2-manifold. In fact, it is a classical theorem that E is homeomorphic to the euclidean 2-space, [V; p. 1531. As an application of the preceding results, we have the following e ) ,
Theorem 17.3. Assume that X i s a connected triangulable space and E i s the universal covering space over B relative to p : E B. Then, for any x, E X, b, E B and e, E $-l(b,), the assignment g + f = p g sets up a one-to-one correspondence between the homotojby classes of the maps g : ( X ,x,) + ( E , e,) and those of the maps f : ( X ,x,) .+ ( B ,b,) with f* sendingn,(X, x,) into the neutral element of nl(B,b,,). --f
Proof. Let us use the usual notation f* = 0 to denote the fact that f* sendsnl(X, x,) into the neutral element of n l ( B ,b,). If f = p g for some g : ( X , x,) + ( E , e,), obviously we have f * = p*g, = 0. Hence, the assignment g + f = p g defines a function of the liomotopy
97
EXERCISES
classes of the maps g : (X, x,) +. ( E , e,) into those of the map f : ( X , x,) +. ( B , bo) with f* = 0. Let f : ( X , x,) +. (B,b,) be a map with f* = 0. According to (16.2) there exists a unique map g : ( X , x,) + ( E , e,) such that fig = f . Hence is onto. Let g, g’: ( X , x,) +. ( E ,e,) be any two maps such that p g N Pg’ re1 x,. Then there exists a homotopy f t : (X, x,) 3 ( B , b,), 0 < t Q 1, such that fo = p g and f , = pg’. According to (iv) of (3.1), there exists a homotopy gt : (X, x,) --f ( E , e,), 0 < t < 1, such that go = g and pgt = f t for every ~ E I Since . gl(xo) = e, = g’(xo) and p g , = f l = pg‘, it follows from the uniqueness part of (16.2) that g, = g’. Hence g I? g’ rel x,. This proves that is one-to-one. I In particular, if X is simply connected, then the assignment g -+f = p g sets up a one-to-one correspondence between the homotopy classes of the maps g : (X, x,) +. ( E , e,) and those of the maps f : (X, x,) .+ ( B , b,). For example, let B = P2,E = S2, and let p : Sz + P2 denote the natural projection. If X is either S2 or Ss, then i t follows that the homotopy classes of the maps f : ( X , x,) .+ ( B ,b,) are in a one-to-one correspondence with the set 2 of all integers. Note. In (17.3), the condition that X is triangulable can be replaced by the weaker condition that X is locally connected. In fact, in the proof of (17.3), one may use Ex. P instead of (3.1).
+
+
EXERCISES A. Sliced fiber spaces Let p : E +. B be a given map. By a slicing structure for p , we mean a collection S = { o,+u } of the following entities: (1) a system o = { U } of open sets of B which covers B, called the slicing
neighborhoods. (2) a system of maps { +U I U E w } indexed by the slicing neighborhoods, called the slicing functions, where each +u, is defined on the subspace U x p - l ( U ) of the product space B x E with images in E in such a way that the following two conditions are satisfied: (SFl) P+u(b, x ) = b, ( b E U,x ~ p - l ( U ); ) (SF2) +u(P(x), 4 = x , ( x E p - w ) . If a slicing structure S = { o,+U } for p exists, we say that @ : E B has the slicing structure property (abbreviated SSP). We shall use the abbreviation Para CHP to stand for the covering homotopy property for all paracompact Hausdorff spaces. Prove the following implications, [Hu 8 ; Huebsch 11 : B P SSP s.Para CHP Hence, p : E -+ B is a fibering if it has the SSP. In this case, E is called a sliced fiber space over B relative to p , and in particular, every bundle space is a sliced fiber space. --f
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111. F I B E R S P A C E S
Let E be a sliced fiber space over B relative to p : E -+ B. Prove the following assertions: 1. The projection p : E -+ B is open, that is to say, it maps open sets onto open sets. 2. If two points a and b of B can be connected by a path in B, then the fibers p-l(a) and p - l ( b ) are homotopically equivalent. 3. E is a bundle space over B relative to p iff the following two conditions are satisfied, [Griffin 11: (GCI) There exists a space D which is homeomorphic with every fiber +-l(b), b E B. (GC2) For each point b E B, there exist a slicing neighborhood U which contains b and a slicing function +u : U x p-l(U) -+ E such that, for every pair of points u and v in U,we always have
4 1 = x, ( x E p-l(U)). 4. If E and B are metrizable then p : E -+ B has the ACHP and hence the + U b , +U(%
PLP, [Curtis 11. Now assume that E and B are spaces such that B x E is a paracompact Hausdorff space and B is an ANR. Let p : E -+ B be a given map. Prove that E is a sliced fiber space over B relative to p iff, for any map g : X -+ E of a paracompact Hausdorff space X into E and any homotopy f t : X -+ B, (0 < t Q l), of the map f = p g , there exists a homotopy g t : X -+E, (0 Q t Q I), of g which covers f t and is stationary with f t , [Fox 21. For the definition of the stationary property, see [S ; p. 501. A sliced fiber space E over d relative to p is said to have a unified slicing fumtion if there exists a slicing structure S = { o,C$U 1 for p : E -+ B such that, for any two slicing neighborhoods U and V in o,we always have +U = +v on the intersection of U x p-l(U) and V x p-l(V). For such a slicing structure S, we may define a unified slicing function C$ on the union W of the subspaces U x p-l(U) for all U E o by taking = +U on each U x P-l(U). The fiber spaces defined by Hurewicz and Steenrod [l] and Fox [2] are those with unified slicing functions. Prove that every metrizable sliced fiber space over an ANR has a unified slicing function, [H ; p. 1701.
+
B. Local path lifting property
A map f : E -+ B is said to have the local path lifting property (abbreviated LPLP) if, for each b E B , there exists an open neighborhood U of b in B such that the map q : w -+ u, = p-yu), q = p W
w
has the PLP. Prove 1. SSP * LPLP => Para CHP. 2. If B is a paracompact Hausdorff space, then LPLP implies PLP. [Hurewicz 21.
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EXERCISES
C. Relations between various notions of fiber space
Let us consider a given map p : E --f B having one of the following properties : PCHP = CHP for polyhedra, ( 3 2), Para CHP = CHP for paracompact Hausdorff spaces, (Ex. A), ACHP = CHP for arbitrary spaces, ( 3 2), PLP = path lifting property, ( 3 12), LPLP = local path lifting property, (Ex. B), SSP = slicing structure property, (Ex. A), B P = bundle property, ( 3 4), where the abbreviation CHP stands for the covering homotopy property, ( 3 2). The first four properties are of global nature while the last three are apparently local properties. A number of implications among these properties are known and can be summarized by the following diagram B P { para }
U
BP
======-
-
=- SSP { para } =- LPLP { para } =- PLP
.v
SSP
v
LPLP
-
.ACHP
U
> ParaCHP
U
PCHP where the attached symbol { para } means that the base space B is assumed to be a paracompact Hausdorff space. Check if all of these implications have been established in the text and in the preceding two exercises.
D. Homogeneous spaces Let E be a topological group and let F be a closed subgroup of E . Define an equivalence relation in E as follows: two elements a , b of E are said to equivalent iff there is an element f E F such that af = b. Thus the elements of E are divided into disjoint equivalence classes called the left cosets of F in E . The left coset containing a E E is obviously the closed set a F of E . According to ( I ; fj 12), we obtain a quotient space B = E/F whose elements are left cosets of F in E and a natural projection p : E +. B which maps a E E onto the left coset a F E B. B = E/F is called the quotient space of E by F ; it will be called simply a homogeneous space. Prove the following assertions : 1. B = E/F is a Hausdorff space. 2. The natural projection p is an open map. 3. E operates transitively on B under the homeomorphisms e :B + B , eE E , defined by e(b) = p [ e . p - l ( b ) ] , ( b E B ) . 4. E is a bundle space over B relative to p iff there is a local cross-section of B in E ; by this we mean a cross-section x : V +. E defined on an open neighborhood V of the point b, = F in B. 5. If E is a Lie group, then a local cross-section of B in E exists and hence E is a bundle space over B relative to p , [Che; p. 1101.
111. F I B E R S P A C E S
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E. Spheres as homogeneous spaces Let Q denote one of the three fields of real numbers, complex numbers, or quaternions. Consider the right vector space Qn whose elements are ordered sets x = (xl, * * * , x n ) of n elements of Q. The inner product x y of x and y in Qn is defined by x y = XIy, * * * + znyn,
+
where Xl denotes the conjugate of xi. The topological group G n of all linear transformations in Qn which preserve the inner product is called the orthogonal, unitary or symplectic grozlp according as the scalars are real, complex or quaternionic. It is a compact Lie group. Let S denote the unit sphere in Q n ; then S is the sphere of dimension n - 1,2n - 1 or 4n - 1 according as the scalars are real, complex or quaternionic. Prove the following assertions : 1. G n operates transitively on S. 2. G n is a bundle space over S relative to the projection p : G n + S defined by P ( f ) = f ( x o ) for every f E G n , where xo = (1,O; * , 0) E S. 3. Let Gnd1 denote the subgroup of G, leaving xo fixed. Then the fibers p - l ( x ) are the left cosets of Gn-1 in G n and hence S can be considered as the homogeneous space Gn/Gn-1. F. Fiberings of spheres over projective spaces
Consider the non-zero elements of the vector space Q" in the preceding exercise.Define an equivalence relation i n X = Qn\O as follows: for any two elements x and y in X, x y iff there is a q in Q such that xq = y . According to (I, 3 12), we obtain a quotient space M called the projective space associated with Qn and a natural projection n : X + M . Let S denote the unit sphere in Qn and p = 7c I S. Prove the following assertions: 1. S is a bundle space over M relative to p. 2. The fibers p - l ( b ) , b E M , are great spheres of dimension 0, 1 or 3 according as the scalars Q are real, complex or quaternionic. 3. The group G n operates transitively on M in some natural way; and hence M can be identified with the quotient space of G n by its subgroup of the elements leaving a given point of M fixed.
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G. Stiefel manifolds
A k-frame, v k , in R n is an ordered set of k linearly independent vectors. Let denote the set of all k-frames in R n . Let L n denote the full linear group, and let L , k be the subgroup of L n leaving fixed each vector of a given frame vok. Then we may identify vn, = L~I L , V'n,k
and hence V ' n , k becomes a homogeneous space called the Stzefel manifold of k-frames in n-space. Let v n , k denote the subspace of V ' n , k consisting of the orthogonal k-frames. Prove the following assertions : 1. The orthogonal group O n Operates transitively on V n , k . If O n - k is represented as the subgroup of O n leaving fixed a given orthogonal k-frame v,k, then one may identify T / ~ =, 0, ~ I
EXERCISES
I01
2. If k < n, the rotation group R n operates transitively on V n , k and hence one may identify k < n. v n , k = R n I Rn-k, 3. V n , k may be interpreted as the space of all orthogonal (k - 1)-frames tangent to 9 - l . In particular, V n , = S n - l and V n , is the space of all unit tangent vectors on S n - 1 . 4.V n , k can be identified with the space of all orthogonal mapsSk-l intoSn-l. 5 . Let v0n be a fixed orthogonal n-frame in R n and let vok denote the first k vectors of Let O n - k be the subgroup of O n leaving vok fixed. Then we obtain a chain of Stiegel manifolds and projections
van.
On = Vn,n+Vm,n-I + * . * + V n , 2 + V n , i
=Sn-l.
Each projection or any composition of them is a bundle map, that is to say, the projection of a bundle space over its base space. H. Grassmann manifolds
Let M n , k denote the set of all k-dimensional linear subspaces (k-planes through the origin) of R n . The orthogonal group O n operates transitively on M n , k . If R k is a fixed k-plane through the origin and R n - k its orthogonal complement, the subgroup of On mapping R k onto itself splits up into the direct product o k x 0 % - k of two orthogonal subgroups the first of which leaves R n - k pointwise fixed and the second leaves R k pointwise fixed. Hence we may identify Mn,k = o n ( o k x on-k). Thus M n , k becomes a homogeneous space called the Grassmann manifold of k-planes in n-space. Prove the following assertions : 1. The natural projection O n + M n , k maps the rotation group R n onto M n , k . Let R k and R n - k denote the rotation subgroups of o k , O n - k . Define
-
Mn,k = Rn
I (Rk
X Rn-k).
Then @ n , k is called the manifold of oraented k-planes in n-space. M n , k is a covering space over M n , k relative to the natural projection with 0-spheres as fibers. 2 . V n , k is a bundle space over M n , k relative to the natural projection induced by the inclusion o n - k c o k x On-k.The fibers are homeomorphic to o k . 3 . If k < n, v n , k is a bundle space over M n , k with fibers homeomorphic to &. 4. The correspondence between any k-plane and its orthogonal (n - k) plane sets up a homeomorphism M n , k t+ M n , n - k . 5 . Mn, is essentially the (n - 1)- dimensional real projective space P - l and A?-,,, , the ( n - 1)-sphere9 - l . 1. Elementary properties of mapping spaces
Let 52 denote the mapping space Yx with the compact-open topology. Prove the following assertions : 1. If Y is a To-,T I - ,T 2 - , or regular space, then so is 52. Conversely, if 52 is a To-,Tl-, Tz-,or regular space, then so is Y .
I02
111. F I B E R S P A C E S
2. If X is a locally compact Hausdorff space and if X,Y are both separable (being separable means having a countable basis), then so is 8.On the other hand, if 8 is separable, then so is Y. 3. Assume that X is a compact metrizable space and Y is a metrizable space. Then 8 is an ANR iff Y is such; similarly, 8 is an AR iff Y is such. J. Admissible topologies
A great variety of topologies may be introduced into the set 8 = Y x making it a space. We shall denote by Qr the space obtained by topologizing 8 with a topology t.A topology T of 8 is said to be admissible if the evaluation w : Q rx X + Y is a map. Thus, by (9.1), the compact-open topology of 8 is admissible provided that X is a locally compact regular space. Prove the following assertions, [Arens 11 : 1. Any admissible topology of 8 contains every open set in the compactopen topology of 8. 2 . If X is a completely regular space and Y is a TI-space containing a nondegenerate path, then a necessary and sufficient condition for the compact-open topology to be admissible is the local compactness of X . K. The topology of uniform convergence
Let Y be a metrizable space and let d be a given bounded metric consistent with the topology of Y.There is a natural metric d* defined on 8 = Y x by
r
a*(f , g) = SUPZ,X d f (4 g ( 4 1 for each pair of map f , g E 8.The metric d* determines a topology of 8 called the d*-topology, or the topology of uniform convergence with respect to d . Prove the following assertions : 1. The d*-topology of 8 is admissible. 2 . If X is compact, then the compact-open topology of 8 coincides with the d*-topology induced by any given bounded metric d on Y . The word “bounded” in this assertion might have been omitted. 3. If X is a completely regular Hausdorff space and Y a metrizable space containing a nondegenerate path, then a necessary and sufficient condition for the compact-open topology of 8 to coincide with the d*-topology induced by a given bounded metric d on Y is the compactness of X [Jackson 11. 4. The d*-topology of 8 depends not only on the topologies of X and Y but also on the metric d chosen for Y.Construct a few examples. 9
L. Maps on topological products Consider three given spaces T. X , Y and the mapping spaces: 8 = yx,@J = yxx T,y = QT. In 5 9, we have defined the association 8 : @ + Y. Prove the following assertions:
EXERCISES
I03
1. If both X and T satisfy the first countability axiom, then 8 sends @ onto Y, [Fox 31. 2. If X and T are Hausdorff spaces satisfying the first countability axiom, then 8 is a homeomorphism of @ Onto Y and hence the exponential law holds in this case. yXxT =
(yX)T
M. Borsuk’s fibering theorem
Let X be a compact metrizable space, A a closed subspace of X , and Y a compact ANR. Consider the mapping space E = Y x and the subspace B of the mapping space YAconsisting of the maps g : A + Y which can be extended over X . Prove that E is a sliced fiber space over B relative to the natural projection p : E + B defined by p ( f ) = f I A for every f E E. By Exercises C and I, show that this fibering has a unified slicing function. [H; p. 1731. N. Change of the boundary sets
There are changes of the boundary sets A and B without changing the homotopy type of the space [ X ; A, B ] of paths. Because of symmetry, we may study only the changes of the terminal set B . Let B , and B , be two subspaces of X . A deformation of B , into B , in X is a homotopy ht : B, + X , (0 Q t < l), such that h, is the inclusion map and h,(B,) c B,. Such a deformation ht induces a map h # : [ X ; A , B , ] + [ X ; A , B,] described as follows: for each f~ [ X ;A , B , ] , g = h#(f) is
B , is said to be a deformation homeomorph of B , i n X i f there exists a homotopy ht : B , + X , (0 < t < l ), such that h, is the inclusion map and h, is a homeomorphism of B , onto B,. Prove that the spaces [ X ;A , B , ] and [ X ;A , B,] are homotopically equivalent if B, is either a strong deformation retract of B , or a deformation homeomorph of B,. Consequently, if X is pathwise connected, then the homotopy type of the space [ X ; a, b] is independent of the choice of the points a and b. In particular, the spaces [ X ; a, b ] , A, and Ab are homotopically equivalent. 0. The space of curves
In our definition of the space of paths, the domain I is the same for all paths. In his studies on Pontrjagin products, J. C. Moore, has found that it is more convenient to allow different domains for different paths. To avoid possible ambiguity, these will be called curves. Precisely, let Y be a given space. Then, a curve in Y consists of a real number a 2 0 and a map f of the closed interval [0, a ] into Y . The points f (0) and f (a) are called the initial point and the terminal point of f respec-
I04
111. F I B E R S P A C E S
tively. A curve f : [0, a] -+ Y is said to be closed if f (0) = f (a); in this case, f (0) is called the basic point of the closed curve f . Now, let r d e n o t e the set of all curves in Y .To topologize I', let us consider the subspace J of the real line R consisting of the real numbers a > 0 and the space Q = YI of all paths in Y. Define a function : + J x 9 by taking C#J( f ) = (a, uf) for every curve f : [0, a ] + Y, where of : I + Y is the path given by q ( t ) = f (at) for each t E I . This function carries onto a subspace of J x Q in a one-to-one fashion. We topologize in such a way that becomes a homeomorphism. Prove that ( J x Q) \ is the subspace 0 x [Q \ i( Y ) ] where i : Y +Q denotes the natural injection j : Y + Q o f $9. Since I = [0, 11, every path in Y is also a curve in Y. The topology of F defined above permits us to consider Q as a subspace of Construct a homotopy ht : + (0 < t < l), such that the following conditions are satisfied : (i) h, is the identity map on (ii) h, is a retraction of r o n t o 52. (iii) ht( f ) = f for every f E 52 and t E I . ( i v ) For every f E I', the initial point and the terminal point of the curve ht( f ) are the same as those of f for each t E I . If f : [0, a ] -+ Y and g : [0, b] -+ Y are two curves in Y such that f (a) = g ( O ) , then we may define a product f - g : [0, a + b] -+ Y by taking
+r + r
+(r)
r +(r)
+
r.
r r,
r.
[ f e d(4 = { f
( 4 1
g(t - 4,
(if 0 < t < a), (ifa Q t < a + b).
This multiplication ( f ,g) -+ f * g of curves is continuous and associative whenever it is defined. Now, let y be a given point in Y and let Ou denote the subspace of consisting of the closed curves with y as the basic point. Then the homotopy ht shows that the space of loops A , is a deformation retract of OY.On the other hand, the multiplication ( f , g) -+ f - g is defined for every pair f , g E Ou with the trivial curve e : [0, 01 + y as a two-sided unit. Hence, if Y is a Hausdorff space, then 0,is a mob with e as a two-sided unit in the sense of Wallace [ 13.
r
P. Generalized covering spaces
A space E is called a generalized coveriizg space over a space B relative to a map p : E -+ B if the following conditions are satisfied: (GCSl) p maps E onto B. (GCS2) For each b E B, there exists an open neighborhood U of b in B such that p - l ( U ) can be represented as a disjoint union of open sets in E each of which is mapped homeomorphically onto U by p . Prove that every bundle space E over B relative to p : E -+ B with discrete fibers is a generalized covering space over B relative to p .
EXERCISES
105
Let E be a generalized covering space over B relative to p : E -+ B. Prove the following assertions : 1. E is a sliced fiber space over B relative to p with discrete fibers. 2. If B is connected, then E is a bundle space over B relative to p with discrete fibers. 3. If E is connected and B is locally connected, then E is a covering space over B relative to p . 4. Let q : X -+ Y be a given map of a space X into a space Y . If g : X -+ E is a map and f t : Y + B, (0 < t < l), a homotopy with foq = p g , then there exists a unique hdmotopy gt : X -+ E , (0 < t < l ) , of g such that ftq = pgt for every t E I and that gt is stationary with f t . In particular, fi : E -+ B has the ACHP. This assertion is also true for sliced fiber spaces with totally pathwise disconnected fibers, [Griffin 11. Q. Local homeomorphisms
A map p : E + B of a space E onto a space B is called a local homeomorphism of E onto B if every point x of E has an open neighborhood which is mapped homeomorphically by p onto an open neighborhood of p ( x ) in B. Prove that, if p : E -+ B is a local homeomorphism of a regular Hausdorff space E onto a space B such that p - l ( b ) is finite for every b E B, then E is a generalized covering space over B relative to p . R. The covering spaces of the torus
As an exercise to determine all equivalence classes of the covering spaces over a given space B, let us study to case where B is the torus S1 x S'. According to (11; Ex. A), the fundamental group n l ( B ) is a free abelian group with two free generators a and b . Among the subgroups of nl(B) there is a doubly indexed system Gm,n, ( m , n = O , 1 , 2 , * * * ) , where G m , , is the subgroup generated by am and bn. Hence, Go,o = 0 and G1.1 =nl(B)* Prove the following assertions : 1. Every covering space over B is regular. 2. Corresponding to G o , o ,we have the universal covering space E = Rz = R x R over B = S1 x S1 relative to the projection Po,, : E -+ B defined by p0,,(x, y) = ( p x , p y ) , where p : R +S1 denotes the exponential map of (11; Q 2). 3. Corresponding to Go,,, n > 0 , we have the covering space E = H x S' over B = S1 x S1 relative to the projection f i o , n : E -+ B defined by Po, n ( x , z ) = (fix, zn) for each x E R and z E S1. Similarly, we may get the covering space corresponding to G,,o, m > 0. 4. Corresponding to G m , n , m > 0, n > 0, we have the covering space E = S1 x S1 over B = S1 x S' relative to the projection p m , n : E B defined by P m , n ( u , v) = (urn,v n ) for each (u, v) E E. --f
106
111. F I B E R S P A C E S
S. Maps of the torus into the projective plane
Consider the torus T and the projective plane P. Pick arbitrary basic points toE T and Po E P,and study the maps f : ( T ,to) + (P, Po). The fundamental group n,(T, to) is a free abelian group with two free generators a and b, while n,(P, Po) is a group of order 2 generated by c. Hence, there are four possible homomorphisms h , n
:n,(T,to) +ni(P, Po),
with m, n running over 0 and 1, defined by h . n ( a ) = m, h . n ( b ) = nc. By considering T as the unit square with the opposite sides identified, construct for each (m, n ) a map f m , n : ( T ,to)
+
(PIPo)
such that (fin,,,)+ = h,,,,,,. Hence, the homotopy classes of the maps f : ( T ,to) --f (P,Po) are divided into four disjoint collections C,, such that / E C m , n i f f f *= h m , n . Next, prove that, for each (m, n ) , the homotopy classes in the collection Cm,n are in a one-to-one correspondence with the homotopy classes of the maps (S2,so) + (P,Po) and hence with the integers. T. Maps of a surface into a surface
Let X and Y be any two closed surfaces other than the sphere and the projective plane. Study the classification problem of the maps f : X + Y as follows. Pick arbitrary basic points xo E X and y o E Y . Two homomorphisms
h, k :n,(X,xo) -jn,(Y,Y O ) are said to be equivalent (notation : h N k ) if there exists an element bE~,(Y y ) ,such that k(a) = b - l . l t ( ~ ) . b holds for every a E n , ( X , x,,). Thus the homomorphisms h :n,(X, xo) + n,(Y,yo) are divided into disjoint equivalence classes. Let C denote the set of these equivalence classes. Prove that the homotopy classes of the maps f : X + Y are in a one-toone correspondence with the equivalence classes C by establishing the following assertions : 1. Every map f : X + Y is homotopic to a map g : X + Y such that d x o ) = Yo. 2. For any two maps f , g : X + Y satisfying f(xo) = yo = g(xo), f N g implies f, N g,. 3. For any ~ E Cthere , exists a map f:X+Y such that f (xo)=yo and f, =h. 4. If f : X -+ Y is a map with f (xo) = y o and h E C is equivalent t o f a , then there exists a map g : X + Y such’ that g(xo) = yo, g N f and g, = h. 5. I f f , g : X + Y are two maps such that f (xo) = yo = g(xo) and f , = g,, then f N g.
CHAPTER I V HOMOTOPY GROUPS
1. Introduction The basic problem which led to the discovery of “homotopy groups” was to classify homotopically the maps of an n-sphere S n into a given space X. In the case n = 1, this was facilitated by pinning down a base point to obtain a group structure as in (11; 5 4). The same trick was found to work in higher dimensions; in fact, if we pinch the equator of S n to a point, we obtain two n-spheres with one point in common. If n > 1, there is a rotation of S n which gives a homotopy interchanging the two hemispheres. This implies the striking feature that the group is abelian. The relation between homotopy and homology groups and the existence of relative homology groups Hn(X, A ) led quickly to the relative homotopy groups nn(X, A , xo), giving a system highly analogous to homology theory. But it differs in several ways: no(X,x,) and n,(X, A , x,) are not ordinarily groups; n,(X, x,) andn,(X, A , xo) are not usually abelian; and the excision property for homology does not hold for homotopy. A very important fact in the minds of those involved in the development of the theory was the result of Hopf in 1930thatns(S2)is infinite. This showed that the groups express some very deep topological properties of spaces. A second important fact is that the definition of n n is not effectively computable; and there was no definition available which led immediately to effective computations as in the case of homology groups of complexes. Successful calculations in special cases came slowly. It is only recently that methods have been found which apply to a reasonably broad variety of cases. These are the subject of much current research. The objectives of the present chapter are to define the groups and related homomorphisms, to establish their main general properties, and to show that certain of these properties are characteristic.
2. Absolute homotopy groups Let ( X ,xo) be a given pair consisting of a space X and a point xo in X. Let no(X,xo) denote the set of all path-components of X. The path-component of X which contains x, will be called the nezctral element of n,(X, xo) and will be denoted by 0. As in (I1; 5 4), we shall denote bynl(X, xo) the fundamental group of X at x,. For each integer n > 1, the definition of the n-th (absolute) homotopy
108
IV. H O M O T O P Y G R O U P S
group n n ( X ,xo) is strictly analogous to that of the fundamental group. We replace the unit interval I by the n-cube In, i.e. the topological product of n copies of I . Every point t E I n is represented by n real numbers t = ( t l ,* * , tn), ti E I , (i = 1 , 2 , * * , n), called the coordinates of t. The number tr is called the i-th coordinate of t. An (n - 1)-faceof I n is obtained by setting some coordinate ti to be 0 or 1. The union of the (n- 1)-faces forms the boundary dIn of I n ; it is topologically equivalent to the unit ( n - 1)-sphere 9 - l . Consider the set Fn = F n ( X ,xo) of all maps
.
f : (I",arn) -+
(x, xo).
These maps are divided into homotopy classes (relative to ill.). We shall denote by n n ( X ,xo) the totality of these homotopy classes. We shall also denote by [ f ] the class which contains the map f and by 0 the class which contains the unique constant map do(In)= xo. Topologize Fn by means of the compact-open topology as in (I11 ; 3 9) ; then n n ( X , xo) becomes the set of all path-components of the space Fn. We may define an addition (usually non-commutative) in Fn as follows. For any two maps f , g in Fn, their sum f + g is the map defined by if 0 < t , Q 4, * *tTb), if 4 < t l < 1, for every point t = (t,, * - ,tn) in In. Obviously, f g is in Fn. The homotopy class [f + g] clearly depends only on the classes [ f ] and [g].Hence we may define an addition innn(X, xo) by taking
(f
( 2 t i , t 2 , . tn), + g) (4 = ( fg(2h - 1, t2; * **
-
[fl
+
+ [gl = [ f + gI.
Just as in (11; tj 4) for the fundamental group, one can easily verify that this addition makes n n ( X , xo) a group which will be called the n-th homotopy group of X at xo. The class 0 is the group-theoretic neutral element of n n ( X , xo), and the inverse element of [ f 3 is the class [fe],where 8 : I n + I n denotes the map defined by O ( t ) = (1 - t,, t2; * *, tn) for every t = ( t l , t 2 ; .,tn) in In. If the boundary aIn of I n is identified to a point, we get a quotient space which is topologically equivalent to an n-sphere Sn with a given basic point so E Sn. It follows that one might equally well define an element of n,(X, xo) as a homotopy class (relative to so) of the maps f : (9, so) -+ ( X , xo). Since the 4 and t, 2 4 respectively, two halves of I n , defined by the conditions t, correspond to two hemispheres of S n , one can clearly see how to define t h e group operation o f n n ( X ,xo) from this point of view. For details, see [Hu 41. Since, when n > 1, there exists a rotation of Sn which leaves so fixed and interchanges the two hemispheres, this definition suggests the following striking property of n n ( X , xo)which however will be proved in a different way.
-
<
2. A B S O L U T E HOMOTOPY G R O U P S
Proposition 2.1. For every n
109
> 1, nn(X, xo) i s an abelian group.
Proof. As in (111; 11), one can prove that F"-' is an H-space with the constant map d,,EFn-l as a two-sided homotopy unit. Hence, by (111; 9.9), we have we have XP = ~ ~ nx I- = l (xI~-~)I.
Then one can easily see that nn(X, xo) and n1(Fn-l, do) are essentially the same group. Hencenn(X, xo) is abelian. I In the precedingproof, we have incidentally obtained an interesting result : nn(X, xO)
= nl(Fn-l,
do).
Hence every homotopy group of a space can be expressed as the fundamental group of some other space. This relation can be used to define higher homotopy groups in terms of fundamental groups; indeed, Hurewicz used this definition when he introduced these groups in 1935. One can say more: let p be any positive integer less than n and let q = n - p . By (111; 9.9), we have XI^ = XIPXIQ = (XI~)IQ. By means of this relation, it is easy to deduce the following result:
+
nn(X, ~ 0 = ) ng(FP, do), p 4 = n, where do denotes the constant map do(lg) = xo. In particular, when p = 1, F p becomes the space W of loops in X with basic point xo and do the degenerate loop at xo. Thus we have proved the following proposition which will be used in the sequel. Proposition 2.2. nn(X, x0) = nn-l(W, do).
Finally, the following proposition is an immediate consequence of the fact that I n is pathwise connected. Proposition 2.3. If X, denotes the path-component of X containing xo, then
nn(X0, xo) =nn(X, xo), 12 > 0. Examples. If X is a contractible space, thennn(X,xo) Next, by (11; 7.1) we have
= 0 for every
no(S', 1) = 0, nl(S', 1) M 2, nn(S1, 1) = 0, if n > 1. On the other hand, by means of (I1; 3 8), one can prove that n,(Sn, so) = 0, (if m nn(Sn, SO) M 2. Finally, by (111; 6.4), we deduce the result n3(S2,so) M 2.
< n),
n
> 0.
I10
IV. HOMOTOPY G R O U P S
3. Relative homotopy groups The objective of the present section is to generalize the notion of homotopy groups in $ 2 by defining the relative homotopy groups n n ( X , A , x,). By a triplet ( X .A, x,), we mealn a space X,a nonvacuous subspace A of X I and a point x, in A. If x, is the only point of A , then the triplet ( X IA , x,) will be simply denoted by ( X , x , ) and may be considered as a pair consisting of a space X and a point x, in X . Let n > 0 and define the n-th relative homotopy set n n ( X , A, x,) as fbllows. Consider again the n-cube I n . The initial (n- 1)-face of 1%defined by tn = 0 will be identified with I n - 1 hereafter. The union of all remaining (n - 1)-facesof I n is denoted by Jn-1. Then we have a I n = I n - 1 U Jn-1
aIn-1 = In-1
n Jn-1.
By a map f : (I%,I n - 1 , Jn-1) + ( X ,A , x0), we mean a continuous function from I n to X which carries 1 n - l into A and Jn-l into x,. In particular, it sends a I n into A and aZn-1 into x,. We denote by F n = F n ( X , A, x,) the set of all such maps. These maps are divided into homotopy classes (relative to the system { In-1, A ; Jn-1, xo } ). We shall denote by n n ( X , A , x,) the totality of these homotopy classes. We shall also denote by [ f ] the class which contains the map f and by 0 the class which contains the constant map d,(In) = x,. Topologize Fn by means of the compact-open topology; then nn(X,A, x,) becomes the set of all path-components of the space Fn. If n > 1, we may define an addition (usually non-commutative) in Fn. For any two maps f , g in F n , their sum f + g E F n is defined by the formula given in $ 2 for the absolute case. The homotopy class [ f g] depends only on the classes [ f ] and k ] and hence we may define an addition innn(A,X,xo) by taking [fl kl = [f + gl. As in Q 2, one can verify that this addition makesnn(X,A, x,) a group which will be called the n-th relative homotopy group of X modulo A at x,. The class 0 is the group-theoretic neutral element of nn(X,A, x,), and the inverse element of [ f ] is the class [/el, where 8 : I n -+ I n denotes the map defined in $ 2. If x, is the only point of A , then we have
+
+
F n ( X ,A , x,) = Fn(X,x,).
Hence, in this case, n n ( X ,A, x,) reduces to the absolute homotopy group nn(X,x,) defined in $ 2.
If Jn-l is pinched to a point so, ( I n , In-1, Jn-1) becomes a configuration topologically equivalent to the triplet (En,9 - 1 , so) consisting of the unit n-cell En, its boundary ( n - 1)-sphereSn-1, and a reference point so E 9 - l . It follows that one might equally well define an element of n,(X, A , x,) as a homotopy class (relative to the system { Sn-l, A ; so, xo } ) of the maps of (En, Sn-l, so) into ( X ,A, xo). Since, when n > 2, there exists a rotation of
3. R E L A T I V E H O M O T O P Y G R O U P S
I11
En which leaves so fixed and interchanges the two halves of En, we see that nn(X,A , x,) is abelian for every n > 2. This commutativity property is also an immediate consequence of the next proposition. n,(X, A, xo) is in general non-abelian. Next, let us introduce the notion of the derived triplet of a given triplet which will be frequently used in the sequel. Let
T
( X ,A, x,)
=
be a given triplet. Consider the space of paths
X' and the initial projection p : X' A' and denote by triplet
E A'
=
[X ; x,X J as defined in (111; 5 10). Let
+X
= P-l(A) =
[X; A , x 0 ] c X'
the degenerate loop x',,(I)
T' = ( X ' , A',
= x,,.
Thus we obtain a
XI,),
called the derived triplet of T , and a map
p
: ( X I ,A',
XI,)
+
( X ,A, x,),
called the derived projection over ( X ,A, x,). Proposition 3.1. For every
> 0 , we have
~t
n,(X, A, xo)
= ~ n -l (A',~
' 0 ) .
Proof. If n = 1, this is obvious since each side can be considered as the set of all path-components of A'. Assume n > 1. Since XIn = bY (111; 9.9)' it follows that P ( X ,A, x,) = Fn-'(A', XI,).
Hencen,(X, A, x,) andn,-,(A', x',) coincide set-theoretically. It is also clear that the group structures are the same for any n > 1. I Another consequence of (3.1) is that every relative homotopy group can be expressed as an absolute homotopy group and hence as a fundamental !PUP* Since aIn is pathwise connected for n > I , the following proposition is obvious. Proposition 3.2. If X , denotes the path-component of X containing x, altd A , that of A, then nn(X,A , x,) = nn(X,, A,, x,), n > 1. Finally, the following proposition will be used in the sequel. Proposition 3.3. If a €nn(X, A , x,) is represented by a map f slcch that f (In)c A, then a = 0 .
E Fn(X, A,x,)
IV. H O M O T O P Y G R O U P S
I12
Proof. Since f E F ~ ( XA ,, x,) and f(P)c A , we may define a homotopy f t E F n ( X , A , x , ) , O < t Q 1, bytaking
+
ft(ti..* tn-1, tn) = / ( t i , . - 8 tn-1, t tn - &). Then we have f o = f and f l ( P = ) x,. Hence a = 0. I As an application of (3.3), let us prove the following Proposition 3.4. If
then n,,,(X, A , x,)
=
( X ,A , xo) is a triplet such that ( X ,A ) is a relative n-cell, 0 for every m satisfying 0 < m < n.
Proof. By the definition of relative n-cells in ( I ; 3 7), X is the adjunction space obtained by adjoining En to A by means of a map g : 9 - 1 + A defined on the boundary ( n - 1)-sphere 3 - l of En. Let e, denote an interior point of En. Then, by ( I ; Ex. S), A is a strong deformation retract of X \ e,. Let a € n m ( X A , , x,) with 0 < m < n and choose a representative map f : (Im, P - l , Jm-1) + ( X ,A , x,)
for a. Applying (11; Ex. C), we can easily prove that we may free e, from the image of f by means of a suitable homotopy of f relative to { Im-l, A ; Jm-1, x, }. Since A is a strong deformation retract of X \ e,, this implies that there exists a homotopy f t (Irn,Im-l, J"-') + ( X ,A , xo), (0 < t < I), c A . By (3.3), we conclude that a such that f o = f and fl(P) n,,,(X,A , xo) = 0 for every m satisfying 0 < m < n. I
= 0.
Hence
4. The boundary operator Let ( X ,A , x,) be a given triplet. For every n > 0,we shall define a transformation a : n n ( X , A , x , )+ n 4 A , x o ) . Let a be any element of n n ( X ,A , x,). Then, by definition, a is a homotopy class represented by a map f : (In,In-1, Jn-1) If n
= 1,
--t
( X ,A , x,).,
f(In-l) is a point of A which determines a path-component
p E Z , - ~ ( Ax,) , of A . If n > 1, then the restriction f 1 In-l is a map of (In-1, W-l) into ( A , xo) and hence represents an element p E nn-l ( A , xo). Obviously, the element Enn-i(A, x,) does not depend on the choice of the map f which represents the given element a Enn(X,A , x,). Hence we may define the transformation a by setting a(a) = b. Hereafter, a will be called the boundary operator. The following two properties of a are obvious from the definition. Proposition 4.1. The boundary operator
n n ( X , A ,~
into that o f n f i - i ( A ,~
0 )
Proposition 4.2. I f n
a
sends the neutral element of
0 ) .
> 1, then the boundary operator a is a homomorphism.
5. I N D U C E D
1x3
TRANSFORMATIONS
5. Induced transformations Let (X,A, xo) and ( Y ,B , yo) be given triplets. By a map of (X, A, xo) into (Y, B , yo),we mean a continuous function X to Y which carries A into B and xo into yo. Consider such a map
( Y ,B, Yo). Since f is continuous, it sends the path-components of X into those of Y . Hence, f determines an induced transformation
f : (XI A, xo)
+
f* :no(X,xo) +no(Y,Yo) which obviously sends the neutral element of no(X,xo) into that of no(Y ,yo). Now let n > 0. For any map q5 E F ~ ( XA , xo), the composition fq5 is in F n ( Y, B, yo) and the assignment q5 + fq5 defines a map f#:Fn(X,A,xo) + F n ( y # B , ~ o ) * The continuity of f# implies that f# carries the path-components of Fn(X, A , xo) into those of F n ( Y , B, yo). Hence it determines an induced transformation f* :nn(X,A, xo) +nn(Y, B, Y O ) which obviously sends the neutral element of nn(X, A, xo) into that of nn(Y9 B , Yo). If n = 1, A = xo, B = yo, or if n > 1, thennn(X, A, xol andn,(Y, B , yo) are groups. For any two maps q5, y in P ( X , A , x o ) , one can easily see that
f(+
+ Y ) = 19 + f Y
in Fn(Y, B, yo). Hence it follows that f* is a homomorphism. Thus we have established the following two properties of f*. Proposition 5.1. If n = 0 , A = xo, B = yo, or if n > 0 , then the induced transformation f* sends the neutral element of nn(X, A , x o ) into that of nn(y , B, Yo). Proposition 5.2. If n = 1, A = xo, B = yo, or if n
transformation f* i s a homomorphism.
> 1, then the induced
In the case of (5.2),f* will be called the induced homomorphism. Now, consider the derived triplets (X’, A ’ , (Y’, B‘, yl0) of the given triplets together with the derived projections %IO),
f~: (X’, A’, ~ ‘ 0 )+ ( X ,A , xo), r : (Y’, B’, y’o)
+
(YpB, Y O ) .
The given map f : ( X ,A , xo) + (Y, B, yo) considered as a map from X into Y induces a map f I : X I + Y’ according to (111; 3 14). Since f (A) c B and f (xo) = yo, it follows that ’f carries X’ into Y’, A’ into B’, and into yl0. Hence ’f defines a map xl0
f ’ : (X’, A‘, do)+ (Y’, B‘, yl0)
IV. H O M O T O P Y G R O U P S
114
which satisfies the relation rf ’ = f p and which will be called the derived m a p of f . Let 7 = 1’ I (A’, do).Then j induces
j*
:nn-1(A’, x‘o) +nn-AB’, Y’o).
Under the identifications in (3.1)it is obvious that f* = j*. Finally, the boundary operator a in 5 4 is essentially a special case of induced transformations. In fact, for any triplet (X, A , xo), consider the pair (A’, do)of the derived triplet and the restriction q = p I (A’,do)of the derived projection p. Then q induces q* :nn-l(A’, x’o)
+%-I
( A , xo).
Under the identification in (3.1),it is obvious that a
= q*.
6. The algebraic properties The homotopy groups n n ( X ,A , x0), the boundary operator a, and the induced transformations defined in the previous sections possess seven fundamental properties which will be given in this and the next few sections. By the definition of induced transformations, the following two properties are obvious. Property 1. If f : ( X ,A , xo) + ( X ,A , xo) i s the identity map, then f* i s the identity transformation on n n ( X ,A , xo) for every n. Property II. I f f : (X, A , xo) + (Y, B, yo) and g : ( Y ,B, yo) + (2, C, zo) are maps, then for every n 2 0 we have ( g f ) , = g, f*. Hence, for any given n, the assignment ( X ,A , xo) + n n ( X , A , xo) and f + f* defines a covariant functor. The next property gives a relation between the boundary operator and the induced transformations. It is an obvious consequence of their definitions. Property 111. I f f : (X, A , xo) + ( Y ,B, yo) is a map and if g : ( A ,xo) + ( B ,yo) i s the restriction of f , then the commutativity relation = g,a holds in the following rectangle for every n > 0 :
nn(X,A , xo)
If*
af,
a
t a nn(Y. B, Yo) -+
nn-,(A, xo) Ig*
t nn-l(B, Yo)
Note that Property I11 is also an easy consequence of Property 11. To see this, let us consider the derived map
f’ : (X’, A ’ , do)+ (Y’, B’, yf0) of f as well as the derived projections
p : (X‘, A‘, x’o) + ( X ,A , xo),
r : (Y’, B’, Y’o)
+
(Y, B, Yo).
7. T H E E X A C T N E S S P R O P E R T Y
115
Let f = f’ 1 ( A ’ , q = p I (A’, do)and s = r I (B’, y f 0 ) .After the various identifications described in (3.1) and 5 5 , the preceding rectangle reduces - form: to the following 4+ nn-1(A’, x’o) nn-l(A, xo) %IO),
Ii;
J.
s+
nn-,(B, Yo) Since f ’ satisfies the relation rf‘ = f p , we have s f = gq. Hence, Property I1 implies that the commutativity relation s,f, = g,q, holds in this rectangle. nn-l(B’J’0)
7. The exactness property Let ( X ,A , xo) be any given triplet. The inclusion maps
i : ( A ,xo) = ( X ,xol, j : ( X ,xo) = ( X ,A , xo) induce transformations i, and j, for each n > 0. Together with the boundary
operators a, they form a beginningless sequence:
*A+n n + l ( X A, , xo) a
a , * nn(A,xo) I , n,(X. xo) A+ n n ( X ,A , xo) ...-+ ir a , n l ( X ,A , xo) n,(A, xo) I+no(X,xo) which will be called the homotopy sequence of the triplet ( X ,A , xo) and will be denoted by n ( X ,A , xo). Every set in n ( X ,A , xo) has a specified element called its neutral element and every transformation in n ( X , A , xo) carries the neutral element into the neutral element. We define the kernel of a transformation in n ( X ,A , x,,) to be the inverse image of the neutral element. Such a sequence is said to be exact if the kernel of each transformation coincides exactly with the image of the preceding transformation. Property IV. T h e homotopy sequence of a n y triplet ( X ,A , xo) i s exact. The proof breaks up into the proofs of the following six statements: (1) j,i, = 0,(2) aj, = 0,(3) i,a = 0. (4) If u €n,(X, xo) and j,a = 0, then there exists an element /? € n n ( A xo) , such that i,p = u. (5) If a €nn(X, A , xo) andaa = 0, then there exists an elementp €nn(X,xo) such that j,p = a. ( 6 ) If aEnn-l(A, xo) and i,a = 0, then there exists an element p € n n ( X ,A , xo) such that ap = a. In the preceding statements, the symbol 0 denotes either the neutral element of the set involved or the transformation which sends every element into the neutral element. Proof of (1). For each n > 0, let a € n n ( A xo) , and choose a map f~ Fn ( A , xo) which represents a. Then the element j*i,(a) in n n ( X ,A , xo) is represented by the composition j i f E F n ( X ,A , x o ) . Since obviously j i f ( I n )c A . i t follows from (3.3) that j,i,u = 0. Since a is arbitrary, this implies j,i, = 0. I * *
---j.
I 16
IV. H O M O T O P Y G R O U P S
Proof of (2). For each n > 0, let cx E n n ( X ,xo) and choose a map f E F n ( X ,xo) which represents u. Then the element aj,a is determined by the restriction j f I In-1 = f I In-1. Since f (Inv1)= xo, we have aj,u = 0. Hence aj* = 0. I Proof of (3). For each n > 0, let u € n n ( X ,A , xo) and choose a map f E Fn(X, A , xo) which represents u. Then the element i,du is determined by the restriction g = f I In-l. Define a homotopy gt : In-l + X , 0 < t < 1, by setting
gt(t,,..-,tn-,)
= j(t1,*.*,tn-1,
4.
Then go = g, g 1 ( P - l ) = x,, and gt E Fn-l(X, xo) if n > 1. This implies i*au = 0. Hence ;,a = 0. I Proof of (4).Choose a map f E P ( X , xo) which represents u. The condition f*u = 0 implies that there exists a homotopy ft : In + X , 0 Q t Q 1, such that f o = f , f l ( I n ) = xo, and f t E P ( X ,A , xo) for each t E I . Define a homotopy gt : In + X , 0 < t < 1, by setting
Then go = f , g,(In) c A , and gt(aIn) = xo for every t E I . Now, g , represents an element p € n n ( Axo) , and the homotopy gt proves that i*p = u. I Proof of (5). First, assume that n > 1. Choose a map f E F n ( X ,A , xo) which represents a.Then the condition au = 0 implies that there exists a homotopy gt : In-1 + A , 0 Q t Q 1, such that go = f I In-1, g,(In-l) = xo, and gt(i31nl.n-1) = xo for every t E I . Define a partial homotopy ht : dIn + A , 0 Q t Q 1, by setting ( s E In-1, t E I ) , gtb) ht(4 = ( s E Jn-l, t E I ) . XO,
{
1
Since h, = f I aIn, i t follows from the homotopy extension property that the homotopy ht has an extension f t : I n + X , 0 Q t Q 1, such that f o = f . Since f , ( a P ) = hl(aI*) = x,,, f l represents an element p in n n ( X ,xo). Since f t E P ( X ,A , xo) for t E I , i t follows that j*p = u. For the remaining case n = 1, u is represented by a path f : I X such that f (0) E A and f (1) = xo. The condition au = 0 means that f (0) is contained in the same path-component of A as xo. Hence there exists a homotopy f t : I -+ X, 0 Q t Q 1, such that f o = f , f t ( 0 )E A , ft(1) = xo, and f l ( 0 ) = xo. Then, f l represents an element /3 E ~ , ( Xxo) , and the homotopy f t implies that j*p = u. I -+
Proof of (6). First, assume that n > 1 . Choose a map f E Fn-l ( A ,xo) which represents u. Then the condition i,u implies that there exists a homotopy f t : In-1 -+ X , 0 < 1 Q 1, such that f o = f , f 1 ( I n - l ) = xo and ft(aIn-l) = xo for every t E I . Define a map g : In + X by taking
g(t1,. *
* I
tn-1, in) =
-
ft,,(ll,' -,in-1).
8. T H E
HOMOTOPY PROPERTY
117
This map g is in Fn(X,A , xo) and represents an element a of n n ( X ,A , xo). Since g 1 In-1 = f , we have aj3 = a. For the remaining case n = 1, a is a path-component of A . The condition i,a = 0 means that a is contained in the path-component of X which contains xo. Pick a point x from a. Then there exists a path f :I + X such that f (0) = x and f (1) = x,. This path f represents an element j3 of n , ( X , A , x o ) . Since f (0) E a, we have aj3 = a. I
8. The homotopy property Consider any two given triplets ( X ,A , xo) and ( Y ,B , yo), and any two given maps f,g:(X,A,x,)+(Y,B,y,). We recall that f and g are said to be homotopic (relative to { A , B ; xo, yo } ) if there exists a homotopy
ht:(X,A,xo)-+(Y,B,yo)0 ,
f and g are homotopic, then their induced transformations
f,, g,
:n n w , A , x g ) +nn(Y,B, Yo)
are equal for every n. Proof. Let a € n n ( X A , , xo). It suffices to prove that f,a = g,a. First, assume that n ; 0. Choose a map+ E F n ( X ,A , xo) which represents a. Then the elements f*u and g,a are represented by the compositions f+ and g+ respectively. The composition ht+ of and a homotopy ht : f g gives a homotopy connecting f+ and g+. Hence f,a = g,a. It remains to settle the case n = 0, A = xo, and B = yo. Here a is a pathcomponent of X . Pick a point x E a. Then f,a and g,a are the path-components of Y containing the points f ( x ) and g ( x ) respectively. Let ht : f z g. Define a path a : I +. Y by taking a(t) = ht(x) for each t E I . Since (I joins f ( x ) to g(x),it follows that f,a = g,a. I We recall that a map f : ( X ,A , xo) + ( Y ,B , yo) is said to be a homotopy equivalence if there exists a map g : ( Y ,B, yo\ + ( X ,A , xo) such that gf and f g are homotopic with the identity maps on the triplets ( X ,A . xo) and ( Y ,B, yo) respectively. As an immediate consequence of the properties I, I1 and V, we have the following
+
Corollary 8.1. If f : ( X ,A , xo) + ( Y ,B. yo) i s a homotopy equivalence, then the induced transformation f , sends nn(X,A , xo) onto n,(Y, B, yo) in a oneto-one fashion.
The significance of (8.1)is that n,(X, A , xo) depends only on the homotopy type of ( X ,A , xo). In particular, if A is a strong deformation retract of X , then i, sends n,(A, xo) onto n,(X, xo) in a one-to-one fashion for every n > 0. Hence, by Property IV, this implies that nn(X,A , xo) = 0 for every n > 0.
118
IV. H O M O T O P Y G R O U P S
9. The fibering property Consider two given triplets (X,A , xo) and (Y, B, yo) and a given map
f
:
w, A , xo)
-+
( Y ,B , Yo).
We are concerned with the notion of fibering as defined in (111; 8 3). Property VI. If f : X
-+
Y i s a fibering and A
= f -1(B), then
the trans-
formation sends n n ( X ,A , xo) onto n,(Y, B , yo) ilz a one-to-one fashion for every n
is onto, let u be an arbitrary element of %(Y, B, yo).
Proof. To prove that f*
By
> 0.
8 3, a is represented by a map
+ : (In,I n - l , In-')
+
( Y ,B, yo).
Since Jn-1 is a strong deformation retract of I", it follows from (v) of (111; 3.1) that there exists a map y : In + X such that f y = I# and y(Jn-') = xo. Since A = f -1(B), f y = implies that y(I*-l) c A and hence we obtain a map p : (In,In-l, Js-') + ( X ,A , xo).
+
This map y represents an element /? of n,(X, A , xo). Since f y = I#, we have f,/? = a.This proves that f* is onto. To prove that f * is one-to-one, let a and /? be elements of n n ( X ,A , xo) such that f*u = fa/?. Choose representative maps
+, y : (In,P1, 1n-l) -+ ( X ,A , xo) for a and /? respectively. Since f*u = f*/?,the composed maps f + and f y represent the same element of nn(Y,B, yo). Hence, there exists a map
F : (In x I,1 n - l x I, such that F(z, 0) closed subspace
= f+(z) and
T
=
F(z, 1)
X
I) + ( Y .B, yo)
= fy(z) for
each z E I". Consider the
( I n x 0) U (I"-' x I ) U (In x 1)
of I n x I and define a map G : T +. X by setting
G(z,t )
+(z), =
xo, Y(Z)>
( z E In, t = O), (2 E p - 1 , t E I ) ,
( z € I n ,t
=
1).
Then we have f G = F I T. Since T is clearly a strong deformation retract of I n x I,it follows from (v) of (111;3.1) that G has an extension G* : In x I + X such that fG* = F. Since F maps In-l x I into B and A = f - l ( B ) , the condition fG* = F implies that G*(In-l x I)c A . Hence we obtain a map
G* : (In x I,1n-l x I,Jn-1 x I ) + ( X ,A , x0)
11. H O M O T O P Y S Y S T E M S
119
+
with G*(z, 0) = +(z) and G*(z, 1) = y(z) for each z E In. This proves that and p represent the same element of n n ( X ,A , x,). Hence a =?!, and f* is one-to-one. I In homology theory, the corresponding fibering property is false in general. Instead of this, we have the famous excision property which does not hold in homotopy theory. As a special case, let us consider the derived projection p : (X’, A‘, x’,) -f ( X ,A , x,) By Properties VI and IV, p , and d in the following diagram
.
n,(X, A , x,) +PI nn(X’, A’, x’,) a4 nn-1(A’, x’,) are both one-to-one and onto. Furthermore, if n n ( X , A , x,) and Z ~ - ~ ( A x’,)’ , are identified as in (3.1) one can easily see that 9, = d. On the other hand, the identification in (3.1) may be considered as being effected by the oneto-one correspondence :n % ( XA, , ~
X =
-+nn-i(A’,~
0 )
‘ 0 )
which will be called the natural correspondence.
10. The triviality property If X is a space which consists of a single point x,, then, for each n, the constant map f ( I n ) = x, is the only map of I n into X . Hence we have the following Property V I 1. If X i s a space consistingof asingle point x,, thenn,,(X, x,) =0
for every n 2 0. This property plays the role similar to that of the dimension property in homology theory. Since it apparently has nothing to do with the choice of the dimension n in n,(X, x,), we propose to call it the triviality property.
11. Homotopy systems In the preceding sections, we constructed geometrically the homotopy groups n n ( X ,A , x,) and established seven basic properties of these groups. In next few sections, we shall show that they are characteristic; in fact, these seven properties, stated in a certain apparently weaker form, together with no(X,x,) for all pairs ( X , x,) determines all n n ( X ,A , x,) for all triplets ( X ,A , x,) up to one-to-one correspondence. A homotopy system H ={n,a,*} consists of three functions n,d and *. The first function n assigns to each triplet ( X ,A , x,) and each integer n > 0 an abstract set n n ( X ,A , x,). The A , x,) and each integer n > 0 second function d assigns to each triplet (X, a transformation :nn(x, A , x,) + n n - l ( ~x,), ,
a
IV. H O M O T O P Y G R O U P S
I20
where, in the case of n = 1, no(A,xo) denotes the set of all path-components of A as in 3 2. The third function assigns to each map f : ( X ,A , xo) + ( Y ,B, yo) and each integer n > 0 a transformation
,
f ,: n n ( X ,A , xo) +nn(Y,B,
YO).
Furthermore, the system H must satisfy the following seven axioms : Axiom I. If f : ( X ,A , xo) -+ ( X ,A , xo) is the identity map, then f , i s the identity transformation on n n ( X ,A , xo) for every n > 0. Axiom 11. I f f : ( X ,A , xo) -+ ( Y ,B, yo) and g : ( Y ,B, yo) += (2,C, zo) are maps, then for every n > 0 we have (gf), = g,f,. Axiom 111. If f : ( X ,A , xo) += ( Y ,B, yo) i s a map and g : ( A , xo) += ( B ,yo) is the restriction of f , then the commutativity relation d f , = g*d holds in the following rectangle for every n > 0 :
a
nn(x,A , xo) -+
1.
nn(Y,B , yo) -+a
nn-I(A, xo)
L*
4
nn-i(B>Y O )
where, in the case of n = 1, g, : no(A,xo) -+no(B, yo) denotes the induced transformation in 3 5. Let ( X ,A , xo) be any given triplet and consider the inclusion maps
i : ( A , x O ) c ( X ,xo), j : ( X , xo) C ( X ,A , ~ 0 ) The transformations i,, j* and d form a beginningless sequence as in
57 which will be called the homotopy sequence of the triplet (X, A , x0) in the system H . Axiom IV. The homotopy sequence of any triplet ( X ,A , xo) is weakly exact. This means that, if n n ( X , xo) = 0 for all n > 0, then d sends n n ( X , A , xo) onto Z , - ~ ( Ax,,) , in a one-to-one fashion for every n > 0. Axiom V. If the maps f g : ( X , xo) + ( Y ,yo) are homotopic, then f , for every n > 0.
=
g,
Axiom VI. If p : ( X I ,A ' , d o )-+ ( X ,A , xo) is the derived projection over ( X ,A , xo), then p , sends n n ( X ' , A', do) onto n n ( X ,A , xo) in a one-to-one fashion for every n > 0. Axiom VII. If X is a space consisting of a single point xo, thennn(X,xo) = 0 for every n > 0. Since the derivedprojectionp : X ' + Xisafiberingby(II1; 13.4),itfollows that the axioms I-VII are weaker than the properties I-VII respectively. Hence, if we neglect the group operation in n,(X, A , xo),the three functions n,a, as defined in 35 3-5 constitute a homotopy system. This proves the
,
12. T H E U N I Q U E N E S S T H E O R E M
I21
existence of homotopy systems. One can also easily construct a homotopy system by induction and without using any geometrical representation, (see Ex. A a t the end of the chapter).
12. The uniqueness theorem Two homotopy systems H = { n , a ,* } and H’ = {n’,d’, # } are said to be equivalent if there exists, for each triplet (X, A , xo) and each n > 0, a transformation h, : n n ( X ,A , xo) +n’,(X, A , xo) satisfying the conditions : ( E l ) h, sends n,(X,A , xo) onto n‘,(X, A , xo) in a one-to-one fashion. (E2) For each triplet ( X ,A , xo) and each n > 0 the commutativity relation hn-l a = d’hn holds in the following rectangle:
a
n n ( X ,A , xo) -+
I”.
n’n(X, A , xo) -+a’
nn-I(A, xo)
(6.-1
4
n’n-AA, xo),
where, in the case of n = 1, h, denotes the identity map. (E3) For each map f : (X, A , xo) +. ( Y ,B, yo), the commutativity relation hnf, = f#h, holds in the following rectangle:
n,(X, A , xo) f*+ n,(Y, B, Yo) lhn
lh*
J. J. n’,(X, A , xo) f#+ n’,(Y, B, yo). A collection of transformations h = { h, } satisfying the conditions ( E l ) through ( E 3 ) is called an equivalence between the homotopy systems H and H’ a i d is denoted by h : H M H‘. Theorem 12.1. Any two homotopy systems are equivalent. [Milnor 11. Proof. Let H = { n,a, * } and H’ = { n‘,a‘, # } be any two homotopy systems. We are going to construct an equivalence h : H w H’ as follows. Let n 1 and assume that we have already constructed the transformations hm:nn(X,A,x,) +n’m(X,A, %I)
A , xo) such that the conditions ( E l ) for each m < n and each triplet (X, through ( E 3 ) are satisfied. Let us construct h, as follows. Let (X, A , xo) be any triplet. Consider the derived projectionp : ( X ’ , A f , x f o ) +. (X, A , x o ) . By Axiom VI, p, sends 7tn(X’,A ’ , do)onto n,(X, A , xo) in a one-to-one fashion and analogously for p#. According to (111; lO.l), X’is contractible to the point xf0. Hence, by Axioms I, 11, V and VII, we obtain
IV. H O M O T O P Y G R O U P S
I22
nm(X’,do)= 0 and nClm(X’,do)= 0 for every m > 0. By Axiom IV, a sends n,(X‘, A,‘ do)onto Z , - ~ ( A ’do) , in a one-to-one fashion, and analogously for a‘. By our assumption of induction, h,-, sends Z , - ~ ( A ’ do) , onto Z’,-~(A’d , o )in a one-to-one fashion. Hence we may define a transformation h9a :n,(X, A , xo)
by taking
+
n’n(X, A xo) 9
h, = p#a’-lh,-,ap;’,
where, in the case of n = 1, h, denotes the identity map. Since ( E l ) is obviously satisfied, i t remains to verify ( E 2 )and (E3). Then we have To check ( E 2 ) ,let q = I ( A ’ ,do). a’h n -- a‘+#a’-lh9a-,ap;l =
q#k,-,dp;l
=
= q#a’a‘-u,-,ap;l
h,-lq*ap;l
= hn-$
by Axiom 111.This proves that 12, satisfies (E2). To check ( E 3 ) , let ( Y ,B , yo) be a second triplet and f : ( X ,A , xo) + ( Y ,B , yo) be any given map. Let r : (Y’,B’,yto)+ ( Y ,B, yo), f ’ : ( X ’ , A ’ ,do)+ (Y’, B’, yl0) denote respectively the derived projection over (Y, B, yo) and the derived map o f f .The relation f p = rf ’is satisfied. Set 7 = f ’ I (A’, d o )Then . we have
f#h,
=
f#p#a’-u,-,ap;l
= r#f’#a’-u,-,ap;1
= r#a’-lf#Izn-lap;l = r#a’-u,-,af’*p;l
=
r#a’-’h,_,f*ap;’
= r#a’-lh,-lar;lf*
= h,f*
by Axioms I1 and 111.This proves that h, satisfies ( E 3 ) . Thus we have completed the inductive construction of an equivalence h = { h, } between H and H‘ which will be called the natural equivalence h : H w H’. I The natural equivalence is the only possible equivalence between H and H‘ as will be seen in Ex. B at the end of the chapter. The uniqueness theorem (12.1) shows that the homotopy system constructed geometrically in $$ 3-5 is essentially the only homotopy system. As a consequence of this, i t follows that the set of seven properties in §$ 6-10 is equivalent to the set of seven axioms in $ 1 1 which are apparently weaker. In fact, one can deduce the seven properties right from the axioms without using any geometrical representation of the sets n n ( X ,A , xo). The details of the proof will be left to the reader as an exercise. See Ex.C a t the end of the chapter.
13. T H E G R O U P
STRUCTURES
123
13. The group structures In the last two sections, we have shown that, apart from the group structures in the homotopy sets n n ( X ,A , xo), the homotopy system constructed in 55 3-5 is completely characterized by the seven axioms in 5 11 and the sets n o ( X ,xo). To complete the axiomatic approach, it remains to determine all the possible group structures that can be introduced into the essentially unique homotopy system
H={n,8,*}.
For this purpose, let us first consider the setsn,(X, xo) in H. According to the uniqueness theorem (12.1), we may assume n l ( X ,xo) to be the underlying set of the fundamental group of X a t xo. The product in n , ( X , xo) as defined in (11; 5 5) will be called the customary product which is denoted by juxtaposition. The reverse of this product, denoted by a dot, is defined by a * @= pa,
(a,@ i n , ( X , xo)).
For any map f : ( X , xo) -+ (Y, yo), the transformation f* : n L ( X ,xo) + n l ( Y ,yo) is a homomorphism under the customary product as well as its reverse. These are the only group structures in all n , ( X , xo) such that f* is a homomorphism for every map f . In fact, we have the following Lemma 13.1. There are exactly two ways of introducing a group structure into the sets n,( X ,xo) in such a way that f., i s a homomorphism for every m a p f : ( X , xo) --f ( Y ,yo). These two group structures are defined by the customary +roduct and its reverse respectively. Proof. Assume that there is a new product, denoted by a 0 @, in each n l ( X ,xo) such that, for each pair ( X ,x o ) , n l ( X ,xo) is a group under this product and that f* is a homomorphism with respect to this product for every map f : ( X ,xo) -+ ( Y , yo). I t suffices to prove that either a o p = ap for any a, p en,(X,xo) of every ( X ,xo) or a o p = pa similarly. Let 2 be the space which consists of two circles, intersecting a t a single point z,. According to (11;Ex. A5),n,(Z, zo) is a free group on two generators a and b. For any two elements a, p of n , ( X , x o ) , obviously there is a map
f : (2,zo) such that f*(a) = a and f,(b) new product, we have
=
p.
( X , xo) Since f* is a homomorphism under the
f*(a o b)
-+
=
a o p.
In terms of the customary group structure of n,(Z, zo), a o b is equal to some word w ( a , b) of the free group. Since f , is also a homomorphism under the customary product, we have f * ( a 0 b) = f * [ w ( a ,41 = w i f * ( a ) ,f * ( b ) l = 4%8,.
This implies that a o p = w ( a , @ ) .Thus it remains to prove that either w ( a , b) = ab or w ( a , b) = ba.
IV. H O M O T O P Y G R O U P S
124
For this purpose, let us first prove that the word w(a, b ) has the following two propeties : (1) w ( a , 1) = a, w(1, b ) = b, w [a, w h c)
(2)
I = w [w(a,b ) , c l ,
where (2) is an identity in the free group on three generators a , b, c. To prove (l), note that the identity element 1 of n l ( Z ,zo) can be defined as the image of the homomorphism
i, :n,(zo, 20) +n,(Z, zo) induced by the inclusion map i : (zo,zo) c (2,zo). It follows that the new product must have this same identity element. Hence W(U,
1)
=uo
1
= a,
~ ( 1b),
=
1o b
=
b.
To prove (2), choose X to be the space which consists of three circles tangent to each other at the same point xo. Then, as in (11; Ex. A5),n l ( X , x o ) is a free group on three generators a , b, c. Since a o b = w(a, b) and b o c = w ( b , c ) , the associative law for the new product implies (2). Finally, we shall complete the proof by showing that, if a reduced word, w(a, b) in the free group on two generators satisfies the conditions (1) and (2), then either w ( a , b ) = ab or w ( a , b) = ba. The proof is a long but easy exercise in the manipulation of reduced words sketched as follows. Let w(a, b) be a reduced word which satisfies (1) and (2). By ( l) , w ( a , b) # a m . More generally, i t is impossible to have w ( a , b) = amlbni. * .UmkbnkUmk+i
--
with non-zero (positive or negative) integers m,,nl, mk,n k , and mk,,. To see this, let us assume that w(a, b) were of this form with k > 1. Then, by ( l ) , one can easily see that the reduced words of (w(a,b ) ) m i and ( w ( b , c))n1 would be of the same form, say, a ,
( w ( a ,b))mi = apibq1.s .a%b%a%+l,
( w ( b ,c ) ) n l = b'lC81. * .brjc8jbrj+l,
with non-zero integral exponents. By ( l ) , i > 1 and have w[a, w ( b , c ) ] = amibric81- - , +
w [w(a, b ) , c ] = api b*i a%
* * *
as their reduced words. This contradicts (2). Similarly, w ( a , b) # bm and it is impossible 'to have W ( U , b) = bm@i* * .bmkd'kbmk+1 with non-zero integral exponents. Next, assume that w(a, b) =
. .amkbnk
i > 1. Then we would
14.T H E R O L E O F T H E B A S I C P O I N T
125
with non-zero integral exponents. If m, < 0, then the reduced word of begins with b-% while that of w [ a , w(b,c ) ] begins with am,. This contradicts (2) and hence we have m , > 0. If n l < 0, then the reduced word of w [ a , w(b, c ) ] begins with amic-nk while that of w [ w ( a ,b ) , c] begins with amibni. This contradicts (2) and so we have both m , > 0 and n1 > 0. Then we have w [ a , w(b, c ) ] = amibmicni. w [ w ( a ,b ) , c ]
- .,
W[W(U,
b), C ]
= amibnl. *
*amkbnk.* *
as their reduced words. By (2), it follows that K = 1 and hence w(a, b) = amibni. Then, (1) implies that m, = 1 and n , = 1. Consequently, we have w ( a , b) = ab. Similarly, we can prove that, if w ( a , b) = bmiani- * 'bmkank
with non-zero integral exponents, then w ( a , b) possible cases. I
=
ba. This exhausts all
Theorem 13.2. There are exactly tzbo ways of introducing a group structure into the sets n n ( X ,A , xo), 12 > 2, and n,(X, xo), i n such a way that the transformations a and f* are homomorphisms. These two group structures are defined respectively by the customary group operation given in 3 3 and its reverse. [Milnor I]. Proof. Let us denote the customary group operation by juxtaposition and assume that there is a new group operation, denoted by a o /?,in the sets n n ( X ,A , xo), n 2 2, and n,(X, xo) such that the transformations a and f* are homomorphisms. We have to show that a o / ? is equal to a/? or /?a. By (13.1), this is true if a and/? are inn,(X, xo). To prove the theorem by induction on n, consider the natural one-to-one correspondence
X =
of
a#;'
:n n ( X ,A , x0) +xn-l(A', do)
6 9. Since X is a homomorphism for both group operations, we have %(a/?)= %(a)X(/?) X(a 0 B) = X(a) 0 W),
for any two elements a, /? in n n (X, A , xo). By the inductive hypothesis, X(a) o X(/?) is equal to X(a) %(/?) or %(/?) %(a).Hence X(a o /?) is equal to %(a/?)or X(/?a).Since X is one-to-one this implies a o /? = a/?or /?a.I The significance of (13.2) is that the group structure in the essentially unique homotopy system H = { z, a, * } is also essentially unique. This completes the axiomatic approach.
14. The role of the basic point In the notion of the homotopy groupsnn(X, xo) andnn(X, A , xo) the basic point xo is explicity used in any geometrical construction of these groups. The objective of the next few sections is to study the role played by the basic
126
IV. H O M O T O P Y G R O U P S
point, to compare the homotopy groups with various basic points, and to free these groups from the basic point wherever it is possible. Let us consider a given space X and two given points xo, x1 connected by a given path a :I-+ a(0) = xo, a(1) = x1.
x,
By the definition in
5 2, we have no(X,xo)
=
P ( X ) = no(X,x1)
as the set of all path-components of X . Moreover, since xo and x1 are contained in the same path-component of X , the neutral element of n o ( X ,xo) is the same as that of n o ( X ,xl). Let us denote by a0 : Zo(X, XI) +no ( X , xo) the identity map on n o ( X ,xl) = n o ( X ,xo).
Theorem 14.1. For each n way an isomorfihism
an :n n ( X ,x1)
M
> 0 , every
path a : I
n n ( X ,xo), xo
= a(O),
-+
X gives i n a natural
x , = a( l ) ,
which defiends only on the homotopy class of the path a (relative to end points). I f a is the degenerate path u ( I ) = xo, then an is the identity automorphism. I f 4, t are paths with t(0)= a ( l ) , then (at), = a n t n . Finally, for each path a : I -+ X and each map f : X -+ Y , we have a commutative rectangle
n n ( X , x1)
If*
A+n n ( X , xo)
I f*
4 4 nn(Y,Yl) A+n n w , Y o ) , where t = fa, yo = f ( x o ) and y 1 = f ( x l ) . As an immediate consequence of (14.l), we deduce the following Corollary 14.2. The fundamental group
as a group of automorphisms.
n , ( X , xo) acts on n n ( X ,x o ) , n
> 1,
To prove (14.1), let us construct un as follows. Let a be any element of n,(X, xl) and choose a representative map f : ( I n , a r n ) -+ ( X ,xl)
for a. The geometrical idea of the construction is to pull the image of i3In along the path u back to the point xo with the image of I n being dragged in an arbitrary way. The map obtained after this homotopy represents an element /3 of n n ( X ,xo) which depends only on a and the homotopy class of a. Then, we define an(a)= ,!I. The details are as follows. First, let us prove that there exists a homotopy f t : I n -+ X , (0 < 1 Q l ) , of f such that ft(dZn) = a( 1 - 1 ) for every t E I . For this purpose, define a partial homotopy+t : 81. -+ X , (0 Q t < I ) , of f by taking+t(aIn) = o ( l -t)
14.T H E R O L E O F T H E B A S I C P O I N T
127
for each t E I . By ( I ; 9.2),aIn has the AHEP in In. Hence the homotopy+t has an extension f t : In -+ X , (0 Q t Q l), such that f , = f . We call f t a homot@y of f along a. Since f l maps aIn into a(0) = x,, it represents an element p of n n ( X , x,). That ,!?depends only on u and the homotopy class of a is an obvious consequence of the following Lemma 14.3. If f , g : ( I n , ill.) + ( X ,x,) are maps homotopic relative to aIn, a, t : I -, X are paths homotopic relative to end points, and f t , gt : In + X , (0 < t Q l ) , are homotopies of .t along a and of g along t respectively, then f l , g, are homotopic relative to aIn. Proof. Define two maps F , G : In x I -+ X by means of the homotopies f t , gt as usual. Consider the subspace A = (In x 0 ) U (din x I ) of In x I . By the hypothesis f N g and a II t, it follows that F I A abd G I A are
homotopic relative to aIn x 0 and aIn x 1. Since A has the AHEP in In x I by ( I ; 9.2), there is a homotopy Ft : In x I -+ X , (0 Q t Q l), such that F , = F , F , I A = G I A , and Ft(aIn x 1') = x, for every t E I . The map F , gives a homotopy ht : In -+ X , (0 Q t Q l), of g along t.Since Ft(aIn x 1) = x, for each t E I , it follows that f l and h, are homotopic relative to aIn. It remains to prove that g, and h, are homotopic relative to a r m . Define a map M : In x I -+ X by taking g,-zg(P),
M(P,q) =
(pEIn,O
< 1). hzg-AP), Then, for each p E aIn, we have M ( p , q) = M ( p , 1 - q). Therefore we may define a partial homotopy Nt : B + X , (0 < t Q l), of M on the boundary B = a(In x I ) = (aIn x I ) U (In x a I ) by taking =
(
ao,
M(P, ( P € In, q E M ( P , 4 - tq), ( P E aIns 0 Q 4 Q t), W P . 1 - q ) , ( P E a I n , Jj Q q < 1). q ) 9
Nt(P, 4)
Since B has the AHEP in In x I , the homotopy Nt has an extension M t : In x I -+ X , (0 Q t Q l), such that M , = M . The map M , gives a homotopy k t : In -+ X , (0 Q t Q l), such that k, = g,, and k, = It,. Since kt(aIn) = M,(dIn x t ) = x,, this implies that g, and h, are homotopic relative to aZn. I Now let us continue the proof of (14.1). We have constructed a transformation an : nn(x,~ 1 +nn(X, ) xg)
which depends only on the homotopy class of the path a. Since the last two assertions of (14.1) are obvious, it remains to prove that a, is an isomorphism. Let u,/J be arbitrary elements of n n ( X ,x , ) represented by the maps I , g : (In,aIn) -+ ( X ,x J . Let f t , gt : In -+ X , (0 Q t < l), be homotopies along
128
IV. H O M O T O P Y G R O U P S
u of f , g respectively; then f l represents un(a) and g, represents un(p). Define a homotopy ht : I n + X, (0 Q t Q l), by taking ht = f t + gt. Then h, represents a p, h, represents an(a) un(j3), and ht is a homotopy along u. This implies that un(a p) = on(a) +On(/?) and hence On is a homomorphism. Finally, let us prove that the homomorphism 0%is an isomorphism. Let t denote the reverse of u, that is to say, t ( t ) = u(1 -t ) for every t E I. Because of symmetry, it suffices to show that is an epimorphism and t n is a monomorphism. For this purpose, consider the product at.Then we have untn = (at)%.Since z is the reverse of u, it is clear that u t is homotopic to the degenerate path at x, with end points fixed. It follows that (0t)nistheidentity automorphism on nn(X, x,). By an elementary group-theoretic argument, it follows that is an epimorphism and tnis a monomorphism. This completes the proof of (14.1). Thus, for a pathwise connected space X, all the groups nn(X, x,) with various basic points x, E X are isomorphic. Hence, as an abstract group, nn(X,x,) does not depend on the basic point x, and may be denoted simply bynn(X).This abstract groupnn(X)will be called the n-th (abstract)homotopy group of the pathwise connected space X. In this terminology, we have
+
+
+
n,(S1)
= 2,
nm(Sn) = 0,
nn(S1)= 0, .(
> 1).
(m< n), nn(Sn)
= 2.
n3(S2) = 2.
Now, let us consider the spaces of loops
w,= [X ; x,> XOI, w,= [ X i x1, x11 as well as the degenerate loops W,E W , and W , E W,. Then a given path : I + X which connects x,, to x1 induces a map l : W , + W , defined as follows: for each w E W , , [(w) E W , is the loop defined by
{
a(3t),
[ l ( w ) l ( t )= w ( 3 t - I), o(3 - 34,
(if 0 Q t Q &), (if 4 Q t < 3, (if 3 Q t Q 1).
Intuitively speaking, t ( w ) is the loop traced by running first from x, to x 1 along the path u, next around the loop w once, and then back to x, along the reverse of 0. On the other hand, u also induces a path r j : I + W , defined as follows: for each s E I, rj(s)E W , is the loop defined by 434, [r(s)I(t)= ( 4 S ) J 4 3 s -3 4 ,
(if 0 Q t Q &), (if & < t < $), (if $ < t Q 1).
Then, obviously we have rj(0) = mo and ~ ( 1 = ) t(wl).
15.L O C A L
SYSTEM OF G R O U P S
129
Proposition 14.4. I f 5 : (Wl, wl)+ (W,, f ( w , ) ) and q : I + W,are induced by the path u, the commutativity holds in the following diagram ‘Jn
nn(X, x1)
-+
n n ( X ,xo)
where n i s any positive integer and X denotes the natural correspondence given by (2.2). Proof. Let a E n n ( X , xl) and choose a representative map f : (In,din) -+ (X, xl) for a. Then it is not difficult to see that both an(&)and X-lqn-lt*X(a) are represented by the map g : (In,i3P) + (X, xo) defined as follows: 43W) (if e ( t ) G g), g(t) = f (3t1- 1; * -,3tn - l ), (if e ( t ) > 4~ where t = ( t l ; - * , tn) is an arbitrar,y point of I n and O(t)denotes the smallest of the 2 n real numbers t 1 ; * - ,tn, 1 -tt,;.., 1 -tn. I
1
13
The property (14.4) of the operations 0%is characteristic. To formulate this fact precisely, let us define the notion of a system of operations as follows. By a system of operations in a homotopy system H = { n,a, * }, we mean for each path u : I -+ X in any space X and each integer n > 0 a transformation an : n n ( X ,X I ) +nn(X, xo), xo = U ( O ) , x1 = u(1) such that a,is theidentityonn,(X, xl) = n,(X, x,j and that (14.4)issatisfied. Since (14.4) implies an = X-lqn-l[*X, the inductive proof of the following theorem is obvious.
Theorem 14.5. I n a n y given homotopy system H = { n,a, * }, there exists one and only one system of operations. Furthermore, for any two given homotopy systems H and H’, the natural equivalence h : H w H’ commutes with the operations in H and H’. Analogously, for each path u : I + A , one can define the operations on on the relative homotopy group and deduce similar results. See Ex.D a t the end of the chapter.
15. Local system of groups The homotopy groups together with the operations un constructed in the preceding section motivated the notion of a local system of groups in a space X , [Steenrod 11. We shall say that we have a local system of groups { G,} in a space X , if the following conditions are satisfied: (LSG1) For each point x E X, there is given a group G,. (LSGS) For each path u : I + X joining x, to xl, there is given a homomorphism a# : GZ1 + G,.
130
IV. H O M O T O P Y G R O U P S
(LSG3) If u is the degenerate path u(I) = xo, then a# is the identity automorphism on Gm. (LSG4) If two paths u, t : I -t X are equivalent, i.e., if u, t have the same end-points and are homotopic with end-points held fixed, then a# = t#. (LSG5) If two paths IT,t : I -t X are consecutive, i.e., if u(1) = t(O), then (at)# = u#t#. According to (14.1), the collection of the homotopy groups { nn(X, xo) I xo E X }, for a given space X and a given integer n > 0, forms a local system of groups in X. Similarly, the collection of the relative homotopy groups { nn(X,A , xo) I xo E A }, n > 1, forms a local system of groups in the subspace A of X. As an easy consequence of (LSG3)-(LSG5), we deduce as in the proof of (14.1) that every a# is an isomorphism. Hence, if X is pathwise connected, then all the groups Gz,x E X, are isomorphic. Since the elements of the fundamental group nl(X, xo) are homotopy classes of the loops [X ; xo, xO] with end-points held fixed, we deduce as a direct consequence of (LSG3)-(LSG5) that, for each xo E X , n,(X, xo) acts as a group of operators (or automorphisms) on , G in the sense defined as follows. A multiplicative group H is said to act as a group of operators on an additive group G, (or, simply H acts on G), if, for every h E H and every g E G, an element hg E G is defined in such a way that
+
+
h(g1 gz) = k l hgtgzt h,(h,g) = (h,h,)g* lg = g where g, g, g, E G, h, h,, h, E H are arbitrary elements and 1 E H denotes the neutral element. Applying this to the local system of groups {nn(X,x,,) I EX}, where 12 > 0, we obtain (14.2) restated as follows. Proposition 15.1. For each xo E X, tlte fundamental group n l ( X , xo) acts on the n-th homotopy group nn(X, xo) as a group of operators.
In the special case n
=
1, one can easily see that, for any two elements
g and h inn,(X, xo),h acts on g as follows:
(15.2)
h(g) = hgh-1.
Similarly, for each xo E A , n,(A, xo) acts on the n-th relative homotopy group nn(X, A , xo), n > 1, as a group of operators. As a consequence of these operations, let us consider two given homotopic maps f,g:X+Y. Let ht : X +. Y ,(0 < t < l ), be a homotopy such that ho = f and hl = g. Choose a point xo E X and denote yo = f and y1 = g(xo). Define a path a : I +. Y by taking 44 = ht(xo), ( t E 4 ;
(xo)
~~
16. n - S I M P L E then a(0) = yo and a( 1) homomorphisms
= y,.
SPACES
According to
3 5, the
131
maps f and g induce
g* : n n ( X , x o )-+nn(Y,Y J ) (Y, YO), f* : n n ( X ,~ 0 -+nn 1. On the other hand, the path a determines an isomorphism for each n an :nn(Y,~ Proposition 15.3. f* = ang,.
-+nn(Y,Y O ) .
1 )
+
Proof. Let a E T C ~ (xo) X , and choose a map : (In, 81.) -+ ( X , xo) which represents a. Define a homotopy y t : I n -+ Y , (0 < t Q l), by taking yt = ht+ for every t E I. Then yo represents f*(a) and y1 represents g*(a). Since yt(8In) = a(t) for each t E I, it follows that f,(a) = crag*(.). I
Corollary 15.4. If f , g : X -+ Y are homotopic maps such that f (xo) = yo = g(xo),then there exists aiz element w ETC,(Y, yo)such that f* = wg,.
As another consequence of (15.3), we have the following Proposition 15.5. If f : X -+ Y is a homotopy equivalence and
then
if f (xo) = yo,
f* :n n ( X , xo) -+nn(Y,Yo)
is an isomorphism for every n > 0. Proof. Since f is a homotopy equivalence, there exists a map g : Y -+ X such that gf and fg are homotopic to the identity maps. Let x , = g(yo). Then g induces g, :nn(Y,Yo) -+nn(X,4 -
Let ht : X -+ X , (0 < t l), be a homotopy such that h, = gf and h, is the identity map on X . Define a path a : I -+ X by a(t) = ht(xo)for every t E I. Then, by (15.3), we have g*f* = an. Since an is an isomorphism, this implies that f* is a monomorphism and g, is an epimorphism. Since g is also a homotopy equivalence, it follows that g, is also a monomorphism. Hence g, is an isomorphism and so is f * = g;lan. I Similarly, iff : (X, A ) -+ ( Y ,B) is a homotopy equivalence and iff (xo) = yo, then f* : n n ( X ,A , xo) +nn(Y, B,Yo) is an isomorphism for every n
> 1.
16. n-Simple spaces A local system of groups { Gz } in a space X is said to be simple, if the homomorphism a# depends only on the initial point a(0) and the terminal point a(1) of the path a : I + X. Let W be a group which acts on a group G. We shall say that W acts simply on G if wg = g for every W E W and g E G .
132
IV. H O M O T O P Y G R O U P S
The proofs of the following two propositions are straightforward and hence are left to the reader. Proposition 16.1. A local system of groups { Gx } in a space X is simple i#, for every xo E X , n , ( X , xo) acts simply on Gzo. Proposition 16.2. A local system of groups { Gx } in a pathwise connected space X is simple i# there exists a point xo E X such that n , ( X , xo) acts simply on Gxo. Corollary 16.3. A local system of groups { Gx } in a simply connected space X is always simple.
Let n > 0 be any given integer. A space X is said to be n-simple if the local system { n n ( X ,xo) I xo E X } of the n-th homotopy groups in X is simple. The following assertions are immediate consequences of the definition and (16.1)-(16.3). Proposition 16.4. A space X is n-simple if, for every xo E X , n , ( X , xo) acts simply on n n ( X , xo). Proposition 16.5. A pathwise connected space X is n-simple i# there exists a point xo E X such that q ( X , xo) acts simply on n,(X, xo). Corollary 16.6. A simply connected space is n-simple for every n
> 0.
Corollary 16.7. A pathwise connected space X is n-simple if n n ( X ) = 0. Corollary 16.8. A pathwise connected space X is 1-simple
commutative.
iff n l ( X ) is
Thus, the m-sphere Sm is n-simple for every m > 0 and n > 0. Now let us consider the unit n-sphere Sn and a given point so E Sn. The geometrical meaning of n-simplicity is given by the following Theorem 16.9. A space X is n-simple i#, for every point, x,, E X and any --f X with f (so) = xo = g(so), f N g implies f N g re1 so.
two maps f , g : S n
Proof. Assume that X is n-simple. Then, by (16.4), n , ( X , xo) acts simply on n n ( X , xo). Since f N g, there exists a homotopy ht : Sn + X , (0 < t < l ) , such that ho = f and h , = g. According to the remark given in the paragraph which precedes (2.1), the maps f and g represent elements a and p of n,(X, xo) respectively. Define a path u : I -+X by taking a(t) = &(so) for each t E I . Since o(0) = xo = a(l),u represents an element w of q ( X , x0). By 5 14 and 4 15, it is easy to see that a = wj3. Since n , ( X , xo) acts simply on n n ( X , x o ) , we have wj3 = j3. Hence a = j3. This proves that f N g re1 so. Next, let us assume that the condition is satisfied. Let w E n , ( X , xo) and choose a loop u which represents w. Let a be any element of n n ( X , xo) represented by a map f : Sn + X with f (so) = xo. Then the element w a of
16. % - S I M P L E
SPACES
I33
n n ( X ,x,) is represented by a map g : Sn + X with g(s,) = x, and satisfying f r g. By our condition, this implies that f N g re1 so. Hence wu = 01. By ( 16.4), X is n-simple. I As a consequence of (16.9), let us prove the following Proposition 16.10. Every fiathwise connected topological group is n-simple
for every n > 0.
Proof. Let X be a pathwise connected topological group and x, its neutral element. Let f , g : Sn + X be any two homotopic maps such that f (so) = x, = g(so). Then these exists a homotopy ht : Sn + X , ( 0 Q t Q l), with h, = f and h1 = g. Define a homotopy kt : Sn -+ X , (0 Q t < l ) , by taking
kt(4
=
[ht(so)l-l. [ht(s)l, (s E S", t E I ) .
Then we have k, = f , k , = g and kt(s,) = x, for each t E I . Hence f N g rel so. By the sufficiency proof of (16.9), n , ( X , x,) operates simply on n n ( X ,x,). By (16.2) this implies that X is n-simple. I This proposition (16.10) can be generalized to the H-spaces as defined in (111; 3 11). See Ex. G. The usefulness of n-simplicity is that, for a pathwise connected n-simple space X , the abstract homotopy groupn,(X) as defined in 5 14 has a natural geometrical meaning as follows. Let us define n n ( X ) to be the set of all homotopy classes of the maps of Sn into X . In other words, n n ( X ) is the set of all path-components of the mapping space @ = XS". Choose an arbitrary basic point x, E X and consider the subspace y / of @ which consists of the maps of (Sn, so) into ( X ,x,). Then the path-components of Y can be considered as the elements of n n ( X ,x,). Hence the inclusion map Y c @ induces a transformation
X :n n ( X , x O ) +nm(X). Lemma 16.11. I f X i s fiathwise connected and n-simple, then X sends n n ( X ,x,) ontonn(X)in a one-to-one fashion. Proof. Let a € n n ( X )and choose a map f : S n + X which represents a. Since X is pathwise connected, there is a path a : I -+ X such that a(0) = x, and a( 1) = x, = /(so). By the method used in the construction of un in 3 14, one can show that there is a homotopy f t : Sn + X , (0 Q t < l ) , such that f o = f and ft(so) = a( 1 - t ) for each t E I . Then f l E !P and represents an element /? of n n ( X ,x,). The homotopy f t proves that X(p) = a. Hence X is onto. Next, let M , /? E n n ( X ,x,) be such that X(a) = X(/?). Choose maps f , g : (Sn, so) ( X ,x,) representing a, /? respectively. Since X(a) = X(@), we have f N g. Since X is n-simple, it follows from (16.9) that f N g rel so. This implies that a = /?.Hence X is one-to-one. I --f
I34
IV. H O M O T O P Y G R O U P S
By means of X , we may define a group structure innn(X)so that X becomes an isomorphism, which will be called the canonical isomorfihism of nn(X, x,,) onto n,(X). To justify our geometrical construction of nn(X)given above, it remains to show that the group structure defined in nn(X) by means of X is independent of the choice of x,, E X. The geometrical meaning of the group operation defined innn(X)by means of X is clearly as follows. Let S: and S! denote the hemispheres defined by 2% < 0 and tn > 0 respectively, where (to, - ,tn) denotes an arbitrary point of Sn. Then the basic point s,, = (1 , 0, * - ,0) is in the equator
--
sn-1 = sn_n s:. Let a and p be arbitrarily given elements of nn(X). Then there exist maps f E a and g E such that f(Sn,) = x,, = g(Sn_). Define a map h : S n --f X by taking
Then h represents the element a + ?!t of n,,(X) which does not depend on the choice of f from a and g from @.We call the attention of the reader to the fact that, in case n = l,n,(X) is commutative by (16.8) and hence the additive notation is preferred. Now let x , E X. Since X is pathwise connected, there is a path a : I -+ X such that a(0) = x,, and a(1) = x , . By a standard method, one can show that there exists a homotopy f t : S n -f X, (0 < t < I ) , such that f o = f and ft(.S!) = a(t) for each t E I. Similarly, there is a homotopy gt : S n + X, (0 Q t Q l ) , such that g, = g and gt(S!) = a(t) for each t E I. Define a homotopy ht : Sn +. X, (0 Q t Q l), by setting
ht(4 for each t E I. Then h,, since
+
=
=h
(if s ES?), (if s E S?).
[ ftbh gt(s),
and h,
N
h. Hence h, represents a
+ 8. Also,
this proves that a does not depend on the choice of the basic point x,, E X. Thus, for a pathwise connected n-simple space X,we have freed the basic point from the definition of the n-th homotopy group. We have just seen that the homotopy classes of the maps of S n into a pathwise connected n-simple space X form a groupnn(X)which is isomorphic with nn(X, x,,) for every x,, E X. If X is pathwise connected but not n-simple this is not true; in fact, the homotopy classes of the maps S n + X are in a one-to-one correspondence with the equivalence classes in nn(X, x,,) under the operations of n,(X, x,,). The proof is left to the reader.
EXERCISES
I35
EXERCISES A. Inductive construction of a homotopy system
a,
Construct a homotopy system H = { n, * } by induction as follows. According to the definition of a homotopy system in 5 11, the homotopy set n o ( X ,xo) and the induced transformation f* : n o ( X ,xo) -+ no(Y,yo) are well-defined for every pair (X, xo) and every map f : ( X ,xo) + ( Y ,yo). Let n 2 1 be a given integer and assume that we have already constructed the homotopy sets n m ( X , A , xo) for each 1 Q m < n and each triplet (X, A , xo), together with the boundary operators and the induced transformations f* on these homotopy sets, such that the seven axions in $ 11 are satisfied. To construct the homotopy set n n ( X ,A , xo) of a given triplet ( X , A , xo), consider the derived triplet (X‘, A’, do)of ( X ,A , xo) and the derived projection p : (X’, A ’ , d o ).+ ( X ,A , x o ) . We define
a
nn
( X ,A , xo)
Next, let q = p I ( A ’ , nn-,(A, xo) is defined by
%IO).
= nn-i(A’,
~’0).
Then, the boundary operator
a :n n ( X ,A , xo)
-f
a = q* :nn-i(A‘,do)+nn-i(A, x0).
Finally, let f : ( X ,A , xo) + ( Y ,B , yo) be a given map. Then, f induces a Then A’, do)+ (Y‘,B‘, yl0). Let 7 = f ’ I (A‘, derived map f ’ : (X’, define f * = :nn-i(A‘, x’o) . + ~ n - ~ ( B~ ’’ ~0 ) . Verify the seven axions of 3 11 for the system H = { n, * } constructed above. %IO).
L
a,
B. The equivalence theorem
Consider two given homotopy systems
H
={n,a,*},
H’
={n‘,a’,#}
together with their natural equivalence h = { hn } : H M H’ constructed in 5 12. By an admissible transformation k = { kn } : H -+ H’, we mean for each triplet ( X ,A , xo) and each integer n > 0 a transformation
kn : n n ( X ,A , xo) -+n’n(X, A , xo) satisfying the conditions : (ATl) For each triplet ( X ,A , xo) and each integer n > 0, we have the commutativity relation kn-$ = a’k,, where, in case of n = 1, KO denotes the identity map. (AT2) For each map f : ( X ,A , xo) -+ ( Y ,B , yo) and each integer n > 0, we have the commutativity relation knf* = knf+ By (El), (E3) and (E4) of 5 12, every equivalence between H and H‘ is an admissible transformation. Conversely, prove the following
IV. H O M O T O P Y G R O U P S
136
Equivalence Theorem. Every admissible transformation k i s a n equivalence between H and H'. I n fact, it coincides with the natural equivalence h, that i s to say, kn = hn for every n > 0 . This shows that the natural equivalence h = { hn } is the only possible equivalence between any two homotopy systems. Furthermore, in order to construct geometrically the natural equivalence between two homotopy systems given by geometric definitions, it suffices to establish an admissible transformation by means of some natural geometric method. C. Properties of the homotopy system
The uniqueness theorem (12.1) implies that the set of seven properties in is equivalent to the set of seven axioms in $ 11. The following is an outline of deducing these seven basic properties together with other properties of the homotopy system right from the seven axioms. Since the properties I, 11, 111, V, VII are stated exactly the same as the corresponding axioms, it remains to prove the properties IV and VI. 1. If f : ( X ,A, xo) +. (Y, B, yo) is a homotopy equivalence, then, for each n > 0 , f* sends n,(X, A , xo) onto n,( Y , B, yo) in a one-to-one fashion. 2. If X is contractible to the point xo, thenn,(X, xo) = 0 for every n > 0. 3. For any given triplet ( X ,A , xo), consider the derived projection p : (X', A', do)+. (X, A , xo). Then in the diagram n n ( X ,A, ~ 0 &nn(X', ) A', ~ ' 0 2-+ ) nn-i(A', d o ) , both p , and a are one-to-one and onto and hence we obtain a natural cowespondence X = d#;' :nn(X, A, ~ 0 +.7~,-1(A', ) do) which sends n n ( X ,A, xo) onto Z ~ - ~ ( A do) ' , in a one-to-one fashion. 4. For every triplet (X, A, xo) and every n > O,n,(X, A, xo) is non-empty. Furthermore, one can uniquely define a neutral element of nn(X, A , xo) in such a way that X,a and the induced transformations send the neutral element into a neutral element. 5. In the homotopy sequence * . * -L~nn+l(X,A,xo) ~ n n ( A , x o ) ~ + . n n ( X , x o ) ~ + . n n ( X , A , xa o ) *- *-. ,
$5 6-10
--
*
4, n , ( X , A ,
x0)
-E+ no(Arx0) I-+ no(X,x0)
of (X, A , xo) every set has a specific neutral element. By the kernel of a transformation in this sequence, we mean the inverse image of the neutral element. Then prove simultaneously the following two theorems : The exactness theorem. T h e homotopy sequence of any triplet i s exact, that is to say, the kernel of every transformation in the sequence coincides with the image of the preceding transformation. The fibering theorem. I f f : X +. Y i s a j i b e r i n g , A = f - I ( B )a n d x o E f - ' ( y 0 ) , then the induced transformation f* carries n n ( X ,A , xo) onto n,(Y, B, yo) in a one-to-one fashion for every n > 0. These two theorems cover the properties I V and VI respectively.
I37
EXERCISES
D. The role of the basic point in the relative homotopy groups
Consider a given space X,a given subspace A of X,and two given points
xo, x1 connected by a path a:I+A,
For each n
a(0) = xo,
u(1)
= x1.
> 0, define a transformation
an:nn(X,A,xI) +nn(X,A,~o) as follows. Let a €n,(X, A , xl). Choose a representative map f : (In,In-1, I"-') -+ ( X ,A , xl) for a. Pull the image of Jn-1 retreating along the path a back to xo with the image of I n being dragged in such a way that the image of In-I is always in A . The map obtained after this homotopy represents an element p of n n ( X , A , x o ) which depends only on a and 0.Then, we define an(a)= p. Give the details of this geometrical construction as in 3 14, and prove the following assertions: 1. For every t z > 2, an is a homomorphism. 2. For every n > 0, an depends only on the class of the path u. 3. If a is the degenerate path u(I)= xo in A , then un is the identity transformation on n,(X, A , xo) for each n > 0. 4. If a, t are consecutive paths in A , i.e. a(1) = t(O),then (at),= antn for each n > 0. 5. For every n > 0, an carries n n ( X ,A , xl) onto n n ( X ,A , xo) in a one-toone fashion. Hence an is an isomorphism for every n > 2. 6. Each rectangle of the following ladder is commutative * *
Lnn(A,xo) 4 , nn(X,x g ) A+ n n ( X , A , xo) --+a
*_a,
kn
n,(A, xl) 2%
bn
n,(X,
XI)
kn
A+ n,(X, A ,
nn-1(A, KO)&
* * *
Ion
x1) -+ a
J. n,-,(A,
XI)
4,
* *
-
7. For any triplet (X, A , x O ) , n 2 ( X , Axo) , is a crossed [ n l ( A ,x o ) , i3l-module. By this, we mean that the following two conditions are satisfied for every w in n l ( A ,xo) and a, p in n 2 ( X ,A , xo) : (i) a(wa) = w(da)w-', (ii) (aa)p = .pa-'. Hence j*n,(X, xo) is contained in the center of n , ( X , A , xo) and i,n,(A, xo) acts as a group of operators on j*n,(X, x o ) . See [Hi, pp. 39-41]. The significance of the assertion (5) is as follows. If A is pathwise connected and n >, 2, then all the groups n n ( X ,A , xo) for various basic points xo are isomorphic. Hence, as an abstract group, n n ( X ,A , xo) does not depend on the basic point xo and may be denoted simply by n n ( X ,A ) . This abstract group n n ( X ,A ) will be called the n-th (abstract)relative lzomoto$y group of X modulo A . For example, we have nm(En,9 - 1 ) = 0, ( m < n ) , n n ( E n , 9 - 1 ) = 2, nm(E2,Sl) = 0, (m > 2), n4(E3,9)= 2.
138
IV. H O M O T O P Y G R O U P S
Next, consider the spaces of paths
w,= [X;A,x,l,
W, = [X;A,x,l as well as the degenerate paths wo E W , and w1 E W,. Then a given path u : I - + A which connects xo t o x , induces a map $ : W, + W, defined as follows: for each w E W,, [(w)E W , is the path defined by i f 0 < t <$, if 4 < t < 1.
[W)l(t) = { a(2-2t), 424,
On the other hand, a also induces a path r] : I each s E I,q ( s ) E W , is the path defined by
Then, obviously we have q(0) = wo and r]( 1) tativity holds in the following diagram an
n n ( X ,A x1) 8
-+
=
W , defined as follows: for
E(w,). Prove that commu-+
n n ( X ,A , xo)
where X denotes the natural correspondences. Hence n > 0. an = X-%ln-,E*X, E. Relative n-simplicity
Let n > 2 be a given integer. A space X is said to be n-simple relative to a subspace A if the local system of groups { n n ( X ,A , x,) I x, E A } in A is simple. In this case, we also say that the pair ( X ,A ) is n-simple. Establish analogues to (16.4)-( 16.7), as well as these assertions: 1. If ( X ,A ) is 2-simple, then n , ( X , A , x,) is abelian for every xo E A . 2. If, for every x, E A , n , ( X , A , xo) is abelian and i, sends nl(A,x,) into the neutral element of n , ( X , xo) then ( X ,A ) is 2-simple. 3. A pair ( X , A ) is n-simple iff, for every point xo E A and any two maps
I , g : ( E n , sn-1, so) ( X ,A , xo), 1 implies that f g re1 { Sn-l, A ; so, x, -+
g rel { Sn-l, A
}. 4. If ( X ,A ) is n-simple and if A is pathwise connected, the elements of the abstract relative homotopy group n n ( X , A ) can be considered as the homotopy classes of the maps of (En,9 - 1 ) into ( X ,A ) . f
N
N
F. The Whitehead product
Consider a given space X and a given basic point xo E X . Let m
n
> 1 be given integers. For any two given elements a E n m ( X , xo),
B E n n ( X , KO),
> 1 and
I39
EXERCISES
we are going to construct an element [a,/?I of nm+n-l(X, xo) which will be called the Whitehead firoduct of a and 8. For this purpose, let us choose representative maps
f : (Im,a r m )
( X ,xo), g : ( I n ,a r n ) -+ ( X ,xo) for a , 8 respectively. Since Zm+n = Im x In, we have aZm+n = ( I m x aIn) U (aim x In). Hence we define a map h : a P + n -+ X by taking for each point (s, t ) in dZm+n if t E aZn,
4%1)
-+
=
('1, { fg(t)
if s E aZm.
Since the point ro = (0;* ., 0) of aZm+n is in aIm x din, we have h(ro) = xo. Since aZm+n is homeomorphic to Sm+n-l, h represents an element y of nm+n-i(X,4. Prove that y depends only on the elements a and 8. So, we may define [at81 = Y . Establish the following properties of the Whitehead products : 1. If a e n , ( X , xo) and B e n , ( X , xo), then [ a , b ] is the commutator a/?a-l/?-l of n , ( X , xo). 2. If a € n m ( X ,xo) and ,!I € n , ( X , xo) with m > 1, then [a,81 is the element @a-a of n m ( X ,x0). 3. If m > 1, then the assignment a -+ [a,83 for a given 8 € n n ( X ,xo) defines a homomorphism p* : n m ( X ,xu) +nm+n-i(X>xo). 4. If m + n > 2 , then, for every a Enm(X, xo) and 8 € n n ( X ,xo) we have
l)mn[a,81. 5. If u : I -+ X is a path joining xo to xl, then, for every a € n m ( X ,x , ) and /?E n n ( X ,x , ) , we have
[S, a1
=
(-
am+n-l[a,81 = [am(a),~ n ( @ ) ] . 6. If : ( X ,xo) -+ ( Y ,yo) is a map, then, for every a € n m ( X ,xo) and / ? ~ n n ( XX O* ) ~we have +*[a,/?]= [+*(a),+,(~)].
+
7. For any a €n,(X, xo), 8 ~ n n ( Xxo), , r € n q ( X xo), , the following Jacobi identity holds:
+ (-
+
l)nm"B, yl, a1 (-- l)@""yja ] ,B1 = 0. Whitehead products may be also defined between relative homotopy groups and between n m ( X ,A , xo) and nn ( A , xo). See [Hu 111 and [Blakers and Massey 21. (-- l)m@"a,83, yl
G. Homotopy groups of H-spaces
Let X be a given H-space as defined in (I11 ; fj 11) and let xo be a homotopy unit of X . Then the group operation in n n ( X , xo) is closely related to the multiplication in X as follows.
IV. H O M O T O P Y G R O U P S
140
Let a,/? be arbitrarily given elements of n n ( X , xo) with n any representative maps f, g : ( ~ n , ( x ,xo)
aw
> 0.
Choose
-+
for a and /? respectively. By means of the multiplication in X , we may define a map h : (In, 81.) -+ ( X ,xo) by taking
h(t) = f ( t )* g ( t ) , 1 E In. Prove that h represents the element a + /? of n,(X, xo). Furthermore, if X is a topological group and xo is the neutral element of X , then we may define a map k : ( I n ,a P ) -+ ( X , xo) by taking
f ( t )* [g(t)]-1, t E In. Prove that k represents the element a -/? of n,(X, xo). Next, let a €n,(X, xo) and 8 € n n ( X ,xo). Choose representative maps
K(t)
f : (P,
=
a r m ) -+
(x, xo),
for a and 8 and define a map h : I m + n h(s, 1 )
=
g ( I %sin)+ , -+
(x,xo)
X by taking
f ( s ) * g ( t ) , s E I m , t E In.
Prove that h I d P + n represents [a,81. This implies that [a,81 = 0.
Hence, it follows from the assertion (2) of Ex. F that every pathwise connected H-space is n-simple for every n > 0 ; in particular, we deduce again that n , ( X , xo) is abelian. H. Semi-simplicia1 complexes
A semi-simplicial complex K is a collection of elements { a } calledcells together with two functions. The first function assigns to each cell u a n integer m > 0 called the dimension of u, m = dim (a); we then say that u is an m-cell. The second function assigns to each m-cell u, (m > 0 ) ,of K and each integer i, (0 < i < m ) , an (m - 1)-cell &u called the i-th face of u, also denoted by di),subject to the condition (SSC)
a,a,u = a,-,aeU
for m > 1 and 0 < i < j < m. We call a, the i-th face operator. It may happen that &a = &a for some i # j. Lower dimensional faces of u may be defined by iterating the face operators. For any two cells u and t of K , we shall write t < u if either t = u or t = .&nu for some set { i1; * * , i, } of integers with 0 < i, < * * * < i n < m. Thus we obtain a proper reflexive partial ordering relation < in K . [Eilenberg and Zilber 13. 1. A simplicia1 complex K is a set whose elements are finite subsets of a given set V , subject to the condition that if U E K and t is a non-empty subset of u, then t E K . Prove that, if V is partially ordered in such a way that every u E K is lineally ordered, then K is a semi-simplicial complex.
a,,. -
EXERCISES
141
2. Given a simplicial complex K , construct a semi-simplicial complex O(K) as follows. The m-cells, m 2 0, are the (m + 1)-tuples (v0; * * , urn), repetitions allowed, of the vertices of some simplex of K ; and &(v0, * * , vm) = (v0;
., &, - .
-
a ,
v,J, where the circumflex over va means that va is omitted.
3. Prove that the singular complex S ( X ) of a space X , [E-S; p. 1851, is a semi-simplicial complex. 4.Construct the homology and the cohomology groups of a given semisimplicial complex K modulo a subcomplex L over a given abelian coefficient group. I. Degeneracy operators
By a system of degeneracy operators in a given semi-simplicial complex K , we mean a function which assigns to each m-cell u, (m 2 0), of K and each integer i , (0 < i < m),an (m 1)-cell &u of K , satisfying the conditions:
+
eae,(u) = e,+,e,(u), (i < j ) ; a,e,(u) = 0 = a,+,e,(a); (m> 0 , i < j ) ; =
aae,(u) ej-lar(u), alej(0)= e,aa-,(u), (m> 0, i > j + 1).
We call Ba the i-th degeneracy operator of the system. An m-cell u, (m> 0), of K is said to be degenerate if u = &(t) for some (m- 1)-cell z of K and someiwitho < i Gm-1. 1. Prove that the semi-simplicial complex O ( K )of a simplicial complex K admits a system of degeneracy operators defined b y
--
- ., va,
-,vm). 2. Prove that the singular complex S ( X )of a space X admits a system of ec(vo,
*,
(vo,
Urn) =
degeneracy operators defined by
Of,
+
vt+,,.
a
--
eao(to, -,L+,) = upo, -,it-,, tt ta+l, -,trn+l). 3. Prove that the semi-simplicial complex obtained from a simplicial complex by a partial ordering of the vertices admits no system of degeneracy. 4.Prove that every semi-simplicial complex of finite dimension admits no system of degeneracy. J. Complete semi-simplicia1 complexes
A semi-simplicial complex K is said to satisfy the extension condition if , urn+lof K such that given m-cells uo,* * , uk-,, ak+,, &a, = d ~ - p g , (i # k , j # k , i < j ) , for the case m > 0, then there exists an ( m + 1)-cell u such that dau = ug for each i # k . A semi-simplicial complex K is said to be complete if it satisfies the extension condition and admits a system of degeneracy operators. A complete semi-simplicial complex K with a given system of degeneracy operators is called a K a n complex, [Kan 11.
--
142
IV. H O M O T O P Y G R O U P S
Prove that the singular complex S(X) of a space X is complete and that the semi-simplicial complex O(K) of a simplicial complex K may fail to be complete. Hence, the class of Kan complexes is rather limited. K. Homotopy groups of Kan complexes
Let K be a given Kan complex. The O-cells and the 1-cells of K will be called vertices and edges respectively. Two vertices vo, v1 of K are said to be equivalent, vo N v l , if there exists an edge e E K with doe = v l and a l e = vo. By means of the extension property of K, prove that this relation is symmetric, reflexive and transitive, and hence the vertices of K are divided into disjoint equivalence classes called the components of K. Let no(K)denote the set of all components of K. Next, let v be a given vertex of K. We are going to define a derived complex.
-
K'
= D(K, U)
+
as follows. The n-cells of K' are defined to be the (n 1)-cellsu of K such that v is the only vertex < u and that a0u = 8,*(v), where O0* is the n-fold iteration of the degeneracy operator 8,. The face operators a', and the degeneracy operators 8'r in K' are defined by
alfu = a,+,u,
e,+lu. Verify that K' is a Kan complex and v' = 8,v is a vertex of K'. Then, for every n > 0, we define nn(K,v ) inductively by 81,
0
=
nn(K, V ) = nn-I(K', v ' ) , no(K,V ) = no(K). Next, let us define the group structure inn,(K, v ) for each n > 0. Because of the inductive definition given above, it suffices to define a group structure in no(K'). Let a,j3 be any two components of K and pick vertices x E a and y E j3. By the definition of K', x and y are edges in K such that aG = v = a1x and a0y = v = a,y. By the extension property of K, there exists a 2-cell u in K such that a,u = x and a0u = y. Then z = a, u is a vertex in K . Prove that the component y of K' which contains z depends only on a andj3. Define a/?= y. Prove that nl(K') becomes a group under this multiplication. Prove the following assertions: 1. nn(K, v ) is abelian for each n > 2. 2. If K is the singular complex S(X) of a space X and v the vertex determined by a given point xo E X, thennn(K,v ) andnn(X,xo) are isomorphic for each n.
CHAPTER V THE C A L C U L A T I O N O F H O M O T O P Y G R O U P S
1. Introduction Neither the geometrical construction nor the axiomatic approach to the homotopy groups in the previous chapter leads to effective computation of these groups. In the present chapter, we shall study a few methods which yield successful calculations of the homotopy groups in various special cases. In the first part of the chapter, we give the celebrated Hurewicz isomorphism theorem. For every integer n > 0, there is a natural homomorphism ha of n,(X, xo) into the integral singular homology group H,(X). If 12 2 2 and if X is (n- 1)-connected, then Hurewicz's theorem states that h, is an isomorphism. Hence the first non-zero homotopy group of a triangulable space is effectively computable. In the second part of the chapter, we give the exact homotopy sequence of a fibering p : E -+ B together with a few direct sum theorems. By employing the numerous known fiberings in conjunction with this exact sequence, many homotopy groups may be computed. In the last part of the chapter, we introduce Freudenthal's suspension together with the notion of triad homotopy groups. These suspensions are crucial in the calculation of the homotopy groups of spheres, some of which will be given in the final chapter of the book.
2. Homotopy groups of the product of two spaces Let X , Y be two given spaces and xo E X , yo E Y be given points. Consider the product 2 = X x Y and the point zo = (xo, yo) in 2. Let
p
(XIxo), 4 : (ZtZO) (Y, Yo) denote the natural projections defined by p ( z ) = x and q(z) = y for each z = ( x , y) in 2. On the other hand, let : (2,zo)
+
+
i : (X, xo) + (2,z0), j : (Y, yo)
+
(2,zo)
denote the maps (called the injections) defined by i ( x ) = ( x , yo) and i ( y ) = (zo,y) for each x E X and each y E Y. Hence pi, qj are identity maps and p j , qi are constant maps. For each n > 0, the maps p , q, i, j induce the homomorphisms
p* :n,(Z,20) -+nn(X,xo), q* :n&, 20) +nn(Y, Yo). i, :n,(X. xo) +n,V, z0), j* : Y,yo) n d Z , zo), +
143
I44
V. T H E C A L C U L A T I O N O F H O M O T O P Y G R O U P S
satisfying the relations
p*i* = 1, q*j* = 1, $*j* = 0, q*i* = 0. Hence i,, j* are monomorphisms and p*, q* are epimorphisms. Let us consider the direct product of the groups n,(X, xo) and nn(Y , yo). If n > 1, these groups are abelian and their direct product is also called the direct sum. We shall use the additive notation
+
n n ( X ,xo) nn(Y,yo), ( n > O ) , for the direct product even in the case n = 1 where the groups are not necessarily abelian. Theorem 2.1. For every n
> 0 , we have
nn(z,2 0 )
M
n n ( X ,xo)
Proof. Define a homomorphism
+ nn(Y,yo).
+
h :n n ( 2 , zo) +.n.n(X, xo) nn(Y,Yo) by setting h(a) = (&(a), q*(a))for every a ~ n , ( Z zo). , It remains to prove that h is an isomorphism. For arbitrarily given a E n,(X, xo) and p E n,(Y, yo),let y = i,a j*p E n,(2,zO).Then we have
+
Nr) = (P*i*a + P*i*b q*i*a + Q*j*B) = (a,/%. Hence 12 is an epimorphism. On the other hand, let 6 € n n ( Z ,zo) be any element such that h(6) = 0. Then, by definition, we have p,S = 0 and q*6 = 0. Let f : ( I n ,81.) +. (2,z,) be any map which represents 6. Since P,S = 0 and q,S = 0, there exist two homotopies gt : (In,azn) ( X ,xo), ht : ( I n , 81.) ( Y ,Y o ) , (0 t Q l ) , such that go = $f,g,(Zn) = xo, h, = qf and h,(Zn) = yo. Define a homotopy ft : Zn -2, (0 t l ) , by taking -+
<
-+
< <
f t b ) = ( g t ( s ) , M S ) 1,
(s E In, t E 1).
Then clearly we have fo = f, fl(l")= zo and ft(8Zn) = zo for each t E I . This implies that 6 = 0. Hence h is also a monomorphism. I The inverse of h is the isomorphism h-' : n n ( X , xo)
+
+ nn(Y,yo) -+nn(Z,zo)
given by h-'(a, /I) = i,(a) i*(j?). By means of finite induction, (2.1) can be easily generalized in an obvious way to topological product of more than two factor spaces. In particular, the homotopy groups of the m-dimensional torus T,, i.e. the product Tm -- S1 x * x S1 of m copies of the unit circle S', are as follows: nl(Tm) is the free abelian group with m free generators andnn(T,) = 0 for every n > 1.
--
3. T H E
ONE -POINT U N I O N O F TWO S P A C E S
I45
3. The one-point union of two spaces Let X , Y be two given spaces and xo E X , yo E Y be given points. Consider the topological sum W = X U Y as in (I, tj 7). If we identify the point xo E X with the point yo E Y , we get a quotient space U of W with a specified point uo which is the class consisting of xo and yo. This space U will be called the one-point union of X and Y,and sometimes denoted by X V Y . Using the notations of 5 2, we can imbed U as the subspace ( X x yo) U (xo x Y )of the product space Z by means of the map
k : (up210) defined by
+
(2,zo)
(if u E X ) , (if zt E Y ) .
(u,Yo) k(u) = ( x o ,u ) , 1
Theorem 3.1. For every
n > 1, we have
nn(U, 240) = n n ( X , xo) Proof. The inclusion maps
+ nn(Y,Yo) + ma+l(Z,u,zo).
i', j' and the map k induce the homomorphisms
i', : n n ( X ,xo) +nn(U, uo), j'* : ndY, yo) + n n f U , uo) k , : n n ( U ,uO) +nn(Z,~o). Define a homomorphism 1 :n,(Z, zo) e n n (U , uo) by taking
+ j',q,(a),
l(a) = i',fi,(a)
Since obviously ki'
k,W
=
=
i and kj'
=
aEdZ,zo).
j , it follows that
k,[i',p,(a) + i',q+(a)l = ;*$,(a) j,q,(a)
+
= =
k * ~ ' , f i & - )+ k,j',q,(a) It-%(a) = a
for each a E nn(Z,zo). Since n > 1, all the groups are abelian. This implies that 1 is a monomorphism, k , is an epimorphism, and nn(U, 240) splits into the direct sum of the image of 1 and the kernel of k,. Since 1 is a monomorphism, we have
+
Image 1 M ndz, zo) M n n ( X ,xo) nn(Y,yo) by (2.1). To determine the kernel of k,, consider the homotopy sequence
"'+nn+l(U,u,) k,7zn+l(Z,zo) +n.n+l(Z,U,zJ-+nn(U,uo) a R,+Xn(Z,Zo)+ of the triplet (2,U , zo). Since k , is an epimorphism, it follows from the exactness of the sequence that d is a monomorphism. Hence we have Kernel k ,
=
Image d
M
n,+,(Z, U , zo). I
In fact, nn(U , uo)decomposes into the direct sum of the images of the three monomorphisms i', if, and d. The condition n > 1 is used only once, namely, to assure that the image of 1 is a normal subgroup of nn(U , uo).In general, this is not true for the case
146
V. T H E C A L C U L A T I O N O F H O M O T O P Y G R O U P S
n = 1. However, we have anyway the weaker assertion that n l ( U , uo) is an extension of n,(Z, U , zo) by the direct product of n l ( X , xo) and nl(Y,yo). If X and Y are polyhedra, we recall that n l ( U , uo) is the free product of n l ( X , xo) and n,(Y, yo) by (11; Ex. A4). This is also true if X and Y are regular and locally simply connected. As an application of (3.1), we have the following Proposition 3.2 For every
p > 0, q > 0
and n
+ q - 1. we have
+ nn(S9). Proof. Since ( S p x SQ, S W S 9 ) is a relative ( p + &ell nn(SPVSg) w nn(SP)
in the sense of
(I; 5 7),it follows from (IV; 3.4) that %(SP
+
x SQ,S W S 9 )
=0
for every m < p q. Hence (3.2) follows from (3.1) immediately. I For example, we have
n,(SWP) = 0. n*(SWS3) = 2, n,(SVS3)
=2
+ 2.
4. The natural homomorphisms from homotopy groups t o homology groups Let ( X ,A , xo) be any given triplet and consider the homotopy set
n,,(X,A,x,), n
> 1.
Let a be an arbitrary element of n , ( X , A , xo) and choose a map
+ : (En,
9-1,
so) -+ ( X , A , xo)
which represents a, where En denotes the unit n-cell in the euclidean n-space Rn, Sn-l the unit (n - 1)-sphere in Rn, and so = (1,0, 0). The natural coordinate system in Rn determines an orientation of Rn and hence a generator &, of the free cyclic homology group Hn(En, Sn-l). As a map of (En, Sn-') into ( X ,A ) ,4 induces a homomorphism
--
+* : Hn(En,Sn-')
a ,
H n ( X ,A ) where H,(X, A ) denotes the singular homology group with integral coefficients. According to the homotopy axiom of homology theory, I$+ depends only on the given element a E n,(X, A , a ) .Hence the assignment a --* I$+([,) defines a transformation xn nn(X,A , ~
-+
0+ )
Hn(X, A ) .
Proposition 4.1. If either n > 1 or A = xo, then x,, is a homomorphism which will be called the natural homomwfihism of n,(X, A , xo) into H n ( X , A ) . Proof. Let a and /Ibe any two elements of n,(X, A , xo). Then we may choose representative maps y : (En,Sn-l, so) + ( X ,A , xo) such that
+,
4. T H E +(tl; * -,tn)
=
NATURAL HOMOMORPHISMS
> 0 and y(t,; * -,t n ) ( X , A , xo) by taking
xo if t ,
X : (En, Sn-l, so)
-+
=
xo if
< 0.
t,
I47 Define a map
for an arbitrary point ( t l ; . * , tn) in E n . Then X clearly represents a By a theorem in homology theory, [E-S; p. 361, we have
+ @.
+
&(En) = + *( E n ) Y*(ln). This implies xn(a @) = xn(a) + xn(p) and hence xn is a homomorphism. I The following proposition is obvious from the above construction of Xn.
+
Proposition 4.2. For any map f : ( X ,A , xo) rectangle is commutative :
nn(X, A , ~
0
Hn(X,A)
For the case A
=
nn( Y
+ . f )
*'
-+
-+
( Y ,B , y o ) , the following
, B. yo)
H n ( Y , B)
xo, we have an isomorphism
j * : H n ( X ) % H n ( X ,xo) and hence we obtain a homomorphism hn
-
.-
~ + ' x n: n n ( X ,xo)
+
Hn(X)
which will be called the natural homomorphism of n n ( X , xo) into H n ( X ) . Clearly, h , coincides with the homomorphism h, of (11; 5 6 ) . Sinced : H,,+,(Enfl, Sn) M Hn(Sn),it follows that q n = a[,+, is agenerator of the free cyclic group Hn(Sn). If we identify the boundary S n - l of En to a single point so, we obtain an n-sphere S n and a point so. Hence there is a relative homeomorphism
p such that
p*([n) =
fi,
: (En,9 - l ) -+ (Sn, so)
j * ( V n ) , where
: H n ( E n , Sn-l) M H n ( S n , SO), j * : Hn(Sn)
are the isoniorphisnis induced by the map (Sn,
3.
p
H n ( S n , SO)
and the inclusion map j : S n c
+
Let a E n n ( X , xo) and pick a map : (En, 9 - l ) + ( X , xo) which represents a. Then there exists a unique map y : (9, so) ( X , xo) such that = y p . We may consider a as represented by y . As a map of S n into X , y induces a homomorphism y + : H n ( W -+ H n ( X ) . --f
+
Then clearly we have hn(a) = y * ( q n ) . This may be used as the definition of the natural homomorphism h,.
V. T H E C A L C U L A T I O N O F H O M O T O P Y G R O U P S
148
Then, the following proposition is obvious. Proposition 4.3.
commutative:
-
For any triplet ( X ,A , xo), the following rectangle is
nn+i(XI A , ~
bn
Hn+i(X,A )
a
0 )
a
nn(A, xo)
-+ H n ( A ) .
Now consider the following homotopy-homology ladder: *
*
Lnn+l(X,A , ~ 0 L ) nn(A, x O ) -5nn(X, ~ 0 L ) nn(X, A , xo) --+a
1.
I n
Hn+l(X, A ) 2+ Hn(A)
I n
I+ H n ( X ) L
P
Hn(X, A )
* * *
5- * * .
By (4.3), the rectangle on the left is commutative. By (4.2), the middle rectangle and the one on the right are also commutative. Hence the whole ladder is commutative. The remainder of this section is devoted to the proof of the Hurewicz theorem for polyhedra. Generalizations will be given in the exercises at the end of the chapter and also in a later chapter. In (11; § 9), we defined the notion of the n-connected spaces. In terms of homotopy groups, one can easily prove that, for a given integer n 2 0 , a space X is n-connected iff it is pathwise connected and n,,(X) = 0 for every m < n. Theorem 4.4. (Hurewicz theorem). If X is an (n - 1)-connected finite simplicical complex with n > 1, then the natural homomorphism hn is an isomorphism. Proof. If X is the n-sphere Sn, then the theorem is given already by the Hopf theorem in (11; 3 8). By (3.2), it follows easily that the theorem holds for the case X = SnVSn. Then, by means of (3.1) and finite induction, one can easily prove the theorem for the case where X is the one-point union of a finite number of n-spheres. Next, assume that d i m X Q r t . If we identify the ( n - 1)-dimensional skeleton Xn-l of X to a single point yo, we obtain a quotient space Y with a natural projection p : ( X , X"-') ( Y ,Yo). +
If Y is different from yo, then it is obviously homeomorphic to the one-point union of a finite number of n-spheres with yo as the common point. Pick a vertex X ~ Xn-1. E Since X is (n - 1)-connected, Xn-l must be contractible to the point xo in X . By an application of the homotopy extension property, it follows that there exists a homotopy f t : X X , (0 t l ) , such that f,, is the identity map, fl(Xn-l) = xo, and f t ( x o ) = xo for each t E I . By (I; 6.1), the map /, determines a map q : ( Y ,yo) ( X , xo) such that the
-.
--+
<<
4. T H E N A T U R A L H O M O M O R P H I S M S
I49
composed map qp =f l is homotopic to the identity map on X relative to xo. , Now, consider the following diagram
where h,, k , denote the natural homomorphisms, p,, p# are induced by the natural projection p , and q,, q# are induced by the map q. By what we have proved above, k , is an isomorphism and q, p,, q#p# are identities. By (4.2) we have p#k, = k,p, and hnq, = q#k,. These imply that k, is an isomorphism. Indeed, let a € n n ( X ,xo) be such that kn(a) = 0. Then we have
k,p,(a)
=
p#hda)
=
P#(O)
=
0.
Since k , and p , are monomorphisms, this implies a = 0. Hence k, is a monomorphism. On the other hand, let P E H , ( X ) . Let b: = q,k,'P#(B). Then we have hn(4 = knq*kn'P#(B) = Q#knk,lP#(P) = 4#P#(B) = 8. Hence h, is also an epimorphism. This proves the theorem for the case dim X < n. Now, assume that dim X = n + 1. Then the natural homomorphism %?&+I
:nn+1(X,xn, xo)
-+
H,+l(X, X.)
=
C,+,(X)
is an epimorphism. Indeed, C,+,(X) is the free abelian group generated by the ( ~ + t 1)-simplexesof X . Let s be any (n 1)-simplex of X.Pick a vertex x1 of s. Since d s c Xn, the inclusion map s c X represents an element a of nn+l(X,Xn, xl). Since X n is pathwise connected, there exists a path u : I -+ X n joining xo to xl. Then clearly sends the element o,(a) of Z , + ~ ( X X, n , xo) into the generator s of C,+,(X). Since s is arbitrary, this proves that is an epimorphism. By ( I V ; 3.4),we have n n ( X , X n , xo) = 0. On the other hand, we have Hn(X,X n ) = 0. Then, in the homotopy-homology ladder of ( X , X n , xo), we have the following diagram
+
j".+,
a
-1".
H n + , ( X , X f i ) --+ Hn(X,) where
3-t
Pn
H n ( X ) A+ 0
is an epimorphism and k , is an isomorphism. These imply that
k, is an isomorphism. In fact, let P E H,(X). Since i, carries Hn(Xn) onto H,(X), there exists an element y E H , ( X ~ )such that i,(y) = p. Let a = i,k;I(y) g n , ( X , xo). Then, we have hn(a) = hni,k,'(y)
=
i,K,k,l(y)
=
i,(y)
=
8.
150
V. T H E C A L C U L A T I O N O F HOMOTOPY G R O U P S
Hence A, is an epimorphism. On the other hand, let LT. enn(X,xo) be any element such that h ( a ) = 0. Choose an element @ ~ n ,(X n ,xo) with i*(p) = a. Since i*k,(@) = hi*@)= h,(a) = 0, it follows from the exactness that there exists a y E H,+l(X, Xn) with a ( y ) = A,(/?). Since x,+~ is an epimorphism, there is a BEZ,+,(X, Xn, xo) with X,+~(B) = y . Then, we have
i$(8)
= i*k,’kna(d)
= i,k;’a(y)
= i,k;;l&n+1(8)
= i*k;lk,,(b) = ;*(/I) = a.
Hence a = i,a(B) = 0. This proves that h, is also a monomorphism. So, the theorem is proved for the case dim X = n 1. Note that the argument used in this paragraph is a special case of the “five” lemma, [E-S;p. 161. Finally, let X be any (n- l)-connected finite simplicial complex. Consider the (n + 1)-dimensional skeleton X,+l of X and the following diagram
+
*,(XR+’, xo) 3+ n,(X, xo)
kn
H,(X,+1)
k n
i#
.
,Ha($
where the natural homomorphism k, is proved to be an isomorphism. By (IV; 3.4), it follows that i, is an isomorphism. On the other hand, i# is also an isomorphism. Hence, h, = i#k,i;l is an isomorphism. I
5. Direct sum theorems In the present section, we shall derive three useful consequences of the exactness of the homotopy sequence. Proposition 5.1. If A
is a retract of X and x O € A , then %(X, xo) w nn(A, xo) %(X, A , xo) for every n > 2 and i, :n,(A, xo) +nn(X, xo) is a monomrfihism for every n > 1. Proof. Let r : X D A be a retraction. Since r i is the identity map on A,
+
it follows that r,i, is the identity automorphism on n,(A, xo) for every
n 2 1. This implies that i* is a monomorphism and r, is an epimorphism
for every n > 1. If n > 2, then n,(X, xo) is abelian. Hence it follows from r,i, nn(X, x,,) decomposes into the direct sum z,(X, xo) = J + K , J = Image i,, K = Kernel r*.
=
1 that
Since i, is a monomorphism, we have J w n,(A, xo). Further, by the exactness of the homotopy sequence of (X, A, xo), it is easy to deduce that j* :n,(X, xo) +n,(X, A, xo) is an epimorphism for every n > 2. By exactness, the kernel of j, is J and so j* sends K isomorphically onton,(X,A, xo). Hence K w n,(X, A, xo). I
5. D I R E C T SUM T H E O R E M S
151
Consequently, if A is a retract of XI then n , ( X , A , x,) must be abelian. Proposition 5.2.
If X is deformable into A relative to a point
+
X,E
A , then
nn(A,xo) !+i nn.(X,xo) nn+l(X,A xo) for every n > 2 and i, : n , ( A , xo) + n f l ( X ,xo) is an epimorphism for every n > 1. : X -t X , Proof. According to the hypothesis, there exists a homotopy (0 < t Q l ) , such that
h,(x) = x , h&) € A , 4 x 0 ) = xo for each x E X and t E I. Define a map h : ( X , x,) + ( A , x,) by taking h(x) = h,(x) for each x E X . Since ih = h, N It, re1 xo, it follows that i,h, is the identity automorphism on n,(X, xo). Hence h, is a monomorphism and i, is an epimorphism for every n > 1. If n > 2, then nn(A,xo) is abelian. Hence it follows from i,h* = 1 that n f l ( A xo) , decomposes into the direct sum nn(A,x,) = J + K , J = Image h,, K = Kernel i,. Since h, is a monomorphism, we have J * n n ( X ,x,). On the other hand, since i, is an epimorphism for every n > 1, it follows from the exactness that d :nn+,(X, A , x,) + n f l ( A ,xo) is a monomorphism. Hence K = Kerneli, = Imaged W : ~ + ~ (AX, ,xo). I For the case n = l , n , ( A , x,) is an extension of n , ( X , A , x,) by n , ( X , xo) if X is deformable into A relative to xo. Proposition 5.3.
If A is contractible i n X relative to a point x O € A , then
+
n n ( X , A , xo) n n ( X ,xo) n n - ~ ( AX O ) for every n > 3 and i, sends n f l ( A xo) , into the neutral element of n n ( X l x,) for every n > 1. Proof. By hypothesis, there exists a homotopy ht : A -+ X , (0 < t Q l), such that h, = i, h,(A) = xo and ht(x,) = xo for each t E I. This implies that i, = 0 for every n 1. Now let n > 2. By the exactness of the homotopy sequence, i, = 0 implies that i., is a monomorphism and a is an epimorphism. This proves A, that n n ( X ,A , xo) is an extension of n n ( X ,xo) by Z ~ - ~ (xo). By means of the contraction ht, we define a homomorphism h, :n,-,(A ,xo) +nfl(X,A,x,)for each n > 2 as follows. Let aEnfl-,(A,xo)be represented by a map f : (Zn-1, d1n-I) -+ ( A , x,). Define a map g : (In,In-,, J n - I ) + ( X ,A , x,) by taking g(t,; * * , tn) = ht, f ( t 1 , . * * , t f l - 1 ) .
Then h,(a) is defined to be the element represented by g. Since g 1 In-,= f , it follows that ah.,, is the identity automorphism on n n - , ( A , xo). Hence h, is a monomorphism for every n > 2.
V. T H E C A L C U L A T I O N O F H O M O T O P Y G R O U P S
1.52
If n
> 3,
then n n ( X ,A , x,) is abelian. Hence, d h ,
=
1 implies that
nn(X, A , x,) decomposes into the direct sum
n n ( X ,A , x,) = J + K , J = Image h,, K = Kernel d. Since h, is a monomorphism, J ~ Z ~ - ~x d(. On A , the other hand, we have K
=
Kernel a
=
Image i, m n n ( X ,x,)
since i, is a monomorphism. I For the case n = 2, the last argument of the preceding proof breaks since n,(X, A , x,) is in general non-abelian. However, it can be proved that n 2 ( X ,A , xo) is isomorphic to the direct product of n , ( X , x,) and n,(A, x,). The assumed relativity with respect to x, in (5.2) and (5.3) is convenient but not essential; for proofs without this assumption see [Hu 11.
6. Homotopy groups of fiber spaces Let E be a fiber space over a base space B with projection p : E + B. Choose a basic point bog B such that the fiber F = p-l(b,) is not empty. CallFthe basic fiber and choose a basic point e,EF.Thus, we obtain a triplet ( E ,F , eo). Since p(F) = b,, the projection p : ( E ,e,) +. (B, b,) defines a map q : ( E ,F , e,) -+ (B, b,) and p = qj, where j : ( E , e,) c IE, F , e,) denotes the inclusion map. According to (IV, 5 9), q, sends nn(E,F , e,) onto nn(B,b,) in a one-to-one fashion for each n > 1. Let
d,
= dq;l
:nn(B,b,) - P ~ ~ - ~e,),( F (, n > 1).
Since p , = q,j,, we can construct from the homotopy sequence of the triplet ( E ,F , e,) an exact sequence A+ Z*+~(B, b,) %nn(F,e,) 2% nn(E , e,) 111, nn(B,b0) ---+ d* ..* *
--.
& nl(B,b,) % n,(F, e,) n o ( E ,e,) which is called the homotopy sequence of the fibering p : E B based at e,. --f
Proposition 6.1. If the basic fiber F is totally pathwise disconnected, then
p, and
p,
:n n ( E ,e,) w n d B , b,), n
i s a monomorphism if n
=
> 2,
1.
Proof. If F is totally pathwise disconnected, then nn(F,e,) = 0 for every n > 1. Hence the proposition is an immediate consequence of the exactness of the homotopy sequence of the fibering fi : E -+ B. I
Proposition 6.2. If the fibering fi : E +. B admits a cross-section X : B then, for every b,E B and e, = X(b,) E F = p-l(b,), we have
+
for each n
nn(E,e,) w nn(B,b,) nn(F,e,) > 2 and p , i s a n epimorphism for every n > 1.
+.
E,
6. H O M O T O P Y G R O U P S O F F I B E R
SPACES
I53
Proof. Since PX is the identity map on B, it follows that p,X, is the identity automorphism on n,(B, b,). Hence, X, is a monomorphism and $* is an epimorphism for every n > 1. If n > 2, then n,&(E,e,) is abelian. Then, p,X, 1 implies that n,(E, e,) decomposes into the direct sum F
+ K,
J = Image X,, K = Kernel p,. Since X , is a monomorphism, we have J w n,(B, b,). On the other hand, since p , is an epimorphism for every n > 1, it follows from an exactness argument that i, is a monomorphism for every n > 1. Hence
nn(E,e,)
K
=J
=
Kernel fi,
=
Image i, w n,(F, e,). I
If n > 1, then (6.2) is a generalization of (2.1). As immediate consequences of (5.1)-(5.3), we have the following three propositions concerning the homotopy sequence of a fibering p : E -, B. Proposition 6.3. If F is a retract of E , then
+
nt@, e,) w n,(B, b,) n d F , 8,) for every n > 2 and p , is a n epimorphism for every n
into F , then
Proposition 6.4. If E is deformable
+ nn+l(B,b,)
nn(F,e,) w nn(E,e,)
for every n
.>2 and
p,
=0
for every n
Proposition 6.5. If F is contractible
> 1.
> 1.
in E , then
+
n,(B, b,) w nn(E,e,) Z ~ - ~ (8,)F , for every n > 2 and p , i s a monomorphism for every n > 1. As an application, let us consider the Hopf fiberings
p
( m = 2,4,8). In each of these fibering, the fiber F is an (m- 1)-sphere S m - l and is contractible in 9 m - l . Hence, by (6.5), we have (6.6) for every m (6.7) When m
=
: s2m-1+ s m ,
n,(Sm) w nn(SZrn-1) + n,-,(Sm-1)
=
> 2. When m = 2, this gives nn(S2)wn,(S3), n > 3.
2 , 4 , 8 and every n
4 or 8, we have
n,(S4) w 2 n15(s8) w
+ n,(S3).
+
n14(s7).
As another application, let us consider the fibering fi : S + A4 in (111;
V. T H E C A L C U L A T I O N O F H O M O T O P Y G R O U P S
154
Ex. F),where S denotes the unit sphere in the space Cn of n complex variables and M the projective space associated with 0. Theti S is an ( 2 n - 1)-sphere and the fiber F is a 1-sphere. If n = 1, M consists of a single point. For the case n > 1, i t follows immediately from the exactness of the homotopy sequence of the fibering p :S --* M that
n,(M) = 0, n,(W M 2, nm(M)w%(S*n-l), m > 2.
7. Homotopy groups of covering spaces As a special case of (6.1), we have the following
a base space B relative to a (B,b,), then p* nn(E, eJ w nn(B,bo), n Z 2, and p* i s a monomorphism if n = 1. In particular, if E is the universal covering space over B, then we have n,(E) = 0, nn(E) m ~ n ( B ) n, 2 2. Therefore, by Hurewicz theorem (4.4 and Ex. C), we have the following Proposition 7.1. If E i s a covering space over
projection
p
: ( E ,e,)
-+
Proposition 7.2. For every connected, locally pathwise connected and semilocally simply connected space B, the second homotopy groupn,(B) i s isomorphic to the second singular homology group H , ( E ) of its universal covering space E. Applying (7.1) to the universal covering p : S m + F of the m-sphere onto the real projective space Pm,we have nn(P”) mnn(Sm), n 2 2.
For another example, since the euclidean 2-space is the universal covering space of any closed surface M r+iier than S2 and Pa,we have n n ( M ) = 0 , n > 2. Now let B be a given space which is connected, locally pathwise connected, and semi-locally simply connected. Let b, E B and n > 1. By ( I V ; 15.1), q ( B ,b,) acts on n,(B, b,) as a group of operators. Using the isomorphisms p , for the universal covering p : E += B and the covering transformations of E , these operators can be interpreted nicely as follows. Ler w E n,( B ,b,). By (111; 16.6), w determines a covering transformation of E which will be denoted by h : E += E for the moment. Pick e, E E with p(e,) = b, and let el = hfe,). Then $(el) = b,. By the construction of h and that of the operator w on nn(B,b,), one can prove that commutativity holds in the following diagram h nn(E,e,) --!+ nn(E, el)
k*
k*
nn(B,b,) . W n n ( B , b,)
7. H O M O T O P Y
GROUPS O F COVERING SPACES
I55
Since h, and p , are isomorphisms, we have
(7.3)
w
=
fi*hz;lp;l.
For example, let B denote the one-point union SnvS1 obtained by identifying so E Sn with 1 E S1,where n > 2. Let b, denote the identified point of so and 1. By (11; Ex. A), we get
nl(B, bo)
M
z
with a generator w represented by the exponential map p : I + S1c B of (11; tj 2). In the product space Sn x R of Sn and the real line R, let us conwhere Z denotes the subspace of R consisting of all integers. This space E is pathwise connected and nm(E) = 0 for every m < n. In particular, since n > 1, E is simply connected and hence n-simple. The n-th homotopy group nn(E) as interpreted in (IV; 4 16) is a free abelian group with a countable set { a( I i E 2 } of free generators, where a( is represented by the map f i : Sn -+ E which is defined as follows: fr(s) =
( s , i ) E S x~ Z C E , ( s E S ~ ) .
For a proof of this result, see Ex. C a t the end of the chapter. Define a map q : E + B by taking
where p : R -+S1 denotes the exponential map of (11; 5 2). One can easily verify that E is a covering space over B relative to q. Hence, n,,,(B, b,) = 0 for each m with 1 < m < n and nn(B,b,) is a free abelian group with a countable set { ,!It I i E Z } of free generators, where ,!It corresponds to at in the natural way. The generator w of nl(B,b,) determines a covering transformation h : E -+ E given by h(s,t ) = (s, t + 1) for every point (s, t ) of E. It follows that h,at = at+1 for each i E Z and hence w/& = bt-l for each i € 2 .This determines the operations of n,(B, b,) on nn(B,b,). Note that B is not n-simple.
8. The n-connective fiberings The interesting property (7.1) of the universal covering space suggests the notion of n-connective fiber spaces as follows. A fibering p : E + B over a given pathwise connected space B is said to be n-connective if E is n-connected and p* :%(E, e,) w nm(B,b,), m > n, where e, E E and +(e,)
= b,,
and hence
156
V. T H E C A L C U L A T I O N O F H O M O T O P Y G R O U P S
Thus, the identity map on B is a 0-connective fibering and the universal covering over B , if i t exists, is a 1-connective fibering. If p : E --f B is an n-connective fibering, then E is said to be an n-connective fiber space over B. The main objective of the present section is to construct an n-connective fiber space over a given pathwise connected space B. Lemma8.1. If B i s a given space, b, a given point in B, and n a given positive integer, then there exists a space X which satisfies the following three conditions : (i) X contains B as a closed subspace. (ii) n n ( X ,b,) = 0. (iii) T h e inclusion m a p i : B c X induces a n isomorphism
i, :nm(B,b,) for every m satisfying 0
= %(X,b,)
< m < n.
Proof. Let A be a set of elements of nn(B,b,) which generates nn(B,b,) and consider A as a space with discrete topology. Let E n f l denote the ( n 1)-cell bounded by the unit n-sphere Sn in the euclidean ( n 1)space. Let W = En+' x A , C = S" x A c W .
+
+
Let so = (1,O;. * , 0) E S ~For . each&€A , choose amapg, : (Sn, so) --f (B,b,) which represents a. Then, define a map g : C --f B by taking g(s, a) = g a b ) ,
(s,
a) E c.
LetXdenote the adjunction space obtained by adjoining W to B by means of the partial map g : C --f B. It remains to verify the conditions (i)-(5). Since C is closed in W , (i) is obvious from the definition of adjunction space in (I; 5 7). To verify (ii) and (iii), let us first prove that
n m ( X ,B , b,)
= 0,
0
<m
Q n.
For this purpose, let 5 be an arbitrary element of %(X,B, b,) and pick a map f : ( I m , P - l , Jm-') + ( X ,B, b,) which represents 5. Since m < n, it follows from the method of simplicia1 approximation that, by a suitable homotopy of f , we can free an interior point e, of En+l x a from the image of f for every ~ E A Hence . there exists a homotopy ft
: ( I m , Irn-l,Jm-l) -+ ( X , B, b,),
(0 < t
< 1).
such that f o = f and f l ( I m ) c B. By (IV; 3.3), this implies that E = 0. Hence nm(X, B, b,) = 0 for every m with 0 < m < n. From the exactness of the homotopy sequence, we deduce immediately that i, nm(B, b,) - ; . ~ m ( Xb,),
8.
T H E n-CONNECTIVE FIBERINGS
1.57
-
is an isomorphism for m = 1,2; -,n - 1 and is an epimorphism for m = n. This verifies (iii). Since g, represents a, i t is clear that &(a) = 0. Since A generates nn(B,b,), it follows that i, sends every element of n,(B,b,) into 0. Since i, is an epimorphism for m = n, this implies that nn(X, b,) = 0. This verifies (ii). I Lemma 0.2. If B i s a given space, b, a given point in B, and n a given nonnegative integer, then there exists a space X which satisfies the following three conditions : (i) X contains B as a closed subspace. (ii) nm(X,b,) = 0 fm every m > n. (iii) T h e inclusion m a p i : B c X induces a n isomorphism
i* :nm(B, bo) for every m satisfying 0 < m Q n.
M
nm(X, bo)
Proof. By recurrent application of the construction given in the proof of (8.l), we obtain a sequence of spaces X,, X,, * * , Xr, * * * such that (1) x, = B , (2) Xr contains Xr-l as a closed subspace for every r > 0, (3) nr+r(Xr,b,) = 0 for every r > 0, and (4) the inclusion map ir : Xr-l c Xr induces an isomorphism (ir)* : %(Xr-1, b,) wnm(Xr,b,) for every m satisfying 0 < m < n + r. As a set, let X denote the union of Xr for all Y > 0. Introduce a topology in X as follows: a set U of X is said to be open iff U n Xr is open in Xr for every r > 0. Thus, we obtain a space X. It remains to verify the conditions (i)- (iii) By (2) and the definition of the topology in X, i t follows that Xr is a closed subspace of X for every r > 0. In particular, (i) is true since X, = B. Next, let us prove that the inclusion map jr : XrC X induces an isomorphism (jr)* :nm(Xr, bo) M nm(X, bo)
-
.
for every m such that 0 < m < n + r. To prove that (jr)*is an epimorphism, let a en,(X, b,) and choose a representative map f : (Im,aim) + (X, b,) for a. Since f ( P ) is compact, there exists an integer q > 0 such that f (P) c Xr+p Hence f also represents an element p of nm(Xr+q,b,) with (jr+q)*(p) = a. According to (4),the inclusion k : XrC Xr+q induces an isomorphism k , :nm(Xr,b,) Mnm(Xr+g,b,). Let y = k;'(p). Then (jr)*(y) = (jr+q)*k*(y)= (&+q)*(p)= a . So, (jr)* is an epimorphism. To prove that (jr)* is a monomorphism, let 6en,(Xr, b,) be an element such that (jr)*(t) = 0. Choose a map g : ( I m ,a r m ) -+ (Xr, b,) representing 6. Then (jr),(l) = 0 implies that there exists a homotopy gt : (Im,dim) -+ (X, b,), 0 < t < 1, such that go = g and g,(Im) = b,. This homotopy gt defines a map G : Im+l+X. Since G(Im+l) is compact, there is an integer q > O such that G(Im+l)c Xr+q.Hence k,(l) = 0, where k : X r C Xr+q. Since k ,
158
V. T H E C A L C U L A T I O N O F H O M O T O P Y G R O U P S
is an isomorphism by (4), this implies that [ = 0. So, (ir),is also a monomorphism. This completes the proof that (jr)* is an isomorphism. By means of this, (ii) and (iii) follow readily from (3) and (4). I Theorem 8.3. For any given pathm'se connected space B and a n y nonnegative integer n, there exists a n n-connective fibering p : E --f B. Proof. Let b, E B and choose a space X satisfying the conditions (i)-(iii) of (8.2). Thus we obtain a triplet ( X , B , b,). Let ( X ' , B', b',) denote the derived triplet of ( X ,B , b,) and q : ( X ' , B', b',) +. ( X ,B, b,) the derived projection as defined in (IV; 9 3). Then x'is a fiber space over X and B' is a fiber space over B relative to q. Let F = q-l(b,), The homotopy sequences of the fiberings are connected by the following commutative diagram
where i,, i', j*, k , are induced by inclusion maps, q,, q', are induced by q, and d,, d', are the boundary operations. Now, since X' is contractible, we have nm(X',b',) = 0 for every m 2 0. Therefore, d', carries n m ( X ,b,) onto nm-I(F, b',) in a one-to-one fashion for every m 1. If m > n, then we have nm(F, b'o) nnt+l(X,bo) = 0, nm-l(F, b'o) M n m ( X , bo) = 0. Hence, q, is an isomorphism for every m > n. On the other hand, if m < n, then k , is an isomorphism and hence d , = d',k, carries nm(B, b,) onto Z ~ - ~ (b',) F , in a one-to-one fashion. Hence, =(B', b',) = O for every m satisfying 0 < m < n. Finally, sincenn(F,a',) M n n + l ( X b,) , = 0 , we obtain zn(B', b',) = 0. Now, let E denote the path-component of B' containing the point e, = b', and let p = q 1 E. In other words, E denotes the path-component of the space of paths [ X ;B, b,] which contains the degenerate path e,(I) = b,, and p :E +.B denotes the initial projection. Then, it follows from what we have proved for q : B' +. B that p : E +. B is a fibering, that E is n-connected, and that p , :nm(E,e,) M nm(B, b,) for every m > n. Hence p : E +. B is n-connective. I Corollary 8.4. If B i s a n ( n - 1)-connected space and p : E B is an n-connective fibering, then the fiber F = p-'(b,) over any given point b, E B has the property: nn-I(F, 80) M nn(B, bo), nm(F,e,) = 0 , m # n - 1, where e, i s any point of F . --f
9.
T H E HOMOTOPY S E Q U E N C E O F A T R I P L E
I59
This corollary is an easy consequence of the homotopy sequence of the fibering p : E + B at the basic point e,.
9. The homotopy sequence of a triple By a triple ( X ,A , B ) we mean a space X together with two nonvoid subspaces A and B such that A 2 B . Hence a triplet is a special case of triple where B consists of a single point. Pick a basic point x, E B . In the homotopy system { n,3, }, the inclusion maps
,
i : IA, B , x,) c ( X , B , x,), 7 : -( X ,B , x,) - c ( X ,A , x,) give rise to the induced transformations i, and j,. On the other hand, the inclusion map k : ( A , xo) C ( A ,B , xo) has the induced transformations k,. Define a boundary operator -
a : nn(X,A , x,)
+nn-1(A, B , xo)
for every n > 0 by taking the composition
a
=
k,a of
nn(X,A , xo) nn-,(A, xg) I-+ nn-,(A, B , KO). Thus we obtain a beginningless sequence
-
*
a *L nn+l(X,A,x,)Lnn(A,B,xo)A+ nn(X,B,xo)LnnIX,A,xo) -+. *
*
* *
?+n,(A,B,x,) A+ n,(X, B , x,) i‘,n,(X,A,x,).
This will be called the homotopy sequence of the triple ( X ,A , B ) a t the basic point x,. Theorem 9.1. The homotopy sequence of a triple is exact. Proof.
Consider the spaces of paths
X‘ = [ X ;X , x O ] , A’ = [ X ; A , x J , B’ = [ X i B , x , ] with x‘, denoting the degenerate path %’,(I) = x,. Then the initial projection p : X’ -+ X defines a fibering q : (A‘,B‘,do)+ ( A , B , x,).
We shall also make use of the inclusion maps i : (B‘,x‘,) c (A’, x‘,) and j : (A‘,x‘,) c (A’,B’, do).Consider the following diagram .i b ~ , , ( A ’ , X -+ ‘~)
ix
a
A , ( A ‘ , B ’ , ~ ’ ~-+ ) z ~ - ~ ( B ’ . x ’ ~ ) n,-l(A’,x’o)
B n,+,(X,A,xo)-+
I
T.
lx
a A+ .. * -+
-
4 qr ., n n ( A , B , x o ) ~ + n , ( X , B , n O ) l ’ + ~ , ( X , A , x ~ ) - * . .“+z 7
no(B‘,xt0)
T.
l(X,B,xo)
no(A’,X’O)
I
ix
n,(X,A,xo)
As the homotopy sequence of the triplet (A’, B’, do),the top row is exact. By (IV; 9 9), X and q, are one-to-one and onto. One can easily verify that
160
V. T H E C A L C U L A T I O N O F HOMOTOPY G R O U P S
commutativity holds in each of the rectangles. These imply that the bottom is exact. I Since the part of the homotopy sequence of ( X ,A , B) which ends with n z ( X ,A , x,) consists of groups and homomorphisms, the exactness of this can also be derived from the algebraic axioms and the exactness of the homotopy sequences of triplets. See [S; p. 771 and [E-S; pp. 24-28]. The proof given above is considerably simplified by the fibering axiom which does not hold in homology theory.
10. The homotopy groups of a triad We recall that a triad ( X ;A , B ) consists of a space X together with two subspaces A and B such that the intersection C = A n B is not empty. Pick a basic point x , E C . We shall define the homotopy groups of the triad ( X ;A , B ) as follows. Let us consider the spaces of paths
P = [ X ;B, 4,Q = [ A ; C, ~ 0 1 and denote by a, the degenerate path a,(I) = xo. Since A c X and C c B, it follows that a, E Q c P and hence we obtain a triplet (P, Q, 0,). According to (IV; 3 9), we have the natural correspondences X : n n ( X , B , x o )wn"n-l(P,0,), X:nn(A,C,xo) w ~ n - i ( Q , a o ) for every n
> 0. Furthermore, we define for every n > 1
n n ( X ;A , B , xo) =nn-l(P, Q, 0 0 ) . If n > 3, then n n ( X ;A , B , xo) is a group which is abelian in case n > 3. This group is called the n-th homotopy group of the triad ( X ;A , B ) a t the basic point x,. For completeness, we shall also call n,(X; A , B , xo), n > 2, the n-th homotopy set of ( X ;A , B ) at x,. Thus, the homotopy sequence of the triplet ( P ,Q, a,) gives rise to the following exact sequence * * Ln,+l(X;A,B,xo) Lnn(A,C,xo)-Lnn(X,B,x,) L z ~ ( X ; A , B , X O )a- - +
...-Lnz(X;A,B,xo)-Ln,(A, C, xo) -Ln , ( X , B , xo) which will be called the first homotopy sequence of the triad ( X ;A , B ) at the basic point x,. The homomorphisms
i, :nn(A, C, xo) + n n ( X , B , x,), n
> 2,
are called the excision homomorphisms, [E-S; p. 2071. Thus the triad homotopy sets n n ( X ;A , B , x,), n 2 2, measure the extent by which theexcision axiom fails for the relative homotopy groups. The above definition of nn(X; A , B, x,), n 2, gives a geometrical representation as follows. An element of n,(X; A , B , xo) is represented by a
10. T H E H O M O T O P Y G R O U P S O F A T R I A D
161
-
map f : I n + X such that, for any point t = (t1; * , tn) in the boundary aIn of I n , f (t)E A if t,-l = 0, f (t)E B if tn = 0, and f (t) = xo otherwise. Precisely, the elements of n n ( X ;A , B , xo) are the path-components of the space of these maps. See [Blakers and Massey 11. Consider the homeomorphism h : I n -+ I n defined by
h(t1,. * tn-2, in-1, in) = (ti,. * * , tn-2, tn, tn-1). Then the assignment f + f k induces an isomorphism e t
h* : n n ( X ;A , B , ~ 0 M) n n ( X ;B , A , ~ 0 ) for every n > 2 and a one-to-one correspondence for n = 2. Hence the first homotopy sequence of ( X ;B , A ) a t xo gives rise to the following exact sequence * *
* L n n + 1 ( X; A,B,xo)a'+,n(B,C,xo)i~~n(X,A,xo)~-+nn(X ;A,B,xo)a'-+ * * * * *
G+ n 2 ( X ;A , B , x0) X+ n,(B, C, x0) .G.+ n l ( X ,A , x0)
which will be called the second homotopy sequence of the triad ( X ;A , B ) at the basic point xo. Another geometrical representation of n n ( X ;A , B, xo), n > 2, can be described as follows. Consider the unit n-cell En in the euclidean n-space Rn and its boundary (n- 1)-sphere Sn-l. Let E?-l and E71-I denote the hemispheres of 9 - l defined by tn > O and tn GO respectively. Let so = (1,O; * * , 0). Then the elements of n n ( X ;A , B, xo) are the path-components of the space of maps f : ( E n ;E:, EE,so) -+ ( X ;A , B, xo).
As an illustrative example of the triad homotopy groups, let us take ( X ;A , B ) to be the triad given by
X =S2, A
=
E:,
B
=
Ea.
Then we have C = S1.Take xo = (1,0,0). Since A and B are both contractible in themselves, it follows from the exactness of the homotopy sequences of the triplets ( A ,C, xo) and ( X , B , xo) that
n n ( A ,C,xo) m nn-I(C, xo), n n ( X ,B , ~ for every n > 0. On the other hand,
0 M )
n n ( X ,xo),
nn(A,C,~ 0 M) nn-l(S1) = 0 for every n 2 3. This implies that j* in the first homotopy sequence of ( X ;A , B ) is an isomorphism for every n > 3. Hence
n n ( X ;A , B, x0) m nn(S2), n > 3. Finally, it is easy to see that in the following part of the first homotopy sequence of ( X ;A , B)
O+ n,(X,B,%o)-tn s ( X ; A,B,xo)-t n 2 ( A , C , x o ) Lnc,(X,B,xo)+~ ~ ( X ; A , B J O ) - + O ,
1b2
V. T H E C A L C U L A T I O N O F H O M O T O P Y G R O U P S
the groupsn,(A, C, xo) andn,(X, B, xo) are free cyclic andi, sends a generator of n,(A, C, xo) onto a generator of n,(X, B, xo). Hence i, is an isomorphism. This implies that n a ( X ;A, B, xg) = 0,
n 3 ( X ;A, B, x0) w n,(X, B, x0) w n3(S2)w 2.
11. Freudenthal's suspension Let (X; A, B) be a given triad such that both A and B are contractible in themselves. Set C = A n B and pick a basic point xo E C. Consider the spaces of paths U = [ X ;A , B ] , V = [C; C, C ] and denote by a, the degenerate path uo(I) = xo. Since a , ~ V cU,we obtain the following exact homotopy sequence of the triplet (U, V , uo): '
- -&nn+Ju,v,a),
Ln,(V,uo) %nn( U,aJ & nn(u,V ,uo)% ... -Ln,(u,v,uo) %no(V,a), 3, no(u,a,).
- *
According to (111; 8 9), there is a homeomorphism 5 : C + V of C onto a subspace 5(C) of V . 5 is called the natural injection of C into V and is defined by taking t ( x ) to be the degenerate path at x for every x E C. By (111; 9. lo), t ( C ) is a strong deformation retract of V and hence we have
5* :n n ( C , xo)
nn(V,no),
2 0. On the other hand, let W = [X; xo, x O ] . Since both A and B are contractible, it follows from (111; Ex. N) that the inclusion map 7 : W c U has a homotopy inverse. This implies that
q* :nn(W, uo)
M
-
nn(
12
u, a o ) ,
z 0.
Tz
According to (IV; 5 9), we have the natural correspondences
x :nn+1(X,xo) W nn(W, uo),
2 0. Next, consider the space of paths 52 = [ U ; V , uo]. By (111; 9.9), 52 can be considered as the space of maps f : I2 -f X such that 12
f ( o , t ) ~ A , f ( l > t ) ~ Bf (, t > o ) ~ Cf (,t S 1 ) 7 x 0 for every t E I . Let Oo denote the constant map Oo(12)= xo. Then, we have the natural correspondences
x :nn+,(u,V , uo)
nn(s2,
eo),
n
> 0.
On the other hand, let P = [X;B, x O ] and Q = [ A ; C, x O ] and consider = [ P ;Q, ao].ThenA can be considered as the space of maps g: I 2 -+ X with g ( O , t ) E A , g(t,O)EB, g ( l , t ) = x o = g ( t , 1)
A
for every t E I . Obviously Oo E A . According t o the definition in 5 10, we have Z ~ + ~ (AX, B, , xo)
Q, uo) & nn-,(n,eo),
= nnp,
12
> 1.
11. F R E U D E N T H A L . S S U S P E N S I O N
Ib3
We are going to prove that Q and A are of the same homotopy type. Let a12 denote the boundary of I 2 and define a map 0 : a I z + a12 by taking for
+(O, t ) = (0,t)l +(t, 1) = (t, I), +(I, t) = (1, 1).
+
On can easily see that is homotopic to the identity map on aI2 and hence has an extension @ : I a + P.For each f E Q, the composed map f @ is in A. The assignment f + f @ defines a map 5 : (Q, 0,) + (A,O0). By constructing another but somewhat similar map Y :I a + I a , one can prove that 5 has a homotopy inverse defined by g g Y for each g E A . Hence we have --f
C*
: Z,-~(Q, 0,)
-
n,-,(A, O0),
n 1. Now let us consider the following composed transformations
r
=
5;1aX-15;1X
:n n + J X ;A , B , xo) +nn(C, xo),
s = %-1q;Ii*t* :n,(C, xo) +nn+1(X,xo),
nn+*(X,xo) +nn+a(X;A , B , xo) for each TZ > 0. If n > 0, then all of these are homomorphisms. The homotopy sequence of the triplet (U,V , u0) gives rise to the following exact sequence t
= X-ll,Xj,q+X:
...A, n + 2 ( X ; A , B , x o ) ~ n n ( C , x o ) ~ 7 d n( +Xi J ~I+nn+,(X;A,B,xol )
. . . J+ne ( X ;A , B , xo) -".no(C,
* *
( X ,xo) which will be called the suspension sequence of the triad ( X ;A , B ) . In particular, the transformations s are called the suspensions. Thus, the triad homotopy sets nn(X;A , B , xo) measure the extent by which the suspensions s, n > 0, fail to be isomorphisms. Proposition 11.1. For any integer m
equivalent :
xo)
> 2, the following three statements are
(i) n,(X; A , B , xo) = 0 for every n < m. (ii) The suspension s :nn(C,xo) +nn+l(X, xo) is an isomorphism for each n = 1; - ,m - 2 and is an epimorphism for n = m - 1. (iii) The excision i , : n,(A, C, xo) + z n ( X , B , xo) is an isomorphism for each n = 2; -,m - 1 and is an epimorphism for n = m. Proof. The equivalence of (i) and (ii) is a consequence of the exactness of the suspension sequence of ( X ;A , B ) , and the equivalence of (i) and (iii) is a consequence of properties of first homotopy sequence of ( X ;A , B). I If X is the r-sphere with r > 2 and A = E:, B = El_,then it will be proved in the sequel that the statement (ii) is true for m = 2 r - 2 . See (XI; 2.1).
V. T H E C A L C U L A T I O N O F H O M O T O P Y G R O U P S
164
Bythe definition s = X--ly;li.,,E.+, it is not difficult to see that a geometrical represention of the suspension s :nn(C, xo) +nn+l(X, xo) can be described as follows.For any u € n n ( C ,x o ) , let us choose a representative map f : (9, so)+ ( C , xo). Since both A and B are contractible, f has an extension F : (Sn+l,so) + ( X ,x,) such that F(E,”+l)c A and F(E?+l)c B. Then s(a) is the element of Z ~ + ~ (xo) X represented , by this map F . This beautiful geometrical representation of s is its original definition given by Freudenthal to the special triad (9; E:, EL). As a generalization of the triad (S‘; E:, ET), let C be a given non-empty space. Consider the space X obtained by joining C to two distinct vertices a and b. Precisely, X is obtained from C x I by identifying the subsets C x 0 and C x 1 into single points a and b respectively. If fl : C x I + X denotes the natural projection, then C can be considered as a subspace of X by the imbedding i : C + X defined by i(c) = fl(c, 4) for each c E C. Let
A
={
f l ( ~t ,) I C E C ,0
Q 4 },
B
=
{ $(C, t ) I c EC,4 Q t Q 1 }.
Then A and B are cones over C and hence are contractible in themselves. Further, obviously we have C = A n B. Thus we obtain a triad ( X ;A , B) which satisfies the conditions given a t the beginning of this section. In particular, if C is the ( Y - 1)-sphere P - l , then ( X ;A , B ) is topologically equivalent to the triad (S’; E:, EC). Finally, let us determine the suspension sequence of the triad (S2;E:, Ea). In this case, we have C = S1; pick a basic point so E 9. Sinceno(S1,so) = 0 = n,(S1, so), we get the following part of the suspension sequence 0 +n3(S2.so) 1+n3(S2; E:, E!, so)J-+nl(S1,so) *n,(S2, so) + O
where each of the four groups is free cyclic. We assert that t is an epimorphism. For, otherwise, the image of r would be of finite order equal to the index of t[n3(S2, so)] in n3(S2;E?, Ea, so). Since nl(S1,so) is a free group, this is impossible. Hence s, t are isomorphisms and Y = 0. On the other hand, since nn(S1,so) = 0 for every n 2, we have
t :nn(S2,SO)
w nn(S2;E t , EZ, SO)
for every n Z 4.
EXERCISES A. The homotopy addition theorem
Consider the unit n-simplex An of the euclidean ( n + 1)-space Rn+l, [E-S; p. 551. For every pair ( X , xo) and any integer n > 0, the elements of x, ( X, xo) may be considered as the hornotopy classes of the maps of (An,8dn) into ( X ,x o ) ; for details of this definition see [Hu 121. Let n > 0 and denote by Kn-1 the (n- 1)-dimensional skeleton of the
165
EXERCISES
+
boundary n-sphere S n of the unit (n 1)-simplex An+, in the euclidean (n 2)-space. Consider any given map
+
f : (9, P -
1 ) +.
( X ,xo).
Since f sends the leading vertex of Sn onto x,, f represents an element [ f ] of nn(X, xo). On the other hand, let (i=O;..,n+
e;:A,,+.An+l,
I),
denote the simplicia1map defined in [E-S; p. 1851. Then, the composed map fe; : ( A n , adn) +. ( X , xo)
represents an element [fe;] of n n ( X , xo) for every i Prove the following Homotopy addition theorem. For any map f : ( S n , Kfi-1)-+ ( X ,x o ) , we alway have
72
= 0,
1;
-
*,
n
+ 1.
>, 2,
n+i
[fl
.x
1)' [fe;I.
= a m 0 (-
For the exceptional case, n , ( X , xo) is not necessarily abelian. However, for any map f : (9,KO) +. f X ,xo), the following relation is obvious : In the remainder of this exercise, we shall give an analog for the maps of cubic boundaries. Let rz > 0 and denote by Kn-l the (12 - 1)-dimensional skeleton of the boundary n-sphere a I n + l of the (n + 1)-cube P + l , i.e., Kn-l consists of all points (t , ; . . , I n + ] ) of I n + l such that tf(l -it) = 0 for at least two indices i. Consider any given map
f : (ain+l, m - 1 )
-+
(x,xo).
Since f sends the point (0; * * , 0) of a P + l into xo, it represents an element [f ] of n,(X, xo). On the other hand, for each i = 1 ;* * , n + 1, let yi and ( r denote the homeomorphism of I n into a I n + l defined by qt(ti,.
* I
In) =
(ti,.
*
*,
ti-],
0, ti,'
* *,
tn),
t t ( L ** * , tn) = (ti,. * * , ti-1, 1, t i , . * * , in). of ( I n , a I n ) into ( X ,xo) represent Then, the composed maps fqc and elements [ f q i ] and [/(*I of nn(X,xo) respectively. Then, prove that nfi
[ f l = 1.x (=1
([/&I - [fytl),
n >2.
For the exceptional case, n , ( X , xo) is non-abelian in general. However, for an arbitrary map f : ( d 1 2 , KO) + ( X , xo), the following relation holds:
166
V. T H E C A L C U L A T I O N O F H O M O T O P Y G R O U P S
B. The relative homotopy addition theorem
For any triplet (X, A , xo) and any integer n 2 2, the elements of the group n n ( X ,A , xo) can be considered as the homotopy classes of the maps of (An,adn,uo) into ( X ,A , xo), whei-e uo denotes the leading vertex of An. Let n 2 2 and denote by Kn-l the ( n- 1)-dimensional skeleton of the boundary n-sphere Sn of An+l. Consider any map
f : (9, Kn-l, uo) -+ (X, A , xo). If we compose f with etn :An +Awl, we obtain a map f e p of (An,adn)into (X, A ) for each i = 0, 1,* ,n + 1. If i # 0, fern maps the leading vertex uo of A , into xo and hence i t represents an element [fern] of n,(X, A , xo). Set x , = f ( u l ) . Then, /eon sends uo into x1 and so it represents an element [/eon] ofn,(X, A , xl). Let u : I + A denote the path joining xo to x1 defined by
--
u(t) == f ( 1 - t , t , O ; . . , O ) ,
tEI.
By (IV; Ex. D), u induces an isomorphism on :nn(X,A , xi) w n n ( X ,A , xo).
Prove the following Relative homotopy addition theorem. For any map we always have
f : (Sn, Kn-l, uo) +. ( X ,A , xo), n > 2,
L[fl
U+l
=dfeonI
+ 3 (*-1
[fan].
where [ f ] i s the element ofn,(X, xo) represented by f as a m a p of (9, vo) into (X, xo) and j* i s the homomorphism induced by the inclusion map j : ( X ,xo) c (X, A xo). State and prove the analogous theorem for the maps of (dIff+l,Kfl-l, uo) into (X, A , xo). 3
C. The Hurewicz theorem
Let X be pathwise connected space, A be a pathwise connected subspace of X , and xo be a given basic point in A . The pair (X, A ) is m-connected if n,(X, A , xo) = 0 for every n satisfying 1 < n < m. By means of the relative homotopy addition theorem in the previous exercise, prove the following Hurewicz theorem. If n 2 and ( X , A ) i s (n- 1)-connected, then the natural homomorphism
in (4.1) i s a n epimorphism and its kernel i s the subgroup generated by the elements of the form u - wa, where u ~ n n ( XA,, xo) and w ~ n , f Axo). , For the special case n = 2, prove that the kernel of x 2 contains the commutator subgroup of n,(X, A , xo).
167
EXERCISES
As corollaries of the theorem given above, we have the following propositions : 1. If ( X ,A ) is (n - 1)-connected and n-simple for a given n 2 2, then 2. If rt
> 2 and X
~n :nn(X,A , xo)
%
Hn(X,A )
is (rc - 1)-connected,then It, :n,(X, X J w H,(X).
-
The last proposition generalizes (4.4). As an illustrative example, let us consider the space E = (9 x 2) U (so x R) of 8 7. Then, E is ( n - 1)connected and hence n.(E) w H n ( E ) . This implies that nn(E)is a free abelian group with a countably infinite set of free generators. Finally, prove
3. If n > 2 and X is [G. W. Whitehead I].
(n
- 1)-connected, then
Itn+l is an epimorphism.
D. The Whitehead theorem
Let ( X ,A , xo) be as assumed at the beginning of the previous exercise. By the aid of the Hurewicz theorem and the following commutative exact ladder " . ~ n n ( A , x o ) 4 , n , ( X , x ~ ) ~ n n ( xa, A , x o ) - n n - ~ (.Ln,(X,A,xo)"-O A,~o)~.. I n
,
lhn
.
I n
kn-1
1.
~ H n ( A ) ~ + . H n ( X , ~ + . H ~" (+X. ,~An) - ~ ( A )&+.fZ,(X,A), ~... prove the following assertions : 1. Let m > 0 be a given integer. If i, is an isomorphism for every n Q m, then so is i#. If i* is an isomorphism for every n < m and is an epimorphism for n = m,then so is i#. 2. Assume that both X and A are simply connected and m > 0 is a given integer. If i# is an isomorphism for every n < m and is an epimorphism for n = m,then so is i,. Furthermore, if i# is an isomorphism also for n = m, then the kernel of i, inz,,(A, X J is contained in that of hn. Next, let X and Y be pathwise connected spaces. Consider a given map f : ( X ,xo) +. ( Y ,yo) and the induced homomorphisms f* :n n w , xo) +.nn(Y,Yo), f# : Hn(W +. Hn(Y). Using the mapping cylinder M f , and ( I ; 12.1 and 12.2), prove the following assertions : 3. Let m > 0 be a given integer. If f* is an isomorphism for every n < m, then so is f#. Iff* is an isomorphism for every n < m and is an epimorphism for n = m, then so is f#. 4. Assume that both X and Y are simply connected and m > 0 is a given integer. If f# is an isomorphism for every n < m and is an epimorphism for
168
V. T H E C A L C U L A T I O N O F H O M O T O P Y G R O U P S
n = m,then so is f,. Furthermore, i ff# is an isomorphism also for n = m, then the kernel of f, in Xn(X, xo) is contained in that of hn : n n ( X ,x0) +. Hn ( X ) Note that the condition that there exists a map f : ( X , xo) +. ( Y ,yo) such that f, is an isomorphism for each n > 0 is much stronger than the condition that n n ( X , xo) m z n ( Y ,yo) for every n > 0. In fact, there are pathwise connected spaces which have isomorphic homotopy groups but some of whose homology groups are different [Wang 11.
-
E. Homotopy groups of adjunction spaces
Let X be a given space and x 0 e X a given point. Consider an indexed family f, : (Sn,so) +. ( X ,xo), p E M . Give M the discrete topology and define a map f:Sn x M + X
by taking f (s, p) = fp(s) for every s E Sn and p E M . Let Y denote the adjunction space obtained by adjoining En+l x M to X by means of the map f, (I; 8 7). The inclusion map i induces the homomorphisms
i, : n m ( X ,xo) + X m ( Y , xo), m > 0. Prove the following assertions: 1. i, is an isomorphism for every m < n. 2. For m = n,i, is an epimorphism and its kernel in n,(X, xo) is the subgroup generated by the elements wa,, where w E X * ( X xo) , and a, is the element of n n ( X , xo) represented by the given map f , for all p E M . [J. H. C. Whitehead 1, p. 2811. 3. If f,, N 0 for every p E M , the i, is a monomorphism for m = n 1 and its image in nn+l(Y,xo) is a direct summand. The complementary summand is isomorphic to the relative homotopy group nn+l(Y , X , xo) which is a free abelian group with free generators wb,, where w € n , ( X , xo) and b, ~ n n + ~X( ,Yx0) , is the element represented by the map
+
gp
x,xo)
: (E%+l,sn, xo) -+ ( Y ,
defined by g,(t) = p ( t , p) where fi : En+l x M projection. [J. H. C. Whitehead 1, p. 2851.
+Y
denotes the natural
F. Spaces of homotopy type (a, n)
Let az be a given group and n > 0 a given integer. If n > 1, we assume that x is abelian. A pathwise connected space X is said to be of the homutopy type (n, n) provided Xn(X) w X , n m ( X ) = 0 if m # n. Thus, by (8.4), if f i : E
+. B
is an n-connective fibering over an (n - 1)-
169
EXERCISES
connected space B , then the fiber F = p-llb,) over any b, E B is of the homotopy type (nn(B),n - 1). Construct a space X of the homotopy type (n,n) as follows. If n = 1, then n can be represented as the quotient group of a free group F over a normal subgroup R of F . If n > 1, then n is abelian and can be represented as the quotient group of a free abelian group F over a subgroup R of F. Hence IC = FIR.
Let M denote the set of free generators of F and let b, be a space consisting of a single point b,. Give M the discrete topology and adjoin En x M to b, by means of the map g :Sn-l x M +b,. Let A denote the adjunction space obtained in this way. Then prove that
nn(A)= F , n,(A)
=
0 if m
< 12.
For each r E R c nn(A),choose a map fr : (9, so) -+ ( A , b,) which represents Let B denote the space constructed by means of the family { fr I Y E R } as in Ex. E. Then we have
Y.
nm(B) = 0 if m < n. Then, by (8.2), there exists a pathwise connected space X which contains B as a closed subspace and nn(B) M FIR
= n,
n,(X) w n, n m ( X ) = 0 if m # n. This completes the construction. Finally, prove that if X is a space of the homotopy type (n,n) with n > 1, then the space of loops A ( X ) at a point x, E X is a space of the homotopy type (n,n - 1). G. The realizability theorems
By means of the spaces of homotopy type (n,n), prove the following Realizability Theorem. Let Zz,'
' '
,nn,'
* *
be a sequence of groups. All groups except possibly the first one are abelian andn, operates Onnn for every n > 2. Then there exists a pathwise connected space X and a point X , E X such that the following three conditions are satisfied, [J. H. C. Whitehead 51 and [Hu 101: (1) There exists, for each n > 0, an isomorphism hn :n n ( X , x O )
M nn.
(2) For any w € n l ( X ,x,) and a Enn(X,x,), n 2 2, we have hn(Wa) = h, ( W )hn(a). (3) For every a € n m ( X ,x,) and head product [a,!] = 0.
Enn(X,x,), m 2 2, n
> 2, the White-
170
V. T H E C A L C U L A T I O N O F HOMOLOGY G R O U P S
. By constructing a cone over X , deduce an analogous realizability theorem for relative homotopy groups.
H. Topological realization of semi-sirnplicial complexes
For a given semi-simplicial complex K as defined in (IV : Ex. H), we can construct a space I K 1 associated with K as follows. To every integer m 0 and each m-cell a of K , let us associate an open m-simplex 1 a 1, called the open m-cell corresponding to a, which is defined to be the topological product I a I = a x Int ( A m ) , Id (Ao) = A,, of u as a single point and the interior of the unit m-simplex Am. Then we define the closed m-cell Cl I a I to be the set Cllal = U t < ~ l t l . There is a natural function Xu of A m onto C1 I a I defined linearly on each open simplex of the complex A m , [Hu, 101. Give CZ I a I the identification topology determined by Xu, ( I ; 3 6). Next, let us denote by I K I the union of all open cells I u 1, a E K , and define a topology of I K I as follows. A set W in I K 1 is called open iff W n CZ I a 1 is an open set of CZ 1 a 1 for every cr E K . This topology of I K I will be called the Whitehead topology of I K I ; it is the largest topology of I K I such that the topology of every closed cell CZ I a I coincides with the relative topology in I K I. Following a usual custom in combinatory topology, we identify K with I K I and a with 1 a 1. Thus, we may consider any semi-simplicial complex K as a topological space which is a union of a collection { a } of disjoint open cells. For example, the singular complex S ( X ) of a space X will be considered as a topological space in this way. For each cell a of K , it follows immediately that the natural function Xo : A m --* C1 a c K , where m = dim (a), is a continuous map of A m onto C1 a and the restriction ;C, 1 Id (Am) is a homeomorphism of Int (Am) onto the open m-cell a. We shall call Xu the characteristic map for the cell a E K . Prove the following assertions: 1. If X is a compact subspace of a semi-simplicial complex K , then X meets at most a finite number of open cells of K . 2. A function f : X + Y of a closed or open subspace X of a semisimplicia1 complex K into a space Y is continuous iff the restriction f I X n C b is continuous for every closed cell Cla of K . 3. A family ft : X + Y , (0 < t < l ) , of functions of a closed or open subspace X of a semi-simplicial complex K into a space Y is a homotopy iff the family fl I X n Ckr, (0 < t < l), is a homotopy for every closed cell Ckr of K . 4. Every subcomplex L of a semi-simplicial complex K is closed and has the AHEP in K.
EXERCISES
17'
5. A necessary and sufficient condition for a semi-simplicial complex K to be a simplicial complex with locally ordered vertices is that, for each m-cell U E K , the characteristic map Xo is a homeomorphism of A , onto Cl u and, for any two cells u and t of K , Cl u f l C1 t is either empty or a closed cell of K . 6. For each n > 2, the n-th barycentric subdivision, defined in the obvious way, of a semi-simplicial complex K is a simplicial complex with locally ordered vertices. Hence, every semi-simplicialcomplex is triangulable. 1. The projection w : S(X) -+ X
For any given space X, there is a natural projection o of the singular complex S(X) onto X as follows. For an arbitrary point p of the space S(X), let 0 denote the unique open cell of S(X) which contains p . Then, u is a singular simplex :A , = dim (a) ~
x,
and the characteristic map X, sends Int A , homeomorphically onto the open cell a of S(X). The natural projection w : S ( X ) + X i s defined b y taking U(P)
= uX,-l(p),
p € u c S(X).
Prove the following assertions: 1. The projection w : S ( X ) + X is a map of S(X) onto X. For every subspace A of X, w carries the subcomplex SIA) onto A . 2. The space X is pathwise connected iff S(X) is connected. Next, let (X, A , xo) be any given triplet and let Po denote the vertex of S(X) such that w(po) = xo. Thus we obtain a triplet (S(X),S ( A ) ,Po), and a map w : ( S ( X )S , ( A ) ,Po) -+ (X,A , xo) defined by the natural projection w . Prove the following important proposition, [Giever 1 and J. H. C. Whitehead 71. 3. The induced transformations W + of w are one-to-one correspondences, namely, w* : nm(S(X),S ( 4 , Po) M n,(X, A , xo), w*
:nm(S(X),Po) w n m w , x0)r
a*
:nm(S(A) P O ) I
= nm(A xo). 9
The significance of this is that, in computing the homotopy groups of a space X, we may assume without loss of generality that X is triangulable and hence locally contractible. In fact, we may replace X by S(X). J. Induced cellular maps
A cellular transformation T : K , + K , of a semi-simplicial complex K , into another such complex K , is a function which assigns to each m-cell u of K , an m-cell t = T(a)of K , in such a fashion that t ( 0 = T ( u ( ~ ) )(i , = 0 , 1;**, m).
V. T H E C A L C U L A T I O N O F H O M O T O P Y G R O U P S
172
K , induces a unique map T(a)E K,. A map f : K , -+ K , is said to be cellular if it carries K,m into K,m for every m = 0, 1; -.Thus, the map ffp : K , + K , induced by T is cellular. Consider a given map : K -+ X of a semi-simplicial complex K into a Prove that a cellular transformation T : K ,
-+
fT : K , -+ K , which carries a E K , barycentrically on t =
+
+
space X. induces a cellular transformation
T+ : K -+S(X) as follows: For each m-cell a E K , m-simplex t
t=
T+(a)is defined to be the singular
=+X,:A,-+X
whereX,denotes the characteristic map of a. Verify that t ( C ) = T+(a(")for everyi=O, - * - , m . This cellular transformation T+ induces a cellular map +# : K -+S(X) which will be called the induced cellular mu# of 4. Prove that w+# = As an application of this, let X = K and take to be the identity map. Then +# maps K homeomorphically and barycentrically onto a subcomplex of S ( K ) .Hence, we may consider K as a subcomplex of its singular complex S ( K ) . Prove that K is a deformation retract of S ( K ) and that the cellular transformation T+ defines a chain equivalence. The second assertion implies that the homology and cohomology groups of a semi-simplicial complex K are topological invariants. Next, let X , Y be spaces and consider a given map f : X -+ Y . f induces a cellular transformation T f :S ( X ) + S(Y )
+
defined by
t =
+.
T f ( a )= fa :dm-+ Y
for every singular m-simplex u E S ( X ) . This cellular transformation T f induces a cellular map f # : S ( X ) + S (Y ) which will be calIed the induced celluEar map of f . Obviously, f# is also the induced map of = f w : S ( X ) -+ Y . Verify that, in the following diagram, we always have f w = w f # :
+
S ( X ) ___ f # -+ S ( Y )
I-
x-
.I
f + Y This implies that, in studying the induced homomorphisms f* of a given map f : X -+ Y on the homotopy groups, we may always assume that X , Y are simplicial complexes and that f is simplicial. K. Admissible subcomplexes of S(X)
Assume that X is pathwise connected space and that a fixed point xo of
X has been selected as basic point.
I73
EXERCISES
Two singular m-simplexes u and t in X are said to be compatible if their faces coincide, that is to say, a(') = t(')for each i = 0; * , m. Hence o and t are compatible iff u I s m - 1 = 7 I sm-1, P - 1 = ad,.
-
Further, two compatible singular m-simplex u and t are said to be equivalent if there exists a homotopy ht : A , -+ X , (0 < t Q l ) , such that h, = (I, h, = T, and ht(d) = u ( d ) for every d E S m - l and every t E I. For m = 0, any two singular 0-simplexes are compatible; and, since X is assumed to be pathwise connected, they are also equivalent. A singular simplex u : A , -f X is said to be collapsed if a(&) = x,. The set of all collapsed singular simplexes constitute the subcomplex S(x,) of S ( X ) associated with x, as a subspace of X . A subcomplex L of S ( X ) is called an admissible subcomplex of S ( X ) if the following conditions are satisfied: (ASl) S(x,) c L. (AS2) If u is a singular simplex such that a(') E L for each i = 0, * ,dim (a), then L contains at least one singular simplex t which is compatible and equivalent with u. If, in the condition (AS2), we require that L contains one and only one singular simplex t which is compatible and equivalent with u, then L is called a minimal subcomplex of S ( X ) . Obviously, S ( X ) is an admissible subcomplex of itself and every minimal subcomplex of S ( X ) is admissible. Let the admissible subcomplexes of S ( X ) be partially ordered by means of inclusion. Then a minimal subcomplex M of S ( X ) is clearly minimal in the sense that, if any admissible subcomplex L is contained in M , then L = M . Prove the following assertions : 1. Every admissible subcomplex L of S ( X )contains a minimal subcomplex it4 of S ( X ) . 2. If L is an admissible subcomplex of S ( X ) ,then there is a function D , called a deformation of S ( X ) into L , which assigns to each singular m-simplex u a singular ( m+ 1)-prismD(u) = P,,[E-S; p. 1931, subject to the following conditions :
--
(i) D(a") = (P,)S, (i = 0; * * , m). (ii) (P,)z= o. (iii) (P,),, is in L. (iv) If u E L, then P,(d, t ) = u(d) for every d E A , and every t E I . Now let A be a pathwise connected subspace of X which contains the basic point x,. An admissible subcomplex L of S ( X ) is said to be relatively admissible with respect to S ( A ) if L n S ( A ) is an admissible subcomplex of S ( A ) . In this case, the deformation D can be so chosen that, for each U E S ( A ) , P, is a singular prism in A .
I74
V. T H E CALCULATION O F HOMOLOGY G R O U P S
3. If L is an admissible subcomplex of S ( X ) which is relatively admissible with respect to S ( A ) , then the inclusion cellular map
1 : (L,L n s ( A ) )--t (s(x), SM)) is a homotopy equivalence and defines a chain equivalence. L. The Eilenberg subcomplexes of S(X)
Let X be a pathwise connected space, A a pathwise connected subspace of X , and x, a given basic point selected from A . For each integer n > 0, define a subcomplex S,(X, A ) of the singular complex S ( X ) as follows. A singular simplex u : d , + X is in S , ( X , A ) iff the following conditions are satisfied: (ESl) u sends the vertices of A , into xo. (ES2) u sends all faces of d, of dimension less than n into A . In particular, if A = x,, then we simply write S,(X) = S,(X, xo). S,(X) is called the n-th Eilenberg subcomplex of S ( X ) ; and, in the general case, S , ( X , A ) will be called the n-th relative Eilenberg szlbcomplex of S ( X ) with respect to A . For completeness, we also define S,(X, A ) = S ( X ) . We thus obtain a descending sequence of subcomplexes of S ( X ) , namely,
S ( X ) = S,(X, A ) 2 S,(X, A ) 2
* * *
2
Sn(X,A ) =I
* *
-
=I
S,(X, A )
= S,(A).
Prove that, if ( X ,A ) is ( n - 1)-connected for a positive integer n, then S , ( X , A ) is an admissible subcomplex of S ( X ) and is also relatively admissible with respect to SIA). In particular, S,(X) is a n admissible subcomplex of S ( X ) provided that X is ( n - 1)-connected.
CHAPTER V I O B S T R UCTlO N TH E O RY
1. Introduction Having studied the homotopy groups in the previous chapters, we are now in a better position to investigate our main problems described in the first chapter. For the sake of simplicity, we shall restrict ourselves to the study of the maps of a finite cell complex K , [ S ; p. 1001, into a pathwise connected space Y . Therefore, we assume throughout the present chapter, that K is a finite cell complex, L a subcomplex of K , and Y a given pathwise connected space with a given point yo E Y . We shall denote by Kn the n-dimensional skeleton of K which consists of the cells of dimension not exceeding n and use the notation Kn = L U Kn. For example, let us study the extension problem. Consider an arbitrarily given map f:L+Y.
As described in (I; tj 2), the extension problem for f over K in the restricted sense is t o determine whether or not f can be extended continuously throughout K . Since L has the AHEP in K according to (I;9.2), this restricted extension problem is equivalent to the broadened problem described a t the end of (I; tj 9). Since K is a cell complex and L is a subcomplex of K , there is a natural approach to attack this problem, known as the obstruction method. We first try to extend step by step the map f over the subcomplexes
En, n
= 0,
1,2;
- ..
We shall carry on this stepwise extension until we meet some obstruction to further extension. Then we propose to measure this obstruction and try to change the already constructed partial extension of f so that this obstruction might vanish and hence further extension would be possible.
2. The extension index In order to study the extension problem, let us define the notion of n-extensibility. A map f : L -+ Y is said to be n-extensible over K for a given integer n > 0 if it has an extension over the subcomplex E n of K . 175
176
VI. O B S T R U C T I O N T H E O R Y
Since Y is assumed to be pathwise connected, the following proposition is obvious. Proposition 2.1. Every map f : L + Y is 1-extensible over K.
For a given map f : L + Y, the least upper bound of the set of integers n such that f is n-extensible over K is called the extension index of f over K. Since L has the AHEP in En, the following proposition is obvious. Proposition 2.2. Homotopic maps have the same extension index.
Now let e : Y + Y' be any map and let g : (K', L') + (K, L ) be any cellular map, [S; p. 1611. Then the following proposition is obvious. Proposition 2.3. If a map f : L + Y is n-extensible ovev K , then the composed map f ' = e f g : L' + Y' is n-extensible over K'. In particular, let ( K ,L) be a subdivision of (K', L') and g be the identity map. If f : L + Y is n-extensible over K, then f' = fg : L' + Y is also n-extensible over K'. By the classical method of simplicial approximation, one can easily deduce from (2.2) and (2.3) the following Proposition 2.4. If (K, L) is a simplicial pair, then the extension index of any map f : L + Y is a topological invariant, i.e., it does not depend on the triangdation of the pair (K, L ). Because of this it seems that we should prefer simplicial complexes in our treatment. However, to unify the study of the extension problem with that of the homotopy problem and to avoid awkward repetitions, we have to consider cell complexes which are not simplicial.
3. The obstruction c"+'(g) Throughout $3 3-12, let n be a given positive integer. For the sake of convenience, we assume that the given pathwise connected space Y is n-simple in the sense of (IV; $ 16). In this important case, every map of any oriented n-sphere into Y determines an element of the homotopy group nn(Y); the latter group is abelian and will be used as the coefficient group of the cohomology groups in these few sections. If Y fails to be n-simple and n > 1, then one can deduce similar results by using local coefficients. See Ex. A a t the end of the chapter. In the present section, let us consider a given map g:Kn+Y. This map g determines an (n + 1)-cochain cn+l(g)of K with coefficients in the homotopy uoupnn(Y) as follows. Let u be any In + 1)-cell of K. Then the set-theoretic boundary du of u is an oriented n-sphere. Since du c Kn, the partial map gu = g I du determines an element [go] of nn(Y). Then cn+l(g) is defined by taking [cn+'(g)l(ul = [gbl Enn(Y)
3.
T H E O B S T R U C T I O N C"+l
+
for every ( n 1)-cell c7 of K. This (n the obstruction of the map g.
(g)
I77
+ 1)-cochain cn+l(g) of K
Lemma 3.1. The obstruction cn+l(g) is a relative (n modulo L ; in symbols, cn+l(g)E Z ~ + ~ L( ;Kn,n ( Y ) ) .
is called
+ I)-cocycle of
K
Proof. Let us first prove that cn+l(g) is in Cn+'(K,L;n,(Y)). For this purpose, let a be any (n 1)-cellof L. Since g is defined on C l o c L , g, has an extension over Cla and hence [g,] is the zero element of nn( Y ).This proves that cn+l(g) is a cochain of K modulo L. Next, let us prove that cn+l(g) is a cocycle. For this purpose, let c7 be any (n 2)-cell of K . It suffices to show [dcn+l(g)](a) = 0. Let B denote the subcomplex i3a of K and Bn the n-dimensional skeleton of B. Then we have the homomorphisms
+
+
Cn+,(B)-LZW(B) = Zn(Bn) = Hn(Bn)+-nn(Bn) h &+nn(Y), where h denotes the natural homomorphism and k , is induced by the partial map k = g I Bn. If n > 1, then Bn is (n- 1)-connected and hence h is an isomorphism according to the Hurewicz theorem in (V; 4.4). If n = 1, then h is an epimorphism and the kernel of h is contained in that of k , sincenn(Y )is abelian. Hence, in either case, we obtain a well-defined homomorphism
+
&h-' : ZW(B)+nn(Y). Since Cn-,(B) is a free abelian group, the kernel &(B) of i3 : CJB) -+ C,-,(B) is a direct summand of Cn(B).Therefore, the homomorphism I# has an extension d : Cn(B)+nn(Y). =
+
For every ( n 1)-cell t in B , the element [cn+l(g)](r) is represented by the partial map k I at. Therefore, it follows that [c"+l(g)](t) = k,h-'(&) = d(&). This implies that
[c"+l(g)](aa) = d(&) = 0. Hence cn+l(g) €Zn+l(K,L ; n , f Y ) ) . I Because of (3.1), cn+l(g) is usually called the obstruction cocycle of g. The following two lemmas are obvious. [dc"+'(g)](a)
=
iff cn+l(g) = 0. + Y has an extension over Lemma 3.2. The map g : Lemma 3.3. Ifgo,g, : Kn-+ Y are homotopic mups, then cn+l(g,) = cn+l(g,). Now let (K', L') be another cellular pair and I# : ( K ' , L') -+ ( K ,L ) be a proper cellular map, [S; p. 1611. For a given map g : Kn -+ Y , we obtain a composed map g' = g+ : K ' n + Y . Since I# is proper and cellular, it induces a unique cochain homomorphism, [S: p. 1611, +# : Cn+l(K,L;nn(Y )) + C*+l(K',L'; n,(Y)).
178
VI. O B S T R U C T I O N T H E O R Y
Lemma 3.4. cn+l(g') = @cn+l(g).
To prove this, one must show that the two sides have the same value on an arbitrary (n 1)-cell u of K'. For this purpose, one should consider the minimal carrier of u and pick an (n + l)-cell, from this minimal carrier; then the lemma is a consequence of a number of trivial commutativity relations. The details of the proof are left to reader, [S; p. 1681.
+
4. The difference cochain In the present section, we are concerned with two given maps
go, g, :9. + Y which are homotopic on En-1; we shall see that the difference of the obstruction cocycles of go and g , is a coboundary. For this purpose, consider a homotopy ht : R*-1+ Y, (0 < t < l), such that h, = go I En-, and h, = g, I En-1. Regard the closed unit interval I as a cell complex composed of two 0-cells 0 and 1 and the lcell I , where 60 = - I and 61 = I . Then the topological product J = K x I is also a cell complex. We shall denote by J n the n-dimensional skeleton of J and use the notation
$. = ( L x I ) U Jn Define a map F :
=
( E n x 0) U (Rn-l x I ) u (Rnx 1).
+ Y by taking
go(x), ( X E R . , t = O), h:(x), (%€En-',t E I ) , ( x €En,t = 1). gl(4, Then, according to 9 3, this map F determines an obstruction cocycle cn+'(F) of the complex J modulo L x I with coefficients inn,(Y). It follows from the definition of F that cn+'(F) coincides with cn+l(g0) x 0 on K x 0 and with cn+l(g,) x 1 on K x 1. Let M denote the subcomplex ( K x 0) U (L x I ) U ( K x 1) ofJ = K x I . Then it follows that F(x,t)
=
(
c*+'(F) - cn+l(gO) x 0 - cn+l(g,) x 1 is a cochain of J modulo M with coefficients in nn( Y).Since u + CJ x I is a 1 - 1 correspondence between the n-cells of K \ L and the (n + 1)-cells of J \ M , i t defines an isomorphism
k : Cn(K,L ; n n ( Y ) )w Cn+l(J,M ; n m ( Y ) ) . Hence there is a unique cochain &(go, gl; h:) in Cn(K,L ;n,(Y)) such that (i)
k&(go, g , ; ht)
= [-
l)m+1{ cn+l ( F )-cn+l(go) x 0 -Cn+l(gl) x 1 }.
4.
I79
T H E DIFFERENCE COCHAIN
This cochain @(go, g,; ht) will be called the deformation cochain. In particular, if go I En-1 = g, I En-1 and ht(x) = go(x) for every x E Rn-1 and every t, we abbreviate &(go, g,; ht) by &(go, g,) and call i t the di#erence cochain of go and g1. The following lemma is an immediate consequence of (3.2). Lemma 4.1. The homotopy ht : En-, -+ Y has an extension ht* : zn-+ Y, ( 0 < t < l), such that ho* = go and h,* = g, i# dn(go, g,; ht) = 0.
The importance of &(go, g,; ht) is shown by a coboundary formula given in the following Lemma 4.2. ddn(go,g,: ht)
= cn+l(g0) -cn+'(gl).
Proof. Since 61 = 0, i t follows that the isomorphism k commutes with 6. Hence, we have k6dn(go, gi ; ht) = 6kdn(go,gi : ht)*
On the other hand, since cn+l(F),cn+l(g,), cn+I(g,) are cocycles and since 60 = - I and 61 = I, we may apply 6 to both sides of (i) and obtain
6kdn(go,g,; ht)
= ~ n +(go) l
x I-ccn+l(g,)
x I.
Hence we have Since k is an isomorphism, this implies the lemma. I Next, let us consider any given map go : Rfi + Y and any given homotopy Y , (0 < t < l), such that h, = go I En-,. We shall establish the ht : following existence lemma. Lemma 4.3. For every n-cochain c in Cn(K,L;nn(Y ) ) ,there exists a map 1 En-1= h, and &(go, g,; ht) = c.
g, : E n -+ Y such that g,
K . Then the baundary = (a x 0) u (a0 x II u (u x 1)
Proof. Let u be any n-cell of
s
of (J x I is an oriented n-sphere and hence there exists a map fa : S -+ Y which represents the element c(u)of nn(Y ) .Since the subspace T = (a x 0) U (80 x I ) is contractible, any pair of maps defined on T are homotopic. Hence, by the AIfEP of T in S , we may assume that fa(%
t) =
(
gob)
(if x E u, t = 0), (if x E au, t E ;).
Thus, we may define a map g, : K* -+ Y by taking
Then it is obvious that @(go, g,; ht) = c. I As an immediate consequence of (4.2) and (4.3) we have the following
I80
VI. O B S T R U C T I O N T H E O R Y
-
Corollary 4.4. For each cocycle z E Zn+l(K,L ;nn(Y))such that z cn+l(go) mod L, there exists a map g, : Rn +. Y such that g, I En-1 = h, and cn+l(g,) = z .
In particular, if the homotopy ht is given by ht = go 1 En-i for every t E I, then the corollary gives the existence of a map g, : En + Y such that go I En-1 = g , I En-1 and cn+l(g,) = z . Next, let us consider three given maps go,g,, g, : an+Y and two homotopies ht, j t : En-1 -+ Y , (0 < t < l), such that go I En-,,
h, = g, I En-, = j,, jl = g, I En-,. Let Kt : En-,+Y , (0 < t < l), denote the homotopy defined by Kt = h Z f if t < 8 and kt = j z t - , if t 2 4. Then we have the following addition lemma the proof of which is left to the reader, [S; p. 1731. h,
=
At) = dn(go, g l ; ht) 4-dn(gl, g 2 ; i t ) . Finally, let us go back to the maps go,g, and the homotopy ht described a t the begnning of the section. Let (K',L') be another cellular pair and : ( K ' ,L') +. ( K ,L) be a proper cellular map. Let
Lemma 4.5. dn(go, g;,
+
g'c
Since
=
gt+ :E'n
+.
Y , h't
=
h d : E'n-1+. Y .
+ is proper and cellular, it induces a unique cochain homomorphism Cn(K,L ; n n ( Y ) )+Cn(K', L';nn(Y)).
The following invariance lemma is an easy consequence of (3.4). Lemma 4.6. dn(g',, g',; h't) = +#dn(g,, g,; ht).
5. Eilenberg's extension theorem As in
5 3, let us consider a given map g:En-+Y.
According to 8 3, g determines an obstruction cocycle cn+l(g) in Zn+l(K,L; nn(Y ) )and hence an obstruction cohomology class
yn+'(g) €Hn+l(K,L ; n n ( Y ) ) represented by cn+l(g). Theorem 5.1. yn+l(g) = 0
iff there exists a map h* :
h*
I xn-1
=g
I if.-,.
-f
Y such that
Proof. Suficiency. Assume the existence of h* - and let h = h* IEn. By (3.2), we have cn+l(h) = 0. Since g 1 E m - 1 = h 1 Kn-1, the difference cochain dn(g, h) is defined. By (4.2),cn+l(h) = 0 implies that cn+l(g)is the coboundary of dn(g, h). Hence yn+l(g) = 0. Necessity. Assume that y"+'(g) = 0. Then cn+l(g) N O mod L. By (4.4), there exists a map h : En + Y such that g I En-1 = h I En-1 and @+'(A) = 0. Then, by (3.2), h has an extension h* : En+,-+ Y . I
6.
T H E O B S T R U C T I O N SETS FOR E X T E N S I O N
181
Now, let us assume that g is an extension of a given map f : L -+ Y.If the obstruction cocycle cn+l(g)is non-vanishing, then it follows from (3.2) that g cannot be extended over and hence our stepwise extension process faces an obstruction. The significance of Eilenberg’s extension theorem (5.1) is that, if cn+l(g) 0 mod L, then this obstruction is removable by modifying the values of g on the open n-cells in K \ L only.
-
6. The obstruction sets for extension Let f : L +. Y be a given map. We are going to define the (n + 1)-dimensional obstruction set On+l(f ) c Hn+l(K,L ;nn(Y)) as follows. I f fis not n-extensible over K , we define O,+l(f) to be the vacuous set. Now, suppose that f is n-extensible over K . Then there exists an extension g : 8, -+ Y of f. The cohomology class yn+l(g)in Hn+l(K,L; n,( Y))is called an (n + 1)-dimensional obstruction element of f . Then, On+l(f) is defined as the set of all (n + 1)-dimensional obstruction elements of 1. The following proposition is obvious. Proposition 6.1. Homotopic maps have the same (n
obstruction set.
+ 1)-dimensional
+
Next, let (K’, L’) be another cellular pair and : (K’, L’) + ( K ,L ) be a proper cellular map. This map induces a homomorphism +* : Hn+l(K, L;n,(Y)) + Hn+l(K‘,L‘;n,(Y)). Then the following proposition is an immediate consequence of (3.4).
+
Proposition 6.2.
If 1‘= f+ : L’ +. Y , then +* sends O.+l(f) into O.+l(f’).
+
In particular, if ( K ,L) is a subdivision of (K’, L‘) and if is the identity map, then +* is an isomorphisni and sends On+l(f)ontoa subset of O,+l(f’) in a 1-1 fashion. Furthermore, if both ( K ,L) and (K’, L’) are simplicial, then the identity map 4-l is homotopic to a simplicial (and hence proper cellular) map. This implies that +* sends O.+l(f) onto On+l(f‘).Hence, the (n 1)dimensional obstruction set is smaller on a subdivision and is smallest if ( K , L) is a simplicial pair. Thus, we have the following
+
Corollary 6.3. If ( K ,L) is a simplicial pair and f : L +. Y is a map, then Oa+l(f) is a topological invariant, i.e., it does not depend on the triangulation
( K ,L ) . Now, let ( K ,L ) be a triangulable pair without any given triangulation and f : L +. Y be a given map. Because of (6.3),it makes sense to talk about On+l(f). Further, if : (K’, L‘) -+ ( K ,L ) is a map of another triangulable pair (K’, L’) into ( K ,L ) , then one can deduce easily from (6.1) and (6.2) that +* sends On+l(f) into On+l(f’),where f’ = f+ : L’ -+ Y. For the remainder of the section, let us go back to the cellular pair (K, L) and the map f : L -+ Y given a t the beginning of the section. The following
of
+
182
VI. OBSTRUCTION THEORY
two fundamental lemmas are easy consequences of the definition of On+l(f) and Eilenberg's extension theorem (5.1). Lemma 6.4. The
empty.
m a p f : L + Y is n-extensible over K iff On+l(f ) is non-
Lemma 6.5. The map f : L + Y is ( n + 1)-extensible over K iff On+l(f) contains the zero element of Hn+l(K,L ; n , ( Y ) ) .
By recurrent application of these two lemmas, one can easily prove the following Proposition 6.6. If Y is r-simple and
H'+'(K, L ; n r ( Y ) )= 0 for every r satisfying n < r < m, then the +extensibility of f : L + Y over K implies its m-extensibility over K . In particular, if K \ L is of dimension not exceeding m, then the hypothesis of (6.6) implies that a map f : L + Y has an extension over K iff it is n-extensible over K . Hence, we have the following Corollary 6.7. If Y is r-simple and H'+'(K, L;nr(Y ) )= 0 for every r > 1, then every map f : L -+ Y has an extension over K.
7. The homotopy problem Let us study two given maps
f:K+Y, g:K+Y agreeing on L , that is to say, such that f I L = g I L. As described in (I; 5 8), the homotopy problem (relative to L) is to determine whether or not f and g are homotopic relative to L, in other words, whether or not there exists a homotopy ht : K + Y , ( 0 < t < I), such that h,, = f, h, = g, ht I L = f I L for each t E I . The most important special case is that L = o. Then the problem is to determine whether or not two given maps f , g : K + Y are homotopic. Since the homotopy problem is a special case of the extension problem, the obstruction method can be applied. Let us begin by defining the notion of n-homotopy as follows. The maps f and g are said to be n-homotopic relative to L if f 1 Rn and g I En are homotopic relative to L. If f and g are homotopic relative to L, then they are obviously n-homotopic relative to L. Since Y is assumed to be pathwise connected, the following proposition is obvious. Proposition 7.1. Every pair of maps f, g : K + Y with f I L O-homotopic relative to L.
=
g I L are
8. T H E O B S T R U C T I O N dn(f,g;ht)
183
The least upper bound of the set of integers n such that f and g are n-homotopic relative to L is called the homotopy index of the pair ( f , g) relative to L. The following two propositions can be proved as in 3 2. Proposition 7.2. If f II f' and g N g' relative to L, then the pair (f', g') has the same homotopy index relative to L as the pair ( f , 9). Proposition 7.3. I f ( K ,L ) i s a simplicia1 pair, then the homotopy index of any pair of maps f , g : K -+ Y relative to L is a topological invariant.
8. The obstruction dn(f,g;ht) Throughout the present section, we are concerned with two given maps
f,g:K+Y, flL=glL which are ( n- 1)-homotopic relative to L. Let ht
:if"-'+
Y , (0 < t Q l),
be a given homotopy such that ho = f I zn-1,h, = g I and ht I L = f I L for every t E I . Since f and g are defined on En, the construction in fj 4 gives a deformation cochain dn( f , g; ht), which will be called the obstruction of the homotopy ht in connection with the pair ( f , g). Lemma 8.1. The obstruction dn( f , g ; ht) is a relative n-cocycle of K modulo L ; in symbols, a n ( / , g; ht) E Z ~ ( KL ,; n n ( Y ) ) . Proof. Since f and g are defined on En+,, it follows that c"+'(f) = 0 = cn+l(g).Hence, by (4.2) we have Gdn(f, g; ht) = c"+' ( f ) - cn+l(g) = 0. I Lemma 8.2. The homotopy ht has an extension ht* : En + Y , (0 < t Q I), such that ho* = f I En and h,* = g I Efl i f dn( f , g; ht) = 0. Proof. Consider the pair ( J ,M ) , where J = K x I and M = ( K x 0) u ( L x I ) U ( K x 1). Let
J . = M U J" Define a map F :
-f
=
( K x 0 ) U (En-, x I ) U (K x 1).
Y by taking
F(x,t) =
1 f (4,
( X E K , t =O),
(xEE"-',tEI), ( x E K , t = 1).
ht(x),
g(x)
Then F determines an obstruction cocycle cn+l(F) of the complex J modulo M. By the formula (i) of fj 4,we obtain ( i)
kdn(f, g; ht)
=
(-
l)n+'c"+l(F).
Since the homotopy ht has an extension ht* iff the map F has an extension
184
VI. O B S T R U C T I O N T H E O R Y
F* :P+l+ Y , the lemma is an immediate consequence of (3.2) and the preceding formula (i) since k is an isomorphism. I The cocycle dn( f, g; ht) represents an obstruction cohomology class
W f , g ;ht) € H n ( K , L ; n n ( Y ) ) . Analogous to (5.l), we have the following Eilenberg's homotopy theorem. Theorem 8.3. & ( f , g; ht) ( 0 Q t Q l), such that
=0
ie there exists
ho* = f I En, hl* ht* I En-' = ht I
=g
a
homotopy ht* : g n + Y ,
I En, (1 E I).
Proof. Consider the map F defined in the proof of (8.2). Then ht* exists Y such that F* 1 p-1 = F I and hence iff there is a map F* : iff cn+l(F)is a coboundary by (5.1). Since the isomorphism k in (i)commutes with the coboundary operator, this implies the theorem. I
p-1
9. The group Rn(K,L;/) Let f : K + Y be a given map. We are going to construct a group Rn(K,L ; f ) as follows. Consider the space B of all maps of En-1 into Y . Let W denote the subspace of D consisting of the maps g : En-l + Y such that g I L = f I L and let woE W denote the restriction w,,= f 1 zn-l.Then we define RnfK,L ; f ) = n,(W, w,,). An arbitrary element u of Rn(K,L ; f ) is represented by a homotopy ht : Y , (0 Q t Q l), such that
%.-I+
ho = f
I En-1 = h1, h t I L = f l L ,
(0 < t < I ) .
Hence we obtain an obstruction cohomology class & ( f , f ; ht) E H@(K,L ; z n ( Y ) )which clearly depends only on a. The following lemma is an immediate consequence of (4.5). Lemma 9.1.
morphism
The assignment u + En(u) = dn(f,f;ht) defines a homo[ n : Rn(K,L
; f ) +Hn(K, L ; n n ( Y ) ) .
Note. If one removes the hypothesis that Y is n-simple, then En becomes a crossed homomorphism.
Let J f n = J p ( K ,L ; n n ( Y ) )denote the image of Rn(K,L ; f ) under &,. Then J f n is a subgroup of Hn(K,L ; nn(Y)).We shall denote the quotient group by Q", = Qfn(K,L ; n n ( Y ) )= Hn(K,L ; n n ( Y ) ) / J f n . Lemma 9.2. The subgroup 3p (and hence the quotient group Q p ) depends only on the (n- 1)-homotopy class of f relative to L ; that i s to say, if f ,g : K + Y are ( n- 1)-homotopic relative to L , then J f n = Jgn.
10. T H E O B S T R U C T I O N S E T S F O R HOMOTOPY
185
Proof. Let kt : En-, +. Y , (0 Q t Q l), be a homotopy such that k, = f I En-,, k , = g I Kn-1, and kt 1 L = f I L for each t E I . Because of symmetry, i t suffices to prove that J p c J p . For an arbitrary element ci of R n (K , L;f ) , the element tn(ci)E J p is represented by the cocycle dn( f , f ;ht) described as above. Define a homotopy ht* : En-, + Y , (0 Q t < l), by taking
( x E ifn-1,o Q t
( hst-,(x), A, (4 kl-Sf(X),
< #,
( x E E n -1 , i Q t Q # , (xEKn-1,# Q 1). -2 Then ho* = g I = h,* and ht* I L = g I L for every t E I . Hence ht* represents an element @ of Rn(K, L ; g). On the other hand, i t follows from (4.5)that ht*(x) =
8
+
+
dn(g. g, ht*) = - a n ( /, g; At) d n ( f p f ht) ; d n ( f ,g; kt) This implies that &(or) = [n(P) E Jgn. Hence J f n c Jon. I
=
an(/, f ; ht).
10. The obstruction sets for homotopy Let f , g : K -+ Y be two given maps such that f I L to define the n-dimensional obstruction set
=g
I L. We are going
O n ( f pg) c H n ( K ,L ;nn(Y)) as follows. If f and g are not ( n- 1)-homotopic relative to L , we define On(f , g) to be the vacuous set. Now suppose that f and g are (n- 1)-homotopic relative to L . Then there exists a homotopy ht : + Y , (0 Q t Q l), such that h0 -- f IXn-l, h, = g IEn-l, ht I L = f I L , ( t E I ) . The cohomology class dn( f , g ; hl) in Hn(K,L ;n n ( Y ) )is called an n-dimensional obstruction element of the pair (f,g). Then, On( f , g) is defined to be the set of all n-dimensional obstruction elements of ( f , g). The following proposition is obvious. Proposition 10.1.
If f
=f '
and g
N
g' relative to L , then On(f,g)
O Y f ', g').
=
+
Next, let ( K ' ,L') be another cellular pair and : (K', L') + ( K ,L ) be a proper cellular map. This map induces a homomorphism +* : H"(K, L ; n n ( Y ) )-+Hn(K',L ' ; n n ( Y ) ) . Then the following proposition is an immediate consequence of (4.6).
+
Proposition 10.2
If f '
=
f+ and g'
=
g+, then
+*
sends On(f , g) into
O Y f ' , g'). Corollary 10.3. If ( K ,L ) is a simfilicial pair, then On( f , g) i s a topological invariant, i.e. it does not depend on the triangulation of ( K ,L).
Lemma 10.4. T h e maps f and g are ( n- 1)-homotopic relative to L i f On(f,g) is a coset of the subgroup J f n ( K ,L ; n n ( Y ) )in the cohomology group Hn(K,L i n n f Y ) ) .
186
VI. O B S T R U C T O N T H E O R Y
Proof. Since the sufficiency is obvious, it suffices to prove the necessity of the condition. For this purpose, assume that f and g are (n- 1)-homotopic relative to L. Let dn( f , g; ht) and 6n(f , g; h't) be any two n-dimensional obstruction elements of ( f , g). Define a homotopy kt : En-1+ Y , (0 < t < l), by taking kt = h,, if t < 4 and kt = h',-8f if t > 4. Then kt represents an element a of R?(K, L ; 1).By (4.5),we have
&(a) = W f ,g; ht) - W f ,g; rt). Since &,(a)E J p , this proves that On(f, g) is contained in the coset W f ,g; ht) Jf". On the other hand, let a be any element of Rn(K,L ; f ) . Then a is represented by a homotopy kt and .&(a) = S S ( f , f : kt). Define a homotopy h't : En-l -+ Y,(0 < t < l), by taking h't = kpt if t Q 4 and h't = hat-1 if t Z 4. Then we have W f ,g; h't) = M a ) W f ,g; hi).
+
+
Hence every element of the coset 6n( f , g; ht)
+ Jfn is in On(f , g). I
Lemma 10.5. (The fundamental homotopy lemma). The maps f and g arc n-homotopic relative to L if
O n ( f p g )= Jfn(K, L ; n n ( Y ) ) .
L. Then there exists a homotopy hf* : -+ Y , (0 < t < l), such that ho* = f I En, It,* = g I En, and ht* I L = f I L for every t E I . Let ht = ht* I En-1. By (8.3),we have &(f, g : ht) = 0. This implies that On(f, g) contains the zero element ofHn(K, L ; n , ( Y ) ) .According to (10.4),it follows that O.(f,g) = Jfn. Proof. Necessity. Suppose that f and g are n-homotopic relative to
SuBiency. Suppose On(f , g) = Jp. Then On(/, g) contains the zero element of Hn(K,L ;nn(Y)). Hence there exists a homotopy ht : -+ Y , (0 < t < l ) , such that h, = f I En-1, A, = g I En-1,ht I L = f I L for each t E I , and 6*( f , g; ht) = 0. By (8.3),there exists a homotopy ht* : E n -+ Y , (0 < t < l), such that ho* = f I En, h,* = g I En, and ht* I En-2= ht I for each t E I . Hence f and g are n-homotopic. I
11. The general homotopy theorem The following general homotopy theorem is an immediate consequence of (10.4)and (10.5). Theorem 11.1. Any two maps f , g : K -+ Y with f I L = g I L which are (n- 1)-homotopicrelative to L determine a unique element Xa( f , g) of the group Q p ( K ,L ;n n ( Y )). X.( f , g) = 0 if f and g are n-homotopic relative to L.
This element Xn( f , g) will be called the n-dimensional characteristic element of ( f , g). By (10.3), X n ( f , g) is a topological invariant if ( K ,L ) is a simplicia1 pair. By recurrent applications of (1 l . l ) , we obtain the following
12. T H E C L A S S I F I C A T I O N P R O B L E M
187
Proposition 11.2. Let f : K -+ Y be a given map and assume that Y i s r-simple and Qfr(K,L ; n r ( Y ) )= 0 for every Y satisfying n < Y Q m. If two maps f , g : K -+ Y with f I L = g [ L are n-homotopic relative to L , then they are also m-homotopic relative to L. Corollary 11.3. If Y is r-simple and Hr(K, L ; nr(Y))= 0 for every r satisfying 1 < r < dim ( K \ L ) , then every two maps f , g : K -+ Y with f [ L = g [ L are homotopic relative to L . As an application of this general result, let us assume that K is connected and L is vacuous. Take Y = K . Then we obtain the following Proposition 11.4. The following statements are equivalent :
(i)K i s contractible. (ii)nr(K) = 0 for each Y > 1. (iii)K is r-simple and Hr(K; n r ( K ) )= 0 for every r satisfying 1 < r dim K . ( i v ) K i s r-simple and Q f ( K ; n r ( K ) )= 0 for every dim K , where i ; K -+ K denotes the identity map.
Y
satisfying 1 < Y
< <
12. The classification problem Let p : L -+ Y be a given map and let us study the set W of all possible extensions of ,u over K . In symbols,
w = { f : K + Y [ f I L = p ). The maps W are divided into disjoint homotopy classes relative to L . As in ( I ; 8 8 ) , the classification problem for these maps W is to enumerate these homotopy classes by means of some convenient invariants. The relation of n-homotopy relative to L among the maps W divides W into disjoint n-homotopy classes relative to L. For each n >, 1, every ( n- 1)homotopy class relative to L contains a certain collection of n-homotopy classes relative to L . For the classification problem, we have to find a reasonable way to count the n-homotopy classes contained in a given ( n- 1)-homotopy class by means of some homology or cohomology invariant. Throughout the remainder of the section, let 8 be a given ( n- 1)homotopy class relative to L of the maps W . We are going to enumerate the n-honiotopy classes relative to L of the maps W which are contained in 8. According to (9.2), 8 determines a subgroup J B n ( K L , ; nn ( Y ) )of the cohomology group Hn(K,L ; n,(Y)) and hence the quotient group Q O w ,L ; ~ ~ ( Y= )HW, ) L ; ~ ~ ( Y ) ) I J PL(;K~ , ( Y ) ) .
Now let us choose a map f : K Y from the class 8 as our reference map. According to (1 l . l ) , every map g E r3 determines a characteristic element X n ( f , g ) in the group QOn(K, L ; n n ( Y ) ) An . element a of Qen(K,L ; n , ( Y ) ) is said t o be f-admissible if there is a map g E 8 with Xn( f , g ) = a. The --f
I 88
VI. OBSTRUCTION T H E O R Y
f-admissible elements of QenfK,L ;nn(Y)) form a set Afn, which will be . following proposition called the f-admissib2e set in Qen(K,L ;nn ( Y ) ) The is obvious. Proposition 12.1. For any two maps f, g : K -+ Y in the ( n- 1)-homotopy class 8, A f n i s the image of A p under the translation determined by their characteristic element Xn(f, g), that i s to say,
+
A/" = X n ( f , g) Agn. Now we are in a position t o establish the following general classification theorem. Theorem 12.2. Given a n (n- 1)-homotopy class 8 relative to L of the maps W , the n-homotopy classes relative to L of W which are contained in 8 are in a 1 - 1 correspondence with the elements of the f-admissible set A f n in Qe"(K, L ;n n ( Y ) )where , f i s a n arbitrarily given m a p in 8. Proof. According to (11.I), every g E 6 determines an element Xn( f, g) in Afn. We assert that Xn( f, g) depends only on the n-homotopy class relative to L which contains g. In fact, if g, h E 8 are n-homotopic relative to L , then i t follows from (11.l) that
Xn( f , h) -Xn( f , g)
=
X"(g, h)
=
0,
and hence Xn( f, g) = Xn( f, h). Therefore, the correspondence g -+ Xn( f, g) defines a function t of the n-homotopy classes relative to L contained in 8 into the elements of Afn. That t is onto follows from the definition of Afn. It remains to prove that T is one-to-one. Suppose that Xn( f, g) = X n ( f , h). Then we have X"(g, h) = Xn( f, h) -X"( f, g) = 0. According to (1 1. l), this implies that g and h are n-homotopic relative to L. I Remark. Since A f n is, in general, not effectively computable, the general classification theorem given above does not solve the problem. In each particular case, i t remains to compute Afn by special methods.
13. The primary obstructions Throughout $ 9 13-17, we shall assume one more condition, namely, that the given space Y is ( n- 1)-connected for a given positive integer n. According to (V ; 4 4), this means that nr(Y ) = 0 for every 7 < 12. If n > 1, then Y is simply connected and hence n-simple. If n = 1, we assume that Y is n-simple and hence the fundamental group nn(Y )is abelian. Let us consider an arbitrarily given map f:L+Y. According to (6.6),f is always n-extensible over K . Hence the first nontrivial obstruction is the ( n + 1)-dimensionalOn"( f ) . We are going to prove
13.T H E
PRIMARY OBSTRUCTIONS
189
that O n + l ( f ) consists of a single element of the cohomology group P + ' ( K ,L ; n,( Y )) * Let 8 : L + Y denote the constant map 8(L) = yo. According to (11.2), f and 8 are ( n - 1)-homotopic grid hence f is homotopic to a map which sends the ( n - 1)-dimensional skeleton Ln-l of L into yo. Since homotopic maps have the same (n 1)-dimensional obstruction set, we may assume that f (Ln-ll = yo. Since f I Ln-1 = 8 I Ln-l, the difference cochain &( f, 8 ) of 5 4 is defined. By (4.2), &(f, 8) is an n-cocycle of L with coefficients inn,(Y). This cocycle dm( f , 8) can be equivalently defined as follows. Let a be any n-cell of L. Since f (au) = yo, the partial map fo = f I a represents an element [ f O ] of n,( Y). Then d n ( f , 8) is given by [dn(f, 8)] (a) = [f,,]. The cocycle dn( f, 8 ) represents a cohomology class x"( f ) of L over nn(Y), which will be called the characteristic element of f .
+
Proposition 13.1. T h e ( n
of a single element
+ 1)-dimensional obstruction set On+l(f ) consists
o""(f) = 6*x"(f) EHn+l(K,L ; n n ( Y ) ) ,
which i s called the primary obstruction to extending f over K , where 6* denotes the coboundary homomor@hism
6* : Hn(L;n,(Y))+H,+l(K, L ; n , ( Y ) ) . Proof. First, let us prove that 6*x"(f) is in O.+l(f). Since f(Ln-l) = yo, f has an extension f * : i?n + Y such that f *(En \ L ) = yo. Let 8* : if, Y denote the constant map 8*(gm) = yo. Then it follows that the difference cochain dn(f*, 8*) is the trivial extension of dn(f,O ) , i.e., for any n-cell u of K , we have --f
Hence 6*x,( f ) is represented by the cocycle
m ( f * , e*) = C,+yf*) - cn+ye*)
= ,n+yf*).
This implies that 6*xn( f ) = yn+l(f *) E On+l(f ) . Next, let a be an arbitrary element of O.+l( f ). Then there exists an extension g* :En Y of f such that yn+l(g*) = a. By (11.2), g* and - f * are ( n - 1)homotopic relative to L. Hence we may assume that g* [ Ks-1 = f * I By (4.2), we have cm+l(g*) = c"+'( f *) + 6d"(g*, f *). --f
Since dn(g*, f *) is in P ( K ,L ; n , ( Y ) ) according to 5 4, this implies that yn+l(g*) = yn+l(f*). Hence a = 6 * x n ( f ) . I Next, let us turn to the homotopy problem and consider a pair of maps
f , g :K
+Y
, f IL
=g
I L.
VI. OBSTRUCTION THEORY
190
Then their characteristic elements %a(/) and xn(g) in D ( K ;n,(Y)) are uniquely defined and depend only on their homotopy classes. Proposition 13.2. T h e n-dimensional obstruction set O n ( / , g) consists of a
single element
o v , g) E Hn(K,L ; n n ( Y ) )
which i s called the primary obstruction to homotopy relative to L of the pair ( f , g) . Moreover x n ( f )- x n ( g ) = j*wn(f,g),
) Hn(K;nn( Y ) )denotes the h o m o m o r ~ h i s minwhere j * : Hn(K,L ; n n ( Y ) -+ duced by the inclusion m a p j : K c ( K ,L ) . Proof. That On+l( f, g) consists of a single element follows from (13.1) and the definition of the deformation cochain in fj 4.For the second part of the proposition, we may assume that
f(Kn-l) = yo = g(K"-l). Let 0 denote the constant map. Then we have an(/,
e) - dn(g, e) = a n ( / ,g).
This implies that x n c f ) -xn(g)
=
j*wn(f,g). I
Corollary 13.3 For any given map f : K
-+ Y , we
have
JPW, L ; ~ , ( Y )=) 0 , e f w ,L ; ~ ~ ( Y= )H)W , L , ~ ~ c , ( Y H . Finally, if ( K ,L ) is a simplicia1 pair, then wn+l(f),x n ( f ) , o n ( / , g) are topological invariants according to (6.3) and (10.3).
14. Primary extension theorems By means of the primary obstructions, we shall be able to strengthen the general results in fj 6 and obtain an effective solution of the extension problem for the case where K \ L is of dimension not exceeding n 1. An element of the cohomology group Hn(L ;G) is said to be extensible over K if i t is contained in the image of the homomorphism
+
i* : Hn(K;G)+ Hn(L;G). induced by the inclusion map i : L c K . Theorem 14.1. For a given m a p f : L equivalent : (1) f i s (n I)-extensible over K. (2) o n + l (f ) = 0. (3) x*( f ) i s extensible over K .
+
Y , the following statements are
+
Proof. The equivalence of (1) and (2) follows from (6.5) and (13.1), and that of (2) and (3) follows from the exactness of the cohomology sequence. I If K \ L is of dimension n 1, then we have the following corollary which includes Hopf extension theorem (11; fj 8) as a special case. See 3 17.
+
15. P R I M A R Y H O M O T O P Y T H E O R E M S
191
Corollary 14.2. A m a p f : L +. Y i s extensible over K i f its characteristic element x n ( f ) i s extensible over K . The following generalization of (14.2) is an immediate consequence of (6.6). Theorem 14.3. If Y i s r-simple and H r + l ( K , L ; n,(Y))= 0 for every r satisfying n < r < dim ( K \ L), then a necessary and suficient condition for a given m a p f : L --f Y to have a n extension over K i s that the characteristic element x n ( f ) i s extensible over K . In particular, the hypothesis of (14.3) holds if ;tr(Y)= 0 whenever n
15. Primary homotopy theorems The following theorem is a n immediate consequence of the fundamental homotopy lemma (10.5) and the corollary (13.3). Theorem 15.1. T w o maps f , g : K + Y with f I L = g I L are n-homotopic relative to L i f wn(f,g) = 0. If L is empty, then this can be given in the following form: Corollary 15.2. For any two given maps f , g : K -+ Y ,the following statements are equivalent: (1) f and g are n-homotopic. (2) w w ,g) = 0. (3) x n ( f ) = x"(g). If K is of dimension n, then we have the following corollary which includes the Hopf homotopy theorem (11;$ 8 ) as a special case. See § 17. Corollary 15.3. Two maps f , g : K +. Y are homotopic i f x n ( f ) = x n ( g ) . The following generalization of ( 15.3)is an immediate consequence of ( 11.2). Theorem 15.4. If Y i s r-simple and H r ( K ; n,(Y))= 0 for each r satisfying n < r < dim K , then a necessary and suficient condition for a given pair of maps f , g : K +. Y to be homotopic i s that xn(f) = xn(g). In particular, the hypothesis of (15.4) is satisfied if nr(Y) = 0 whenever n
16. Primary classification theorems Lemma 16.1. If Y i s r-simple and H r f l ( K , L ; n r ( Y ) )= 0 for every Y satisfying n < r < dim ( K \ L ) , then, for each m a p f : K + Y and each element a E Hn(K,L ;n n ( Y ) ) ,there exists a map g : K -+ Y such that f I L = g I L and wn(f,g) = a. Proof. Choose a cocycle z E Zn(K, L;n n ( Y ) )which represents a. According to (4.3), there is a map h : g n +. Y such that f I xa-l = h I Zn-l and h) = z. By (4.2\, we have c""(f) -c"+'(h) = W(f,h) = 6~ = 0.
192
VI.
OBSTRUCTION THEORY
Since f is defined over K 3 En+1,we obtain c"+l(f) = 0. This implies that cn+'(h) = 0 and hence h has an extension h* : + Y . By (6.6), one deduces that h has an extension g : K + Y . Then it is clear that f I L = g I L and w n ( f ,g) = a. I If L is empty, then the preceding lemma can be formulated as follows.
z#+l
Lemma 16.2. If Y i s r-simple and Hr+l(K;nr( Y ) )= 0 for every r satisfying n < r < dim K , then, for each element a in H n f K ; n n ( Y ) )there , exists a m a p f : K + Y such that x n ( f ) = a. Now let f : K -+ Y be a given map and let us consider the totality W of the maps g : K + Y such that f 1 L = g I L. Since n,,(Y) = 0 for every m < n, there is only one ( n--1)-homotopy class of the maps W relative to L. We are going to enumerate the n-homotopy classes of the maps W relative to L. Theorem 16.3. If Y i s r-simple and Hr+l(K, L;nr(Y))= 0 for each r satisfying n < r < dim ( K \ L ) , then the n-homotopy classes relative to L of the maps W are in a one-to-one correspondence with the elements of the group H n ( K , L;n,(Y)).The correspondence i s giuen by the assignment g -+ wn(f, g). Proof. By (13.3), we have
Q fn ( KL; , n n ( Y ) )= H (K , L , n n ( Y ) ) . On the other hand, (16.1) implies that every element of Q f n ( K ,L;n,(Y)) is f-admissible. Hence, we have Afn
= H n ( K ,L ; n n ( Y ) ) .
Then, (16.3) follows from (12.2). The second assertion is obvious. I If L is empty, then (16.3) can be conveniently restated as follows. Corollary 16.4. If Y i s r-simple and H r + l ( K ; n r ( Y ) )= 0 for each r satisfying n < 7 < dim K , then the n-homotopy classes of the maps f : K + Y are in a one-to-one correspondence with the elements of the group H n ( K ; nn(Y)). T h e correspondence i s determined by the assignment f -+ xn(f ) . If K is of dimension n, then we have the following corollary which includes the Hopf classification theorem (11; § 8) as a special case. See § 17. Corollary 16.5. The homotopy classes of the maps f : K -+ Y are in a one-toone correspondence with the elements of the group H n ( K ;nn(Y)).The correspondence i s determined by the assignment f -+ x n ( f ) . The following generalization of (16.5) is an irnmediate consequence of (11.2) and (16.3). Theorem 16.6 If Y as r-simple and
H'+'(K, L; n r ( Y ) )= 0
=
H ' ( K , L;nr( Y ) )
for every r satisfying n < r < dim ( K \ L ) , then the homotopy classes of the maps W relative to L are in a one-to-one correspondence with the elements of
EXERCISES
I93
the group H n ( K , L; n,( Y ) ) .The correspondence is determined by the assignment g ma(/, g). In particular, the hypothesis of (16.6) is satisfied if nr(Y)= 0 whenever n < r
17. The characteristic element of Y Throughout this last section of the chapter, we assume that Y is a finite cell complex in addition to the conditions assumed a t the beginning of 5 13. Then, according to 5 13, the identity map L : Y --f Y determined a unique characteristic element x n ( L ) = On(L, 0) y y :nn(y)), where 8 denotes the constant map e( Y ) = yo. This will be called the churucteristic element of Y and will be denoted by x n ( Y ) . For example, if Y is the n-sphere, thenn,(Y) PV 2.It can be easily verified that the characteristic element xn( Y ) is a generator of the free cyclic group H n ( Y ) which depends on the given orientation of Y. Now, consider an arbitrary map f : L + Y . This map f induces a homomorphism
/*: H n ( Y ; n n ( Y ) -) H n ( L ; n n ( Y ) ) .
The following lemma can be proved by means of (4.6) and simplicia1 approximation. Lemma 17.1. x n ( / ) = f * [ x n ( Y ) ] .
This lemma justifies the assertions of $8 14-16 that (14.2). (15.3) and (16.5) include the Hopf theorems given in (11; 5 8).
EXERCISES A. Minor generalizations of the theory.
1. Make use of the characteristic maps X, in (V; Ex. H) and establish the obstruction theory for the case where K is any topologically realized semisimplicia1complex instead of a finite cell complex and L is a subcomplex of K. 2. Study the notion of aCW-complex, [J.H. C. Whitehead41, and generalize the obstruction theory to the case where K is any CW-complex and L is a subcomplex of K . 3. By the use of local coefficients, construct the obstruction theory for the case where Y is not necessarily n-simple. 6. The fundamental group of a remi-simplicial complex.
Let K be a given connected semi-simplicial complex and vo a given vertex of K . Then, as in (11; Ex. A) for finite simplicia1complexes, the fundamental group n,(K, v,,) may be given in terms of generators and relations as follows. A broken line joining a vertex of K to another is a path which consists of
I94
VI. O B S T R U C T I O N T H E O R Y
a finite number of edges of K . Since K is connected, vo can be joined to any vertex v of K by a broken line P f v ) . Suppose that these broken lines PCv) have been chosen and that /3(vo) consists of only a single point v,. To each edge e of K , let us consider the loop
1, = B(eco,)*e-"ecl))l-l which represents a element ge of n , ( K ,vo). Prove that the set { ge I e E K 1 generates n,(K, wo). It remains t o find the relations among the generators { ge }. For each edge e E K , the loop ;Icconsists of a finite number of edges of K ; more precisely, Ae = e"l e%. eep 1
where ad
= f 1 for each i =
1,2;
* *,
1
.
n. Then, deduce the relation
(Re) ge = C: g z * * *gZn". On the other hand, for each 2-cell CT of K,verify the following relation
(%I
Gpd) g O ( l l 2 ) = gU(o72)' Now prove that n l ( K ,vo) can be defined as the group with generators { ge ) and relations { Re } and { R, }. By means of this expression of n , ( K , vo), establish the following results: 1. Condition for 2-extensibility. Let L be a connected subcomplex of K containing vo and f : (L,vo) -+ ( Y ,yo) be a map into a pathwise connected space Y .Then f and the inclusion map i : L c K induce the homomorphisms
f* :n,(L, vo) +n]CY,yo), i* : n,(L, vo) + n , ( K , ~ 0 ) . Prove that the map f is 2-extensible over K iff there exists a homomorphism h:n,(K,vo) -+nl(Y,yo) such that f* = ki,-and that, for any such homomorphism h, there exists an extension g : K a + Y of f such that g, = k. 2. Condition for 1-homotopy. Let us consider any two maps f, g : ( K ,vo) -+ ( Y ,yo). They induce the homomorphisms
f** g* :n1(K,vo) -+n,(Y,Yo). Prove that f and g are l-homotopic iff there exists an element 5 €n,(Y,yo) such that g*(a) = E-'.f*(a).t, a E n l ( K ,vo). For an indication of the proof, see [Hu, 71. V
C. Generalizing the obstruction theory by using Ccch cohomology theory
The obstruction theory studied in this chapter is formulated for finite cell complexes so as to avoid the complications involved in limiting processes. One can extend the theory to more general spaces by using tech cohomology theory as follows. Assume that X is a compact Hausdorff space, A a closed subspace of X , and Y a given ANR. For every finite open covering a = { a,; - ,(lr } of X , the nerve K. of a is a finite simplicia1 complex. The non-void sets A f l at,
EXERCISES
I95
--
(i = 1, , r), form an open covering of A whose nerve La is a subcomplex of K,. With the aid of the bridge theorems in (11;Ex. B), we can formulate an obstruction theory as follows. A map f : A -+ Y is said to be n-extensible over X if it has a bridge u and a bridge map y , : La -+ Y which is n-extensible over K , in the sense of $ 2. Prove the following assertion which gives set-theoretic meaning to the n-extensibility defined above. 1. If X is metrizable with dim (X\A) Q q and if f : A + Y is n-extensible over X , then there exists a closed subspace B of X contained in X \A with dim B < q - n and such that f has an extension g : X \ B -+ Y . [Hu 5 ; p. 3441. Next, assume for sake of convenience that Y is n-simple and let nn = nn(Y).Define the (n + I)-dimensional obstruction set On+l(f ) of a given map f : A -+ Y in the tech cohomology group Hn+l(X,A ; nn)as follows. Iff is not n-extensible over X , define @+I( f ) to be the vacuous set. Assume f to be n-extensible. There exists a bridge 0: with a bridge map ya : La -> Y which has an extension y', : Y . The obstruction cocycle cn+l(y',) of K, modulo La represents an element y"+l(y',) of H%+l(X,A ; n n ) , called an (n I)-dimensional obstruction element of f . Then, @+l( f ) is defined to be the set of all (n 1)-dimensional obstruction elements of f. Homotopic maps have the same (n 1)-dimensional obstruction set. Prove the following assertion from which one can deduce further results as in $5 6, 13 and 14. 2. A map f : A .+ Y is ( n + 1)-extensible over X iff On+l(f ) contains the zero element of Hn+l(X,A ; nn). [Hu 5 ; p. 3461. Now let us turn to the homotopy problem. Two maps f , g : X -+ Y are said to be n-homotopic if there exists a bridge ci for both f and g with bridge maps C$, : K -+ Y and y , : K --t Y respectively such that C$, I Kn = y, I Kn. Prove the following assertion which gives set-theoretic meaning to n-homotopy. 3. If X is metrizable with dim X Q q and f , g : X -+ Y are n-homotopic, then there exists a closed subspace B of X such that dim B < q - n and f I X\B N g I X \ B. [Hu 5 ; p. 3491. Next, assume that Y is n-simple and let n n = nn(Y).Define the ndimensional obstruction set On(f, g), of a pair of given maps f, g : X -+ Y in the tech cohomology group H n ( X ; n n ) and prove the following assertion from which one can deduce further results on honiotopy and classification as in $5 10, 11, 12 and 15. 4. The maps f, g : X + Y are n-homotopic iff On+'( f, g) contains the zero element of H n ( X ;n n ) . [Hu 5 ; p. 3501.
xa* --f
+
-+
+
D. Generalizing the obstruction theory
by using the singular complex
The generalizations of the obstruction theory by using tech cohomology as illustrated in the preceding exercise are very satisfactory except that one
196
VI. O B S T R U C T I O N T H E O R Y
must assume either the domain X or the range Y to be reasonably smooth. To avoid these conditions, one can also generalize the obstruction theory by using the singular complex S ( X ) of the domain X as follows. [Olum 1 and Hu 71. Consider two spaces X , Y and a subspace A of X which might not be closed. According to (V; Ex. I),we have a projection o : ( S ( X ) S, ( A ) )* ( X , A ) .
A map f :A + Y is said to be n-extensible over X if the map fo:S ( A )+ Y is n-extensible over S ( X ) in the sense of Ex. Al. Prove that a map f : A +. Y is n-extensible over X iff, for every map + : ( K ,L) +. ( X ,A ) of a semisimplicia1 pair ( K ,L) into ( X ,A ) ,the map f+ : L -+ Y is n-extensible over K . Then, prove that, if ( X ,A ) is a semi-simplicial pair, the definition of n-extensibility defined here is equivalent to that of Ex. Al. Pick xo E A and yo = f (x,) E Y . Then the map f : A +. Y and the inclusion i :A c X induce the following homomorphisms f* : n , ( A ,xo) + n , ( Y , Yo), i, : n , ( A , xo) * n 1 ( X , xo). Assume X , A , Y to be pathwise connected and prove that f is 2-extensible over X iff there exists a homomorphism h :n , ( X , xo) +n,(Y,yo) such that f + = hi,. [Hu 7 ; p. 1771. Assume Y to be pathwise connected and n-simple with n n = n n ( Y , yo). Define for any given map f : A -+ Y the (n 1)-dimensional obstruction set On+l(f)in the singular cohomology group Hn+l(X,A ; nn)in obvious way, and prove that f is (n 1)-extensible over X iff On+l(f ) contains the zero element of the group Hn+l(X,A ;En). Deduce further results on the extension problem. Now, let us turn to the homotopy problem. Two maps f , g : X -+ Y are said to be n-homotopic if the maps f o , go : S ( X )+ Y are n-homotopic in the sense of Ex. Al. Prove that f, g are n-homotopic iff, for every map #I: K - t X of a semi-simplicial complex K into X , the maps f+, g+ are n-homotopic. Then, prove that, if X is a semi-simplicial complex, the definition of n-homotopy given here is equivalent to that of Ex. Al. Pick xo E X and yo E Y and assume f (xo) = yo = g(xo). Then, the maps f , g : X 3 Y induce the following homomorphisms
+
+
f * , g, 1 % ( X , xo) -+%(Y,Yo). Assume X , Y to be pathwise connected and prove that f , g : X I-homotopic iff there exists an element 5 E ~ , ( Y yo) , such that g*b)
=
-+
Y are
5-'*f*(a)*5, a E n l ( X ,xo).
Assume Y to be pathwise connected and n-simple with n n = nn( Y , yo). Define for a given pair of maps f , g : X -+ Y the n-dimensional obstruction set On(f, g) in the singular cohomology group H n ( X ;nn) in the obvious way, and prove that f , g : X -+ Y are n-homotopic iff On(f,g) contains the zero
I97
EXERCISES
elementof Hn(X; nn). Deduce further results on the homotopy problem and the classification problem. E. The obstruction theory of deformation
Throughout this exercise, let ( Y ,B ) be a given pair and (K,t)a given finite cellular pair. A map f : ( K ,L) + ( Y ,B) is said to be deformable into B if there exists a homotopy f t : (K, L ) + ( Y ,B ) , (0 < t Q l), such that fo = f and fl(K)c B. By the aid of the AHEP, one can easily prove that , in this definition, we may require that f&) = f ( x ) for every x E L and t E I . In order to apply the obstruction method, let us define the notion of ndeformability. Let E n = Kn U L. A map f : ( K ,L)--f ( Y ,B) is said to be n-deformable into B if the partial map f I (En, L ) is deformable into B. A map f is said to n-normal if f ( R n - l ) c B. Hence f is (n - 1)-deformable into B iff it is homotopic (relative to L) to an n-normal map. Hereafter, let us assume that both Y and B are pathwise connected and pick yo E B. Then i t is obvious that every map f : ( K ,L)+ ( Y ,B) has to be 0-deformable into B. Prove that every map f : [ K ,L) + ( Y ,B) is l-deformable into B provided that nl(Y,B, yo) = 0. The latter is equivalent to the condition that the induced homomorphism
i* :n,(B, Yo) - + n d YYo) , is an epimorphism. Next, let n > 1 and assume that ( Y ,B) is n-simple in the sense of (IV; Ex. E).Then the n-th relative homotopy groupnn = nn(Y,B) is abelian and may be used as coefficient group. For each n-normal map f : ( K ,L ) + ( Y ,B ) , define an n-cochain cn( f ) of K with coefficientsinnn in the obvious way. Prove the following assertions, [Hu 9; p. 1941: 1. c"(f) is a cocycle of K modulo L. 2. c @ ( f ) = 0 iff there exists a homotopy f t : K Y , (0 < t < l), such that fo = f , fl[Efi) and t E I. c B, and f&) = f ( x ) for every x E 3. cn(f) is a coboundary of K modulo L iff there exists a homotopy f t : K --f Y , (0 < t Q l), such that f o = f, c B, and f t ( x ) = f ( x ) for every x E En-2 and t E I. Next, define the n-dimensional obstruction set On(f ) in the n-dimensional cohomology group Hn(K,L ; nn)in the obvious way and prove the following assertions : 4. The map f : ( K ,L) + ( Y ,B) is (n - 1)-deformable into B iff O*(f) is non-empty . 5. The map f : ( K ,L) + ( Y ,B) is n-deformable into B iff On( f ) contains the zero element of Hn(K, L ; n n ) . 6 . Every map f : ( K ,L) + ( Y ,B ) is deformable into B if n,(Y, B, yo) = 0 and if, for each n > 1, (Y, B ) is n-simple and Hn(K, L;nn) = 0, where n n == nn(Y,B ) . --f
fl(zn)
19s
VI. O B S T R U C T I O N T H E O R Y
7. Assume that K and L are pathwise connected and xo E L. Then, L is a deformation retract of K iff n,(K, L ; xo) = 0 for every n > 1. Develop this obstruction theory of deformation as far as possible and then generalize this theory as in Exs. A, C and D. In particular, deduce a proof of the following assertion, [Hu 9; p. 2161. 8. If Y and B are ANR's and if B is closed, then the following statements are equivalent : (i) B is a deformation retract of Y . (ii) There exists a homotopy ht : ( Y ,B) + ( Y ,B ) , (0 < t Q l), such that h, is the identity map and h,( Y )c B. (iii)nn(Y,B , yo) = 0 for every n > 1.
F.
(n,n) Let Y be a space of homotopy type (n,n) as defined in (V; Ex. F) with a given point yo E Y, and let K be a finite cellular complex with a given subcomplex L and a given vertex vo E L. If n = 1, we assume that both K and L are connected. The extension, homotopy and classification problems Maps into a space of homotopy type
are completely solved for this special case as follows. 1. The case n = 1. Let f : ( L ,vo) + ( Y ,yo) be a given map. Then f and the inclusion map i : L c K induce the homomorphisms
f* : nl(L, vo) -+n,(Y,Yo), i* : nl(Lvo) +Zl(K, 210). Prove that f is extensible over K iff there exists a homomorphism h : nl(K,vo) +nl ( Y ,yo) such that f* = hi.,, and that, for any such homomorphism h, there exists an extension g : K + Y o f f such that g, = h. For the homotopy problem, let f, g : K -+ Y be two given maps. Without loss of generality, we may assume that f (v,) = yo = g(vo). Then, we have the induced homomorphisms
f*, g* :n1(K vo) -+Zl(Y,Yo). Prove that f and g are homotopic iff f* and g, are equivalent, that is to say, there exists an element 5 i n n , ( Y ,yo) such that g*(a)= t-'.f*(a) . 5 , a €n1(K,yo). For the classification problem, let us consider the set H of all homomorphisms of n , ( K , vl) inton,( Y ,yl). The equivalence relation defined above divides H into disjoint equivalence classes. Prove that the homotopy classes of the maps K + Y are in a natural one-to-one correspondence with the equivalence classes of the homomorphisms H. 2. The case n > 1. Then Y is n-simple and we may write n = n,(Y). The following assertions are special cases of f 14.3), (15.4) and ( 16.4). A map f : L + Y is extensible over K iff its characteristic element x n ( f ) is extensible over K . Two maps f , g : K + Y are homotopic iff x n ( f ) = x n ( g ) .
I99
EXERCISES
The homotopy classes of the maps K + Y are in a one-to-one corespondence with the elements of the cohomology group Hn(K ; n) determined by the assignment f --f xn( f ) . 3. Generalize the preceding results as in Ex. A and Ex. C. G. Homology groups of (n,n)
Let K and L be two semi-simplicial complexes of the homotopy type (n,n) and (T,n) respectively. Then, according to the previous exercise, the classification problem of the maps K -+ L is solved as follows: the homotopy classes of the maps K + L are in a natural one-to-one correspmdence with the equivalence classes of the homomorphismsn + t.If n > 1, then n and t are abelian and therefore these homotopy classes are in a natural one-to-one correspondence with the homomorphisms n + t themselves. An important direct consequence of this result is that K and L are homotopically equivalent iff n and t are isomorphic. It follows that, for any space X of the homotopy type (n,n) and any coefficient group G, the singular homology group H m ( X ; G) and the singular cohomology group H m ( X ; G) depend only on the integers m, n and the groups n,G. They will be called the m-th homology group and the m-th cohomology group of the pair ( n , n ) with coefficients in G, denoted by H m ( n , n G) ; and H m ( n , n ;G) respectively. The integer n will be simply deleted from the notation if n = 1. Similarly, we shall omit the group G if it is the group 2 of all integers. In particular, if n = 1 and G = 2, these are denoted by H m ( n ) and H m h ) and called the m-th homology group and the m-th cohomology group of the group n. Finally, let n = 1 and let G be an abelian group on which n acts as left operators. Choose a space X of the homotopy (n,1). Since the fundamental group of X is n which acts on G, the singular groups H m ( X ; G), H m ( X ; G ) with local coefficients in G are defined. Prove that these groups do not depend on the choice of the space X and hence may be denoted by H m ( n ; G) and H m ( n ;G). H. The complex
K ( n )of a groupn
Let n be a given abstract group written multiplicatively. Define a semisimplicia1 complex K ( R )as follows. [Eilenberg and MacLane I]. 1. The homogeneous definition. For each n > 0, consider the set @n of all ordered sets (xo,* * , x,) of n + 1 elements of n called the n-sets of n. Two n-sets ( x 0 ; . * , xn) and (yo;-*, y,) of n are said to be equivalent if there exists an element x of n such that yt = xxt for each i = 0, * * , n. The n-sets @n of n are thus divided into disjoint equivalence classes, called the n-cells of n. If n > 0, the i-th face of the n-cell 0 = [x0; *,x,] of n,0 < i < n, is defined to be the following (n - 1)-cell
-
-
UQ) =
[x0;
*
a ,
xt-1, X f + l , '
- %,I. *,
EXERCISES
201
([q,n). In this way, n acts on K as a group of left operators. If [ # 1, prove that the homeomorphism [ has no fixed point. Hence n acts freely on K . J. The influence of the fundamental group
Let X be a pathwise connected space and x,, E X a given point as the common basic point for all homotopy groups. Study the influence of the fundamental group n = n l ( X ) upon the structure of homology and cohomology groups of the space X as follows. According to (V; 8.2), X can be imbedded in a space Y of the homotopy type (n,1) such that the inclusion map i : X C Y induces an isomorphism i, : n , ( X ) w n l ( Y ) . Let n > 1 be a given integer such that n,(X) = 0 whenever 1 < m < n. Verify the following assertions: 1. By the exactness of the homotopy sequence, n m ( Y , X ) = 0 for m Q n and 8, : n,(Y, X ) w n m - I ( X ) for m > n. 2. By the relative Hurewicz theorem (V; Ex. C), H,(Y, X ) = 0 form Q n and the natural homomorphism g :nn+l(Y ,X ) + Hn+l( Y , X ) is an epimorphism. 3. By the exactness of the homology sequence, the induced homomorphism i# Hm(X) -+ H,(Y) w Hm(n) is an isomorphism for m < n and is an epimorphism for m 4. In the following commutative ladder * * *
* * *
= n.
-+nn,](Y, X ) 2 a + n n ( X )A+nn(Y)--+
--+
1.
% Hn+,(Y, X ) 2
h i k
H n ( X )2 % &(Y)
--j
-
* *
* *
-
where g, h, k are natural homomorphisms, the spherical subgroup x n ( X ) = Im(h)of H n ( X )coincides with Im(d#) = Key(+). 5. The induced homomorphism
P ( X ) +z#Hm( Y ) w Hm(n) is an isomorphisni for m < n and is a monomorphism for m = n. 6. The subgroup A n ( X ) = Im(i#) of H n ( X ) consists of all elements which annihilate Z n ( X ) . Summarizing these assertions, we obtain the classical theorem : If n l ( X ) w n and n m ( X ) = 0 whenever 1 < m < n, then H m ( X )w H,(n), H m ( X ) H,(X)/C,(X) w H n ( n ) ,
-
H m ( n ) , m < n, nyx)W Hfl(n).
K. Computation of H m ( n ;G ) and H m ( n ;G) Let G be a given n-group; that is to say, G is an abelian group on which az acts as left operators. A n-group G is said to be n-free with { g,, } as a
VI. OBSTRUCTION THEORY
202
n-basis if the elements .$ga for all .$E n and gaE { g.} form a basis of G . A homomorphism f : A +. B of n-groups is called a n-homomorphism i t i t commutes with the operators. A n-group G is said to be n-injective if, for every n-group B and every n-subgroup A of B, every n-homomorphism f : A +. G has a n-extension g : B +. G. For any n-group G, we shall denote by I,(G) the subgroup of G which consists of the elements g E G such that 5g = g for all .$ ~ n and , denote by J,(G) the quotient group of G over its subgroup L,(G) generated by the elements &--g for all .$enand g E G. Prove the following assertions : 1. H,(n;G) w J,(G), H o ( n ; G )M I,(G). 2. If G is n-free, then H,(n; G) = 0 for each m > 0; if G is n-injective, then H m ( n ; G) = 0 for each m > 0. 3. If n = 1, then Hm(n;G) = 0 = H m ( n ; G) for each m > O ; if n is a free (non-abelian) group, then H,(n; G) = 0 = Hm(n;G) for each m > 2. 4. If n is a free abelian group with r generators and if n operates simply on G, then H&; G) = 0 = H m ( n ; G ) , form > r,
~,(nG ; )W
~(1) w ~ m ( nG; I ,
form
-
< r,
where GI denotes the direct sum G + * * + G of j terms of G. Now let n be a cyclic group of finite order r and G a n-group. Let be a generator of n and define a n-homomorphism d : G +. G by taking d(g) = g + tg + + P-lg for each g E G. Denote the image of d by M,(G) and the kernel of d by N,(G). Then M J G ) c I,(G) and L,(G) c N,(G). 5. If n is a cyclic group of finite order r, then for each 9 > 0 we have
---
Hzp+l(n;G) w In(G)/Mn(G)M H'P+'(n; G ) , Hzp+'(n;G) w N,(G)/L,(G) w H2P+l(n;G). In particular,
H,p+l(n) M n M H2P+2(n), H,p+2(n)= 0
=
HQ+'(n).
6. Universal coefficient theorem. If n operates simply on G, then
Hm(n;G) R+Hm(n)8 G
+ Tor (Hm-l(n),G ) ,
H m ( n ; G) w Hom (Hm(n),G) + Ext ( H m - l ( n ) G). , 7. Kunneth relations. For the direct product n x t of two groups n and t, we have H m ( n x t) = Hp(n)@Hq(t) 2 TOY( H p ( n ) H , q(d). P+q=m
+
fi+q = m-1
The assertions 4-7 allow for a complete computation of the groups
H,(n; C) and H m ( n ; G) for a finitely generated abelian group n which operates simply on G.
EXERCISES
203
L. The Eilenberg-MacLane complex K ( n , n) Let n be a given abelian group written additively and n a positive integer.
Define a semi-simplicial complex K(n,n ) as follows. An m-cell u of K(n, n) is an &-cocycle of the unit, m-simplex A , with coefficients in n. To define the i-th face a(') of u, 0 < i < m, consider the cellular map d, :Am-l + A m as defined in [E-S; p. 1851. It induces a homomorphism (efm)# : Zn(A, ;n) + Zn(Am ;n). Define u(f)= (dn,)#a.If m > 1 and i < j , verify the condition (SSC).Thus we obtain a semi-simplicial complex which is called the Eilenberg-MacLane complex K ( n ,n), [Eilenberg-MacLane 11. Prove that K ( n , 1) is essentially K ( n ) and that K ( n ,n) is a space of the homotopy type (n,n). M. The influence of nn(x)
Let n > 2 be a given integer and X an ( n - 1)-connected space. Study the influence of n = n n ( X ) upon the structure of homology and cohomology groups of X as follows. According to (V; 8.2), X can be imbedded in a space Y of the homotopy type (n,n) such that the inclusion map i : X C Y induces an isomorphism i, : n n ( X )w n , ( Y ) . Let q > n be a given integer such that n,(X) = 0 whenever n < m < q. Verify the following assertions as in Ex. J : l.n,(Y,X) = O f o r m < q a n d d , : n , ( Y , X ) =nm-](X) f o r m > q . 2. H,( Y, X )= 0 for m < q and the natural homomorphism g : Y,X) + Hg+,(Y,X )is an isomorphism. 3. The induced homomorphism
i# : H m ( X )+ Hm(Y) M Hm(n.n) is an isomorphism for m < q and is an epimorphism for m 4. In the following commutative ladder * * *
-+ng+l(Y)
I
A,3dq+](Y,X+) . a
1
n * ( X )>+
= q.
%(Y) -+
1 " P
''*
- - -.- - - t H g + ] ( Y ) 1 # , H * , ] ( Y , x )a#+Hg(X)E#,Hg(Y)-+..* where f , g , h, k are natural homomorphisms, the spherical subgroup & ( X ) = Im(h) of H p ( X ) coincides with Im(d#) = Ker(z#). 5. The induced homomorphism z# : H m ( Y ) + H m ( X ) is an isomorphism for m < q and is a monomorphism for m = q with A g f X ) as image. Summarizing the assertions (1) - ( 5 ) , we obtain Hm(X)M H,(n, n),Hm(X) M Hm(n,n), m < q, H 4 ( X ) / Z g ( X M) Hgfn,n ) , & ( X ) M H*(n,n).
204
VI. O B S T R U C T I O N T H E O R Y
6. The composed homomorphism
d*g-lj# : Hg+1(n,4 + n g ( X ) , considered as an element of Hq++l(n,n ;n p ( X ) )does , not depend on the choice of Y and is called the invariant &+l(X) of the space X . The following conditions are equivalent: (i) &+'(X) = 0. (ii) h :n g ( X )-+ H g ( X )is a monomorphism. (iii) i# : H g + , ( X )-+ Hg+l(n, n ) is an epimorphism. 7. If q = n + 1, then h :n g ( X )+H,(X) is an epimorphism by (V; Ex.C). This implies that Hn+l(n,n) = 0. Furthermore, K",+2(X)= 0 iff h is an isomorphism.
CHAPTER V I I C O H O M O T O P Y GROUPS
1. Introduction Let ( X ,A ) and ( Y ,B ) be any two pairs each consisting of a space and a subspace. As in ( I ; ij 8),we denote by the symbol
n ( X ,A ; Y , B ) the set of all homotopy classes of the maps of ( X ,A ) into ( Y ,B) relative to { A , B }. As described in (I; 5 8),n ( X ,A ; Y , B) is a functor contravariant in ( X ,A ) and covariant in (Y, B). If X is the n-sphere Sn, A consists of a single point so of Sn and B consists of a single point yo of Y , then n ( X ,A ; Y , B ) is the underlying set of the n-th homotopy group n n ( Y , y o ) of Chapter IV. As shown in the preceding two chapters, the group operation in Itn( Y ,yo) is very helpful in studying the extension and classification problems especially for the maps of Sn into Y . On the other hand, if Y is the m-sphere Sm and B consists of a single point so of Sm, then n ( X ,A ; Y , B ) will be simply denoted by nm(X,A ) and called the m-th cohomotopy set of ( X , A ) . Under suitable conditions on ( X ,A ) , we shall see in 5 5 that nm(X,A ) forms an abelian group which will be called the m-th cohomotopy group of ( X ,A ) .This group operation was first sketched by Borsuk [3] and later studied in detail by Spanier [l]. It is useful in studying the extension and classification problems for the maps of X into Sm. Between the homotopy groups and the cohomotopy groups, there is an informal duality given in 3 12 by means of the usual composition operation. Precisely, for every pair (m,n) such that nn(X,A , xo) and nm(X,A ) are abelian groups, the composition operation yields a homomorphism
nn(X,A , xo) @ n m ( X A , ) +nn(Sm, so). In particular, if m = n , then the right side can be replaced by the additive group 2 of integers.
2. The cohomotopy set n m ( X , A )
+
Let Sm denote the unit m-sphere in the ( m 1)-dimensional euclidean space and let so denote the point (1,O; * * , 0). Then, as described in the introduction, the m-th cohomotopy set nm(X,A ) of a pair ( X , A ) is defined to be the set of all homotopy classes of the maps of ( X ,A ) into (9, so) relative to A . I t is a contravariant functor in ( X , A ) . If A is empty, then it will be 205
206
VII. COHOMOTOPY GROUPS
simply denoted by nm(X) and called the m-th cohomotopy set of the space X . The set nm(X, A ) has an exceptional element, namely, the homotopy class of the constant map 0 : X + so. This exceptional element will be denoted by the symbol 0 and called the zero element of nm(X, A ) . Since { so } is a closed set of Sm, one can easily see that
n m ( X ,A) = n"(X, A ) , where A denotes the closure of A in X. Thus, we may assume that A is a closed subspace of X ; however, we will not so assume unless explicitly mentioned. Finally, nO(X, A ) can be considered as the set of all open and closed subspaces of X not meeting A . If so considered, then the zero element 0 of nO(X, A ) is the empty subspace of X.
3. The induced transformations Sincenm(X, A ) is a contravariant functor in (X, A ) , every map f : (X, A ) + (Y, B ) induces a transforniation
f * :nm(Y, B) +nm(X, A ) , called the induced transformation of f on the m-dimensional cohomotopy sets. For any element a in nm(Y, B ) , choose a representative map : (Y, B ) + (Sm, so), then f *(a)is represented by the composed map+f : ( X ,A ) + (Sm,s0). The zero element of nm(Y, B ) is evidently mapped into that of nm(X, A ) by f * ; in other words, f * preserves the zero element 0. Furthermore, f * depends only on the homotopy class off relative to { A , B }. In the remainder of this section, we shall prove some important properties of the induced transformations. Let us consider a pair ( X ,A ) , where the subspace A of X is non-empty. Identifying A to a point q A , we obtain a space XA. Let
+
f
:
(x, A)
+
(XAS qA)
denote the natural projection. Then f is a map which sends X A \ morphically onto XA \ q A .
homeo-
Lemma 3.1. The induced transformation f* carries n m ( X ~g, ~ )onto nm(X, A ) in a one-to-onefashion.
C#J
Proof. Let a ~ n m ( XA, ) be an arbitrary element represented by a map : ( x , A ) -+ (Sm,so). We may define a map y : (XA,q A ) + (Sm, so) by
+
The continuity of y follows from that of by (I; 8 6 ) .The map y represents an element p of n m ! X ~q,A ) . Since yf = #, it follows that f*(p) = a. Hence f* is onto.
3.
T H E INDUCED TRANSFORMATIONS
Next, let 6,q : ( X A ,q A ) + (Sm, so) be two maps such that homotopic relative to A . Then there exists a map
207
tf
and qf are
F : ( X x I,A x I) + (Sm, so) such that F ( x ,0) = &) and F ( x , 1) = q f ( x ) for every x E X . Define G : ( X A .X I , q A x I ) -+ (Sm, so) by taking
Then G is continuous and gives a homotopy between t and 7 relative to q A . This proves that f* is one-to-one. I The significance of (3.1) is that, in the computation of the cohomotopy sets of a given pair (X, A ) , we may assume that A consists of a single point a if A is non-empty. For the case m > 1, since S m is simply connected, one can prove without difficulty that the inclusion map i : X c (X, a) induces a one-to-one transformation i* of nm(X, a) onto nm(X)provided that a certain homotopy extension properties are satisfied. In particular, these homotopy extension properties are satisfied if X is a paracompact Hausdorff space and a is any given point of X. Theorem 3.2. (The map excision theorem). If f : (X, A ) + ( Y ,B) i s a relative homeomorphism, that is to say, f maps X \A homeomorphically onto Y \ B, and if X is a compact space, A is non-empty, Y is a regular Hausdorf s$ace and B is closed, then the induced transformation f * sends nm(Y, B ) onto nm(X,A ) in a one-to-one fashion. Proof. Since A is non-empty, so is B. It follows that we may identify A and B to single points q A and q E respectively. Let
(XA, q A ) , v : ( y , B ) (YE, q E ) denote the natural projections of identification. One can easily verify that f induces a one-to-one map g : ( X A , q A ) * (YE, q E ) of (XA,q A ) onto (Ye, 46) such that qf = g t . Since X is compact, so is X A . Since Y is a regular Hausdorff space and B is closed, it follows that Y Eis a Hausdorff space. Hence, g is a homeomorphism. By a property of contravariant functors, we have f*q* = l*g*. Since g is a homeomorphism, i t is obvious that g* is a one-to-one correspondence. On the other hand, [* and q* are one-to-one correspondences according to (3.1).Therefore, f * = t+g*q*-lsendsnm(Y, B ) ontonm(X, A ) in a one-to-one fashion. I The auxiliary conditions in (3.2) are essential. In fact, we have the following counter examples. First, let Y denote the unit m-cell of the euclidean m-space and B its boundary (m- 1)-sphere. Let yo, y1 E B where y o # yl. Let X = Y \ yo
t :(x,A )
+
+
VII. COHOMOTOPY G R O U P S
208
and A = B \yo. Then clearly there is a deformation retraction of ( X , A ) into the pair (yl, yl). Therefore nm(X,A ) = 0 although nm(Y,B) m 2.This shows that the induced transformation i* :nm(Y ,B) -+nm(X,A ) of the inclusion map i : ( X ,A ) c ( Y ,B ) , which is a relative homeomorphism, cannot be one-to-one. In this example, X fails to be compact. Next, let Y denote the unit ( m + 1)-cell of the euclidean ( m + 1)-space and X the boundary m-sphere. Let A consist of a single point of X and let B = ( Y \X) U A . Then nm(X,A ) m 2 while nm(Y,B) = 0. This shows that the induced transformation i* :nm(Y , B ) + f l ( X , A ) of the inclusion map i : ( X ,A ) c ( Y ,B ) , which is a relative homeomorphism, cannot be onto. In this example, B fails to be closed. Corollary 3.3. (The excision theorem). Let ( X ,A ) be a puir where X i s a compact Hausdorfl space and A is a closed subspace of X . If V is any open set oi X contained i n A , then the induced transformation e* of the inclusion map e:(X\V,A\V)c (X,A) carries nm(X, A ) onto nm(X \ V ,A \ V ) in a one-to-one fashion. Proof. Since e is a relative homeomorphism, (3.3) follows from (3.2) if A \ V is non-empty. If A \ V is empty, then A = V and hence A is both open and closed in X . Therefore, every map I# : X A \ + Sm has a unique extension y : ( X ,A ) -+(Sm, so). This proves that e* is a one-to-one correspondence. I
Remark. The condition in (3.3)that X is a compact Hausdorff space can be removed if we assume that the closure of V is contained in the interior of A . The verification is left to the reader.
4. The coboundary operator In the present section, we are going to construct a coboundary operator 6 for cohomotopy sets which will be analogous to the coboundary operator for cohomology groups. Unfortunately, this coboundary operator 6 is not defined for a completely arbitrary pair ( X ,A ) . Since, in the construction of 6, one has to use some forni of Tietze's extension theorem, it is necessary to assume a normality condition on the pair ( X ,A ) . By a binormul pair ( X ,A ) , we mean a binormal space X , (I; 8 9),and a closed subspace A of X . Then, by definition, both X and X x I are normal spaces. In particular, ( X ,A ) is a binormal pair if X is a semi-simplicial complex and A is a subcomplex of X , or more generally, if X is a paracompact Hausdorff space and rl is a closed subspace of X . Throughout the present section, we assume that (X, A ) is an arbitrarily given binormal pair. Let Sm+l denote the ( m + 1)-sphere obtained by joining Sm to two points, the north pole and the south pole. Denote by ET+l and E?+l the north and the south hemispheres respectively. Then we have Sm+l Em+l U Em+l s o ~ S m= Em+l + n Em+l. ~
4. T H E C O B O U N D A R Y O P E R A T O R
209
Consider the following diagram
n ( X ,A : Sm+', so)
n(A:S m )
Iy
i'
n ( X ,A ; E?+l, Sm) y-.+n ( X ,A ; Sm+l, E?+l) where the transformations are natural, namely, a is defined by restriction while t9 and y are induced by inclusions. Since E?+l, ET+l are contractible to the point so and (X, A ) is a binormal pair, it is straightfonvard to verify that both u and y are one-to-one correspondences. Thus, we obtain a transformation
6 = y-1pu-1: n y A ) + n m + l ( X , A ) , which is called the coboundary operator. Geometrically, the coboundary operator 6 can be determined as follows. Let e € @ ( A ) . Choose a map : A + Sm which represents e. Since ET+l is contractible,
+ + has an extension
+# : ( X ,A ) + (ET+l,S m ) . Then, +# represents u-l(e). Let tt
: (EY+l,Sm) + ( S m + l , Etn+1), 0
Q 1,
be a homotopy such that tois the inclusion map and 6 , maps S m into so and ET+1\ Sm onto Sm+l\ so honieomorphically. I t is obvious that such a homotopy & exists and that El is essentially unique, i.e., unique up to a homotopy relative to Sm. Then the map tl+#: ( X , A ) + ( S m + l , so) represents the element 6(e) = y - l P u - l ( e ) . Obviously, 6 sends the zero element of n m ( A ) into that of f l + l ( X ,A ) ; in other words, 6 .preserves the zero element. Now let ( X ,A ) and ( Y ,B) be two binormal pairs and f : (X, A ) + (Y, B) a given map. Let g : A + B denote the map defined by f . Since 6 = y-lpa-' and since a, 8, y obviously commute with the induced transformations, we obtain the following Proposition 4.1. The commutativity relation f*6 = 6g* holds in the rectangle:
nm(A) 2nm+1 (XP A )
18 n m ( B ) A+
n m + 1 ( Y ,B).
5. The group operation in n m ( X ,A ) For convenience in using the obstruction method, we shall assume throughout this section that ( X ,A ) is a cellular pair, that is to say, X is a finite cell complex and A is a subcomplex of X.Generalizations to more
VII. COHOMOTOPY GROUPS
210
general pairs will be indicated in the exercises a t the end of the chapter. A cellular pair ( X ,A ) is said to be n-coconnected if the cohomology group Hq(X,A ; G) = 0 for every integer q > n and every coefficient group G. In particular, ( X ,A ) is n-coconnected if X \A is of dimension less than n. However, the latter condition is undesirable because dimension is not a homotopy invariant. By the universal coefficient theorems, it follows that (X, A ) is n-coconnected iff the integral cohomology group Hq(X,A ) = 0 for every q n. For any given pair ( Y ,yo) consisting of a space Y and a point y o € Y, consider the set n ( X ,A ; Y , yo) of all homotopy classes of the maps ( X ,A ) -+ ( Y ,yo). Then every map f : ( Y ,yo) +. (Y',Y ' ~ induces ) a transformation f* :n ( X ,A ; Y , yo) +.n(X, A ; Y ' , Y'o) by means of composition. In particular, let us consider a triplet ( Y ,B, yo) where Y and B are pathwise connected. Then the inclusion map i : ( B ,yo) c ( Y ,yo) induces a transformation i* :n(x,A ; B, yo) -+n(x, A ; Y , yo). We recall that ( Y ,B ) is said to be n-connected if nq(Y,B) = 0 for every q < n. Then the following lemma can be easily established by the obstruction method described in (VI; Ex. E). Lemma 5.1. If ( X ,A ) is n-coconnected and ( Y ,B) is n-connected, then i, sends n ( X ,A ; B , yo) onto n ( X ,A ; Y ,yo) in a one-to-one fashion. Throughout the remainder of this section, let m be any given positive integer and assume that ( X ,A ) is (2m- 1)-coconnected. Under this assumption, we shall define a group operation in nm(X,A ) . For this purpose, let us consider the triplet Y = Sm x Sm, B = S m V S m , yo = (so, so). Then, by (IV; 3.4), ( Y ,B) is (2m- 1)-connected and hence i, is a one-toone correspondence according to (5.1). Next, consider the map j : ( B , y o )-+ (Sm, so) defined by j ( s , so) = s = j(so, s) for every s in Sm. Then j induces a transformation j , :n ( X ,A; B, yo) +.nm(X, A ) . Finally, we shall define a transformation h, :nm(X,A ) x nm(X,A ) - + n ( X A , ; Y , yo) as follows. Let (a,/?) be any pair of elements in nm(X,A ) . Choose representative maps +, y : ( X ,A ) -+ (Sm,so) for a, /? respectively. Then the map 4 x y : ( X ,A ) +. ( Y ,yo) represents an element of n ( X ,A ; Y , yo) which obviously depends only on the ordered pair (a,p). We define h,(u, p) to be this element. Composing these three transformations, we obtain a transformation
K
= j,i+'h,:
n m ( X ,A ) x n m ( X ,A ) - + n m ( XA, ) .
5. T H E
GROUP OPERATION IN
n m ( X ,A )
211
For any pair (a,p) of elements in'zm(X, A ) , the element k ( a , 8) of n m ( X , A ) will be called the s u m of a,@ and denoted by a p.
+
Theorem 5.2. If ( X ,A ) i s a (2m- 1)-coconnected cellular pair, then @ ( X , A ) forms a n abelian group with the addition a + p as its group operation and 0 as its group-theoretic neutral element. If a ~ n m ( XA, ) i s represented by a map+ : ( X ,A ) +. (Sm, so), then its negative - a i s represented by the composed m a p 7+, where r : (Sm, so) -+ ( S m , so) is a n arbitrary m a p of degree - 1. Proof. (a)Commutativity. Let us consider the homeomorphism : (Y, yo) + (Y, yo) defined by t ( s , t ) = ( t , s) for every pair of points (s, t ) in Sm. Since 5 maps B onto itself, it defines a homeomorphism q : (B,yo) -+ ( B ,yo).
Obviously we have iq commutative:
=
E i and jq
=
j . Therefore, the following diagram is
I"*
it*
n(x,A ; Y,yo) A n ( X ,A ; B, yo) >nm(X,A)
n ( X ,A ; Y ,yo) + b z ( X , A ; B, Yo) Let a, B be any two elements of n m ( X ,A ) . It follows from the definition of h, that E*h*(a*B) = h*(BS a). Hence
@ -ta
j*i;lh*(B, a) = j,<&*(a, B) = j*q& h,(a, B) = j,i;'h,(a,B) =
=
a
+ @.
(b) Associativity. Consider the triplet (2,C, zo), where
z = s m x Sm x
Sm,
c =s m v s m v s m ,
zo = (so, so, so).
Then, by (I; Ex. S), it is easy to verify that ( Z ,C) is ( 2 m - 1)-connected and hence, by (5.1), the inclusion map x : (C, zo) c (2,zo) induces a oneto-one transformation x* :n(X, A ; C, zo) - + n ( X ,A ; Z, zo). Let A : (C, zo) -+ (Sm, so) denote the map defined by (if s, = so = s3), (if s1 = so = s3), A(s1, s,, s3) = s2, sl' (if s1 = so = sz). s3
1
1
Then A induces a transformation
A* :n ( X ,A ; c,zo) + n m ( X , A ) .
Now, let a, @, y be any three elements of n m ( X , A ) and choose representative maps +> y,x : ( X ,A ) (Sm,so) -+
for a,P, y respectively. Then the map
#x
ly
x X represents an element
VII. C O H O M O T O P Y G R O U P S
212
,u*(a, /?,y) of n ( X ,A ; 2, z,) which depends only on the triple ( a , /?,y ) . Thus we obtain an element A,x&u,(a, @, y ) of nm(X,A ) . We are going to prove that For this purpose, let p : (2,zo) -+ (Y, yo) denote the map defined by Since p sends C onto B, it defines a map 8 : (C, zo) + ( B ,y o ) .Then the rectangle
p(s,, s2, s3) = (sl, s,).
n ( X ,A ;2, z0) +% n ( X ,A ; C , z0)
I
I
J
J
n ( X ,A ; Y , yo) A n ( X ,A ; B, yo) is commutative. Furthermore, by the definition of p, i t is clear that p*,u*(a*@> y ) = h*(% B). Choose a map f : ( X ,A ) + (C, zo) representing the element x;l,u*(a, /?,y ) . Then Af represents theelement &x;lp*(a, /?,y ) and j0f represents the element a + /?.Let w : (2,zo) -+ (Sm, so) denote the map defined by w(sl, s, s3) = s3. Then the map wf clearly represents y. Since f sends X into C, it is easy to see that j0f x wf carries ( X , A ) into ( B , y o ) and that j ( j 0 f x w f ) = Af. This implies that (a + @) + y = A*xil,u*(a,/?,y ) . Similarly, we can prove a + (/? + y ) = I,x;l,uu,(a, @, y ) . This completes the proof of associativity. (c) Existence of neutral element. Let a be any element of n m ( X , A ) and choose a representive map+ : (X, A ) + (Sm, so) of a. Let [ : (X, A ) + (Sm, so) denote the constant map [ ( X ) = so. Then x 5' sends ( X ,A ) into (B,yo) and clearly j(+ x 5) = This implies that a + 0 = a for everya € n m ( X , A ) . (d) Existence of negatives. Let a ~ n m ( XA, ) be represented by : (X, A) + (Sm, so) and let r : (Sm, so) + (Sm, so) be any map of degree - 1. Then r+ represents an element /? ~ n m ( XA,) which depends only on a. We are going to prove that a + /? = 0. Without loss of generality, we may assume that r is defined by
+
+.
r(xo,' ' * ,
+
xm-1, xm) = (xo,.
-
->
xm-1, -xm).
Let ET and Etn denote the hemi-spheres of Sm defined by xm > 0 and xm < 0 respectively. Then r maps E$ onto Em and E!? onto E$. Let ht : (Sm, so) -+ (Sm, so), 0 < t < 1, be a homotopy such that ho is the identity map and hl(E?) = so. Then hl+ represents a and h,r+ represents /?. Let M , = ++(E?) and M , = +-l(Etn),then we have
A c M , n M,, hl+(M2) = so = hlr+(Ml). Hence the map hl+ x h,r+ sends X into B and the composed map y = j(h,+ x h,r+) represents the element a + /?.Furthermore, it is obvious that y(x) = h,+(x) if x E M I andy(x) = h,r+(x) if x E M,.Then, define a homotopy yt : ( X ,A ) + (Sm, so), 0 < t < 1, by taking
X
= MI u M,,
5. T H E G R O U P O P E R A T I O N
IN
n m ( X ,A )
+
Then yo also represents u /I. Since h, is the identity map, it follows that yo sends X into ET and hence yois homotopic (relative to A ) with the constant map e ( X ) = so. This proves that u j3 = 0. I For the special cases m = 1 or 3, Sm is a topological group with so as neutral element. Then there is a natural multiplication in the cohomotopy set n m ( X , A ) of any pair ( X , A ) defined as follows. Let u,j3 be any two elements of n m ( X ,A ) represented by 4, y : ( X ,A ) -+ (Sm,so) respectively. Let X : ( X ,A ) --f (Sm,so) denote the map defined by X ( x ) = + ( x ) y ( x ) . Then X represents an element [ X I of nm(X,A ) which obviously depends only on u and j3. This element [XI will be called the product of u and j3, and will be denoted by up. Obviously, n m ( X ,A ) forms a group with this multiplication as group operation.
+
Proposition 5.3. If m = 1 or 3 and if ( X ,A ) i s a ( 2 m - 1)-coconnected cellular +air, then uj3 = u + j3 for a n y pair (a,j3) of elements of n m ( X ,A ) .
Proof. Let ,u : ( Y ,yo) +. (Sm,so) denote the map defined by p(s, t) = st for any pair (s, t) of elements in the topological group Sm. By (5.1), there exists a homotopy f t : ( X ,A ) +. ( Y ,yo), 0 < t < 1, such that fo = x y and f l ( X )c B, where 4, y are representative maps of u,j3 respectively. Then ,ufo represents cij3 and j f l represents u j3. Since f l ( X )c B, it follows that pf, = ifl. Hence the homotopy ,uft implies that aj3 = u + j3. I In the remainder of the section, we shall establish that the induced transformations and coboundary operators are homomorphisms.
+
+
Proposition 5.4. If ( X ,A ) and ( X ' , A') are two ( 2 m - 1)-coconnected cellular pairs and if f : ( X ,A ) + ( X I ,A ' ) i s a map, then the induced transformation f * : @(XI, A') +.nm(X,A ) i s a homomor#hism.
Proof. Let U ,/?beany two elements ofnm(X', A ' ) and choose representative maps y : ( X ' ,A ' ) + (Sm, so) for u,j3 respectively. By (5.1) there exists a homotopy gt : ( X ' , A') + ( Y ,yo), 0 < t Q 1 , such that go = x y and g,(X) c B. Then jg, represents u + j3 and hence j g l f represents f * ( u + j3). On the other hand, since+/, yf represent /*(u), f*(B) respectively and since go/ = +f x y f , it follows that j g l f also represents /*(a) + /*(/I). Hence / * ( a j3) = /*(a) /*(PI. I
+,
+
+
+
Proposition 5.5. If ( X ,A ) i s a ( 2 m + 1)-coconnected cellular pair such that A is ( 2 m - 1)-coconnected, then the coboundary operator 6 :nm(A) -+nmtl ( X ,A ) is a homomorphism.
Proof. Let a, @ be any two elements of @ ( A ) and choose representative maps 4, y : A +Sm respectively for u,B. By (5.1), there exists a homotopy
214
VII. COHOMOTOPY GROUPS
=+
g t : A + . S m x Sm, 0 < t < 1, such that go x y and g l ( A ) cS m V S m . Then j g , : A +.Sm represents the element a B. Since ET+1 v ET+1 is contractible over itself to (so, so), g , has an extension
+
v ET+l,Sm v Sm).
gl# : ( X ,A ) +. (ET+l
Let j # : ET+l v E T + l + . ET+l denote the map analogous to j : Sm 'd Sm + Smn; then j#gl# is an extension of jgl. Hence, Elj#gl# represents 6(a+B). Let pi : ET+l x ET+l +. ET+l, (i = 1,2), denote the projection defined by P<(yl,y e ) = yt. Then the map plgl# : ( X ,A ) -+ (ET+l,Sm) is an extension p1gl which obviously represents a. Similarly, Pegl# is an extension of psgl which represents @. Therefore, tlplgl# represents 6(a)and [l#agl# represents
s(B)' Let
p : (ET+1V ET+1, Sm
v Sm)
denote the map defined by Y ( y , , ys)
=
+.
v Sm+l, (so, so) )
(Sm+l
(El(yl),t1(y8)). Then we have
51P1g,# x 51Psg,# = R l # . Let J :Sm+1v Sm+l+. Sm+l denote the map analogous to j : Sm Then the map J?Pg,# represents 6(a) S(B). Since clearly JYgl# = [,j#g,#, i t follows that
+
&a
v Sm + Sm.
+ B) = 6(a) + S(B). I
6. The cohomotopy sequence of a triple B y a binormal triple ( X ;A , B ) , we mean a binormal space X and two closed subspaces A and B such that A 3 B. Throughout the present section, let ( X ,A , B) be a given binormal triple. Then ( X ,A ) , ( X , B) and ( A ,B ) are binormal pairs. By 9 3, the inclusion maps i : ( A ,B) c ( X ,B ) and j : ( X , B ) c ( X ,A ) induce transformations i* :n m ( X , B) +.nm(A, B ) , j* :n m ( X ,A ) +.@ ( X ,B) defined for every m 0. On the other hand, the inclusion map k :A c ( A , B ) induces a transformation k* : n m ( A ,B) +.nm(A).
If we compose k* with the coboundary operator 6 :n m ( A ) +nm+l(X, A ) , we obtain a transformation 6* :nm(A,B ) +.nm+l(X,A ) which will be called the coboundary operator for the triple ( X ,A , B). Thus, 6* = 6k*. Therefore, we obtain the following sequence of sets and transformations :
-
5 + ( A , B ) c+ n l ( X , A )z+n l ( X ,B ) n m ( X , A )Z+ nm(X, B) X+ nm(A.B ) T-+nm+l(X,A ) G
n?(X,A) ?+# ( X , B ) *
-
*
d*
*
--
* *
6. T H E
COHOMOTOPY SEQUENCE OF A TRIPLE
21.5
which will be called the cohomotopy sequence of the triple ( X ,A , B). In particular, if B = n, it is called the cohomotopy sequence of the pair ( X ,A ) . As usual, we define the image of the transformation S* : n m ( A , B) -+ nm+'(X, A ) to be the set Im(S*) = 6*(nm(A,B ) ) and the kernel Ker(S*)of 6* to be the inverse image S*-l(O) of the zero element 0 of nm+l(X, A ) . Analogously, one may define the images and the kernels of i* and j*. This being done, i t makes sense to ask whether or not the cohomotopy sequence is exact in the usual sense, namely the image of each transformation should coincide with the kernel of the subsequent transformation. As usual, to prove its exactness, one has to prove the following six statements:
I m (j*) c Ker (i*), I m (i*) c Ker (S*), I m (6*) c Ker ( j * ) , I m ( j * ) 3 K e y (i*), I m (i*) =- Ker ( S*) , I m (6*) 2 K a (j * ) . Theorem 6.1. The statements (1)-(4) hold for every binormal triple ( X , A ,B). Proof. To prove ( l ) , let
a ~ n m ( XA, ) be represented by
: ( X ,A ) -+
(Sm, so). Then, by definition, z*i*(a) is represented by the partial map+ 1 A .
Since + ( A ) = so, we get i*j*(a) = 0. This proves (1). To prove (2), let @ ~ n m ( XB) , be represented by y : ( X ,B ) -+ (9, so). Then the partial map = y I A represents i*(@). Define an extension +# : ( X ,A ) -+ (E?+l, Sm) of by taking +#(x) = y(x) for each x E X . Then, +#(X) c Sm and hence [,+#(X) = so. Since l,+# represents S*i*(P), we deduce S*i*(/j) = 0. This proves (2). To prove (3), let y € @ ( A , B) be represented by X : ( A , B ) -+ (Sm, so). Take an extension x# : ( X ,A ) -+ (ET+',Sm) of X . Then EIX# represents the element 6*(y) of nm+l(X,A ) . Since X#(B) = X(B) = so, we may define a m a p F : ( X x 0) U ( B x I ) U ( X x 1) -+E?+'by
+
w ,t )
+
x,
=
so, X#(x)D sot
(x E t = O), ( x E B, t E I ) ,
( X E X , t = 1).
Since ( X , B) is binormal and ET+l is solid, F has an extension F# : X x I + ET+l. Define a homotopy f t : ( X , B) -+ (Sm+l, so), (0 < t < l), by taking f t ( 4 = t m x , t ) , ( x E x,t E 4. Then f o = [,X#j and f l ( X ) = so. Hence j*S*(y) = 0. This proves (3). To prove (4),let a be an element of nm(X, B ) such that i*(a) = 0. Pick a map 4 : ( X ,B ) -+ (9, so) which represents a: then $i = I A represents i*(a) = 0. Hence there exists a partial homotopy yt : ( A , B ) -+ (Sm, so) (0 < t g l), of such that yo = C$ I A and y l ( A ) = so. Since ( X , A ) is a
+
+
216
VII. COHOMOTOPY G R O U P S
binormal pair and S m is a compact ANR, it follows from ( I ; Ex. N) that yt has an extension 4t : (X, B) + (Sm, so), (0 < t < I ) , such that +o = +. Hence a is also represented by +l. Since +,(A) = yl(A) = so, 4, represents an element /? ~ n m ( X ’A) , and j*(/?) = a. This proves (4). I There are examples which show that ( 5 ) and ( 6 ) may fail (see Ex. B a t the end of the chapter); under additional hypotheses, ( 6 ) and ( 5 ) will be proved in $ 8 and $ 9 respectively.
7. An important lemma Lemma 7.1. Assecme that A i s a (finitely) triangulable space which is 2mcoconnected and that X denotes the conc over A with vertex v . Then the cobozcndary operator 6 carries nm(A) onto nm+’(X, A). Proof. First, let us prove the lemma under a further condition that either dimA < 2 m - 1 o r m = 1. For this purpose, let a be any element of nm+l(X,A), then a is represented by a map f : (X, A ) + (Sm+l,so).
Give S m + l a finite triangulation such that so is a vertex, S m is a subcomplex and the north pole u of S m + l is an interior point of some (m 1)-simplex which is not in the star of so. By the usual process of simplicia1 approximation, we may assume that f is a simplicia1 map on some finite triangulation of the pair (X, A ) such that the cone K over the (2m - l)-dimensional skeleton B = A2m-l of A is a subcomplex of X . Let
+
+ : (K,B)+ (Sm+l,so)
denote the restriction off on the pair ( K ,B). Since4 is a simplicial map and u is an interior point of an (m 1)-simplex, the inverse image C = +-l(u) is of dimension less than m. It is obvious that B and C are disjoint. Let 0 : B x I -+ K denote the identification map which defines the cone K over B. Then 0 carries B x 1 on the vertex v and carries the remainder of B x I homeomorphically onto K \ v . Since C is compact, disjoint from B , and of dimension less than m, there exist a positive real number Y < 1 and a closed subset D of B with dim D < m such that every point of C \ v can be expressed in the form B(d, t ) with d E D and r < t < 1. Let E denote the set of all points 8 ( d ,t ) with d E D and r Q t < 1. Then we have
+
C c E, Bll E = o , d i m E < m .
+
The proof depends on the construction of a few homotopies of sketched as follows. The details are left to the reader, [Spanier 1, p. 2291. Since dim E < m, is homotopic to a map y : ( K ,B ) + (Sm+l, so) relative to B such that y-’(~)= C and so 4 y(E). Since so 4 y(E), there is an open set W of K such that E c Wand so 4 y ( W ) . Pushing y(E) to u along line-segments in the euclidean space Sm+l\ so by
+
7. A N
I M P O R T A N T LEMMA
2x7
the aid of a continuous real function p : K + I such that p-l(O) = E and p-l( 1) = K \ W , one can construct a map X : ( K ,B) + (Sm+l,so) which is homotopic with y relative to B and such that X-l(u) = E. By means of a continuous real function q : K,+ I ,such that q-l(O) = D , one can construct a map M : ( K ,B) + ( K ,B) which sends E into the vertex v , carries K \ E homeomorphically onto K \ v , and is homotopic with the identity map relative to B. Then x = XM-l : ( K ,B) -+ (Sm+l,so) is a map homotopic with X relative to B and such that x-'(u) = v. Since x ( v ) = u, there exists a positive real number s < 1 such that xO(x, t ) is in the north hemisphere E:+l for every x E B and s Q t Q 1. Let
P = { O ( x , t )I x E B a n d s Q t G I } , Q ={O(x,t)IxEBandO Q t 9 s ) . Then x ( P )c EY+l and x ( Q ) c Sm+l\ so. Pushing x ( Q ) along geodesic arcs from u until Q is mapped into the south hemisphere Em+l, one can easily see that x is homotopic with a map 1 : ( K ,B) + (Sm+l,so) relative to B such that A(P)c ET+l and 1(Q) c EP+l. Since C# N y N X N x N il relative to B, it follows from the homotopy extension property that the representative map f of a can be so chosen that f I K = 1and hence we may assume that / ( P )c E Y + ~ , c E Z P + ~f ,( P n Q ) c s m . Now we are going t o construct a map g : A + S m as follows. First, define a map p : B + S m by taking p(x) = f O ( x , s) for every x E B. If dim A Q 2m-1, then B = A and we set g = p. On the other hand, if m = 1, thenp determines an obstruction cocycle c2(p)of A . Since f is defined throughout X , it follows from (11; Ex. D) that c2(p)= 0 and hence p is 2-extensible over A . Then, by (11; 7.4), p has an extension g : A + Sm. This map g represents an element /I of @ ( A ) . We are going to prove that S ( / I ) = a. Define a map h : ( K ,B) +(ET+l,Sm) by setting hO(x, 1) = fO(x, s +t-st) for every x E B and t E I . Since h I B = g [ B, we may define an extension k : K U A + ET+l by taking k ( x ) = g ( x ) if x E A and k ( x ) = h(x) if x E K . By Tietze's extension theorem, k has an extension g# : ( X ,A ) -+ (ET+',Sm). Composing with the relative homeomorphism E , of 8 4, we obtain a map f# = t,g# : ( X ,A ) + ( S m + l , .so). Since g# I A = g, f# represents the element S(/I) according to 5 4. We have to prove that f and f # are homotopic relative to A . Since gltt I K = h, we can easily see that f I K N Elh = f # I K relative to B. Since f ( A ) = so = f # ( A ) , this implies that f I K U A N f # 1 K U A relative to A . The cones over the simplexes of A constitute a triangulation of X . In this special triangulation of X , we have K U A = Xam U A . Hence f and f # are 2m-homotopic relative to A . Since A is Pm-coconnected and X is
218
VII. COHOMOTOPY GROUPS
+
contractible to a point, i t follows that (X,A) is (2m 1)-coconnected. Hence, by (VI; 11.2), f and f#are homotopic relative to A. Thus we have proved the lemma for the special case that either dim A <2m-lorm= 1. Next, let us prove the lemma with the condition that m > 1 andA is Simply connected. I n this case, we shall first prove that A is dominated by its (2m- 1)dimensional skeleton B. Since 2m - 1 > 3 and A is simply connected, so is B. Hence the pair (A, B ) is n-simple for every n 2 2m. Consider the inclusion map e : (A, c (A, B ) ,
and apply the obstruction method described in (VI; Ex. E) for deforming e into B. Since Hn(A, A h - 2 ) = P ( A ) = 0 for every n > 2m, it follows that e is deformable into B with AW-2 held fixed. In particular, there is a map i: A + B such that the composition ij with the inclusion map i : B c A is homotopic to identity map on A. This proves that B dominates A. Since X and K are the cones over A and B respectively, the maps i, i extend to maps : ( K ,B ) + (X, A), ( X ,A) + ( K ,B)
a
/:
A
A
in the obvious way so that i j is homotopic to the identity map on (X, A). This gives a commutative diagram
, then i*(ct)~ n m + l ( KB). , Since dim B < 2m - 1. Given a ~ n m + l ( XA), we have prove that 6 sends @(B) onto nm+l(K, B).Hence there exists a n element y ~ n m ( Bsuch ) that 6(y) = i*(a). Let /I = i*(y) ~ n m ( A ) Then . we have A
Thus we have proved the lemma also for the case that m > 1 and A is simply connected. Finally, let us prove the lemma for the general case m > 1. Let A' c X denote the union of A and the cone over its 1-skeleton. Then A' is simply connected and 2m-coconnected. Consider the inclusion maps k : A c A' and b : (X, A) c (X, A'). We obtain the commutative rectangle nm(A') I
k*
nm(A) I
8. T H E S T A T E M E N T (6)
219
+
Since dim (A'\A) < 2 and m 1 > 2, it follows that i* is an epimorphism. Let X denote the cone over A'. Then ( X ,A') is homotopically equivalent to ( X ' , A ' ) . Hence S sends nm(A') onto nm+l(X,A') since A' is simply connected and 2m-coconnected. Then it follows from the commutativity of the rectangle that 6 cames nm(A) onto nm+l ( X ,A ) . I
8. The statement (6) A triple ( X ,A , B ) is said to be (finitely) triangulable if there exists a finite triangulation of X such that A and B are both subcomplexes.
L e m m a 8.1. If ( X ,A , B) i s a triangulable triple such that ( A , B ) i s 2mcoconnected, then the statement ( 6 ) of 5 6 holds. More explicitly in the sequence nm(A,B) ??+nm+l(X,A ) i'-+ nm+l(X,B ) the image of 6* contains the kernel of i*. Proof. We shall be dealing with subsets of X x I and will use the following abridged notation. If Y is a subset of X , we define Y I = Y x I , Y,=YXO, Y,=YXl. ( X I ,X , U A , U BI, X , U BI) denote the map defined by Let f : ( X ,A , B ) -, f ( x ) = ( x , 0) for every x E X . Let fl =
According to
f2
=
f I (X, A).
4 3 and 5 4, commutativity holds in the following diagram:
nmfA,B)
I
f I ( A ,B ) ,
t
6.
-+ nm+l(X,A )
TI**
'8
1 , nm+l(X,B )
fl*
nm(X, U A , U BI, X , U BI)L n m + l ( X r ,X , U A , U BI) Since f is essentially the excision map obtained by excising ( B I\ B,) U X , from X , U A,, U Bx, it follows from (3.3) that f l * is one-to-one and onto. By the definition of homotopy, one can easily see that Im ( f2*) = Ker (i*). By the commutativity relation S*f,* = f2*6,*, it suffices to prove that the coboundary operator 6,* is onto. Let g : ( X , U A I , X , U A , U BI)c ( X I ,X I U A , U B I ) denote the inclusion map. Since X , U A I is a deformation retract of X I according to (I; lO.l), it is easy to see that the induced transformation g* is one-to-one and onto. By the commutativity of the triangle
,
n m ( X , U A , U B I , X , U Bx) 61++ nm+'(xr, X , U A , U BI)
$/
k!
nm+'(X, U A I , X , U A , U B I ) , it remains to prove that the coboundary operator 6,* is onto.
VII. C O H O M O T O P Y G R O U P S
220
Identify B to a point q and X , to a point r, and let
h : ( X , U Az, X , U A , U Bz, X , U Bz) + (LE,A E U
1,i)
denote the identification map, where if Y is a subspace of A , = A , Y denotes the join of Y with the point r , and A g denotes A , with B, identified to a point q. Let
hl
=
h I ( X , U A , U Bz, X, U Bz), h2
=
h I ( X , U Az, X , U A , U Bz).
According to tj 3 and tj 4, the following rectangle is commutative :
n m ( X , U A , U Bz, X I U Bz) L6 -*+ n m + l ( X ,U AT,X , U A , U Bz)
7
h,*
nm(Ag U
4,<)
63*
I"*
+ n m + l ( a g , Ag U
4).
Since h, and h, are obviously relative homeomorphisms, it follows from (3.2) that both h,* and h,* are one-to-one and onto. Hence, it remains t o prove that 6,* is onto. Consider the inclusion map
(A,,
k : (213,A E , 0)c A E U i,i ) and let k , = k I ( A g , o)and k , = k I ( A g ,A g ) . Then, we obtain a commutative rectangle : 6* nm(Ag U G,i)~ - + n m + l ( ~Agg, U
4)
1k1*
lkz*
-+ z , + ~ ( ~ E A , E).
nm(Ag)
+
Let a ~ n m ( A g )be an arbitrary element represented by : A g + Sm. Since Sm is pathwis! connected, we may assume that $(q) = so. Then has a unique extension : ( A B U i )+ (Sm, so). Hence k,* is onto. Since A e is astrong deformation retract of A B U 1,i t follows that k2* is one-to-one and onto. Since ( A , B ) is 2m-coconnected, so is A g . Hence, by (7.1), 6,* is onto. This implies that 6,* = k:-1d4*K,* is also onto. I
+
4,
+
9. The statement (5) Lemma 9.1. If ( X , A , B ) is a triangulable triple such that ( X ,A ) is 2mcoconnected, then the statement ( 5 ) of tj 6 holds. More explicitly, in the sequence
n m ( X , B ) ?+ nm(A, B ) the image of i* contains the kernel of S*.
x+nm+l ( X ,A ) ,
Proof. Let o r ~ n m ( AB, ) be an arbitrary element with d*(a) = 0. Let $ : ( A , B ) + (Sm, so) be a map which represents a. We are going to extend 4 throughout X by means of stepwise inductive construction over the ndimensional skeletons = X'a U A of X .
9.
THE STATEMENT
+
(5)
221
I t is obvious that can be extended over x m . Assume that n is an integer with m < n < 2m - 1 and that has an extension $n : ( X n , B ) + ( S m , so). Consider the inclusion maps
( A , B ) A+
+
(In,
B ) %.( E n , A ) 5 ( X ,A ) -L( X ,x n ) k
and the coboundary operators 6*, 8,*, S2* given in the following diagram:
This diagram has the following properties: (a) d*in* = kn*6g* by 5 3 and $ 4 . (b) 62*jn* = 6,* by the definition of 6,* and d2*. (c) I m (a,*) = Ker [kn*) by 5 6 and 5 8. (d) I m ( i n * ) = Ker ( i n * ) by 5 6. The map +n represents an element @ of n m ( X n , B ) . Since $n is an extension of we have in*(@) = a. Hence
+,
kn*62*[@) = d*in*(@) =
6*(a) = 0.
By (c), there exists an element y in n m ( X n , A ) such that 6,*(y) = S,*(@). Since dim (1.\A) = n < 2m - 1 and ( X , x n ) is Pm-coconnected, it follows that n m ( X n , A ) and n m + l ( X , 1%) are abelian groups according to (5.2). By the proof of (5.5), 6,* is a homomorphism. Hence
+ 62*in*
&*(@) +6,*(--) = &*(@) - 8,*(y) = 0. Let y : (En, A ) -+ ( S m , so) represent - y . Consider f = $n x y . Since f ( A )c S m x s 0 c S m V S m , it follows from (5.1) that f is homotopic to a map g relative to A such that g ( 1 n ) c S m V S m . Consider the composed map
Sz*(@)
(-7)
=
x
= jg
: ( B , B ) -+ ( 9 , so),
where j : S m V S m - + S m denotes the map in 5 5. Since g I A = f I A and y ( A ) so, we have X I A = $n I A = +. Hence X is also an extension of over X n . X represents an element @' of n m ( X n , B). As in the proof of (5.5), we have S,*(@') = S,*(@) 62*jn*(-y) = 0.
+
+
Let ln* : n m ( X , x n ) + n m ( X n + l , x n ) be induced by inclusion and 6*, : n m ( X n , B) + n m + l ( I n + l , 1%) the coboundary operator. Then da* = 1n*62* and hence a,*(@') = 0. According to Ex. D a t the end of the chapter, X has an extension over i3n.l. This completes the inductive construction of an extension $ 2 m - 1 of over X 2 m - l . Since ( X ,A ) is Pm-coconnected, it follows that has an extension +# throughout X . Hence a is contained in the image of i*. I Summarizing the results of (6.1), (8.1) and (9.1) we state the following
+,,+,
+
+
VII. COHOMOTOPY GROUPS
222
If ( X ,A , B) is a triangulable triple such that H y A , B) = 0 = H q X , A ) for every n > 2m, then the following part of the cohomotopy sequence i s exact: '* m X,B) "',nrn(A,B) -+nm+l(X,A) S* L'*n m + l ( x , B )---+ i* * * nm(X,A)Ln ( Theorem 9.2.
Corollary 9.3. If ( X , A ) is a triangulable pair such that Hn(A) = 0 = H n ( X , A ) for every n > Zm, then the following part of the cohomotopy sequence i s exact: '* '* nyx, A ) L nyx)5 n m f A )---+6 n m + y x , A ) L n m + l ( x )5 * .
10. Higher cohomotopy groups Theorem 10.1. If ( X ,A ) i s an m-coconnected cellular pair, t k n n m ( X , A )=O. Proof. Since Sm is (m- 1)-connected, every two maps q5, y : ( X ,A ) -+ (Sm, so) are ( m- 1)-homotopic relative to A . Then, by (VI; 11.3), q5 and y are homotopic relative to A . I The significance of this theorem is that, for a (finite) cellular triple ( X ,A , B ) , the cohomotopy sequence of ( X ,A , B) ends with a term 0.
X i s an m-coconnected triangulable space, t k n n m ( X ) = 0. I n particular, the corollary includes the obvious special case : If X consists of a single point, then n m ( X ) = 0 for every m > 0 ; on the other hand, n"(X) clearly consists of two elements. Corollary 10.2. If
11. Relations with cohomology groups I n (V; 9 4), we constructed a natural homomorphism hm from the homotopy group n m ( X , A ) into the homology group H m ( X , A ) over integral coefficients. In the present section, we shall define a similar operation
h m : n m ( X , A ) + H m ( X , A ; n m ( S m , ~ o ) ) ,m > 0 , which will be a homomorphism if ( X , A ) is an ( 2 m - 1)-coconnected cellular pair. Since nm(Sm, so) is free cyclic and the identity map on S m represents a generator, it can be identified with the group Z of integers in a natural way. Hence we may denote the cohomology group H m ( X , A ;%(Sm, so) ) simply by H m ( X ,A ) . I n order to construct the natural homomorphism hm, let us denote by Xm the characteristic element of the cohomology group Hm(Sm, so) as defined in (VI ; 9 17). This characteristic element Xm can also be defined by the natural homomorphism hm :nm(Sm, so) -+ Hm(Sm, so) as follows. By Hurewicz's theorem, (V; §4),hm is an isomorphism and hence the inverse hG1 is a welldefined homomorphism of Hm(Sm, so) into nm(Sm,so) = 2. Therefore, hi' determines a generator of Hm(Sm, so) which can be easily proved to be Xm.
11. R E L A T I O N S W I T H C O H O M O L O G Y G R O U P S
Now let us define the natural operation hm as follows. Let be an element represented by a map 4 : ( X ,A ) -+ (Sm, so). homomorphism +* : H m ( s m , so) +. P ( X ,A ) .
223 01
4
~ n m ( XA,) induces a
If we use singular cohomology, than #* depends only on u according to the homotopy axiom. In this case, we define hm(a) = #*(Xm).
This completes the definition of the natural operation hm. Proposition 11.1. If ( X ,A ) is a (2m - 1)-coconnected cellular pair, then hm i s a homomorphism.
Proof. Let fit : Sm x S m +.Sm, (i = 1,2), denote the projections defined by Pt(y,, y z ) = y t ; let i : S m V S m + S m denote the map defined in 3 5 ; and consider the inclusion map k : S m V Sm c Sm x Sm. These maps induce the homomorphisms pt* : Hm(Sm, so)
+.
x
Hm(Sm
s m , (so, so)),
i* : Hrn(Sm,so) - + H m ( S m V S m , (so, so)), k* : Hm(Sm x Sm, (so, so)) + H m ( S m V S m , (so, so)).
+
It is easy to see that i*(Xm) = k*Pl*(Xm) k*pz*(Xm). Now let a, /?~ n m ( XA, ) be represented by#, y : ( X ,A ) -+ ( S m , so) respectively. By (5.1), 4 x y is homotopic to a map g : (XIA ) + (Sm V S", (so, so)) relative to A . Then we have # r p,kg and y N fizkg relative to A . Hence hm(a) hm(p)= #*(xm) y*(Xrn) g*k*p,*(Xm) g*k*pz*(xm)
+
+
+
= g*i*(X,).
On the other hand, since ig : ( X ,A ) +. (Sm, so) represents a we have This implies that hm(u
+
b y definition,
+ /?) = hm(a) + hm(p).I
proposition 11.2. For a n y m a p f : ( X ,A ) +. ( Y ,B ) , the following rectangle
i s commutative :
nyx,A ) + L n m ( Y , B )
lhrn
4.
lhrn
4.
H y X , A ) tf' Hm( Y , B ) . Proof. Let u ~ n mY(, B) be represented by f *(a)is represented by # f and hence
4 : ( Y ,B) +. (Sm,SO).
hmf *(a) = (+f)*(Xm) = f *#*(Xm) = f *hm(a).I
Then
VII, COHOMOTOPY GROUPS
224
Proposition 11.3. For any binormal pair ( X ,A ) , the following rectangle i s
k
commutative :
p
nm(A)d nm+1 ( X ,A ) Hm(A)2 4Hrn+l(X,A ) .
Proof. Consider the map
El of 5 4. Then we have
Hm+l(E?+l, Sm) +-!.L Hm+l(Sm+l,so). Hm(Sm, so) By the definition of the characteristic elements Xm, i t can be seen that 6*(Xm) = 51*(Xm+,). Now let a be any element of n"( A ) . Choose a representative map : A +Sm. Then, by 5 4, 6(a) is represented by E1y: ( X ,A ) + (Sm+l,so), where y : ( X ,A ) 3 (E?+l, Sm) is any extension of +. Hence we have hm+l&a) = (5,y)*(X,n+,) = w * E I * ( x m + l ) = y*6*(Xm) = &$*(Xm) = Ghm(a). I As a consequence of the last two propositions, we have the following Proposition 11.4. For every binormal triple ( X ,A , B ) , the natural operators
hm, m = 1,2; -, define a transformation of the cohomotopy sequence of ( X ,A , B ) into its integral cohomology sequence, that i s to say, each rectangle of the following ladder i s commutative: nl(X,A)
i ,
..*
i"l
Hl(X,A) +
* * .
1"-
nrn(X,A)
i ,
+
p
@ ( X , B ) + n"(A,B) /hm
-+
k+.
nm+l(X,A) +
+ Hm(X,A) + Hm(X,B) + H m ( A , B ) -+ Hm+l(X,A) +
*..
*
Theorem 11.5. (Hopf theorem). If m i s a positive integer and ( X ,A ) i s a n (m + 1)-coconnected cellular pair, then hm sends nm(X, A ) onto Hm(X,A ) in a one-to-one fashion. This is merely a special case of (VI; 16.6).The significance of this theorem is that, for an ( m + 1)-coconnected cellular pair ( X ,A ) , we have @ ( X , A ) w Hm(X,A ) , n n ( X ,A ) = 0 for n > m.
12. Relations with homotopy groups If we neglect the group operations, nn(Sm,so) and nm(Sn, so) are identical. If n < 2 m - 2, nm(Sn,so) has a group operation as defined in 5 5 ; on the other hand, nn(Sm, so) is always a group. We are going to see that these two group operations are the same. Let a, be any two elements of nm(Sn, so). Let E: and E!! denote the north and the south hemispheres of 9.As in (IV; $2), we can choose representatives y : (9, so) + (Sm, so) of a, p respectively such that
+,
+(Elf)= SO
= y(ET).
12. R E L A T I O N S W I T H H O M O T O P Y G R O U P S
225
Then the map g = C$ x y sends X into Sm V S m and jg represents the sum of u and t9 in &(Sn, so), where j denotes the map defined in 4 5. On the other hand, since if x E ET, ( C$(X)I =
jg also represents the sum of following
if x E E!, 01
and t9 in nn(Sm, so). Hence we obtain the
Proposition 12.1. The cohomotopy group nm(Sn, so) i s exactly the homotopy group nn(Sm, SO), that is nm(Sn, SO) = nn(Sm, SO).
Now let us consider a pair ( X ,A ) and a given point xo E A . Let u ~ n m ( X ,A , xo) and /?~ n n ( XA,) be represented by the maps
q5 : ( I m , Im-1, I"-') + ( X ,A , xo), y : ( X ,A ) (Sn,so). The composition y$ is a map of (Im,dim) into (Sn,so) and therefore represents an element [y$] of nm(Sn,so) which obviously depends only on u and 8. -+
We denote
PO.
=
[y$l
and call /I0a the composition of u and t9. If m = n, then 10u ~ n m ( S mso) , and hence is an integer. The following proposition is an immediate consequence of the definition of addition in n m ( X ,A , so) and in nm(Sn,so).
B ~ n n ( XA,) , then we have B 0(a1 + a2) = (B 0El) + (B 0a2). we assume that n n ( X ,A ) forms a group according to 4 5, then
Proposition 12.2. If u l , u 2 ~ n , ( X ,A , xo) and
Now, if we have the following
Proposition 12.3. If
0:
+
e n m ( X ,A , xo) and
0 Proof. Let+ : (Im, I m - l , I"-') (81
B2)
= (PI
pl, /I2~ n n ( XA,) , then
we have
04 + ( B 2 04.
-+ ( X ,A , xo) represent u andy,, y 2 : ( X , A ) + (9, so) represent PI, B2. Let g : ( X ,A ) -+ (Sn V Sn, (so, so)) be a normalization of y1 x y 2 ;then gq5 is a normalization of
(y1 x y!J+ = y14 x yz+ (Irn, aim) (Sn,so). This and (12.1) imply the proposition. I Combining (12.2) and (12.3), we obtain that the composition operation defines a homomorphism -+
n m ( X ,A , xo) @ n n ( X A > ) +nm(Sn, so). The following properties of the composition operation can be easily verified.
2 26
EXERCISES
Proposition 12.4. Let f : (X,A,x,) + (Y,B,y,) be a map and aEnm(X,A,xo),
B ~ n nY(, B). Then we have
( f *B) 0a that is, the induced transformations f
=
B 0( f * a ) ,
* and f* are "dual" to each other.
Proposition 12.5. Let ( X ,A , B ) be a binormal triple with x, a €nm(X, A , x,) and fi ~ n n - ' ( AB , ) , then we have
(d*B)
0a
=
E B.
If
x [B 0@*a)1,
w k r e Z :n,,,-l(S*-l) +nm(S") denotes the suspension of Freudenthal, (V; 9 11). Thus, in case X is an isomorphism, a, and d* can be looked on as dual operations.
EXERCISES A. Generalizations to compact pairs
For the sake of simplicity, most of the operations and results in this chapter are formulated and proved for triangulable pairs. However, by using Cech cohomology theory, these can be generalized to the finite dimensional compact pairs ( X ,A ) , that is to say, X is a finite dimensional compact Hausdorff space and A is a closed subspace of X . The pair ( X ,A ) is said to be n-coconnected if the integral Cech cohomology group Hg(X, A ) = 0 for every q > n. Prove that, if ( X , A ) is a (2m - 1)-coconnectedfinite-dimensional compact pair, then it is possible to define a commutative group operation in nm(X,A ) by the method of fj5. See [Massey 1, p. 2831. Also prove that the induced transformations f * and the coboundary operator 6 are homomorphisms for finite-dimensional compact pairs satisfying similar conditions concerning coconnectivity . Now let ( X ,A ) be a compact pair with dim X < 2m - 1. Consider any finite open covering a = { U ) of X . Let K,, denote the geometric nerve of a and La the geometric nerve of the open covering a n A = { U n A } of A . Then (K,,,La) is a finite simplicia1 pair. Since dim X < 2m - 1, the finite open coverings 3: of X such that dim K,, < 2m - 1 form a cofinal subset M of the directed set of all finite open coverings of X . For eacha E M , nm(K,,,La) is an abelian group; and hence we obtain a direct system of abelian groups { nn,(Ka,La) I a E M }. Prove that nm(X,A ) is isomorphic to the limit group of this direct system, [Spanier 1 ; p. 2271. Consequently, the cohomotopy groups satisfy the continuity axiom. This can be used to extend results from finitely triangulable pairs to compact pairs. For example, generalize (9.3)to compact pairs.
EXERCISES
B. Connection with
227
Freudenthal's suspension
Consider the inclusion maps #:(Sm+l,s o ) c (Sm+l,E??+l)and q:(Ey+l,Sm) c (Sm+l,E??+l).By (3.3), q* is one-to-one and onto and hence q*-l is welldefined. Prove the commutativity of the following diagram:
I(=)
nn(Sm,sol
E+. nn+l(ET+l,Sm)4+-1-+ nn+l(S"+1,E!?+') e+ n"+'(Sm+l, so) [(=I z
nm+l(Sn+l,so) where 2 denotes the suspension. Since p* is also one-to-one and onto, this shows that the coboundary operator 6* is essentially the suspension X. Now consider the following part of the cohomotopy sequence of the triple (E?+', 9,so): nn(E?+l,so) 2 nn(Sm,so) E+nn+l(Ey+l,Sm)i'+ zn+l(E~+l,so). Since E?+' is contractible, we have Im(i*) = 0 and Ker(i*) = nn+l(EY+l,Sm). If m = 2 and n = 1, then (6) of 9 6 is false sincen,(S) = 0 whilen3(S2)M 2. On the other hand, if m = 3 and n = 2, then (5) of tj 6 is false since n,(S3) M 2, while n3(S2)M 2. %n(Sn, so)
-+
C. Connection with cochain groups
Let ( K ,L ) be a finite cellular pair. Denote E m = K m U L. Let a be any element of @(Em, Em-1) and pick a representative map : ( E m ,f f m - l ) + (9, so) for a. For each simplex u p of K , the partial map +I = I ( u p , sup) represents an element [+t] of nm(Sn,so) which depends only on a and u p . Hence a determines a n m-dimensional cochain y ( a ) of K modulo L with coefficients in nm(Sn,so). Prove that @(Em, Em-1) forms an abelian group with addition defined as in tj 5 and that the correspondence a -+y(a) defines an isomorphism y :n n ( E m , Em-1) M C y K , L ;nm(Sn,so)).
+
+
Furthermore, verify that y commutes with the induced homomorphisms. Prove also that the following rectangle is commutative:
C y K , L;nm(Sn, so)) A+Cm+1(K,L;nm+l(Sn+l, so)). D. Connection with the obstruction
Let ( K ,L ) be a pair as in Ex. C. Let a be any element of n n ( E m ) and +Sn be a representative map of a. By (VI; 9 4), determines an obstruction cocycle C"+l(+) E Cm+l(K,L ; nm(Sn,so)),
+
r j :E m
which depends only on a. Prove that
y6(a) = Z (cm+'(+)),
EXERCISES
228
where 6 :n n ( E m ) +nn+l(Em+l,E m ) is the coboundary operator, y denotes the isomorphism in Ex. C, and X is the homomorphism determined by the suspension in the coefficient group. Hence, if X is an isomorphism, @(a) is essentially the obstruction cocycle cm+l($). Therefore, has an iff 6(a)is the zero element of nn+l(Em+l,E m ) . extension over
+
E. The structure of n n ( X , A )
Let ( X ,A ) be a given (2n - 1) coconnected cellular pair. Define a sequence of subgroups nn(X, A ) = Dn-13 Dn 2 . . 3 p n - 2 = 0
.
as follows: an element of n n ( X ,A ) is in Dm iff i t can be represented by a map f : ( X ,A ) --f (9, so) such that / ( E m )= so. As in [Hu 61 and [Chen 13, define the #resentable subgrou# and the regular subgroup of H m ( X ,A ; nm(Sn,so) ) and consider their quotient group
J*"(X, A ;nm(Sn,so)) = P*m(X, A ;%(Sn, ~ g ) ) / R * ~A( ;xnm(Sn, , so) ). Then prove that Dm-llDm w J*"(X, A ;nm(Sn, SO) ) for every m
=
n,
--
a ,
2n - 2 .
F. Relations between induced homomorphisms
Let ( X ,A ) be a triangulable pair such that H n ( X ) = 0 = Hn(A) for every > 2m - 1, where m is a given positive integer. Prove that the following four statements are equivalent: 1. The induced homomorphism i* : H n ( X ) -+ Hn(A) of the inclusion map i : A c X is an isomorphism for every n > m and is an epimorphism for n = m. 2. H n ( X ,A ) = 0 for every n > m. 3 . n n ( X , A ) = 0 for every n > m. 4. The induced homomorphism i* : f l ( X )+nn(A) of the inclusion map i : A c X is an isomorphism for every n > m and is an epimorphism for n = m. Next, let X and Y be any two (2m - 1)-coconnected triangulable spaces and f :X + Y a map; prove that the following two statements are equivalent. 5. The induced homomorphism f * : H n ( Y ) + H n ( X ) is an isomorphism for every n > m and is an epimorphism for n = m. 6. The induced homomorphism f* : nn( Y ) + n n ( X ) is an isomorphism for every n > m and is an epimorphism for n = m. n
CHAPTER V l l l E X A C T C O U P L E S A N D SPECTRAL S E Q U E N C E S
1. Introduction Spectral sequences were originally devised by Leray to exhibit relations among the Cech groups of the various spaces of a fibering; but they have proved useful, indeed crucial, in numerous other investigations. Serre, for example, established the same results for the singular groups, and, using relations between homotopy groups and singular groups, he was able to obtain important information about the homotopy groups of spheres; and Eilenberg, Massey, and Spanier found spectral sequences to be useful in the study of the homotopy groups of a complex. Leray’s original device involved imbedding the homology or cohomology groups in question in a much larger system of groups and homomorphisms, namely the spectral sequence; this sequence, after the first term, is itself an invariant of the fibering. Being quite complicated, it is not amenable to successful computation except in special cases ; but it always provides much information, and it furnishes a pattern for deeper investigation. Massey improved Leray’s device by imbedding in a still larger but much more versatile system which he called an exact couple. The present chapter will be devoted to this formulation of the machinery of spectral sequences; an outline of Leray’s direct construction appears in Ex. A. There follows in Chapter IX Serre’s version of homology and cohomology theory of a fibering, together with several immediate applications ; results from Chapter IX will then be used in the computation of certain of the groups n,(Sn).
2. Differential groups Let A be an abelian group. An endomorphism
d:A+A is said to be a di#erential operator on A if dd
=
0,
i.e., d[d(a)] = 0 for each a E A . An abelian group A furnished with a given differential operator d is called a differential group, or simply a d-group. With the d-groups as objects and the homomorphisms which commute with d as mappings, we obtain a category 93’a called the category of differential groups. For the definition of a category, see [E-S: p. 1091. 229
230
If f :A
V I I I . E X A C T C O U P L E S AND S P E C T R A L S E Q U E N C E S
B is a mapping in g d such that A c B and f ( a ) = a for each A is called a subgroup of the d-group B. Let C = B/A : then we can define a differential operator on C so that the projection g : B 3 C -+
a E A , then
-
becomes a mapping in g d . When furnished with this differential operator, C is called the quotient d-group. We obtain an exact sequence (2.1)
0-
A f,B A+C
0.
Conversely, if an exact sequence (2.1) of d-groups and mappings is given, then A can be identified with a subgroup of the d-group B b y means of the monomorphism f and the quotient d-group BIA can be identified with C by means of the epimorphism g. Given a d-group A, we shall denote b y S ( A ) the kernel of d (called the group of cycles of A), and by g ( A ) the image of d (called the group of boundaries of A). The condition dd = 0 implies that g ( A ) c S ( A ) and that they are subgroups of the d-group A with d = 0 on each of them. Hence we may define the quotient d-group
&'(A) = d ( A ) / N A ) with d = 0 which is called the derived group of the d-group A. Let f : A -+B be a mapping in g d . Then the commutativity df --- fd implies that f m a p sd (A) intoZ(B) a n d a ( A ) intoB(B) and hence f induces a mapping S ( f:&'(A) ) -+&'(B). One can easily verify that the operation &' is a covariant functor from g d to itself, [E-S; p. 1111. Two mappings f , g :A + B in are said to be homotopic (notation: f a? g) if there is a homomorphism 6 : A -+ B such that
f -g
=
d6
+ 6d.
The homomorphism 6 is called a homotopy (notation: 6 : f N g). One can = &'(g). easily prove that f cxg implies &'(I) Let an exact sequence (2.1) in g d be given. We shall define a mapping
a :&'(C)
+&'(A)
as follows. Let aE&'(C) and choose x ~ a ( Cwhich ) represents a. There is some Y E B with g(y) = x . Since gd(y) = d(gy) = 0, there is a Z E A with f ( z ) = d(y). Since f d ( i ) = d( f i ) = 0 , d(z) = 0 and z represents a @ €#(A) which depends only on a. Then a is defined bya(a) = /I. Since the differential operators on&' (C) and&'(A) are trivial, a is a mapping in g d . Thus we obtain a triangle in Yd: &'(A) --+ Wf) X ( B )
3. G R A D E D A N D B I G R A D E D
GROUPS
231
One can verify that this triangle is exact in the sense that the kernel of each homomorphism is precisely the image of the preceding.
3. Graded and bigraded groups An abelian group A is said to be graded, or to have a graded structure, if there is prescribed to each integer n , (positive, zero, or negative), a subgroup A , of A such that A can be written as a (weak) direct sum
A
==
ZAn. U
The elements of the subgroup An are said to be homogeneous of degree n . Similarly, an abelian group A is said to be bigraded, or to have a bigraded structure, if there is prescribed to each (ordered) pair (m, n) of integers a subgroup Am,,, of A such that A can be written as a direct sum
A
= Z Am,%. m,n
The elements of the subgroup Am,, are said to be homogeneous of degree (m, 4. When dealing with graded or bigraded groups, only a certain limited class of homomorphisms are of interest, namely, the homogeneous homomorphisms. If are bigraded groups, then a homomorphism f : A + B is said to be homofor every pair (m, n). With bigraded groups as objects and homogeneous homomorphisms as mappings, we obtain a category %a called the category of bigraded groups. Similarly, one can define the category ggof graded groups. If f : A + B is a mapping in g b of degree (0,O)such that A c B and f (a) = a for each a E A , then A is called a subgroup of the bigraded group B . Let C = B / A . Then one can verify that C is isomorphic to the direct sum of the groups Bm,n/Am,n. We agree to identify these naturally isomorphic groups. Then C is bigraded and the projection g : B + C is a mapping in g b of degree (0,O). With this bigraded structure, C is called the qiiotient bigraded group. Note that both the kernel and the image of a mapping in are subgroups of the corresponding bigraded groups. One can easily formulate the analogous concepts for graded groups. In algebraic topology, we have to deal with graded (or bigraded) groups with a differentialoperator which is homogeneous. In this case, the derived group is also graded (or bigraded). For example, let us consider the group C ( X )of all singular chains in a space X with integral coefficients as defined in [E-S; p. 1871.C ( X )has a natural graded structure
C ( X ) = X Cn(X). n
V I I I . EXACT C O U P L E S A N D S P E C T R A L SEQUENCES
232
where C,(X) is the group of n-chains if n > 0 and C,(X) = 0 if n < 0. The boundary homomorphism a : C,(X)-+ C,-,(X) extends t o a differential operator d = : C ( X ) +C(X)
a
which is homogeneous of degree - 1. Hence the kernel Z ( X ) of d and the image B(X)of a' are subgroups of the graded d-group C ( X ) with graded structures Z ( X ) = Z Z , ( X ) , B(X)= Z B%(X). n
n
Furthermore, the derived group H ( X ) of C(X)has a graded structure
H ( X ) = ZH,(X), n
where H n ( X ) is the n-dimensional singular homology group of X if n and H,(X) = 0 for n < 0.
>0
4. Exact couples By an exact couple, we mean a system
Y =(D,E;i,j,k> which consists of two abelian groups D and E , and three homomorphisms
i:D-+D, j:D+E, k:E+D such that the following triangle is exact:
D
i
+D
There is an operation which assigns to an exact couple V another exact couple V' = < D',E'; i', j', k' > called the derived couple of morphism
Y constructed as follows. Define an endod:E+E
b y d = jk. Since k j = 0 byexactness, i t follows that ad = j k i k = j ( k j ) k = 0 . This implies that d is a differential operator on E . Let D' = i ( D ) , E' = & ( E ) ; then D' is a subgroup of D and E' is the derived group of the d-group E. Since D' c D, i(D')c D'. We may define an endomorphism i' : D' + D'
i I D'. Since one can easily verify that k [ J ( E ) ]c D', K[&'(E)j = 0, k induces a homomorphism k' E' D I . by i'
=
~
233
4. E X A C T C O U P L E S
Let x E D' and choose a y E D with i ( y ) = x . Then i ( y ) is in 9 ( E )and the coset of j ( y ) mod B ( E ) does not depend on the choice of y. Denote this coset by it(%). The assignment x + j ' ( x ) defines a homomorphism
j' : D'
-+
E'.
This completes the construction of W. The verification that i', j' k' form an exact triangle is straightforward and is left to the reader. This process of derivation can be applied to v' to obtain a second derived coztple V", and so on. In this way, we obtain a sequence of exact couples
Vn = ( D n , E n ; i n , j n ,kn>, n
=
1,2;..,
defined inductively by
V 1= V, V n = ( W - I ) ' ,
n > 1.
The sequence { W n } has two important properties. Firstly, the groups Dn form a decreasing sequence D = Dl 2 0 2 2 . . =I Dn =I Dnil I>
.
. ..
with i n : Dn -+ Dn defined by the restriction of i to Dn. The intersection of the groups Dn is denoted by Dm. Secondly, i t has been shown above that the endomorphisni dn = jnkn : En -+En is a differential operator on En and En+' is the derived group of En with respect to dn. Hence we obtain a sequence of differential groups E = E l , E 2 , . . . , En , ... such that En+l = .%'(En) for each n > 0. This will be called the spectral sequence associated with the exact couple V. In the spectral sequence { En } there exist natural homomorphisms
It, : I ( E n ) -+ En+' defined by assigning to each element of 9 ( E n ) its coset modulo .%f(En). Thus hn is an epimorphism of a subgroup of En onto En+l. We can define an epimorphism ht of a subgroup of En onto En+P by the formula hi
=
ha+p-lhn+p-2
* * *
hn.
Then hf, = hn. The precise domain 9% of definition of hf: can be defined inductively as follows: 3; =%"(En), 9f:= { u E 9 2 - l I h$-'(a) E Z(En+P-l) },
p > 1. Let E n denote the intersection of all subgroups9{, p = 1,2; -., then, for each a E E n , ht(a) is defined for all values of p . Define 2%: E n B n + l -+
to be the restriction of hn to E n . Then the sequence of groups { } and homomorphisms { i n } constitutes a direct sequence of groups in the usual
234
V I I I . EXACT COUPLES A N D SPECTRAL S E Q U E N C E S
sense, [H-U'; p. 1321. The limit group Ew of this direct sequence of groups will be called the limit grozcp of the spectral sequence { En }. In particular, if there is an integer r > 1 such that dn = 0 for each n 2 r, then I ( E n ) = En, hn : En M En+' and therefore E n = En and En = hn for each n > r. This implies that E" w Er. Returning to the general case again, we can express the limit group E* in terms of the original exact couple V as follows. Consider the epimorphisms i(n) = inin-1. .il : D + Dn+l c D.
.
Since i n = i 1 Dn for each n > 1, i(n) is actually the n-fold iteration of the homomorphism i : D + D . Let 01
Dw = n Dn n=i
03
=
W
n Im [icn)], Do = n U- 1 Ker [;(")I, n=i
where Im [ W ]and Key [icfi)]denote the image and the kernel of i(n) respectively. Then it is easy to verify that
E"
M
k-lD"ljDO.
In the following chapters as well as in the exercises a t the end of this chapter, we shall give detailed accounts of some of the important exact couples. For the moment, let us be content with a simple illustrative example. Consider a (finitely) filtered space, i.e., a space X furnished with a finite increasing sequence of subspaces
(0)
0 =
X-,C
x o cx , c * . *
c
x, = x.
Using total singular homology groups over a given coefficient group G,we define D = X H ( X p ) , E = I:H ( X p , Xp-1). P
P
Then the homology sequence of the pairs ( X p ,X p - l ) give rise to an exact couple VH(@) = < D,E ; i, j , k > called the homology exact couple of the filtered space X .
5. Bigraded exact couples In the exact couples V = < D , E ; i, j , k > which we shall deal with in the sequel, the groups D , E are usually bigraded and the homomorphisms i, j , k are homogeneous. In this case, in the successive derived couples %n of V, the groups Dn, En are also bigraded, i.e., and the homomorphisms in, i n , kn, and dn
=
jnkn are homogeneous. The
5. B I G R A D E D EXACT C O U P L E S
235
elements of D;,q and E:,q are said to be homogeneous of degree (p, q). We shall call p the primary degree, q the complementary degree, and p q the totd degree. It is easy to verify the following relations about the degree of homogeneity:
+
deg (in) = deg (i), deg (kn) = deg (k), deg (in) = deg (j) - ( n - 1) [deg (i)].
Hence the degree of the differential operator deg (an)
=
deg ( j )
dn
is given by the formula:
+ deg (k) - (n- 1) [deg (41.
The following two special classes of bigraded exact couples are important. A bigraded exact couple V = < D , E ; i, j , k > will be called a a-couple if i is of degree (1, - I), j is of degree (0,O)and k is of degree (- 1,O). In this case, the degrees of in, jn, kn, and dn are listed as follows: ( dl ) i ni s of deg ree(l ,-1 ); (d2) jn is of degree (-n 1, n- 1); (d3) kn is of degree (- 1,O); ($4) dn is of degree (- n, n - 1).
+
Similarly, V will be called a 6-couple if i is of degree (- 1, I), j is of degree (0,0), and k is of degree (1,O). Then the degrees of in, in, kn, and d n are listed as follows: (61) i n is of degree (- 1, 1); (62) jn is of degree (n- 1, - n 1) ; (63) kn is of degree ( 1, 0) ; (& dn is I) of degree (n,- n 1).
+
+
A %couple V = < D , E ; i, j , k > is a quite elaborate structure; it can be developed into a “lattice-like’’ diagram as follows:
ii
ii
*
*
-li k
k
* * *
* *
’
k
EP,*il--
DP+I,Q
EPil,Q---+
-li k
i
DP,4+1
DPi2,q-I
li
k
i
-+
k
Ep+z,q-1-+
ii k
i
DP-1,*+1-2, Ep-I,Q+l-
DP-2,(1+1---+
li
4-1.4
DP*Q
li
DPil*Q-l’+
li
-li
i > EP,Q
EP+I,P-I
k
*
DP,,-l
li
*
i ...
- li k
-
i
* * *
236
V I I I . EXACT C O U P L E S A N D S P E C T R A L S E Q U E N C E S
The steps from upper left to lower right are exact sequences, for example,
...-R
i i R i i D,-r,,+1D**,- E,.,-+ DP-l,,-+ D*.q-1-+ " ' is an exact sequence. Similarly, one can develop the derived couples W, n = 2, 3; -,into analogous diagram. It is more or less evident that the notions of &couple and 8-couple are dual to one another. Thus, let %?= < D, E , i, j , k > be any given bigraded exact couple. We shall construct another bigraded exact couple %f* = < D*, E * ; i*, j*, k* >, called the dual of W by merely reindexing the groups D,,, and E p , g as follows:
.
D;,, = D-,.-,, E;,, = E-,,-q. Then we obtain D* = D and E* = E ; therefore, we may take i* j* = j , and k* = k. Their degrees are given by deg (i*)
= - deg
[i), deg ( j * )
= - deg
=
i,
( j ) , deg (k*) = - deg ( k ) .
Hence the dual of a &couple is a &couple and vice versa. For example, the homology exact couple VH(@)of $ 4 is a &couple Indeed, the groups D and E are bigraded, namely,
DP,, = % * ( X P ) , E,,, = %*(XP> X P - 1 ) ; and the homomorphisms i, j , k are obviously homogeneous of degree ( 1, - l), (0, 0), (- 1,O) respectively.
6. Regular couples A %couple 5f = < D , E ; i, j , k > is said to be regular provided: (Rdl) DP,, = 0 if p < 0; (Ra2) E,,, = 0 if q < 0.
(Ral) and the exactness of
i
R
DmlEP,, DP-14 imply that Ep,q = 0 if p < 0. Hence, for any positive integer +
1z,
we have
En = 0 = Egg, if p < 0 or q < 0. (Ra3) P.4 If n > p, each element of E;A is a cycle since dn is of degree (- n, n - 1). If n > q + 1, no non-zero element of E;,q can be a boundary under dn. Hence En = E n f l =...= Egg if n > max (p, q I),
maq
P4
+
Ps4
Since each ir is of degree (1, - l ) , we have Dn9.4 = in-1 in-2 . . . a'1 (D p - n + i , q + n - J . W5) This and (Rdl) imply
(Rd6)
D"Pd = O
=
Dw ifn>p P,9
+ 1.
6. R E G U L A R Consider the homomorphismi : D,,, k
237
COUPLES
-+
D p + l , q - l (Rd2) . and theexactnessof
i
i
EP+1,4 DP,, DP+l,,-l-+ EP+199-l imply that i is an epimorphism if q < 0 and an isomorphism if q we obtain a sequence -+
+
< 0. Thus
where i : Dm, + D,+,,-, is an epimorphism. This suggests defining a graded
group
S ( V ) = ZXm(V), x m ( V ) = Dm+l,-] m
=
D&+1,-1.
Then D:+l,-l is a subgroup of Sm(%) and (Rd5) gives a homomorphism
-
A,,, : DP,,
-+.@P+,(%)9
+
(q
> 019
where A,,, = ig+1 iq * il is actually the ( q 1)-fold iteration of the homomorphism i : D + D. Denote the image of A,,, by
D;$;+l,-l. If p < 0, then S P , , ( V )= 0 since D,,, = 0. If q = 0, then X P , , ( V )= .x?~,($?)since A,,, reduces to the epimorphism i. Hence, for each m 2 0, we obtain a finite decreasing sequence: = AP,Q(DP,,) =
=@P,,(W
(Rd7) X m ( v ) =
~ m , o ( 3~x )m - 1 ,
i ( g )2 *
*
* = S o , m ( W = ' S - i , m + i ( g ) = 0.
The first row is exact since it is a part of W . Since i maps Dp+q+r,-r isomorphically onto Dp+q+r+l, - - I - l for each 7 > 1, we obtain two natural isomorphisms u
[email protected] denotes the inclusion and the rectangle is commutative. Since n is greater than max ( p , q 2), we get
+
-
E$+n-i,q-n+Z = 0, Eg,q = E$,qr Dg-1,q according to (Rds), (Rd4), and (Rd6). Hence we have
=
0
- - @ P , P ( ~ ) I ~ P - l , , + l ( ~ ) E&.
(Ra8)
In other words,XP,,(V) is an extension of X p - l , p + l ( % by )E&. Similarly, a &couple V = < D , E ; i, j , k > is said to be regular provided:
< 0; + < 0.
(R61) D P , , = 0 if q (R62) E,,, = 0 if
By methods dual to those used above, one can prove the following assertions.
(R63)
E;.,
=
0
=
E&
if
p < 0 or q < 0.
238
VIII. EXACT C O U P L E S A N D S P E C T R A L S E Q U E N C E S
E;,q
(RW
.--
E;;' = = E;,p i f n > ma x (p ,q Dpn,q = in-1 in-2 . 2'1 (D p + n - i , q - n + i ) .
=
..
+ 1).
(RS5) (RW DRq = 0 = D;,,if n > q 1. The homomorphism i : D p , q+ Dp-1,9+1is an isomorphism if we obtain a sequence
+
i
i
i
i
i
i
p < 0. Thus i
O + D m , o - + D m - l , 1 - - + * * * - - - + D O , m w D - 1 , m + l ~w D-r,m+rw *
*
a
*
a
*
This suggests defining a graded group
x ( v )= xxm(v), xm(v) m
=
Do,, w D4i,m+l-
denote the subgroup D;,;+, ofSPM(V).Then, for each m we obtain a finite decreasing sequence:
> 0,
(R67) x m ( v ) = x o , m ( v ) I x i , m - i ( q ) I * * . = ' ~ m , o ( v I) x m + i , - l ( % ) = O which satisfies the relations :
X P ,,(a)IS,+] .Q-lW)= E;, Q' Let us consider again the &couple W H ( @ )of 3 5. (R8l) is obviously true while (R82) is false in general. However, in case X is a finite simplicia1complex and X, is a subcomplex of X containing the p-dimensional skeleton of X, then (R82) is also satisfied. Hence, in this case, 'ip~(@) is a regular 8-couple. If, in particular, X , is the p-dimensional skeleton of X , then Ep,q = 0 for each q # 0 and E,,o = H p ( X p ,X,-J is the group of p-chains of X over G. One can verify that the differential operator d : E p , o+ E p - l , o is precisely the boundary operator on p-chains. Hence we obtain
(RW
E&o = H,(X), E i , q = 0 if q # 0. If n > 2, dn = 0 since it increases the complementary degree q. Hence we obtain H ( X ) = E2 = E3 = * * * = Em.
7. The graded groups It(%')and S(%) Let Q = < D , E ; i, j , k > be a regular 8-couple. Define a graded group R(W) =
x4 Rq(Q),
Rq(W
=
El),,.
Since dn is of degree (- n,n - l), each element of E t , 4 is a cycle. Thus we obtain epimorphisms Rq(W) = Ei,q?k+ E:,,?!, . %+EQ+z = Em O*Q 0.4.
--
3 6, we have EEQ * xO*Q(v) =*dW*
Let x denote the composition. By (R88) of Denote the composed monomorphism by
(Rag)
1.
Then we have
Rq(5f)-X+ E C Q L Xg(%?).
7. T H E
GRADED GROUPS
R(V) A N D S(V)
239
On the other hand, define a graded group
S(V) = ZSS,(V), S,(V) P
,
=
JZ;,,.
If n 2 2, no non-zero element of Ep"* can be a boundary under dn. Thus we 0btai.n the monomorphisms
EZ,,
=
E g 7 t - L EpP,,
* * *
lJ,E;,,%
Let L denote the composition. By (Rd8) of Eg.0
7 w
E;,,
= SP(V).
5 6 , we have
J f P , O ( ~ ) / Z I J - l l(%) . =s P ( ~ P ) / ~ P - l * l ( ~ ) .
Let x denote the natural epimorphism. Since
X*(W= S,(W = E;,,. we have j z : J f P ( V )-+SP(V).One can verify commutativity in the triangle q + 1 * - 1 9
(RalO)
Thus is is factored into the composition of an epimorphism and a monomorphism. If V is a regular &couple, then we define the graded groups R ( V )and S(V) exactly as above. Since dn is of degree (n,- n + l ) , no non-zero element of E:,, can be a boundary under dn and every element of Epn,o with n 2 2 is a cycle. Hence, one may establish the epimorphisms x and the monomorphisms L in an analogous way as above with the roles of R(%)and S(V) interchanged. Furthermore, since Jfg(V)= and &('if) = E,,,, we have j : Z q ( V )-+ I?,(%'). Thus we get
(R69)
S p ( V ) -If+ EEo LX p ( V )
and a commutative triangle (R610)
Note. One might define I?,(%) to be E ; , , instead of EO,g.Then all the results in this section stand as they are except that the j in (R610) should be replaced by j'. Our choice of E,,, is based on the fact that it gives a natural expression for %(a)if V is defined by a filtered graded differential group. See 3 1 1 below.
240
VIII. EXACT COUPLES A N D SPECTRAL SEQUENCES
8. The fundamental exact sequence Let Q = < D, E ; i , j , k > be a regular a-couple and v > 0 an integer. Under certain conditions on the bigraded group E”,we shall obtain a useful exact sequence which will be referred to as. the fundamental exact sequence. Theorem 8.1. If Ea has only two rows which might be non-trivial, more precisely, if there are two integers a < b such that
E;4 = 0 if p # a and p # b. then we obtain a fundamental exact sequence
.- .+~ i A!+ , ~ .sr ~n(Q L ) E&,,.b
Xm
Ehm-1.a
&L+ sm-l(Y)-+
---
Proof. The hypothesis implies that E;,q = 0 for each n > 2 and hence E& = 0 if p # a and p # b. Then (Rd7) and (Rd8) of 9 6 give rise to an exact sequence
Since dn is of degree (- n, n - 1) and a < b, every element E:,m-a is a cycle if n > 2. Hence we obtain epiniorphisms xr Ei,m.a+ %* Ei,m.a+ . . . E+En+’ a,m-a = E*a,m-a)
where n = max (a,m - a + 1). Similarly, if n > 2, no non-zero element of Et,m-bcan be a boundary under dn and we obtain monomorphisms
In
n
ECm.6 = GLtb Eb,m.) -+ * ’ 4 E&& where n = max (b, m - b+ 1). Thus we obtain an exact sequence
~ i ,““-*rn(Q) ~ +
bn-1
S
19
Ei,m-b*
To determine the kernel of +m, we have to study that of x r , r = 2, * * -,n. The kernel of xr consists of those elements of which are boundaries under dr. There are only two terms of Er of total degree m + 1 which might be non-trivial, namely, EL,m+l-aand Ei,m+l-b,The elements of the first are cycles; and dr maps the second into Ei,m-aiff r = b - a. Hence & is a monomorphism if b - a = 1. Now let r = b - a > 2; we have an exact sequence Furthermore, E&,,a = Ei,m-aand ELZa = E&,. A similar argument shows that EL,,+,, = Elm+,,. Thus we obtain an exact sequence
-
E:,m+l-, Xm+i E:,m-a A zrn(Q) where&,,+, = O i f b - a = 1 andXrn+, = d r i f b - a = r >2. By similar methods, one can prove the exactness of the sequence sm(a)
tpm,E:,m.b
%m -+
I
Ea,m.l-a*
The fundamental exact sequence is obtained by putting the various parts together. I
8. T H E
FUNDAMENTAL EXACT SEQUENCE
I t was proved above that Xm+, following
=
0 if b - a
=
Corollary 8.2. If in the hypothesis of (8.1), b - a we have a n exact sequence
o+
241
1. Hence we have the =
1, then, for each m,
h z m ( V )2E ; , ~+ . ~0.
H e n c e Z m ( V ) i s a n extension of E:,m-aby Ei,m.b. Analogously, one can prove the following Theorem 8.3. If E2 has only two columns which might be non-trivial, more precisely, if there are two integers a < b such that E;,¶ = 0 , if q # a and q # b,
then we obtain a fundamental exact sequence
-
-bm-1 + x m - i ( w )+ * '* Ern-l-b,b I Corollary 8.4. I f , in the hypothesis of (8.3), b - a = 1, then, for each m, we have a n exact sequence *
9
-+ E;+,b
h z m ( V ) 3E;-,,& -+Xm
0 + Ek-bsbh z m ( V ) EL.,, -+ 0. Hence S m ( % ' ) i s a n extension of EL-b,bby EL-,,,. Although the preceding two theorems cover most of the applications, it is sometimes necessary to deal with fundamental exact sequences obtained under weaker conditions. T h e two-term condition. Let 2, p, v be integers such that 1 < p and v 2 1. We shall say that V satisfies the two-term condition { I , p ; v } if the bigraded group E' has the following properties. For each integer m such that il < m < p, E;,c = 0 if p + q = m and ( p , q ) is different from two given pairs (am, bm) and (cnl, dm), where am bm = m = Cm dm, am < Cm.
+
+
Moreover, we require that the following two conditions should also be fulfilled : = 0 if p q = m - 1, p Q am - v , and il < m Q p ; (1)
(2)
+ E;,¶ = 0 if p + q = m + 1, p > cm + v, and I
Q m Qp.
With some obvious modifications of the proof of (8.1) one can prove the following theorem which includes (8.1) and (8.3) as special cases. Theorem 8.5. If V satisfies the two-term condition { 1,p ; v }, then we have a fundamental exact sequence
E&,bp
-+
* * *
-+
Eim,bm - + m ( % ) Erm9dm Eim.l,bm.1
Corollary 8.6. Let v
+
> 1 and { a,,
+
+
'''
-+
&A,dA
} be a sequence of integers such that
< am-1
+v
242
VIII. EXACT COUPLES A N D SPECTRAL SEQUENCES
for each m 2 0 . If Ei4
p # am, then
=0
Jfm(Y)
Proof. Put Cm = a,
El,,b,,
+q = m
( p , q) such that p
for every pair bm
=
and
m-arn.
+ 1 and dm = bm - 1. Then V satisfies the two-term
condition { 0, p ; v } for every p > 0. Since Erm,d, = 0 for each m > 0, the fundamental exact sequence implies the conclusion of this corollary. I If U is a regular &couple, then the two-term condition is stated by precisely the same words. The fundamental exact sequence in (8.5) becomes
E:,,,b,, * * ' f Eim.bm+ J f m ( q ) + EIm.dm+ Elrn-1,brn.l f ' * *+EZn,di and similarly in (8.1) and (8.3). Regular 3- and &couples are considered further in Ex. B at the end of the chapter. f
9. Mappings of exact couples Let 'ip, = < Dr, Er; if, jr, mapping
Kf. >, r
=
(+, y ) : Vl
we mean a pair of homomorphisms +il = i2+,
If dr = jrkr, then y d , a homomorphism Since
=
1,2,, be two exact couples. By a
yi1
+
v,
+ : D,+ D , and y : E ,
-+ E ,
= i2+>
+kl
=
such that
k2y.
d,y and y is a mapping in the sense of
3 2. y induces
y' : E ; -+EL.
+(D;) = +il(ol) = i2+(DJ C
i2(D2) =
0;)
+
we may define a homomorphism +': D;-+D; by +' = 10;. It can be readily verified that the pair (+',I$) of homomorphisms constitutes a mapping of the derived couples. This mapping y') : U; + U; is called the derived mapping of (+, y ) . Thus, the set of all exact couples and their mappings forms a category, and the operation of derivation is a covanant functor. If we iterate this process, we obtain a sequence of successive derived mappings ( r p , yn) : v,'z + V$, (n = 1,2; -). (+I,
In particular, the homomorphisms yn : Eln + E2n, ( n = 1, 2; -), of the spectral sequences commute with the differential operators dn, and yn+' is the induced homomorphism of yn. If the exact couples Wl and V, are bigraded, then only a limited class of mappings are of interest, namely, the homogeneous mappings. The mapping (4, y ) is said to be homogeneous of degree ( p , q) if both and y are homogeneous of the same degree ( p , q ) . By reindexing one of the given exact couples, we may always assume that (+, y ) is of degree (0,O); in this case,
+
9.
M A P P I N G S O F EXACT C O U P L E S
243
we say that (+,y) preserves degree. Then each of the derived mappings (+n, yn) also preserves degree. Now let Vl, V, be regular d-couples, and (+, y ) : V, -+ V, a mapping which preserves degree. Then the homomorphism +z defines a homomorphism
(4,Y)* :%(V1) - + S ( @ z ) which carries Xm(V1)into X m ( V 2and ) S p , q ( V l )into 2f?p,q(V2). Furthermore, y and y2 define homomorphisms
RWI)
+
W f Z ) ,
S(53-1)
-+
S(53-2).
These obviously commute with the epimorphisms x and the monomorphisms I in 3 7. Proposition 9.1. (4, y)* is a n isomorphism if either of the following two eqzcivalent conditions is satisfied:
(i)+2 : D12M Dz2; (ii)y2 : E l 2 w Ez2. Proof. I t is obvious that (i) implies that (+, y)* is an isomorphism. That (i) e- (ii) is an immediate consequence of the "five" lemma, [E-S; p. 161. Finally, that (ii) e- (i) follows from an easy induction on the primary degree by using (Rdl) and the "five" lemma. I Note. (9.1) remains true if we replace the superscripts 2 in (i) and (ii) by any positive integer n. In fact, it can be easily seen from 3 6, that (i)implies that (+, y)* is an isomorphism while the equivalence of (i) and (ii) is proved exactly as above. Obviously, similar results can be obtained for regular d-couples. Now, let us go back to the general case where V, V, are any two exact couples and consider two mappings
(+, y ) , (a, z) : Vl
+
V,.
They are said to be homotopic (notation: (+, y ) 3~ (a,z)) if there is a homomorphism 5 : E l + E , such that
4%) -+@) 4 Y ) -Y(Y) for every x E D,and y but important
E E,.
=
=
kZEjl(4,
Eddy)
+ dzE(Y)
Then one can easily prove the following obvious
Proposition 9.2. If the mappings (4,y ) and (a, z) are homotopic, then the (+I, y') and (a', z') are equal.
derived mappings
Hence, (p, y") = (a",7%) for every n 2. In particular, if V, W, are regular &couples or regular &couples and if (4, y ) , (a, t)preserve the degree, then (4, y ) N (a,z) implies that (+, y)* = (a, z)*.
244
VIII. EXACT COUPLES A N D SPECTRAL S E Q U E N C E S
10. Filtered differential groups Let A be a d-group. By an increasing filtration in A , we mean an increasing sequence of subgroups { A D } of the d-group A , p ranging over all integers, with their union equal to A ; in symbols, we have UpAP
=A,
AD c AP+',
d(AP)c A p .
If an increasing filtration { Ap } is given in A , A will be called an increasing filtered d-group. Let A be an increasing filtered d-group. For each a E A , the greatest lower bound of the integers p such that a E AD is called the weight of a and is denoted by w ( a ) ,The following properties are obvious : w ( a - b)
< max [w(a),w ( b ) ] ,
w(da)
< w(a).
Conversely, if there is given a function w defined on a d-group A with integral values (including - 03) which has the properties given above, we can define an increasing filtration { Ap } by taking
A ~ = ( u E A I w ( u<) p } . Let A and B be increasing filtered d-groups. A homomorphism f : A + B is called a mapping if i t commutes with d and preserves the filtration, that is, fd = df, f ( A p )c B p . The set of all increasing filtered d-groups and their mappings constitutes a category$*. Subgroups and quotient groups of groups in S g are defined in a manner analogous to that of 3 2. They are also groups in St.In particular, for each A in $*, the derived group X ( A )is also in $f. One can verify that the operation .% of § 2 defines a covariant functor from 9 6 to itself. To each A i n s { , there is an associated graded group B ( A ) of A defined by
8 ( A )=
A
=
X P
Alp = Ap/Ap-'
Ap,
and each mapping f : A -+ B induces in an obvious way a homogeneous homomorphism of degree 0
8(f )
=
f": %(A)-+ 9 ( B ) .
One can verify that the operation Y is a covariant functor from the category ,F{ to the category Bgof 5 3. The associated graded group of the derived groupX(A) is important and will be denoted by
%#(A)
=.+(A)
=
X2p(A). P
Since Ap-l, Ap, ADare d-groups, the exact sequence
o - + A P - ~ + A P + Ap + O gives rise to an exact triangle
10. F I L T E R E D D I F F E R E N T I A L GROUPS
245
according to fj 2. Define two graded groups
D
=
X Dp,
D p =&‘(AP),
E
=
Z Ep,
Ep
P
P
=#(AD).
Then the homomorphisms i , j , k of the preceding exact triangle define homomorphisms i :D + D , j : D +E , k : E + D and we obtain an exact couple
%‘(A)= < D,E ; i , j , k > which will be referred to as the exact couple associated with A . The groups D and E are graded and the homomorphisms i, j , k are homogeneous of degree 1,0, - 1 respectively. Now let f : A + B be a mapping in F i . Then f defines two mappings f p :A p + B p and : Alp + Bp in 9 g . One can easily verify that, by passing to direct sums, the derived homomorphisms
ip
2 ( f:p X ()A P ) - . * ( B p ) ,
&‘(fp)
:%‘(A,) + 2 ( k p )
define a mapping in the sense of 3 9. One verifies that the operation $9 is a covariant functor from 9tt to the category of 3 9. Similarly, a decreasing filtration in a d-group A consists of a decreasing sequence of subgroups { AP } of A such that
npA p
= 0,
A p
Z I
AP+l,
d(AP) c
Ap
The weight w ( a )of a E A is defined to be the least upper bound of the integers p such that a E AP. By an obvious analogy, one can formulate all of the preceding concepts for the categorysd of decreasing filtered d-groups and their mappings. By a filtered d-group A , we mean either an increasing or a decreasing filtered d-group. The associated exact couple V ( A ) constructed above and its associated spectral sequence { En, n = 1,2; * } will be referred to as those of the filtered d-group A .
-
11. Filtered graded differential groups By an increasing (or decreasing) filtration i n a graded d-group A , we mean an increasing (or decreasing) filtration { A , } of the d-group A such that
VIII. EXACT COUPLES A N D SPECTRAL SEQUENCES
246
each Ap is a subgroup of the graded group A. Such a filtration {AD } is said to be regular if 0 < w(a) 6 deg (a) (RF1) for every non-zero homogeneous element a of A . In other words, the weight and the degree are both non-negative, and the weight does not exceed the degree. The filtered d-groups with which we have t o deal in the sequel usually belong to two limited classes, namely, the filtered &complexes and the filtered 8-complexes. A filtered 8-complex is a graded d-group, where d is of degree - 1, with a regular increasing filtration; a filtered 8-complex is a graded d-group, where d is of degree 1, with a regular decreasing filtration. Theorem 11.1. The associated exact couple
“(A) of a
=
< D, E ; i, j , k >
regular &complex A is a regular a-cmrple.
Proof. Since A p and A, = Ap1Ap-l &‘(Ap)
are graded d-groups, the derived groups andX ( & ) are also graded. So we may put
DP,P = x P * ( A P ) , E P , P =&+g(&). Thus the groups D and E of Q(A) are bigraded. One can easily verify that i is of degree (1, - l ), j is of degree (0,0 ) ,and k is of degree (- 1,O). This proves that Q(A) is a &couple. For an increasing filtration { Ap } of A , the regularity condition (RFl) is equivalent to the following two conditions: (RF82)
Ap=O
ifp
(RF83)
Apc
for all p.
AD
(RF82) implies Dp,g= 0 if p < 0; (RFd3) implies Ep,g = 0 if q < 0. Hence %(A) is regular. I Next, let us investigate the graded groups S(Q), R(Y), and S(Q) of the regular 8-couple V = Q(A) associated with a given regular %complex A. The inclusion Ap c A induces homomorphisms hP,(I : Dp,q = ZP+(I(AP) +-#P+P(A). is an epimorphism if q By (RFa3), one can verify that morphism if q < 0. Hence
< 0 and an iso-
&+I, -1 : s m ( q w #m(A). Passing to direct sums, we obtain a homogeneous isomorphism of degree 0
h : X ( Q )w X ( A ) . According to 5 lO,X(A)is filtered; hence.Xp+q(A)is also filtered. It is clear that the image hP,g(Dp,g)is the subgroup of#p+q(A) with filtration p. This will be denoted by&‘=,,(A). Then it is easy to show that hm+l,-l maps
11. F I L T E R E D G R A D E D D I F F E R E N T I A L G R O U P S
.#‘,,*(W) isomorphically onto &‘,,*(A) if m
=
theorem is a consequence of (Ra7) and (Ra8) in
p + q. Thus,
247
the following
4 6.
Theorem 11.2. The filtration o f X m ( A )is given by the finite sequence
(RFd4) #m(A) = s m , o ( A )I z m - 1 , , ( A )2
‘ I X o , m ( AD&‘-I,m+i(A) ) =0 and its associated graded group g X m ( A ) is isomorphic with X p + g - m Egg as given by the relation (RFa5) &‘P**(A)/JfP-l,r?+l(A) Egg. Since A-’ = 0 by (RFa21, we have R*(%‘)= Eo,p = .#*(Ao) * *
-
and hence R(V) =&‘(Ao). Then one can verify that the composition (Rd9) in 3 7 gives the derived homomorphism
LX
of
&‘(I) :&‘(Ao) +&‘(A) of the inclusion f : A o c A . Thus X ( f )is factored into the composition of an epimorphism and a monomorphism. Define a graded group A = Z A, by A, = Ep,o. Since d is of degree (- 1,O) on E , A is a subgroup of the d-group E and
S.p(V) = E:,o
=
&‘,(A).
Hence we have S(V) =&‘(A). There is a natural epimorphism g : A + A defined as follows. Let x E A,. Then x E AS by (RFa3), and dx E c Ap-l. This implies that x is a cycle of A, and represents an element g(x) of A, = The assignment x + g(x) defines an epimorphism g : A + A which preserves the degree and commutes with d. One can verify that the derived homomorphism
X(g): # ( A ) +&‘(A) reduces to the homomorphism i2 :&‘(%‘) +S(W) in (RalO) of 4 7 after identifying &‘(A)with&‘(%‘)and &‘(A)with S(V). Thus &‘(g) is factoredinto the composition of an epimorphism and a monomorphism. As a consequence of this, the kernel of #(g) in &‘,(A) is &‘p-l, , ( A ) and the image in &‘,(A) = Eia0is the subgroup Ego. One can establish the foregoing results for regular &complexes analogously. The theorems (1 1.1) and (1 1.2) can be stated and proved with some obvious modifications. The graded groups &‘(V), R(V),and S(V) are identified to be &‘(V) =&‘(A), R(V) =&‘(Ao), S(V) =&‘(A).
where A, = Ao/A1 = A/A1 and A = Z E p , o .After these identifications, the homomorphism j : X ( V ) + R(W)in (RdlO) of 3 7 reduces to the derived homomorphism &‘(I) :&‘(A) +&‘(A,) of the natural projection f : A +Ao. There is a natural monomorphism
248
VIII. EXACT COUPLES A N D SPECTRAL SEQUENCES
g : A + A which preserves degree and commutes with d. The derived homo-
morphism
W g ) :#(A) + W A )
is given by the composition cx in (R69) of 5 7. As an example of regular &complexes and &complexes, let us consider a filtered space
(@I
0 =
x-,c X,C x , c
a * *
c
x,= x
where X is a finite simplicia1 complex and X , is a subcomplex of X . Assume that X , contains all $-simplexes of X . Then, for any given coefficient group G , the group A = C ( X ) of chains form a regular d-complex with
A,
= C(X,),
A,
=
C,(X).
One can easily see that %(A) is precisely the exact couple W H ( @ ) of 5 4. Hence WH(@) is also a regular &couple in this case. Similarly, the group of cochains of X over G forms a regular &complex.
12. Mappings of filtered graded d-groups Let A and B be two regular d-complexes. A homomorphism f : A -+ B is said to be a mapping if it commutes with d and preserves both degree and filtration, i.e. fd = df, / ( A m ) c Bm, f (A,) c B’. The set of all regular d-complexes and their mappings constitutes a category Xa which is a subcategory of 9f. Now let f : A + B be a mapping in Xa. According to 5 10, f induces a mapping
W(f)
in the sense of
=
hjbf, yf) : %(A)
-+
WB)
5 9. On the other hand, we have a derived X ( / ): # ( A )
homomorphism
+ &‘(B).
The following proposition is a corollary of (9.1) and its note. Proposition 12.1. For any mapping f :A + B in . f a , # ( f ) is an isomorphism if there exists an integer n >, 1 such that either of the following two equivalent conditions is satisfied :
(i)$7 : D ( A ) M D ( B ); (ii)y7 : En(A) M En(B). Similarly, one can define the mappings of regular &complexes and the category Xa of regular d-complexes and their mappings. The preceding proposition is also true for any mapping in . f a .
249
EXERCISES
EXERCISES A. Direct construction of the spectral sequence of a filtered differential group
Let A be a filtered d-group with an increasing filtration { Ap }. Introduce the following notations : Zp = { x E Ap I d ( X ) = 0 },
Z ; = { X E A P I ~ ( X ) E A P - ~(n} >, O ) , B;
=
d(z;+,,),(n >O).
Verify the following inclusions and equalities :
ZPC ZP+l> d(AP) = B ~ c B i c . . . c B ; : c B ~ l c . . . c Z p , 2, c - * cZ; , z ; - l ~ * . * ~ z i ~ z ~ = A P ,
z;-,
=
z;+1 n AP-1
c
z;+l,
n A P - ~ C~ z - 1 .
=~ z - 1
Then define the following quotient groups:
D;
E;
= Zp-n+l/B;2+1, = Z;/(Z$I;
(n > I),
+ Bg-'), ( W > l ) ,
and the graded groups
D"
=
X D;,
En
P
=
E E;. P
The inclusions Zp-n C Zp-n+l and B$:A c B $ Z ~ define + ~ a homomorphism in : Dn + Dn which is homogeneous of degree 1. The inclusions Zp-n+l c Z;-n+l and B $ I ~ c+ (2;:: ~ B ; Y ~ +define ~ ) a homomorphism j n : Dn +En which is homogeneous of degree - (n - 1). Finally, since d(Z$)c Zp-n, d(Zg:i) = B$:k, and d(B;-]) = 0, the differential operator d induces a homoniorphism kn : En + Dn which is homogeneous of degree - 1. Prove that < Dn, Elk; in, i" kn, >
+
is an exact couple for each n = 1,2, and in fact is precisely the associated exact couple @ ( A ) defined in 5 10. The differential operator d also induces the homomorphism dn = jnkn 1 En +En a ,
of degree -n. Thus one obtains Leray's direct construction of the spectral sequence { En } of the filtered d-group A . See [Leray 11. Formulate a similar construction for the case that the filtration { Ap } is decreasing.
250
VIII. EXACT COUPLES A N D SPECTRAL S E Q U E N C E S
B. Regular couples satisfying two-term conditions
Let V = < D, E ; i,j , k > be either a regular &couple or a regular &couple. Prove the following assertions, [Moore 1 ; p. 3271: 1. L e t p > Oandv >2. IfE;,q = 0 wheneverp # 0 , q # 0 , a n d p p, then V satisfies the two-term condition { 0, p ; v } with
a,
=
- 1, 6,
=
1,
C,
=
+q <
0, do = 0,
m, cm = m, dm = 0 , (0 < m G p ) . 2. Let v, Po,pl, p 2 and qo be integers such that v > I , @ , > 9, > p , > 0, and qo > 0. Then V satisfies the two-term condition { 0, 9, + qo - 2; v } am
=
0, bm
=
if the following two conditions are satisfied:
(i) E;,q = 0 if p < pol p # pl, and p # 15,; (ii)EXq = 0 if q < q,. 3. Let v, fi,, pl, qo and q1 be integers such that v > 1, Po > p1 > 0 and qo > q1 > 0. Then V satisfies the two-term condition { 0, Po + qo - 1 ; v } if the following two conditions are satisfied:
(i)E;,q = 0 i f p < p , and p # p l ; (ii)E;,q = 0 if q < qo and q # q l . C. Multiplicative structures
Let A be a regular &complex. A multiplication in A which makes A a ring is said to be allowable if it satisfies the following three conditions : (Ml) If x E A , and y E Aa, then x y E A p + q . (M2) If x E A , and y E A,, then x y E A,+*. (M3) d is an anti-derivation, i.e., for x E A , and y
+
EA ,
we have
d ( x y ) = ( d x ) y (-- I)Px(dy). Assume that an allowable multiplication has been given in A . By using (M l)-(M3), define multiplicative structures in the bigraded groups D n and E n of the exact couples V n ( A )= < Dn, E n ; in, jn, kn >
-
for each n = 1,2; * , Prove the following assertions, [Massey, 21 : 1. If x is homogeneous of degree ( p , q) and y is homogeneous of degree (7, s ) , then x y is homogeneous of degree ( p + r, q s). 2. i n is a transducer of Dn, i.e., in(xy) = x [ i n ( y ) ]= [in(x)]y.
+
3. jn is multiplicative, i.e. jn(xy) = P ( x ) jn(y). = jnkn is an anti-derivation of En, i.e., for x E E,,, we have dn(xy) = [dn(x)]y (- l)Ptqx [dn(y)]. 4. dn
+
and y
E En,
EXERCISES
5. For x E D;,, and y E En, we have kn[jn(x).y]= (- I)P+qx.kn(y), k n b . j n ( ~ )=] kn(y).x. D. The exact couples of a bundle space over a finite simplicia1 complex
Let X be a bundle space over a base space B with projection o : X + B and fiber F as defined in (111, 4). Assume that B is a finite simplicia1 complex with B p denoting its #-dimensional skeleton and that G is an abelian group. Define the bigraded groups D and E by taking the singular homology groups Dp,q = H p + q ( X p ; G), Ep,q = Hp+p(Xp,X p - 1 ; GI, where X , = m-l(BP), and the homomorphisms i : D + D , j : D -+ E , and k : E + D as in the example of 5 5. Thus one obtains an exact couple %f
=(D,E;i,j,k>
which is called the homology exact couple of the bundle space X over B. Consider the derived couples ben = < Dn, En; in, i n , kn> of C and prove the following assertions : 1, W n does not depend on the triangulation of B if n > 2. 2. Ei,q is naturally isomorphic to the homology group H,(B; H,(F; G)) of the complex B with local coefficients in H,(F; G). 3. V is a regular a-couple with D,,, M H,+,(X; G) if q < 0. Similarly, one can construct the cohomology exact couple and prove the analogous assertions. E. The homotopy exact couple
Let K be a connected cellular complex and v be a vertex of K . Denote by Kn the n-dimensional skeleton of K. Then, for each pair (KP,K p - l ) , there is an exact homotopy sequence
...
--f
R nm(KP-l,u)2+ nm(KP,u) L nm(KP.KP-1,u) + n*1(KP1,u) +
Define two bigraded groups D and E as follows:
DP,,
=
E,,,
=
4, [ n,+,(KP, o,
(
n,+,(KP, Kp-l, v ) , j[n,+*(KP,v ) ] , 0,
* * *
.
+
i f p ZOandp q >2, for other values of p and q ;
+ +
if p Z 1 a n d p q Z 3, if p Z 1 a n d p q = 2, for other values of p and q.
The homomorphisms i, j , k in the homotopy sequences of the pairs ( K p ,Kp -I) may be extended in a unique way to define homomorphisms
i:D+D, j:D+E, k:E+D.
252
V I I I . EXACT C O U P L E S A N D S P E C T R A L S E Q U E N C E S
Prove that the following triangle is exact :
D--+D i
Thus we obtain an exact couple Q ( K ,v ) = < D, E ; i, j , k > which is called the homotopy exuct couple of K at v. Prove the following assertions [Massey 11: 1. The homomorphisms i, j , k in Q ( K ,w ) are homogeneous of degree (1, - l), (0,0), (- 1,O) respectively. Hence Q ( K ,v ) is a &couple in the sense of 5 5. 2 . D P , , = 0 if p < 2 or p + q < 2 ; D p , , w n , , ( K , v ) if q < O and p+q>1. 3. E P , , = 0 if p < 2 or q (0 or p q < 2. 4. Q ( K ,v ) is a regular &couple in the sense of 5 6. 5. If vo and v1 are any two vertices of K , and if u : I +. K 1 is a path joining v,, to v l , then [ induces a natural isomorphic mapping
+
Q(8= (+€, yd : Q(KV J
-
V ( K ,Vo)
in the sense of 3 9. Furthermore, if two paths [, 7 : I +. fl from vo to w1 are homotopic in K with endpoints fixed, then Q([) = Q(7).These facts can be concisely expressed by saying that the set of homotopy exact couples V ( K ,v ) for various vertices v of K constitutes a local system of exact couples in K . The same remarks hold for the successive derived couples @ ( K , v l . 6. Let K , L be connected cellular complexes, v, w be vertices of K , L respectively, and f : ( K ,v ) +. (L, w ) be a cellular map. Then f induces a mapping Q ( f ) = (+j, yr) : Q ( K ,v ) +. WL, w ) . Let X denote the category in which the objects are all pairs ( K ,v ) of a cellular complex K with a vertex v E K and the mappings are all cellular maps. Let Va denote the category of all regular &couples and their mappings. Then the operation ( K , v ) + Q(K,v ) and f --f Q ( f ) is a covariant functor from .fto %‘a. 7. Let k be a connected covering complex over K with projection o :I? -+ K and 3 be a vertex of I? with o(3)= v. Then g(o) :
W(k,Z )
w Q ( K ,v ) .
The significance of this result is that there is no essential lack of generality if we assume that K is simply connected when discussing the properties of V ( K ,v ) and its derived couples. 8. The derived couple Q2(K,v ) = < 0 2 , E2; i2, j 2 , k2 > is an invariant of the homotopy type of K . 9. Di,q = 0 if p < 3 or p + q < 2 .
253
EXERCISES
+
10. D;,4 w nP+,(K,v ) if q < 0 and p q > 1. 11. E;,q =O if p < 2 or q < 0. 12. If K is simply connected, then Eb,o is isomorphic to the homology group H,(K) of the complex K for each p > 2. 13. If K is 7-connected, then Db,q = 0 for p < 7 1 and E;,, = 0 for < 7. 14. The exact couple V ( K ,v ) contains the following exact sequence of J. H. C . Whitehead: 'k-f D ; , o S D$+1,-1& E i 3 0 S D&-I,o-i' . . * .
+
...
If K is simply connected, then, by 10 and 12, there are natural isomorphisms Db+,, -1 w n p w ,4, Eb, 0 = H p ( K ) for each p 2 2. Hence we obtain the following exact sequence for a simply connected I<;
-
... 'k
0 5 , ?+n p ( K , V ) & H p ( K ) E+. DP-], ?+ * * . *
As a consequence of this and 12, we obtain the result that, if n > 2 and K is (n - 1)-connected, then j 2 :n p ( K ,v ) -+ H p ( K ) is an isomorphism for p < n and an epimorphism for p = n 1. This includes essentially the Hurewicz theorem. 15. Since V = V ( K ,v ) is a regular &couple we may apply the results of 3 6 and 5 7. Thus we obtain that
+
s p ( V ) % ~ p ( K vp) , R p ( W ) = 0, sp(v)w H p ( K ) for every p 2 2 and that these groups are all trivial whenever p (RalO) of 5 7 gives a commutative triangle for each p 2 2:
< 2. Then,
np(K, v ) .-JL Hp(K)
This implies that Eg, ,is isomorphic to the image of n p (K,v ) inH,(K) under ia. 16. The inclusion K p c K induces homomorphisms 5,q
:ZP+q(KP,
-+np+q(K,v ) .
Let us denote the image of 12p,g byn,,,(K, v ) . Then
+
X p , q ( YM) Z ~ , ~ (v )K ,if p 2 2 and p q > 2; i@p,q(V)= 0 if p < 2 or p + q < 2. Hence (Rd7)of fj 6 gives a finite sequence n,(K, v ) = nm,o(K, v ) = Z,-1,l(K, v ) 2 * 3 n,,m-JK. v ) for every m > 2. Furthermore (Rd8) of Q 6 implies that
--
=0
254
VIII. EXACT COUPLES A N D SPECTRAL S E Q U E N C E S
If q = 0, the triangle in 15 shows that np-l,l(K, v ) is the kernel of the homomorphism jz. 17. Generalize the foregoing results to semi-sixnplicial complexes or, more generally, to CW-complexes. 18. Finally, let X be a pathwise connected space and xo a given point in X . Consider the singular complex K = S ( X ) . Then K is a connected seniisimplicia1 complex and xo determines a vertex v of K . Q ( K ,v ) is a homotopy invariant of the pair ( X ,xo) and will be called the homotopy exact couple of X at xo, denoted by W ( X ,x,,). By means of the relations nrn(K,v ) w nrn(X, x ) , Hm(K)= Hrn(X), deduce results concerning n r n ( X ,xo) and H m ( X ) analogous to those given above for the pair ( K ,v ) . The usefulness of the homotopy exact couple and i t s associated spectral sequence is limited by the fact that we know very little about the groups E p , q . F. The cohomotopy exact couple
Let K be a finite cellular complex of dimension 12. Let r denote the least integer greater than t ( n 1). Then we have the following exact cohomotopy sequence of the triple ( K ,K p + l , Kp) :
+
.
It'(K,KP+l) l-+ *
*-+
Itm
A+
(K,KP+1) 272" (K,KP)i,xn, (KP+l,KP) It"+I(K,KP+1)
+. . .
where i and j are homomorphisms induced by the inclusions and k denotes the coboundary operator. Define two bigraded groups D and E as follows : DP4
=
Ep,q =
{ no,p + q ( K ,KP),
if p if p
[
ifp
nP+Q(KP+', KP),
r ( 0 c) nr(K,Kpfl),
p if p
if
+ q >r, + q
+q
2 7 ,
+ q = r - 1, + q c r - 1.
The homomorphisms i, j , k in the cohomotopy sequence may be extended in an obvious way to define homomorphisms i, j, k of the triangle
D
i
+D
which are homogeneous of degrees (- 1, l ) , (0,O)and ( 1 , O ) respectively. Prove that the triangle is exact. Thus we obtain a &couple W*(K)= < D,E ; i, j , k > which is not necessarily regular in the sense of fj6; i t is called the cohomotopy exact couple of K , [Massey 11 and [Peterson 11.
EXERCISES
255
Study %‘*(K)and its derived couples as in the previous exercise. In particular, the exact couple %‘*2(K) contains the following exact sequence
... --+ ha Db-1,1 -LDb-2,2 -LEb-,,,+ !
Prove that, for each
p > r, we have
Db,l
L . *
Hp(K), where r p ( K )denotes the image of the homomorphism i :f l ( K , Kp) -+ nP(K,K p - l ) . Hence the preceding exact sequence becomes Db-l,l =
P-
*..
rp(K),
--+ r p - 4
D5-2.2 = n p ( K ) , Eb-1.1
nP(K)5 H q K ) A+
w
rp+1-
- * *
where It denotes the natural homomorphism defined in (VI; 5 11). Prove that 0 for all p n. This implies that h is an isomorphism for p = n and is an epimorphism for p = n - 1. Thus we obtain the Hopf theorems as corollaries.
r p =
G. The Gamma functor of abelian groups
A function f : A + B of an abelian group A into another B is said to be quadrutic if f(a+b+c) --f(b+c) -/(c+a) -f(a+b)+f(a)+f(b~+f(c) = 0 for all elements a , b, c in A . Let c$f : A x A + B denote the function defined by
b) = / ( a + 4 -f ( 4 - f ( 4 for every pair of elements in A ; in other words, c$f is the deviation of f from a homomorphism. Prove that f is quadratic iff c$f is bilinear. Hence every homomorphism is quadratic. Also prove that compositions of quadratic functions with homomorphisms are quadratic. A quadratic function f : A + B is said to be homogeneous if
f(-4
=
f(4
for every a E A . Prove that every quadratic function is the sum of a homomorphism and a homogeneous quadratic function. Define an abelian group r ( A ) as follows. Let F ( A ) denote the free abelian group generated by a set of generators in a 1 - 1 correspondence with the elements a E A . Let R ( A ) denote the subgroup of F ( A ) generated by the elements of the forms
+ +
+
./(a)-4-
4,
+ +
+
+
w(a b c) - w(b c) - w(c + a ) - w(a b) w(a) w(b) w(c) for all elements a, b, c in A , where w(a) denotes the generator of F ( A ) which corresponds to the element a E A . Then the group P(A) is defined by
r ( A ) = F(A)/R(A). For each a E A , let [a] denote the element w(a) + R ( A )in r ( A ) .Prove that the assignment a -+ [a] gives a homogeneous quadratic function y : A r ( A ) . --f
256
VIII. EXACT COUPLES A N D SPECTRAL SEQUENCES
If f :A -+ B is any given homogeneous quadratic function, prove that there exists a homomorphism h : F ( A ) -+ B such that hy = f . Next, let f : A -+ B be a homomorphism. Prove that the assignment [a] +. [ f a ] gives a homomorphism
T(f1 : W
) +. T ( B )
called the homoniorphism induced by f . Verify that the assignments A - + r ( A )and f f ) define a covariant functor from the category of abelian groups and homomorphisms into itself and hence f :A FW B implies T(f ) : F(A) M I'(B). Prove the following elementary properties, [J. H. C. Whitehead 71 : 1. If A is additive group of rationals, then T(A)M A . 2 . If every element in A is of finite order and also divisible by its order, then r ( A ) = 0. In particular, if A is the group of rationals mod 1, then r ( A ) = 0. 3. If A is a free abelian group with a free basis { a t } indexed by an ordered set { i }, then r ( A ) is the free abelian group freely generated by the elements y(ar),+,,(a,, ar) with j < k . 4. If A is a finite cyclic group of order h and if a is a generator of A , then r ( A ) is a cyclic group of order h or 2h, according as h is odd or even, and is generated by ~ ( a ) . 5. If A is the direct sum Z A , of its subgroups { A , } indexed by an ordered set { p }, then
-+r(
r
Properties (3)-(5) provide a complete computation of the group r ( A ) for A an abelian group with a finite number of generators. 6. If f is an epimorphism, so is I '( f ) . 7. If A is finitely generated, then r ( f :)r ( A ) M r ( B ) implies that f:AwB. 8. If n is a group of operators for A , then n also operates on r ( A )according to the rule &(a) '= y(&) for 6 ~n and a E A . 9. If n operates on A and B and if f commutes with the operators, then so does r(f ) . 10. r commutes with the direct limit functor. Finally, prove that the group D& in (14) of Ex. E is isomorphic to r ( n * ( K ,4 1. H. Transgression and suspension
Let A be a regular &complex. Consider the natural epimorphism g : A -+A defined in 5 11. Let A: = Ao fl A,. If p < 0, then g maps A: into 0 and hence defines a homomorphism
h :&/A:
-+
&.
257
EXERCISES
If p > 1, then h commutes with d and hence induces a derived homomorphism Prove that the images of g,
X ( g ) and h, in X P ( A are )
=
Ego. Next, consider h, together with the boundary homomorphism d : g*[zP(A)I
=
EgJlt k,[.@P(A/AO)I
=
h a X P ( A )+ XP(A/AO) -+ IX p 1 ( A O ) , ( p > 2). Denote by J the image of h,, K the kernel of h,, L the image of d, and M
the kernel of 3. Let X E J . Choose Y E X ~ ( A / A O withh,(y) ) = x , and consider d(y). It is an element ofXP-l(AO) which, when y varies, describes a coset mod d ( K ) . Hence we obtain a homomorphism
T :J %p-1(AO)/d(K) called the transgression. As the image of h,, we have J = EPP,o.By 3 7, E&,-l is a quotient group of i@P-l(Ao).Denote the projection by x . Prove that the rectangle -+
is commutative and d(K)is the kernel of x . Hence x induces an isomorphism ($ > 2). After this identification, the transgression T reduces to the differential operator d P : EPP,o -+ E&-l. x*
i@p-I(A)/a(K)M
P
E0.p-I'
For an arbitrary d-couple, we may define the transgression to be this differential operator d P . Analogous to the transgression, define a homomorphism
x :L
3
Xp(A)/h*(M)
called the suspension. If XP(A)= 0 = i @ P - l ( A ) , then d becomes an isomorphism and the suspension reduces to
x = h,W
: XP-l(AO)+ X p ( A ) .
In this case, the transgression diagram
dp =
T is an isomorphism and in the
258
VIII. EXACT COUPLES A N D SPECTRAL S E Q U E N C E S
the image of X coincides with the image of the monomorphism i and the kernel of Z is precisely the kernel of the epimorphism x . Furthermore, prove that the following relation holds:
z = I (dq-1 x. Establish analogous results for a regular &complex A . 1. Properties of Steenrod squares
Let (X, A ) denote any pair consisting of a space X and a subspace A of X , and 2, denote the cyclic group of order 2. Reproduce the definition of the Steenrod square operations Sq' : @ ( X , A :2,)+ Hn+'(X, A ; Z,),
i
> 0,
and prove the following properties of these operations, [Steenrod 2 and Cartan 11: 1. Sq' o f* = f* o Sq' for any map f : ( X , A ) -+ (Y, B). 2. Sq' 0 6 = 6 0 Sq', where 6 denotes the coboundary operator in the cohomology sequence. 3. Sq'(a
u B) = j+k=s I: . S q W u sqk(B).
4. Sqf(a) = 0 if dim (a) < i.
5. Sq{(a)= a u a if dim ( a ) = i. 6 . SqO(a)= a. 7. Sql coincides with the coboundary operator induced by the exact sequence 0
+z,
-2,
+z,-+o
and hence we have an exact sequence * *
. - + H n ( X , A ; z ~ ) + H n ( X , A ; ZH1n) +~l~( X , A ; Z z ) - + ~ n + l ( X , A ; Z* '4 ) ~ .
CHAPTER IX T H E SPECTRAL S E Q U E N C E O F A FIBER SPACE
1. Introduction We turn now to the study of the relations among the homology and cohomology groups of the various spaces of a fibering. As already indicated, the principal tool is the machinery of spectral sequences developed in the preceding chapter. Since we shall follow Sene, we use the singular groups; our principal hypotheses will be that the fibering have the covering homotopy property and that the fiber be pathwise connected. I t is convenient to use cubes rather than simplexes in defining the singular groups, and hence we begin with an outline of this construction of the singular complex of a space. The associated exact couple of a fibering is then introduced, and the term E2 of this exact couple is computed ($9 3-10; the main result appears in §$ 5-6). A variety of applications follow. Kelations among the Poincark polynomials of the fiber space, base space, and fiber are deduced in $ 11; and several exact sequences, including those of Gysin and Wang, are constructed in the next three sections. In the final sections of this chapter ( $ 5 15-18), regular covering spaces are treated: there is a difficulty, namely that the fiber fails to be connected, but this is avoided by the introduction of certain auxillary fiberings. A spectral sequence due to H. Cartan is constructed and used to deduce certain facts about the action of finite groups on a sphere and the celebrated results on the determination of certain homology and cohomology groups in terms of the fundamental group.
2. Cubical singular homology theory In the traditional simplicia1singular homology theory, [E-S; pp. 185-2 1 11, the unit 12-simplex An is used as the anti-image in defining the n-chains of a space. However, to study the spectral sequence of a fiber space, it will be more convenient to use the cubical definition in which the n-cube 1" plays the role of An. In the present section, we shall give a sketch of the cubical theory. As to the equivalence of the cubical theory and the simplicia1 theory, a proof is given in [Eilenberg and MacLane 21. By a singular n-cube in a space X , we mean a map u : I n + X.If n = 0, then u is interpreted as a single point in X.If fa > 0, we define the i-th lower and @per faces Afou and A& of u to be the singular (n - 1)-cubes given by 259
260
IX. T H E SPECTRAL S E Q U E N C E O F A F I B E R S P A C E
(2r".u) (t1; for every i we have
=;
-
1, 2;
a ,
tn-1)
* *,
n, E
= u(t,;
= 0,
* *,
1, and (tl;
ti-1,
E,
ti;
., tn-,)
*
', tn-1)
*
E
In-1. Then, for i
< i,
1; 17 = 45 1;
where E and 7 may be either 0 or 1. Define Qn(X)t o be the free abelian group generated by all singular n-cubes in X if n 2 0 and Qn(X) = 0 if n < 0. Then the operation
au =
n
;r; (- i)yjzeU - 1pU)
i=I
determines a homomorphism
a
e n ( X ) -+ en-](x)
aa
= 0. This yields a chain for every n. It is straightforward to verify that complex { Q n ( X ) , }, [E-S; p. 1241. Unlike the simplicia1 theory, this chain complex { Qn(X), } does not give the "correct" homology groups of X ; for example, if X consists of a single point, a simple computation reveals that the n-th integral homology group of { Qn(X), } is infinite cyclic for every n 2 0. Hence, we have t o "normalize" { Q n ( X ) , }. For each singular ( n- 1)-cube u in X , n > 0, we define a singular n-cube D u in X by taking (Dzc) (t1; * * , tn-1, tn) = u(t,; * - ,t n - 1 ) .
a
a
a
a
A singular n-cube v in X is said to be degenerate if v = Du for some u. In other words, v is degenerate iff it does not depend on the last coordinate tn of the point (tl; * t n )in In. The degenerate singular n-cubes in X , n > 0, generate a subgroup D n ( X ) of Qn(X).Since a ,
it follows that
i < n, An'D
=
n- I
ItODu) = Z (i=I
1,
l)t(Dl"u - DL&)
and hence carries D n ( X )into Dn-,(X). So, the degenerate singular n-cubes for all n form a subcomplex { D n ( X ) , } of the chain complex { Q n f X ) ,a }. For each integer n, the quotient group
a
C n ( X ) = Qw(X)/Dn(X) is obviously a free abelian group and will be called the group of normalized cubical singular n-chains in X . Since carries D n ( X )into D n - l ( X ) ,i t induces a homomorphism
a
a : c,(x)+ cn-,(x)
for every n, which will be called the boundary homomorphism. Since aa = 0, we obtain a chain complex { C,(X), a}. For an arbitrarily given abelian group G,the homology group H n ( X ; G) and the cohomology group H % ( XG) ;
2. C U B I C A L S I N G U L A R H O M O L O G Y T H E O R Y
261
of this chain complex over G are called respectively the n-dimensional cubical singular homology and cohomology group of the space X over the coeficient. group G . Let C ( X )denote the direct sum of the groups C,(X) for all n. Then, C ( X ) is a graded differential group with a homogeneous differential operator
a : C ( X )+ C ( X ) of degree - 1. We shall call C ( X )the group of all normalized cubical singular chains in X . For any subspace X , of X , the group C,(X,) can be considered as a subgroup of the group C,(X). The quotient group
C,(X, X,)
=
C,(X)/C,(X,)
is obviously isomorphic to the free abelian group generated by the nondegenerate singular n-cubes in X not contained in X,. Since i3 cames C,(X) into C,-,(X) and C,(X,) into C , - l ( X o ) , it induces a boundary homomorphism a : C,(X, X,) + C,-,(X, X,). Since di3 = 0, we obtain a chain complex { C,(X, X,), a } and hence the relative homology groups H n ( X ,X,;G) and the relative cohomology groups P ( X ,X,; G) over a n arbitrarily given abelian group G. Furthermore, if G = { G, I x E X 1 denotes a local system of abelian groups in X , then one can define the groups H,(X,X,; G) and H n ( X , X , ; G) with local coefficients in G in a way similar to that in the traditional singular homology theory. See [Steenrod 11 and [Eilenberg 31. The direct sum C ( X ,X,) of the groups C,(X, X,) is a graded differential group, called the group of all normalized cubical singular chains in X modulo X,. The direct sum H ( X , X,; G) of the groups H n ( X ,X,; G) will be called the total singular homology group o! X modulo X , over G , and the direct sum H * ( X , X,; G) of the groups P ( X , X,; G) will be called the total singular cohomology group of X modulo X , over G . For the important special case that both X and X,, are pathwise connected, pick a point x, of X as basic point and assume x, E X , if X , is non-empty. Just as in the traditional singular theory, it may be proved that we may consider only the singular cubes with all vertices a t x,. Let G = { G , I x E X } denote a local system of abelian groups in X . Since all vertices of the singular cubes are a t x,, the coefficients of the chains and the cochains with local coefficients in G are all in the group , G on which the fundamental group n , ( X , x,) operates. For this particular case, the homology groups with local coefficients can be simply defined as follows. Let G denote an abelian group on which n , ( X , x,) acts as left operators. For each n, consider the group Cn(X,Xo; G) = C n ( X , Xo) 8 G,
262
T H E SPECTRAL S E Q U E N C E O F A F I B E R S P A C E
where Cn(X,X,) denotes the group of normalized cubical singular n-chains in X modulo X , (defined by the n-cubes with all vertices at x,). Define a boundary homomorphism
a : Cn(X,Xo; G )
-+
Cn-i(X, Xo; G)
as follows. Consider a generator u 8 g of the group C n ( X ,X,) 8 G, where u : I n -+ X is a nondegenerate singular cube not in X , with all vertices a t x, and g is an element of G. For each i = 1, * , n, we define a loop afu:Z + X
--
by taking
where ti = t and = tj0 for every j # i. The loop uiu represents an element [atu] of n , ( X , x,,). Then is defined by
a
n
a ( u 8 g)
=
x
l){ { Ir'u 8 [Ufulg- IrOu €3g }.
(-
i- 1
It can be easily verified that ad = 0. Hence we obtain a chain complex { Cn(X,X,; G ) , }. The n-th homology group of this chain complex is defined to be the n-dimensional homology group H n ( X , X,; G) of X modulo X , with local coeficients in G. Analogously, one may define the n-dimensional cohomology group H n ( X , x,; G) of X modulo x, with local coeficients in G. In the sequel, we shall need a slight refinement of the notion of the degeneracy of a singular cube. A singular cube u :I n -+ X is said to be of degeneracy q iff there exists a non-degenerate singular cube v : Zn-q -+ X such that u = Dqv, where Dq denotes the q-fold iteration of the operation D. Hence the degeneracy of a non-degenerate cube is 0, and degeneracy of u : I n -+ X is not less than q if
a
~ ( t , ,*. * , for every point (ti, *
*
a ,
tn)
In) = U(t1,. *
*,
.*,
t n q , 0,.
0)
of In.
3. A filtration in the group of singular chains in a fiber space Throughout $9 3-14, let w : X -+ B be a given fibering as defined in (111; $ 3). In other words, X is a fiber space over a base space B with w as projection. Pick a point x, E X and let b, = w(xo)E B. The subspace
F
=
w-'(b,)
of X will be called simply the fiber. We assume that both B and F are pathwise connected. As an immediate consequence of the existence of covering paths, X is also pathwise connected. Hence, according to a remark of $ 2, we may consider only the singular cubes with all vertices at x, or b, when dealing with the cubical singular theory. We assume once for all that every singular cube considered in the sequal is of this limited kind.
4. T H E
263
A S S O C I A T E D EXACT COUPLE
A singular cube u : In X is said to be of weight p (in notation : w(u) = p ) if the singular cube wu : In + B is of degeneracy n - p ; thus u is of weight fi iff u(t,, * * *, t n ) remains within a fixed fiber as tPtl; * * , tn vary, but moves from fiber to fiber as t p varies. For each singular cube u in X , the weight w(u)satisfies the condition --f
0
(3.1)
< w(u) Q dim (u).
It is also straightforward to verify the following two relations : (3.2) (3.3)
w(&"u)< w(u)if 1
< i < w(u)and E = 0, 1 ;
w(&"u) = w(u)if w(u)< i
=
0, 1.
Let B, be a pathwise connected subspace of the base space B which contains b, if B, is not empty. Let X , = w-l(B,) ; then X , is also pathwise connected. Let C = C ( X ,X,) denote the group of all normalized cubical singular chains in X modulo X,. Then there is a natural isomorphism between C and the free abelian group generated by the non-degenerate singular cubes in X not contained in X,. Let us identify these two groups by means of this , = natural isomorphism. Then C is a graded differential group with C C,,,(X, X,) and with a homogeneous differential operator d of degree - 1. Furthermore, we may define an increasing filtration { C p } by taking CP to be the subgroup of C generated by the non-degenerate singular cubes of weight not exceeding p . I t is obvious that
UPCP
=
c,
c p c
Of', d ( 0 ) c c p
and that C P is a subgroup of the graded group C. In fact C$ = CP n C,. If we extend the weight function w over C by defining w(c),c E C,to be the greatest lower bound of the integers p such that c E C p , then (3.1) implies
0
(3.4)
< w(c)
(c)
for every non-zero homogeneous element c E C. Furnished with this filtration { C P } , C becomes a regular d-complex in the sense of (VIII; 11).
4. The associated exact couple Consider the regular d-complex C = C ( X ,X,) of the previous section. According to (VIII; 1 l.l), the associated exact couple
U(C) = < D,E ; i, j , k > is a regular &couple. We shall determine the structure of the bigraded differential group E. Since
E,,,
=%'p+p(CP),
C P = CP/CP-l,
we have to analyze the graded differential group &.
264
T H E SPECTRAL SEQUENCE O F A FIBER S P A C E
By definition, ck C f l C f - l . Let u be a non-degenerate singular cube in Cf, then u represents an element [u] in By (3.2) and (3.3), it is evident that the differential operator on c P sends [u] into the following element of C$-,: 2
cf.
a
m
a[@]= i--p+z
(4
(-
i)r { [ A ~ ~ [~ ~ It u l ) .
1
Now let u : Im -+X be a singular cube with w ( u ) < p . Set q and define two singular cubes
=m
-p
B P u :I p - + B , F p u : I q - + F as follows:
B p u ( t l , * * 't p, ) FpU(t1; *
= ~ ~ ( t l , . t. p., ,O , * . ' , O ) ;
.,tq)
=~
( 0* * ,* , 0, t1;
*
., tq).
For the second, since
W u ( o , " ' , o , t l , ~ ~ ~=, t Wq )u ( o , ~ * ~ , = o )b,,
-
we have Fpu(tl, * * , tq) E F. Thus, for a given u, we can define Bpu and Fpu for all f~ such that w ( u ) < @
0, then Fpu is degenerate. (4) If i > p , then BP&% = Bpu and FpA& = A;+FPu for E = 0, 1. Let K p = Cp(B,B,) 8 C ( F ) ; then K p is a graded group with KP, = Cp(B,B,) 8 Cm-p(F).Define a differential operator d F on Kp by
d p ( b 8 1) = (-
(ii)
for each generator b 8 f of KP. Then morphism : C P -+ K p by taking
+
i)p b 8
dF
a/
is of degree - 1. Define a homo-
+(u)= Bpu 8 Fpu for every generator u of C p . This definition is justified by the properties (1) and (3) listed above. According to (2),(b carries C p - 1 into the zero element of K p and hence induces a homomorphism
y : &' -+ K
p
by passing to the quotient group C P = C p / C p - l . By using (1) through (4), one can verify that y commutes with the differential operators in &' and d F in Kp. Hence y is a mapping in the sense of (VIII : 5 12). Since Cp(B,B,) is free, we have
a
Xp,q(KP) =
Cp(B,B,) 8 &(F)
4. T H E
A S S O C I A T E D EXACT COUPLE
265
where Hg(F) denotes the q-dimensional singular homology group of F. Hence y induces a homomorphism
x,,q : E,,g for each pair of integers
+
C,(B, Bo) €3 HdF)
( p , q).
Theorem 4.1. Xp,g i s a n isomorphism of Ep,qonto C,(B, B,) €3 Hg(F).
To prove this theorem, it suffices to construct a mapping p :KP -+ d p such that yp is the identity endomorphism of KD and py is homotopic to the identity endomorphism of &'. Lemma A. For every pair of singular cubes u : Ip --f B and v : Ig + F , we can construct a singular cube
z
=
M(u, v) : rp+q -f
x
satisfying the conditions : (Al) W ( Z ) < p . (A2) B p = u and Fpz = v. (A3) G+iz = M(u, &'v) for each i Q q and E = 0, 1. (A4) If v is degenerate, so is z. (A5) If u is in B,, then z i s in X,. The proof of this lemma will be given in 3 8. The singular cube z = M ( u , v ) represents an element M [ u ,v ] of 6, because of (Al). If u is degenerate, then (A21 implies that w ( z ) < p and hence M [ u , v ] = 0. If v is degenerate, then so is z and hence M [ u , v ] = 0. If u is in B,, then z is in X,and hence M [ u ,v ] = 0. Therefore, we may define the homomorphism p by taking p ( u @u ) = M [ u , v] for each generator u €3 u of C,(B, B,) €3 C ( F ) . The condition (A3) implies that p commutes with the differential operators. Hence p is a mapping. The condition (A2) shows that wp is the identity on Kp. I t remains to prove that py is homotopic to the identity of &. Lemma B. For every singular cube u : Ip+Q + X with w ( u ) < p , we can construct a singular cube v = D,u : IPfq+l+ x satisfying the conditions : (B1) w(v) Q P. (B2) B,v = Bpu. (B3) v ( O ; * * , O t , t l ; . . , t g ) = u ( O ; * * , O , t , ; . * , tq). = M(B,u, Fpu). (B4) A:+,v = u and (B5) A;+lv = D,A& for each i > p and E = 0, 1. (B6) If q > 0 and u i s degenerate, so is v. (B7) If u is in X,, so i s v. The proof of this lemma will be given in 5 9. Now the proof of the theorem can be completed. The correspondence
266
T H E SPECTRAL SEQUENCE O F A FIBER S P A C E
u -+ (- 1)9Dpudefines a homogeneous endomorphism t : & + (Y of degree 1. To justify this, it suffices to note the facts: if u is degenerate and q > 0 , then D,u is degenerate by (B6); if u is degenerate and q = 0 , then (B2) implies that w(D,u) < p ; if u is in X,, then D,@ is also in X , by (B7); and if w(u) < p , then w(Dpu)< p according to (B2). By means of the formula (i)and the conditions (B4) and (B5), one can easily verify that
a(&) + t@)= c -p y k ) for every c E &. Hence py is homotopic with the identity endomorphism of c p . I The isomorphism X p , q : E,,, w Cp(B, B,) 8 Hq(F) indicates the significance of the various degrees. We shall call the primary degree p the base degree, and the secondary degree q the fiber degree. The total degree p + q is usually called the dimension.
5. The derived couple In the preceding section, we studied the associated exact couple U(C)=
< D,E ; i, i, k > of the regular &complex C and established an important isomorphism
X :E
w C(B,B,) 8 H ( F )
which is homogeneous of degree (0,O).According to (VIII; 5 4), U ( C ) has a derived couple V2(C) = < D2, E 2 ; i2, j2, k2>. The key to all applications is the bigraded group E2, and hence we shall determine its structure. To this end, we must analyze the differential operator d = jk on E . If we identify E and C(B, B,) 8 H ( F ) by means of the isomorphism X, then our problem is to determine how d operates on C(B,B,) 8 HW). For this purpose, let us first show that the fundamental groupn,(B, b,) acts on H ( F ) as a group of left operators. Lemma C. For each loop u : I + B with u(0) = b, = o ( l ) , there exists a homogeneous homomorphism D, : C ( F )-+ C ( X ) of degree 1 which assigns to each singular n-cube v in F a singular ( n + 1)-cube D,v in X satisfying the conditions : (Cl) ill0D,v = V . (C2) (wD,v) (t. t1; * * , tn) = ~ ( t ) . (C3) D,ilt"v = il:+lDav, ( E = 0 , 1). (C4) If v is degenerate, so is D,v. The proof of this lemma will be given in 9 10. Such a homomorphism D, will be called a deformation of C ( F ) covering 0. By (C2), D, determines a homogeneous endomorphism J,:C(F) -+ C(F) as follows. For each singular n-cube v in F , J,,v is the singular n-cube in F defined by (1,~) (tl; * * , tn) = (D,v) (1, t,; * *, tn).
5.
267
THE DERIVED COUPLE
By (C4), it follows that Jazi is degenerate if v is degenerate. By (C3), we have = of H ( F ) which, by the next lemma, depends only on the element [u] of n l ( B ,b,). Lemma D. I/ two loops u, t represent the same element o/n,(B, b,), then J a and J z are chain homotopic. The proof of this lemma will be given in fj 10. Thus, it is clear that n,(B, b,) acts on H ( F ) as a group of left operators. So, we may define a differential operator d B on C(B,B,) 8 H ( F ) by taking dB to be the boundary operator on chains with local coefficients in H ( F ) as defined in fj 2. Now we can settle our problem by saying that, if we identify the groups E and C(B,B,) 8 H ( F ) by means of X , then the differential operator d coincides with dB. Precisely, this can be stated in the form of the following
Ja aJ0. Hence Ja induces an endomorphism
Lemma 5.1. The isomorphism X is a mapping, that is to say, dB% = Xd. Proof. We shall prove that dB = XdX-'. Consider the generator u @h of C(B,B,) 8 H ( F ) described above. The homology class h E H,(F) can be represented by a singular q-cycle
-z r
f
j- 1
apj,
where the v j ' s are q-dimensional singular cubes in F and the aj's are integers. Consider the singular ( p + q)-chain r
z
=
x UjM(24,
Vj)
j= 1
in X . By the definition of p and X , z represents the elements p(u 8 f ) E &,
X-l(u 8 h) E ED,,.
The boundary of the chain z is given by
dz
P+rl
=
z z
(-
1)'Uj
{
&'M(U, V j ) - IrOM(u, V j )
}.
j=1 i - 1
If i > fi, it follows from (A3) of fj 4 that Ir"M(24,V j )
=
M ( u ,& p j ) ,
(E =
0, 1).
Since f is a cycle, the expression r r l
X X
i=l
y-1
(-
l)taj(il?vj-
ilrOvj)
is equal to a linear combination of degenerate singular cubes in F . Hence, i f we neglect the degenerate terms, we can write dz in the following form:
dz
=
r
P
j-1
i-1
X X
(-
1)raj{ Ir'M(u,
vj) -
i l r o M ( ~v j,) }.
268
IX. T H E S P E C T R A L S E Q U E N C E OF A F I B E R S P A C E
According to (3.2), d z is a cycle in CP-l. Projecting to the quotient group az represents a cycle of &'-I and hence an element y of E p - ],g. By the definition of d , we have =
&-I,
Hence the element X ( y ) X =
r
P
j=1
i-1
I:
(-
=
XdX-l(u 8 h) is represented by the cycle
{ Bp-llt'
1)'Uj
BP-lhO
M(U,V j )
8 Fp-l&.l
hf(24, V j )
-
M ( f 4V j ) 8 FP-lhO M ( u , V j ) 1
of CP-,(B,Bo)8 C ( F ) under dF. We have to consider the singular cubes BP-,&' M(u, v j ) and F p - , V M(u,v j ) for i < p. The first is evidently A&; the second is defined by (FP-llt8M(%vl))( 4 , * ' . , t * ) = M ( u , v j ) ~ O ; ~ ~ , O , E , O ; " , 0 , t ] , " ' , t g ) '
where E is at the i-th place. Obviously we have Fp-,&oM(u, v j ) r
x
=
I: (-
1y { 2t'u
@3 gt
= vj;
hence
- It024 8 f },
i- 1
where gi denotes the cycle r
gt =
of C ( F ) .
i- 1
ajFp-l&'M(u,
vj)
To detennine the homology class of gt, let us construct for each q-dimensional singular cube v in F a ( q + 1)-dimensional singular cube Dv in X defined by
(Dv) (t, t,;
* *,
t*) = M ( u , v ) (0;
.)0,t, 0;
*)
0,t,;
*,
t*),
where t is at the i-th place. Since ( d v ) (1, ti,.
., tg)
=
(GP) (t),
where otu : I + B denotes the path defined as in 8 2, one can easily verify that the assignment 2: + Dv determines a deformation covering the path uiu. Since
D : C(F) - + C ( X )
vj(h, * - * , t*), t)= F p - l A t 1 M ( u , "I) (4,.
(Dud (0,tl, *
(Dq)(1, t1,. * *,
-
*
,t*,
=
*
* I
f*),
it follows that gt represents the element [atulh of H,(F). Therefore, we obtain %dX-'(U
8 h)
I
=
2 (a-
1)' {
& ' U
8
[ U t U ] h - &O.U
8 h } = dB(U €3h).
1
Hence we have XdX-1 = da. I As an immediate consequence of the lemma, we have the following
6.
HOMOLOGY WITH ARBITRARY COEFFICIENTS
Theorem 5.2. The isomorphism X induces for each pair an isomorphism x p , g : Ei,q w Hp(B, Bo; Hg(F))
( p , q)
269 o/
integers
of E& onto the $-dimensional singular homology group of B modulo B, with local coeb’icients in H g ( F ) .
The isomorphisms x p , greveal an important fact, namely that the bigraded group Ea is the same for all fiber spaces having the same base space B, the same subspace B, of B , the same fiber F , and the same operations ofn,(B)on H ( F ) . In particular, if n , ( B ) operates simply on H ( F ) ,the bigraded group Ea is the same as that of the product space B x F . The deviation from the product structure is bound up in the deviation of the differential operator d2 from 0.
6. Homology with arbitrary coefficients In the preceding two sections, we restricted ourselves to integral coefficients for the sake of simplicity. However, the results can be easily generalized to arbitrary coefficients as follows. Let G be an abelian group and C denote the regular &complex C ( X , X,) of the previous sections. Consider the group
A=C@G of normalized singular chains of X mod X , with coefficients in G. Then A is a graded group with Am=Cm@G and a differential operator { Ap } by taking
a of degree - 1. Define an increasing filtration G.
A p =C P @
Thus, A becomes a regular &complex. We shall study the associated exact couple %‘(A)= < D,E ; i, j, k > and its derived couple %?(A) = < D2,E z ; i2, j2, ka >. Since C p is a direct summand of C, it follows that = APIAP-1 = t P
8 G.
Next, let us consider the group K p =
Cp(B,B,) 8 C ( F )@ G.
Define a differential operator dF on K
dF(b8 f 8 g)
=
p
(-
by 1)pb 8
a/ 8 g
for every generator b 8 f 8 g of Kp. Then
s p + g ( K P )= Cp(B, Bo) 8 Hq(F; G),
270
IX. T H E SPECTRAL SEQUENCE O F A F I B E R SPACE
w k r e H,(F; G) denotes the q-dimensional singular homology group of F with coefficients in G. The homomorphism p of 5 4 defines in this case a mapping y : A p + Kp and hence induces the homomorphisms &,q
The lemmas of
: E p , q +Cp(B, Bo) 8 Hq(F;G).
5 4 imply the following
Ep,q onto C,fB, B,) 8 H,(F; G). By the lemmas C and D in 5 5, n l ( B ,b,) acts on H ( F ; G) as a group of left operators. Let d B denote the boundary operator in C(B,B,) 8 H ( F ;G) as chains with local coefficients in H ( F ; G). Then, one can easily verify that Theorem 6.1. Xp,, is an isomorphism of
(5.1) is also true in this general case. Hence we have the following Theorem 6.2. The isomorphism X induces for each pair
isomorphism
xp,q
( p , q) of integers an
: Ej,q * Hp(B, Bo; Hq(F; G))
of E;,q onto the $-dimensional singular homology group of B modulo B, with local coeficients in H,(F; G).
An important special case of (6.2) is that A = C ( X )8 G. Then B, is empty and x,,, is an isomorphism of E;,q onto the homology group H,(B; Hq(F;G)) with local coefficients in H,(F; G) . Analogous results hold for cohomology with arbitrary Coefficients. See Ex.A at the end of the chapter. In most of the applications, we shall deal with the case where n , ( B , b,) operates simply on the homology and cohomology groups of the fiber F . In particular, this is the case if B is simply connected or if X is a fiber bundle over B with a pathwise connected structural group, [Serre 1; p. 4451. Now assume that the coefficient group G is either the additive group of integers or a field. Then the isomorphism x P , , in (6.2) is actually an isomorphism of G-modules. If n,(B, b,) operates simply on H,(F; G), then the group Hp(B,B,; Hq(F,G)) in (6.2) reduces to the singular homology group with coefficients in H,(F; G) in the usual sense. Hence, by the Universal Coefficient Theorem, [E-S; p. 1611, we obtain the following Theorem 6.3. If n,(B. b,) operates simply
+
011
H,(F; G), then
Ei,q * Hp(B, B,; G) @cHq(F;G) T o Y G ( H ~ - ~B,; ( BG), , H & F ; G)). The torsion product TOYG has the important property that Torc(L,M ) = 0 if L or M is a free G-module, [E-S; p. 1341. Hence we have the following Corollary 6.4. If H P - ] ( B ,B,; G) or H,(F; C ) is a free G-module, then
Ei,q w Hp(B, Bo; G) @GHq(F;G). Note that the hypothesis of (6.4) is always true if G is a field.
7. T H E
S P E C T R A L HOMOLOGY S E Q U E N C E
271
7. The spectral homology sequence Let us consider the regular 8-complex
A
=C8
G, C = C ( X , X,) of the preceding section. As in (VIII; 3 4), we denote by
@(A)
= < Dn,En; in,
p,kn >
the successive derived couples of the associated exact couple % ( A ) = W ( A ) of A . Then, En is a bigraded differential group with dn = jnkn as differential operator and is the n-th term of the associated spectral sequence {EnIn=l,2,-.-} of Y = % ( A ) , which will be called the sfiectral homology sequence of the fiber space X modulo X , over the coefficient group G. Since A is a regular &complex, we may apply the results of (VIII ; §§lo-1 1). In particular, #(%) = # ( A ) = H ( X , Xo; G).
Then H ( X , X,; G) is filtered with E* as its associated graded group; more precisely,
Hm(X, x,; G) = Z m , o ( % )
= %m-1,1(%)
2 ''
*
= X o . m ( V ) = 2P-I,rn+1(%)
= 0,
(U) = EZq. Hereafter, we shall use the notation H p , q ( X X,,; , G) = 2Pp,q(%). For further studies in this section, we have to specify whether or not X , is empty. Let us first consider the case that X , is empty. According to (VIII; 3 I l ) , Rq(%)= S q ( A 0 ) . A singular cube G in X is of weight w(u) = 0 if and only if the image of wu is a single point. Since all vertices of u are assumed a t x,, this single point must be b, and hence u is in F . Therefore, Ao = C(FI 8 G. Then it follows that /%P-1,9+1
#,*(1(%)
I?*(%)
=
Hq(F; G).
Next, S,(%) =#,(A) with A, = ED,,. However, by means of Xp,o, we may identify E p ,o with C,(B) 8 H,(F; G). Since F is pathwise connected, H,(F; G) may be identified with G on which n l ( B ) operates trivially. Hence A = C(B)8 G and S p ( % ) = H p ( B ;G ) . Then, we may apply the results of (VIII; two commutative triangles
Hq(F;G)
\
'\EZq
& ( X , G)
/
3 7)
and obtain the following
H p ( X ;G) W. H P P , G)
\ J/ Ego
where 8*, o* are induced by the inclusion 8 : F c X and the projection o : X + B, the x's are epimorphisms, and the L'S are monomorphisms. It is
272
IX. T H E S P E C T R A L S E Q U E N C E O F A F I B E R S P A C E
an immediate consequence of these triangles that H,,,(X; G) is the image of 8, and H p - l , l ( X ;G ) is the kernel of we. It remains to study the case that X , is not empty. Then F c X,. One can easily see that A0 = 0 and hence E:.q = 0 = EC,; in particular, Rq(U) = 0. On the other hand, we have S p ( V ) = Hp(B, Bo; G). Thus, we still have a non-trivial commutative triangle
Hp(X, Xo; G) 0.Hp(B, Bo; G)
\
EZO
J/
The kernel of w* is H p - l , l ( X ,X,; G ) and the image of o* is isomorphic to EZo. Similarly, one can study the spectral cohomology sequence. See Ex. B.
8. Proof of Lemma A We shall prove the lemma by induction on the integer q. First, assume that q = 0.Then the singular cube v reduces to the point x, ; and the problem reduces to constructing, for each singular cube u : I p -+ B , a singular cube z = M(u, v ) : I p +. X such that wz = u. Let V,denote the leading vertex (0; * ., 0) of I p . Since V , is a strong deformation retract of I p , we may apply (111; 3.1) and obtain a map y :I p --f X such that wy = u and y(Vo) = x,. Let V , denote the various vertices of Ip. Since u(Va)= b, and wy = u, it follows that y(Va) is a point of F. Since F is pathwise connected, there exists a path a. : I +.F such that a,(O) = y(Va) and aa(l) = x,. Let Q denote the subspace of I p which consists of all vertices of I p . Define a homotopy f t : I p +. B, (0Q t Q I ) , and a homotopy gt : Q +. X , (0 Q t Q l) , by taking ft =
gt(Va) = 4
for every t E I and every vertex V , of go = Y I
e,
Ip.
4
Then we have
wgt
= ft
I
e
for every t E I . According to (111; 3.1), there exists a homotopy gt* :I p + X , (0 Q t Q l ) , such that
e,
Wgt* = f t go* = y , gt = gt* I for every t E I . Let z = g,*. Then z is a singular cube in X with all vertices a t x, and such that wz = u. This proves the lemma for q = 0. Next let q > 0 and assume that we have constructed M ( u , v ) for all u and all v with dim (v) < q satisfying the conditions of the lemma. We are
8. P R O O F O F
LEMMA A
273
going to construct a singular cube z = M(u, v ) for a given pair of singular cubes u : I P -+ B and v : I q -+ F as follows. To construct the singular cube z = M ( u , v ) , let us first dispose of the special case where v is degenerate. Then v does not depend on the last variable and Aqo(v) = Aql(v). Define z = M(u, v ) by means of the formula: z(t1,.
tp+g) =
*
M ( u , &Ov) (ti). * *
t
tptq-1).
We have to verify the conditions (Alk(A5) of the lemma. The conditions (Al), (A4) and (A5) are obviously satisfied. The condition (A2) is verified as follows: B p ~ ( t , ; * * , t p ) = wz(t,, * * * , tp, 0, * * * , 0 ) = wM(zt, A ~ O V ) ( t , ; . * . t p , O ; * * , O ) = up,, * , t,) ;
--
Fpz(t,,
*
*,
tq) =
z(0, * ' - ,0, t,,
* * *,
tq)
M ( u , A*%) (0; * -,0 , t,; * tpl) = &O"(t,, * * * , = "(t,, * * * , tq). =
* )
To verify the condition (A3), we consider two cases. If i
=
q, then we
have
G+k (tl, If i
* * *
,tp+q-1)
=
< q, we have
A;+$
(t,, *
= Z(t1,
* *,
tp+q-J
* * *
I
tp-q-1, E )
M ( u , &%) (t,;
= Z(t,, '
* *
,tp+i-,,
* *,
tp+q-l).
E , tp+a,
-
.
* *>
tp+q-J
M ( u ,AqO") (t,, * * , t p + ; - , , E , t p + z . * = n;+i M ( u , &On) (4,* * * , t p + q - 2 ) = M ( u , /?a"*") (tl,* * * ) tp+*-J. =
* *
,t p + q - , )
On the other hand, since i < q and v is degenerate, it follows that A~'v is also degenerate and so is M(u,A(ev\. Therefore, we have
M ( u , Atv) (tl, * * * , t p + * - 1 )
M ( u , Afv) (4,* * * , t p + q - 2 , 0 ) = A;+4-l fif(.u, A t 4 ( 4 , . * * , tp+*-2) = M ( u , il;,La"v) (11,* * * , t p + q - 2 ) . =
Since AaeAqOv= A:-lAzev, this completes the verification of the condition (A3). It remains to construct z = M ( u , v ) for a non-degenerate v . Let P = Z p + q = Z p x I q and Q = V , x Zq U Z p x a P , where V , denotes the vertex (0; * . , 0) of I p and d Z 4 the set-theoretic boundary of IQ.ThenQ is contractible, because it is clearly deformable into V , x I 4 U V , x = V , x Zq which is contractible. Hence, Q is a strong deformation retract of P and we may apply (111; 3.1) once again.
IX. T H E S P E C T R A L S E Q U E N C E O F A F I B E R S P A C E
274
Define maps f : P -+ B and g : Q -+ X as follows: f(tl;
g(0; g(t,,
*
tp, s1;
*,
- -,0,s]; * * *
= M(u,
, t p , S,'
* *)
sg) =
uft,;
*,
sg) =
v(sl;
*
* * *
Ar8v) (t,, *
*
9
si-1, -5, Sf,
., tp, s1;
*
*,
- .)t p ) ;
-
sg),
a ,
* * *,
on V , x Ig;
+I)
sg-J, on I p
x i3D.
I t is straighforward to verify that g is one-valued and hence, by (I; 5.1), g is continuous. Besides, it is obvious that wg = f 1 Q. Therefore, according to (111; 3.1), there exists a map z :I p + g -+ X such that wz = f and z 1 Q = g. Since q > 0, Q contains all vertices of I p + g . This implies that z = M(zl,v) is a singular cube in X with all vertices at x,. The conditions (Al) through (A5) are all obviously satisfied. This completes the inductive construction of M(u, v ) and proves the lemma.
9. Proof of Lemma B The proof of this lemma is analogous to that of Lemma A and is based on a construction u + Dpu by induction on the integer q. It will be sketched as follows. First assume that q = 0. Let
P
= I,+, = I p
x I, Q
= (Ip
x 0) U (V, x I) U ( I D x 1) c P .
-
where V , denotes the leading vertex (0; - ,0) of I p . Then Q is a strong deformation retract of P. Define two maps f : P -+ B and g : Q -+ X by taking
g(t,;
f (4,* * * ,
t p , t ) = wu(t,,
g(t1, *
*,
t p , 0) =
g(0,
',
0,t )
=
I
tp),
onIp x I ;
* *>
tp),
o n I p x 0;
* * *
u(4, *
onV, x I ;
x,,
- ., tp, 1) = M(Bpu,Fpu) ( t l ;
+, tp), on I p
x 1.
Applying(III;3.1),weobtainamapv:P - + X s u c h t h a t w v = f andv IQ=g. Since Q contains all vertices of P, v is a singular cube with all vertices x,. Let Dpu = v and the conditions (Bl)-(I37) are obvious. Next let q > 0 and assume that we have constructed Dpu for every u with w(u) Q p and dim(u) < p + q. Now let u be a singular cube in X with w(u) Q P and dim(u) = p + q. To construct v = Dpu, let us dispose of the special case where u is degenerate. Then u does not depend on the last variable and Ai+qu = A;+,. Define v = D,u by taking "(t,, *
-
*
,tp+g+J
=
--
DpA;+qu (4, - ,t p + g ) .
By means of the hypotheses of induction, one can easily verify the conditions (Bl)-(B7).
10. P R O O F O F LEMMAS C A N D D
275
It remains to construct v = D,u for a non-degenerate u with w ( u ) and dim(u) = p + q. For this purpose, let p = I , x I x 14 = IP+(l+l, Q
=
(I, x o x 14)
u (I, x
1 x IQ) u
Qp
(v,x I x IQ) u ( I P x I x 3141,
where V , denotes the leading vertex of Ip and 319 the set-theoretic boundary of [email protected] two maps f : P +. B and g : Q -+ X by taking
f (t,;
t, SIP-* * ,
BpN1,' * *, t p ) . g(tl,...,1,,O,Sl,...,Sq) = U(tl,"',tp,Sl,...,Sq), g ( t 1 , . * * , t p , 1, ~ 1 , '* * , Sq) = M(Bpu, Fpu) (ti,. * t p , ~ 1 , ' . Sq), g(0; * * , 0 , t, s]; * *, sq) = ~ ( 0 *;* , 0 , sl; * * , ~ 4 )= F ~ U ( S *, ;* , Sq), - 8
tp,
Sq) =
' 8
t, S1" * * , S t - 1 , E , .Q>* B&+,,iu ( t i , .* t p , t, ~ 1 * , *
g(t1,.
*,
tp,
* t
*
* )
-,e - 1 )
=
Sq-1).
Since Q is a strong deformation retract of P and wg = f I Q, there exists a map v : P +. X such that o v = f and v 1 Q = g. Since Q contains all vertices of P , v is a singular cube with all vertices at x,. Let v = D,u and the conditions (Bl) through (B7) can be easily verified. This completes the proof.
10. Proof of Lemmas C and D To prove Lemma C, we consider u as a singular 1-cube in B . For each singular n-cube v in F , we take D,v = M(u, V ) given by Lemma A . Then D,v is a singular (n + 1)-cube in X . The assignment v -+ D,v defines a homogeneous homomorphism D, : C(F)+. C ( X ) of degree 1. The conditions (Al)-(A4) imply the conditions (Cl)-(C4). This proves Lemma C. To prove Lemma D, let u, t be two loops in B representing the same element of n , ( B , b,). Then there exists a singular 2-cube u : I 2 -+ B such that u(0, t)
=
b,
=
u(1,t ) , u(t, 0)
=
up), u(t, 1)
= t(t)
for each t E I . Let D, and D, be deformations of C ( F ) covering the paths u and z respectively. For each singular n-cube v in F , we shall construct a singular (n + 2)-cube Qv in X satisfying the conditions: (D1) ~ ( Q v <) 2 . (D2) B,Qv = U. (D3) AIOQv(t,t,; -,In) = ~ ( 1 , ; * In). (D4) A,OQv = D,v; il21Qv = D,v. (D5) QA& = A;+,Qv, ( E = 1 ; i = l;..,n). (D6) If v is degenerate, so is Qv. a ,
9'
IX. T H E S P E C T R A L S E Q U E N C E O F A F I B E R S P A C E
276
The cube Qv will be constructed by induction on the dimension n of v. First assume 1z = 0. Then v reduces to the point x,,. Consider the subspace
A
=
( I x 0) U ( I x 1) U (0 x I )
of I 2 and define a map f : A
-+
X by taking
f ( t ,0) = Dav(t), f ( t , 1) = Drv(t), f(0, t ) = xo
for every t E I . Since of = u I A and since A is a strong deformation retract of 1 2 , f has an extension g : I 2 -+ X covering u. We define QV to be the singular 2-cube g . The conditions (Dl)-(D6) are obviously satisfied. Next assume that n > 0 and that Qv has been constructed for every singular cube v in F of dimension less than n. Let v be a singular cube in F of dimension n ; we are going to construct Qv. If v is degenerate, we set
Qv(s, t, t,;
*
*,in) =
Q&'v(s, t, t,;
* *,
in-,).
If v is non-degenerate, let us consider the subspace A
=
of I"+2 setting
ox
(I x
*
u (0 x
x 1 x
I x I%)u ( I x I x aIn)
x I n and define two maps f 1 A
= I2
+(s, t, t,,
u
~ n )(I
-+ X
and
+ : In+z
-+
B by
- ' , tn) = 4%0 ,
f (s, 0, t,,* * > tn) = Dav(s,t i > ** * > i n ) , f (s, 1, ti,. * tn) = Drv(s, ti,* * ) tn), f (0, t, t,; * tn) = "(ti,.- ., t n ) , *
*
- 9
*>
f ( s , t, t,,.
* *)
=+
4-3,
E , tt,.
* )
tn-1) =
Q&'v(s, t, ti;
* *,
in- 1 ) .
Since of I A and A is a strong deformation retract of In+a, f has an extension g : In+2 -+ X covering We define QV to be the ( n 2)-cube g. The verification of the conditions (Dl)-(D6) is left to the reader. The inductive construction of Qv is complete. Now consider the endomorphisms
+
+.
Ja, J r
:C(F)
-+
C(F)
defined by the deformations Do, D, respectively. Let K denote the homogeneous endomorphism of C ( F ) of degree 1 defined by
( K v ) ( t , t i , . * * , t n ) = (Qv)( 1 , t , t , , * ' * , t n ) for each singular cube v : I n
-+
aKv
F . Then, one can easily verify that
+ Kav = JIv -Jav.
Hence, Ja and J , are chain homotopic. This completes the proof of Lemma D. Remark. If X is a bundle space over B with respect to the projection w : X --f B , then the proofs of the lemmas A-D can be simplified as follows.
11. T H E
POINCARBPOLYNOMIALS
277
For a given singular cube u : I p 7B, one considers the bundle space U over IP induced by u. By a theorem of Feldbau [S; p. 531, U is equivalent to
the product space I p x F and hence the constructions can be easily carried out in U instead of X .
11. The Poincard polynomials Throughout this section, let G be a field. We shall consider vector spaces Mover G graded by the spaces M p ; the dimension of M p is called the p-th Betti number of M and is denoted by R p ( M ) Then . M is of finite dimension iff the Betti numbers { R p ( M )} are all finite and only a finite number of them are different from zero. We assume once for all that, unless there is a statement to the contrary, a given graded vector space M is of finite dimension over G and M p = 0 if $ < 0. For such an M , we define the Poincart! polynomial y ( M ) and the Euler characteristic X(M) by
y ( M ) = 2: Rp(M)tP, X f M ) = P
x (P
l)PRp(M).
A few elementary properties of the PoincarC polynomials are listed as follows. Proofs are left to the reader. (1) If L is a subspace of M and N = M / L , then y ( N ) = y ( M ) - y ( L ) . (2) For any two graded vector spaces M and N , we have y ( M @ G N )= y ( M ) y ( N*) (3) If M is a graded vector space with a linear differential operator d : M + M o f d e g r e e - I , thenwe havey(&'(M)) = y ( M )- ( l + t ) y ( d ( M ) ) . For any two polynomials f and g, the symbol f Q g will mean that g - f is a polynomial with non-negative coefficients. (4)Let M be the same as in (3). If L is a subspace of the graded vector space M such that L n d(M) = 0, then N = L n Z ( M ) is isomorphic to the subspace of X ( M ) represented by the cycles N and y ( L )-y ( N ) < t y ( d ( M ) ) . (5) Let M be the same as in (3). If L is a subspace of Z ( M ) and N denotes the subspace of X ( M ) represented by the cycles L , then we have y ( L )y ( N ) Y ( d ( W). Let ( Y , Yo)be a pair of a space Y and a subspace Y oof Y . If H ( Y , Y o ;G) is of finite dimension, then its Betti numbers, its Poincark polynomial, and its Euler characteristic will be defined to be those of the pair ( Y , Yo) over G denoted by R p ( Y ,Y o ;G ) , y ( Y ,Y o ;G ) , and X(Y, Y o ;G) respectively. Now let us consider the spectral homology sequence of 5 7. Assume that n,(B, b,) oprates simply on H ( F ; G) and that H(B, B,; G) and H ( F ; G ) are of finite dimension. Then we have the following two theorems: Theorem 11.1. R,(X,
X,; G) Q X Rp(B, B,; G)R,(F; G ) . P+q=m
Theorem 11.2.
X(X, X,; G) = X(B, 23,; G)X(F;G).
In words, (1 1.1) states that the Betti numbers of the fiber space X cannot be greater than those of the product space B x F and (1 1.2) states that the
278
IX. T H E SPECTRAL SEQUENCE O F A FIBER S P A C E
Euler characteristic of the fiber space X is the same as that of the product space B x F. These two theorems are immediate corollaries of the general theorem below on PoincarC polynomials. Since H(B, B,; G ) and H ( F ; G) are assumed t o be of finite dimension, their PoincarC polynomials are of the form :
y(B,B,; G) = bata + ba+lt"+l+
- - - + bptp,
y ( F ;G) = 1 + c1t + c2t2 +
+
(@ > a
> 0),
( y >O).
~$7,
We shall also consider the following four polynomials:
P
=
b,+2ta+2 + *
u = Clt + C2t2+
* *
*
+ bptp,
+cy,
+ bp-2tfl-2, v = 1 + C1t + + CV-ltY-'. Q
=
bata + *
* *
* * *
If M is a vector space over G bigraded by the subspaces M p , p ,then M can be graded by means of the total degree p + q and its PoincarC polynomial y ( M ) is defined in terms of this grading. Now let
A
=
Z dn(En). ?I>;!
Then A is a bigraded vector space over G of finite dimension. The general theorem mentioned above can be stated as follows. Theorem 11.3. If H ( B , B,; G ) and H ( F ;G) are of finite dimension, then so
i s H ( X ,X , ; G) and its Poincart polynomial i s given by
+ t)y(d).
y ( X , X o ; G ) = y ( B ,Bo; G)y(F;G)- (1
Furthermore, the polynomial ~ ( dsatisfies ) the following inequalities:
y(A) < t - ] W , y ( A ) G Q U . Proof. Consider the spectral homology sequence of
3 7. By (6.4), we have
E2 w H ( B ,B,; G ) @ G H ( FG). ; This implies that Em is of finite dimension and so is H ( X , X,: G ) . According to (2), we have y ( E 2 ) = y ( B ,Bo; G ) y ( F ;G ) . By (3), we obtain y(En+l) = y(En)- f 1 + t)y(dn(En)). Hence i t follows that
y ( X , Xo; G)
= y(Em)= y ( E 2 )- (1
+ t)y(d).
This proves the first part of the theorem and it remains to establish the two inequalities. For each n 2 2, consider the subspace
11. THE
PO IN CAR^
POLYNOMIALS
279
of En. Since dn is of degree (- n, n - l ) , every element of M n is a cycle and hence we have the epimorphisms M2-+M3+...
By (5), we have
+MQ.
+Mn+Mn+l+...
y(Mn)-y(M"+') Q v(dn(En))
for each n Z 2 and hence
Y ( M 3 < Y W r n ) + Y ( 4
M 2 % [Ha(B,Bo;G) +Ha+l(B,Bo;G)] @ G H ( F G) ; +H(B,Bo;G) @ c H Y ( F ; G ) , it follows that y ( M 2 )> y(E2)- PV. Hence we deduce y ( d )
Next, let us consider, for each n 2 2, the subspace
N"
=
C EZ-l,q + C E;,q + C E;,o P
I
P
of En. No non-zero element of Nn can be a boundary under dn and hence we have the monomorphisms
N"+ By (4), we have for each n
* * *
+ N n + l - + N n + * * - +N3-;"2,
y ( N n )-y(Nn+l)< ty(dn(En))
2 and hence we obtain
Y(N2) Y ( E 2 )-Y ( 4 as before. On the other hand, we have
y(N2) > y(E2)- Qu-
Hence we obtain y ( d ) Corollary 11.4.
< QU. I
R P + ~ (XXo, ;G)
=
R,@, Bo;G) R y ( F ;G).
Rp+y-l(X,Xo; G) = Rp(B, Bo; G)R A - ~ ( F G);
+ R@-i(B,B,;G)Rn(F;G).
As an application of (1 1.3), let us study the fiber spaces where the base space and the fiber are homology spheres over G. For this purpose, let us assume that y ( B ;G) = 1 t P , y ( F ; G) = 1 tQ
+
+
with p > 1 and q > 0. Leaving aside the critical case q = p - 1, the PoincarC polynomial of X over G is completely determined by B and F : Proposition 11.5.
y ( X ;G)
=
(1
+ t") (1 + tQ) if q # fi - 1.
IX. T H E S P E C T R A L S E Q U E N C E O F A F I B E R S P A C E
280
Proof. P = tP, Q = 1, U = tg, V = 1. According (1 1.3), we have
y(X;G) = (1
+ tP) (1 + t*) - (1 + t)y(d)
with y(d) < tp-l and y(d) < tQ. Since q # p - 1, the latter conditions on y(d) imply that y(d) = 0. I For the critical case q = p - 1, the conditions on y(d)imply that either p(d) = 0 or y(d) = tp-l. Hence we obtain the following Proposition 11.6.
+ t2p-1.
or 1
If q
=
p - 1, then y ( X ; G) i s either
(1
+ tp) (1 + tp-l)
A consequence of (11.5) and (11.6) is that the only possibility for a homology n-sphere X to be a fiber space over a simply connected homology p-sphere B with a homology q-sphere F as fiber is the critical case q = p - 1 and n = 2 p - 1, [S; p. 1451. Examples for p = 2, 4,8 are the Hopf fiberings, (111; 5 5).
12. Gysin's exact sequences Let G denote either the additive group of integers or a field. Assume that the fiber F is a homology r-sphere for some r > 1 and that q ( B , b,) operates simply on the total homology group H ( F ; G) and the total cohomology group H * ( F ; G). Then we have the following Theorem 12.1. There i s an exact sequence
*H m ( X , X , ; G )
i. Hm+l(B,BO;G)__* Hm-r(B,Bo;G)
HmfB,Bo; G) ( . ,* * *
called Gysin's homology sequence, where w* i s induced by the projection w : (X, X,)+ ( B ,Bo). Proof. Since F is a homology r-sphere, i t follows that E2 contains only two columns which might be non-trivial, namely
Hp(B,B,:G), Ei,, M Hp(B,B,;G) @G Hr(F;G) w Hp(B,B,;G). Applying (VIII; 8.3) and the commutative triangle at the end of § 7, we obtain the theorem. I Similarly, one can establish the following M
Theorem 12.2. There i s an exact sequence a
*
.
-+ 6.
H m ( B , B o ; G ) " l ~ H m ( X , X , ; GH)m~- r ( B , B , ; G ) L
Hm+l(B,B,;G)
x-+ --
called Gysin's cohomology sequence, where w* is induced by the projection : (X,X,) + (B,B,). If B, = u, then Ho(B;G) w G. Let s = +*(1) E H ~ + ~ (G). B ;Then it can be proved that +*(x) = x u s = s u x
12. GYSIN'S E X A C T S E Q U E N C E S
281
for every x E Hm-r(B; G) and that 2s = 0 if r is even. Since this result will be used only in the present section, the proof is omitted. See [Serre 1; p. 4701. As an application of Gysin's sequences, let us consider the fiberings of spheres by spheres and study the structure of the integral cohomology ring of the base space. For thls purpose, assume that X itself is a homology n-sphere for some n > r and consider the cohomology ring H*(B). By the exactness of Gysin's cohomology sequence, we deduce that the homomorphism +* : H"-'-l(B) -+ H y B ) is a monomorphism if m = 0 or m = n, is an epimorphism if m = 1 or m = n + 1, and is an isomorphism for other values of m. It follows immediately that
+
Hm(B) m 2, m = O mod (r l ) , 0 Q m < n, Hm(B) = 0 , m =/SOmod ( r + l), 0 < m < n. The structure of Hm(B) with m > n depends on the relation between n and Y. If 12 = p ( y 1) q, 0 Q q < 7,
+ +
then it is easy to verify that the cohomology ring H*(B) is generated by three elements 1 E Ho(B) w 2, s = +*(1) E Hr+l(B),and t E Hn(B) with o*(t) as a generator of H n ( X ) w 2. More precisely, the cohomology group H*(B) is free abelian and has a free basis (12.3)
{ 1, s, s2;
-
* )
t, st, s2t;
* *
},
where juxtaposition denotes cup product. If n = p(r
+ 1) + r ,
we have to consider the exact sequence
o LH ~ ( B2 ) H ~ ( xL ) HP(r+yB)LHn+l(B)-%0, where we have H n ( X )w 2 and HP(r+l)(B)m 2. Since Hn(B) is isomorphic with a subgroup of H n ( X ) ,it follows that either Hn(B) M 2 or Hn(B) = 0. If Hn(B) w 2, it follows that y* = 0 and hence w* and +* are both isomorphisms. In this case, H*(B)is free abelian and has (12.3) as a free basis. If H n ( B ) = 0, then i t follows that Hn+l(B) is a cyclic group of finite order k 1. In this case, the groups Hm(B),m > n, are given by
Hm(B) M z k , m=O mod ( r + l), H y B ) = 0, m+O mod ( Y + 1).
Hence H*(B) has a system of generators (12.4)
{ 1, s, s2;
* *
>,
where si is free if i Q p and s{ is of finite order k if i
> p.
282
IX. T H E SPECTRAL S E Q U E N C E OF A F I B E R S P A C E
13. Wang's exact sequences Let G denote either the additive group of integers or a field and assume that the base space B is a simply connected homology r-sphere over G for some r 2 2. First, let us consider the case B , = o. Theorem 13.1. There is an exact sequence
.
* *-%-
Hm-r+l(F;G ) P++ Hm(F;G ) 8 ' + H m ( X ;G)'+
Hm-r(F; G)
*
*
called Wang's homology sequence, where 8* is induced by the inclusion 8 :F c X . Proof. Since B is a homology r-sphere, it follows that E2 contains only two rows which might be non-trivial, namely
(p = 0 , ~ ) . the commutative triangles i n 5 7, we obtain
w H p ( B ; G) @GHq(F;G ) w Hq(F; G),
Applying (VII; 8.1) and one of the theorem. I Similarly, one can establish the following
Theorem 13.2. There is an exact sequence T* H ~ XG);-e* + H ~ ( FG ;) P', ~ m - r + y G) ~ ;5 H ~ + I ( xG) , B',
. . .+
--
called Wang's cohomology sequence, where 8* is induced by the inclusion 8 : F c X. The homomorphisms p* define an endomorphism p* : H * ( F ;G ) + H * ( F ;G ) .
According to Leray, this endomorphism p* is a derivation if r is odd and an anti-derivation if r is even, that is to say, p*(x
u y ) = p * ( x ) u y + (-
l)@+l)P x u p*(y)
for each x E HP(F;G ) and y E H* ( F ;G ) .The proof of this result is left to the reader. [Serre 1 ; p. 4711. Next, consider the case B, = b, and hence X , = F . Since B is a homology r-sphere, Hr(B, b,; G) m G and Hm(B,b,; G) = 0 for m # r. Then E2 and E*2 contain only one row which might be non-trivial, namely
E:A w H,(F; G ) , E;; w Hg(F; G ) . Hence, by (VIII; 8.1) we obtain the following Theorem 13.3. For every integer m, we have
H m ( X , F ; G ) w Hm-r(F; G ) , Hm(X, F ; G) w Hm+(F; G). As an application of these theorems, let us consider the fiber space
X
=
[ B ;B , b,]
13. WANG’S E X A C T S E Q U E N C E S
283
over B with the initial projection w : X .+ B as defined in (111; fj 10). Then we have F = [B;b,, b,]. Since n,(B) = 0, it follows from ,(IV; fj 2) that F is pathwise connected. Therefore, we may apply (13.1) to this case. Since X is contractible, we obtain Hm(F) M Hm-r+l(F) for every m > 1. Together with H,(F) theorem of Morse. Theorem 13.4. If
[B; b,, b,l, then
M
2, this implies the following
B i s a simply connected homology r-sphere and F
=
H m ( F ) - 2 ifm=Omod(r-1), Hm(F) = 0 if m 0 mod ( r - 1).
+
For another application, let us consider the sphere bundles over spheres. Assume that the fiber F is a homology s-sphere for some s > 1 and consider the integral homology groups H m ( X ) . Assume that s > 2. Then, by the exactness of Wang’s homology sequence, we deduce that the induced homomorphism
e* : H ~ ( F.+) H ~ ( X ) if m = Y - 1 or m = r + s - 1, is a monomorphism if
is an epimorphism m = Y or m = 7 + s, and is an isomorphism for other values of m.Therefore, it remains to compute H m ( X )for the four critical values 7 -1, r, r + s - 1, r s of m. First, let us compute Hr-l(X).If Y - 1 # s, then it follows that
+
Hr-i(X) = e*[Hr-i(F)]= 0 since F is a homology s-sphere. If 7 - 1 = s, we define a numerical invariant k of this fibering as follows. Consider the homomorphism
p* : Ho(F) + Hs(F).
If p* = 0, then k is defined to be zero; otherwise, the quotient group Hs(F)/ p*[H,(F)] is a finite cyclic group and k is defined to be the order of this finite cyclic group. Hence, if Y - 1 = s, we have if k = 0, ifk # O .
Next, let us compute H,(X). From the exact sequence e
it follows that
0 PI.+ Hr(F)A+ & ( X ) A+H,(F) A+0,
Hr(X) M
2,
(Z.2.
ifr # s , if Y = s.
284
IX. T H E SPECTRAL S E Q U E N C E O F A F I B E R S P A C E
Then, let us compute Hr+a--](X).Since r > 1, we have Hr+a-I(F) = 0. Hence, we obtain Hr+a-I(X) = O*[Hr+a-,(F)I = 0. Finally, let us compute Hr+a(X).In the following exact sequence ~r+a(x) A+H ~ ( FA ) Hr+a-l(F),
Hr+s(F)
we have Hr+,(F) = 0 = &+8-1(F)and hence
Hr+a(X)M Ha(F) M 2. For the remaining case s
=
1, the induced homomorphism
0, : H,(F) -.H,(x)
is an epimorphism if m = r - 1, is a monomorphism if m = r + 1, and is an isomorphism if m < r - 1 or m > r + 1.Therefore, it suffices to compute the group H m ( X ) for three critical values of m,namely, m = r - 1, r, and r + 1. The computation is similar to the case s > 1 and hence is left to the reader. Note that the same results can be obtained by using Gysin's homology sequence instead of Wang's homology sequence.
14. Truncated exact sequences Let G denote either the additive group of integers or a field. Assume that X , = and that n,(B, b,) operates simply on H ( F ; G ) . Theorem 14.1.
0
If H,(B, G) = 0 for 0 < m
< m < q, then we have a n exact sequence
Hp+Q-l(F; G)*. +
e
*
and H m ( F ;G) = 0 for T
T
*--+Hm(F;G)B'+Hm(X; G ) b H m ( B G)--+Hm-l ; ( F ;G )
. . .5.+H , ( B ; G )
T --j
H , ( F ; G ) 54 H , ( X ; G ) %-+ H l ( B ; G )?+ 0
where O* and w* are induced by the inclusion O : F c X and the projection w : X + B, and T i s called the transgression. Proof. According to (6.3), we have
E:,j w H t ( B ;G) @ G H ~ ( F G);
+ TOYG[Ht-I(B;G),H j ( F ;G)].
Therefore, Ef, = 0 if i # 0, j # 0, and i + j < p + q - 1. I t follows that, for each total degree m with 0 < m < p q - 1 , E2 contains only two terms which might not be zero, namely,
+
EE,, rn H,(B; G ) , E& m H,(F; G). According to (VIII; 8.5), we have a fundamental exact sequence in which, by the results in (VIII; § 7), one can verify the assertions about the homomorphisms 6*, w * , and T . I
15. T H E S P E C T R A L S E Q U E N C E O F Corollary 14.2. I f H,(F; G)
Hm(B;G ) for each m
>0.
Proof. Apply (14.1) with
p
=
Corollary 14.3. I f H,(B; G) H , ( X ; G) for each m Z 0 . Proof. Apply (14.1) with
0
p
=
A REGULAR COVERING SPACE
=
= 00
> 0 , then w* : H m ( X ;G ) w
0 for each m
1 and q
= 00.
I
0 for each m and q
=
285
> 0, then 8, : Hm(F;G) w
1. I
Corollary 14.4. If H m ( X )= 0 for each m > 0 and Hm(B;G ) = 0 for < m < p , then we have T :Hm(B:G ) w H m - I ( F ;G) for 2 Q m Q 2p -2.
Proof. Applying (14.1) with q = 1, we find that H m ( F ;G) O<m<~-ll.Then,apply(14.1)againwithq = p - l . I
=
0 for
Similar results hold for cohomology groups.
15. The spectral sequence of a regular covering space Let X be a pathwise connected regular covering space over a locally pathwise connected base space B with a projection
w:X+B in the sense of (111; 5 16). Then it follows readily that B is pathwise connected and X is locally pathwise connected. Pick a basic point x, E X and call b, = w(xo)the basic point in B . Then the basic fiber
F
= o-'(b,)
is discrete and w induces a monomorphism w+ :n i ( X ,xo) +ni(B, bo).
Since X is a regular covering space over B , the image
w,xo)
=
w*[n,(X,x0)l
is an invariant subgroup of n,( B , b,) and the quotient group 72 = n,(B, bO)/X(X,x,)
acts on X as a group of right operators: see (111; 16.6). If 5' # 1 is any element in n,then 5' is a homeomorphism of X onto itself without fixed point. Let W be a contractible (finite or infinite) simplicial complex (with weak topology) on which x acts freely as a group of simplicia1 homeomorphisms. For the existence of such a complex, see (VI; Ex. I). Let P denote the orbit space W l n with projection t : W -+ P. Then P is a space of the homotopy type (n,1) and t is the universal covering. Following a procedure introduced by Ehresmann, we shall construct a space Y as follows. Let n operate on the left of the product space W x X by
t ( w ,x )
=
(tw, x t - I ) , 5 En,w E w ,x E x .
286
IX. T H E SPECTRAL SEQUENCE O F A F I B E R S P A C E
Then Y is defined to be the orbit space (W x X ) / n . Define two maps
P
+ '
W x X'.
B
by taking l(w, x ) = t ( w ) and r(w, x ) = w ( x ) . Since obviously l t r l = r for every 5 EZ, they induce two maps
=
I and
P-"YLB One can verify that the maps A : Y +- P and p : Y +- B are fiberings in the sense of (111; Q 3). In fact, Y is a bundle space over P with X as fiber and A as projection, and Y is also a bundle space over B with W as fiber and p as projection. These will be called the bundle spaces associated with the regular covering space X over B . Let G be an abelian group on which n acts as left operators. Since P is a space of the homotopy type (z, l ), it follows from (VI; Ex. G )
Hm(n;G)
= H m ( P ;G ) ,
H m ( n ; G ) = H m ( P ; G)
with local coefficients in G. Throughout the remainder of the section, let G denote an abelian group on which the group n operates simply. First, consider the associated fibering p : Y -+ B. Since the fiber W in this fibering is contractible and hence acyclic, it follows from (14.2) that p* Hm(Y) w Hm(B)for each m > 0 and therefore (15.1)
p* : H m ( Y ; G )
Hm(B; G ) , (m Z 0).
Next, consider the associated fibering A : Y +- P with fiber X . According to Q 6 and Q 7, this fibering determines a homology exact couple
V =(D,E;i,j,k> and a spectral homology sequence { E n } which will be referred to as those of the regular covering space X over B . According to Q 5, the fundamental group n,(P,Po) m az acts on H ( X ; G) as a group of left operators. Hence, by (6.2), we have (15.2)
w H p ( P ; H g ( X ; GI) = H p ( n ; Hp(X;GI);
and, by Q 7, the group #(%) = H ( Y ; G) H ( B ;G) is filtered with E" as its associated graded group. More precisely, we have
Hm(B;G)= J f m , o ( V )
3
*m-l,,(Y)
2 + . * 3 Jfo,m(V)
I x - i , m + ~ ( V= ) 0,
2P,*(W2P-l,4+1(W w q q .
Furthermore, we have Rg(w) = H g ( X ; G ) ,
sp(w) = H p ( n ;G )
together with two commutative triangles
16. A
T H E O R E M O F P. A. S M I T H
H p ( X ;G) A+H,(B; G) \\
/
287 H,(n; G)
H,(B; G)
\ / EZO
EEq
where w* is induced by the projection o : X -+ B, ,u* = A,p;l, the x's are epimorphisms, and the L'S are monomorphisms; p* will be called the natural homomorphism. As immediate consequences, Zo, *(V) is the image of w* andXp-l,,(Q) is the kernel of ,u*. Similarly, we have the cohomology exact couple Q* = < D*, E*; i*, j*, k*, > and a spectral cohomology sequence { E*n } of the regular covering space X over B. In particular, (15.3) E*2 M H*(P;H * ( X ; G)) = H * ( n ; H * ( X ;G)) and Z ( V * ) = H * ( B ;G) is filtered with E*m as its associated graded group.
R,(Q*) = H g ( X ; G), S,(V*) and we have two commutative triangles
HU(B;G) *-+ Hg(X; G)
\ /
=
H P ( n ; G) 2HP(B; G)
\\
E;;
HP(n; G)
E;;
/
with properties analogous to those in the homology case. If G is a commutative associative ring with unity element, then (15.3) is a ring isomorphism.
16. A theorem of P. A. Smith In the present and the following two sections, we shall give a few of the applications of the spectral sequences obtained in 5 15. Theorem 16.1. If a discrete groul, n acts freely on a locally contractible acyclic space X of finite dimension, then n has no element of finite order other than 1. Proof. Assume, on the contrary, that n has an element 6 of order Y # 1. The subgroup of n generated by 6 is a cyclic group of order Y and acts freely on X.Thus we may assume that n itself is a cyclic group of order Y . Let B = X ! n denote the orbit space. Then X is a regular covering space of B with n as quotient group. This implies that B is locally contractible and finite dimensional. Let G = 2 be the additive group of integers and let n operate simply on G. Consider the spectral homology sequence of 3 15. Since X is acyclic, it follows that E& M H,(n) and E;,q = 0 if q # 0. Then, by (VIII; 8.3), we obtain H,(B) H p ( n ) , ( P > 0).
-
288
IX. T H E S P E C T R A L S E Q U E N C E O F A F I B E R S P A C E
Since R is locally contractible, H p ( B )= 0 if p > dim B. However, H,(n) w x if p is odd. See (VI ; Ex. K5). This is a contradiction. I
17. Influence of the fundamental group on homology and cohomology groups Let B denote a pathwise connected space. Since the singular complex S ( B ) has the same homotopy and homology structure as R, (V; Ex. I), we may assume without loss of generality that B is locally contractible. Then the universal covering space X of R is defined and the fundamental group n = n,(B) operates freely on X . Theorem 17.1. If
morphisms
np(B)= 0 for 1 < p < r, then we have the natural iso-
p*:H,(B; G) M H,(n; G ) , ,u*:HP(n;G) M HP(B;G) for each p
< Y and every coeficient group G o n which x operates simply.
Proof. Consider the spectral homology sequence of 3 15. Then the assumption implies that & ( X ; G) = 0 if 0 < q < r. By (15.2),we have
E:,q = 0, if p + q < r and q # 0. Then it follows from (VIII; 8.5) and one of the commutative triangles in 9 15 that ,u* is a n isomorphism whenever p < r. By considering the spectral cohomology sequence, one can also show that ,u* is an isomorphism whenever p < r . I Thus we see that the fundamental group of B determines the homology groups and the cohomology groups of B for all dimensions less than r if n p ( B )= 0 for every p satisfying 1 < p < Y . To study the groups of the critical dimension r, let us consider the homomorphisms ( o , : H r ( X ;G) + H r ( B ;G ) , w*:Hr(B;G) - + H r ( X ;G) induced by the projection o : X + B. Denote by & ( B ; G) the image of o* and by k ( B ;G) the kernel of o*. Theorem 17.2. If n p ( B )= 0 for 1
< p < r, then we have the exact sequences
(a)
H r ( X ;G) a Hr(B, G) PI+ H,(n;G) +. 0 ,
(4
W ( X ;G ) +f-W ( B ;G ) +-"-I-W ( n ;G) c 0 ,
H r ( B ;G)/&(B;G) w H r ( n ;G ) , Ar(B;G) M H r ( n ; G ) . Proof. Consider the spectral homology sequence of the terms of E 2 with total degree r -- 1 arid r. Let
am
= 0,
bm
=
m, C,
=
m, dm
=
3 15 and in particular
0 , (m = Y - 1, Y ) .
17.I N F L U E N C E O F T H E
FUNDAMENTAL GROUP
289
Then it is easily verified that the two-term condition { r - 1, r ; 2 } of (VIII; 4 8) is satisfied. Since E&l = 0 and E:,o w H r ( B ;G ) , we obtain an exact sequence Hr(B;G) 2-tH r ( n , G) -+ 0. (4
According to 3 15, the kernel of !I* isXr-l,l(%’) and the image of w * :Hr ( X ; G) +- Hr(B;G) is X o , r ( g ) . Since E m contains only two terms of total degree r which might be different from zero, namely, E& and E20, it follows that Xr-.l,l(%) =Zo,,-(5f). Hence we obtain the exact sequence (a). By considering the spectral cohomology sequence of 3 15, one can also establish the exactness of the sequence ( b ) . I In the proof of (17.2), we did not make full use of the two-term condition { Y - 1, r ; 2 }. Indeed, we actually have longer exact sequences given in the following Theorem 17.3. I f n p ( B ) = 0 for 1
< p < r, then we have the exact sequences:
G)%+J,(Hr(X; G ) ) L H r ( B G)&+Hr(n; ; G)+-0,
( d ) Hr+l(B;G)”+Hr+l(n;
(el H‘+~(B;G) +@:~r+l(n ;G)L I , J G) H -+CH~(B ~(x ;G) s;~ r ( nG)+o. ; For the definition of the operators I , and J n , see (VI; Ex. K). Proof. Applying the two-term condition { r - I , r ; 2 } as in the proof of ( 1 7.2), we obtain instead of (c) a longer exact sequence
(4
E20.7 Hr(B;G) 3-t H r ( n ;G) + 0, where Ei,rw H o ( n ;H r ( X ;G ) ) is isomorphic with J,(Hr(X; G)) according to (VI; Ex. K l ) . For further extension of this exact sequence, we have to consider the terms of E 2 with total degree Y + 1. There are only three such terms which might be different from zero, namely -+-
E;,r+l, E ? p G + , , O w &+An; GI. Let n > 2. Since the differential operator dn in En is of degree (- n, n - l), the elements of E$+, and ET,r are cycles and dn sends EY+l,,, into E& iff n = Y + 1. Since every element of E& is a cycle, we get an exact sequence Er+I EY+1 c-tET+2 + O , (ii) r+1,0
-
+
0,r
Since E,2+1,0= E:$:,,, E;,, = E$‘, and EZr and (ii) so that we obtain an exact sequence
o,r
=
E;;t2, we can combine (i)
*+
Hr(n; G) -+o E,2+1,2A+ ~2 on*%+ Hr(B; G) with +* = &+I. Since the kernel of dr+l is obviously EL$?,o = E,*;,,, and since E:+I,Ow Hr+l(n;G ) , one of the commutative triangles in 8 15 gives an exact sequence H ~ ( +B ;G) ~ 3+Hr+l(n;G) A+ (iv) Combining (iii) and (iv), we obtain the exact sequence ( d ) . Similarly, one can deduce (e). I
(iii)
~2,~.
290
IX. T H E S P E C T R A L S E Q U E N C E O F A FIBER S P A C E
Note. If G is the group of integers, then the groups&(B; G) andAr(B; G) are essentially the groups Zr(B) and k ( B ) as defined in (VI ; Ex. J). See also [Eilenberg-MacLane 13.
18. Finite groups operating freely on s' Let n be a finite group operating freely on the r-sphere X = Sr. Let B = Srln denote the orbit space and w : S + B the projection. By the finiteness of n,one can easily show that Sr is the universal covering space of B with o as projection and that n can be considered as the fundamental group of B. To apply the results of 3 17, let us take the additive group 2 of integers as the coefficient group and let n operate simply on 2. By (17.1) and (17.3), we have the natural isomorphisms (i) ,u* : Hp(B) H p ( n ) , /A*: HP(n) M HP(B) for each p (ii)
< r and the exact sequences 0 + Hr+l(n)-!L J,(Hr(Sr))>-+ Hr(B)Ifi+ Hr(n) + 0,
+c
Hr(B) +f-Hr(n)+- 0, 0 c Hr+l(n) I,(Hr(Sr)) (iii) since dim (B) = r and therefore Hr+,(B) = 0 = H*+l(B). Proposition 18.1. If there is an element 6 EZ which changes the orientation of Sr, then r must be even.
Proof. Since 6 changes the orientation of Sr, 5 is of order h > 1. The subgroup of n generated by 6 is a cyclic group of order h and operates freely on S'. Therefore, we may assume that n itself is the cyclic group of order h generated by 6. Since 6 changes the orientation of Sr, we have t ( x ) = - x for every element x of the free cyclic group Hr(S*).It follows that I,(Hr(Sr))= 0 and the exact sequence (iii) reduces to
0 c Hr+l(n)+ 0 c Hr(B)
Hr(n)c 0.
This implies that (iv) Hr+l(n) = 0, p* : Hr(n) w Hr(B). By (VI; Ex. K5),the first equality implies that r must be even. I Proposition 18.2. If n contains ait element 6 # 1 which preserves the orientation of Sr, then r must be odd.
Proof. We may assume that n is the cyclic group generated by 5. Since 6 preserves the orientation of Sr, n operates simply on Hr(S*).It follows that ],(Hr(Sr)) = Hr(Sr)and the exact sequence (ii) becomes 0 Hr+l(n)+ Hr(Sr) + Hr(B) + Hr(n) + 0. (4 This implies that Hr+l(n)is isomorphic to a subgroup of the free group Hr(Sr)and hence it is also free. According to (VI; Ex. K5), we conclude that Hr+1(n)= 0 and so r must be odd. I -+
18. F I N I T E
GROUPS OPERATING FREELY O N
sr
291
Proposition t8.3. If r is even and n contains more than one element, then n is a cyclic group of order 2 and, for each p with 1 Q p < r, we have:
H,(B)
= 0,
HB(B) M n, if p is even;
H,(B) w n, HP(B) = 0 , i f p is odd. Proof. Let 6 # 1 and 7 # 1 be any two elements inn, then, by (18.2), both t and 7 must change the orientation of S r and hence t7-l preserves the orientation of Sr. Using (18.2) once more, we deduce that 6q-l = l. This proves that n is cyclic of order 2. For the rest of the proposition, (i)and (iv) give all the homology and cohomology groups except Hr(B).The latter can be computed either from the relations between homology and cohomology groups or as follows. In the exact sequence (ii), since both Hr+l(n)and J,(Hr(Sr)) are of order 2, +* must be an isomorphism. On the other hand, we have Hr(n) = 0. Therefore, it follows easily from the exactness that Hr(B) = 0. I Proposition 18.4. If Y is odd a n d n is abelian, thenn must be cyclic, Hr(B) M 2 m Hr(B),and for each p with 0 < p < r we have:
H,(B)
= 0,
H,(B)
M
p i s even; if p is odd.
HP(B) w n, if
n, HP(B) = 0,
Proof. According to [ 18.1), every element of n preserves the orientation of 9. I t follows that J,(Hr(Sr)) = Hr(Sr),
and hence the eFact sequence (ii)becomes
(4
H,(s~)2-tHr(B)&-+Hr(n) + 0. Hr+i(n) Since every element of n preserves the orientation of Sr, B is an orientable manifold of dimension r and hence we have Hr(B) w 2 M Hr(B). Assume 0
-+
that n is of order h. Then, by the definition of o*, one can easily see that we can choose generators a E Hr(Sr) and B E Hr(B)such that o * ( a ) = hp. Hence the exact sequence ( v i ) implies that Hr(n)is a cyclic group of order h. According to ( i )and (VI; Ex. K), it remains to prove that n is cyclic. Let us assume that n is not cyclic. Then there is a prime number p such that n contains a subgroup of the form Z, + Z,, where Z, denotes the cyclic group of order p . We may assume that n itself is of the form Z, 2,; then n is of order Pz. Therefore, Hr(n) is a cyclic group of order pz. Since Hr(Zp) w Z,, i t follows from (VI; Ex. K) that 2, is a direct summand of Hr(n).This is impossible. I
+
Note. 1. If n is a discrete group operating freely on the r-sphere 9,then the compactness of Sr implies that n is finite. 2. One can deduce all the homology and cohomology groups of the real projective spaces from (18.3) and (18.4).
292
IX. T H E S P E C T R A L S E Q U E N C E O F A F I B E R S P A C E
EXERCISES A. Cohomology with arbitrary coefficients
Let G be an abelian group and C denote the regular &complex C ( X ,X,). Consider the group A* of singular cochains of X modulo X,with coefficients in G. Then A* is a graded group with A: = Hom(C,, G) and a differential operator 6 of degree 1. Since Hom (C, G) is the direct product of the groups Horn(&, G ) , [see E-S, p. 1471, A* is a subgroup of Hom(C, G). Define a decreasing filtration { A*P } in A* by taking
A*P={f€A*If(CP-l) = o > . Then A*P is a subgroup of the graded group A* with A t 9 = A*P n A:. Define a weight function w on A* by taking w ( f ) , f E A*, to be the least upper bound of the integers p such that f E A*p. Prove that 0 < w(f ) < dim ( f ) for every non-zero homogeneous element f E A*. Therefore, with the filtration { A*P }, A* becomes a regular &complex. Study the associated exact couple
%'(A*)= < D*, E * ; i*, j*, k* > and its derived couples, as follows: prove that
E;,,
Hom(Cp(B,B,), Hq(F;G ) ) , E *2m M HP(B, B,; Hq(F;G)), M
where HP(B, B,; H @ ( FG; )) denotes the $-dimensional singular cohomology group of B modulo B, with local coefficients in H @ ( FG) ; ; and that if G is a commutative associative ring with a unity element, then these are ring isomorphisms.
B. The spectral cohomology sequence Consider the regular &complex A* of the preceding exercise. Denote by
$+(A*)
=
< D*n, E*n; i*n, j*n, k*n >
the successive derived couples of the associated exact couple V = %'(A*). Then the associated spectral sequence { E*n I n = 1,2; * } of W will be called the spectral cohomology sequence of X modulo X , over G. Prove that &'(%') = &'(A*) = H * ( X , X , ; G) is filtered with E** as its associated graded group; more explicitly,
-
Hm(X,Xo;G)=*o,m(%')
3 s i , m - i ( % ' ) 3 * . - 3*m,o(%')
&'PA
%'I /&'P
+ I . q -1
(%')
= q,;.
3
zm+i,
-i('+4= O ,
Hereafter, we shall use the notation Hp,q(X,X , ; G) = &'p,q(%'). Next, assume X , = and prove that &(W) = H@(F;G ) and Sp(V) = HP(B; G). Then we obtain the following commutative triangles
H q ( X ;G )
e*
293
EXERCISES
HU(F;C)
\ J E;P
HP(B; G) "-'+
\\
H p ( X ;G)
J
Ego
where 8*,w* are induced by 8 : F c X and o : X +. B, the x's are epimorphisms, and the i s are monomorphisms. Prove that H 1 ~ g - l ( XG) ; is the kernel of 8* and Hpv0(X; G) is the image of w * . Now, consider the transgression. According to (VIII; Ex. H), there are equivalent definitions : (1) The transgression is the differential operator d*m : E:,; +. E:,:, (2) In the following homomorphisms
H ~ - I ( Fc) ; 2-tH ~ XF ,; G) + C H ~ ( B G), , (m 2 21, let M denote the image of w* and K the kernel of w *. Then the transgression is the homomorphism T : d-l(M) -+ H m ( B ;G)/K defined as in (VIII; Ex. H). A comparison of these definitions shows that E:; is isomorphic to the image of Hm(B; C) in H m ( X ,F ; G) under the induced homomorphism w*. Let G be the group of integers mod 2. Prove that the transgression T commutes with the square operations and the reduced powers of Steenrod. Investigate analogously the transgression in the spectral homology sequence of Q 7. C. The maximal cycle theorem
In addition to the assumptions of 9 3, assume that G is a field. Then prove the following Theorem. If Hm(B, B,; G) = 0 for each m > fi and Hm(F;G) = 0 for each m > q, then we have:
+
H,(X, X,; G) = 0, (m> fi q ) ; H p ( B , Bo; G) @G Hq(F; H p + q ( x ,xo;G) Deduce the following assertions : 1. If G # 0 and H m ( X ;G) = 0 for each m > 0, then a t least one of the following three statements must be true: (a) H m ( B ;G ) = 0 = H m ( F ;G) for each m > 0. ( b ) H,(B; C) # 0 for infinitely many values of m. (c) H m ( F ;G ) # 0 for infinitely many values of m. 2. If a euclidean space X = Rn has a bundle structure over a base space B with a connected fiber F , then both B and F are acyclic.
D. An isomorphism theorem In addition to the assumptions of Q 3, assume that X , # B, # o. Prove the following
and hence
294
IX. T H E S P E C T R A L S E Q U E N C E O F A F I B E R S P A C E
Theorem. If Hm(B,B,; G)
0
< m < q, then we have
=0
for each m
and H m ( F ;G)
o* : H m ( X , X o ; G) M Hm(B, B,; G), (m< p and an exact sequence
=0
for
+ q),
a H p + q + ] (B,Bo;G) H p B , B,; G ) @G Hg(F;G) Hp+q(X,X o ; G) a Hp+g(B,Bo; G) 0.
H p + q + ] ( X ,X o ;G) I _+
+
Establish a similar theorem for the spectral cohomology sequence. E. Some special cases of 5 15
Prove the following assertions : 1. If n is a free cyclic group and acts simply on the homology groups H , ( X ; G ) , m Z O , thenHm(B;G)isanextensionof H m ( X ; G )by H m - , ( X ; G ) . Hence, in particular, if G is a field, then H m ( B ; G )is the direct sum of H m ( X ;G) and H,-,(X; G ) . 2 . I f n is a finite group of order h and G is a field of characteristic zero or prime to h, then
H m ( B ; G )m J n ( H m ( X ; G ) ) , H m ( B : G ) m I , ( H m ( X ; G ) ) . 3. If n is the direct product of a free cyclic group and a finite group of order h, G is a field of characteristic zero or prime to h, and n acts simply on the homology groups H m ( X ;G), m > 0 , then Hm(B; G) M H m ( X ;G) + Hm-I(X; G ) .
F. Relations between the cohomology algebras of a space and
i t s space
of loops
Let B be a pathwise connected and simply connected space and b, a given point in B . Let X = [B; B, b,]. Then, X is a contractible fiber space over B with fiber F = A(B),the space of all loops in B with b, as basic point. Prove the following assertions: 1. If K is a field and the cohomology algebra H * ( F ;K ) is isomorphic to an exterior algebra over K generated by an element of odd degree n, then the cohomology algebra H * ( B ;K ) is isomorphic to a polynomial algebra generated by an element of even degree n + 1. [Serre 1 ; p. 5011. 2. If K is a field of characteristic zero and the cohomology algebra H * ( F ;K ) is isomorphic to a polynomial algebra generated by an element of even degree n, then the cohomology algebra H*(B;K ) is isomorphic to an exterior algebra generated by an element of odd degree n + 1. [Serre 1 ; p. 5011. 3. If K is a field and the cohomology algebra H * ( B ;K ) is isomorphic to a polynomial algebra generated by an element of even degree n > 2, then the cohomology algebra H * ( F ;K ) is isomorphic to an exterior algebra generated by an element of odd degree n - 1. [Serre 1 , p. 4921. 4. If K is a field of characteristic zero and the cohomology algebra
EXERCISES
295
H * ( B ;K ) is isomorphic to an exterior algebra generated by an element of odd degree n, then the cohomology algebra H * ( F ;K ) is isomorphic to a polynomial algebra generated by an element of odd degree n- 1. [Serre 1; p. 4891. 5. If K is a field of characteristic zero and the cohomology algebra H * ( B ; K ) is isomorphic to an exterior algebra generated by an element of even degree n > 2, then the cohomology algebra H * ( F ;K ) is isomorphic to the tensor product of an exterior algebra generated by an element of degree 1z - 1 and a polynomial algebra generated by an element of degree 2(n - 1). [Serre 1; p. 4891. 6. If K is a field of characteristic p # 0 and Hm(B; K ) w Hm(Sn; K ) for every m < p(n - 1) + 1, where n > 3 is an odd integer, then the subspace of the cohomology algebra H * ( F ; K ) formed by the elements of degree not exceeding p ( n - 1) admits a homogeneous basis which consists of the elements { 1, y, y2; ., y p - l , z }, where [Serre 1 ; p. 4941 deg ( y ) = n - 1, deg (z) = p(n- l), y p = 0. 7. If K is a field of characteristic p # 0 and the subspace of the cohomology algebra H*(B;K ) formed by the elements of degree not exceeding pq, where q > 2 is an even integer, admits a homogeneous basis which consists of the elements { 1, y, y2,* * .,y p - l , z }, where deg (Y)= q, deg (4= pq, yp = 0, then the subspace of H * ( F ;K ) formed by the elements of degree not exceeding p q - 2 admits a homogeneous basis which consists of the elements { 1, u, v }, where [Serre 1 ; p. 4951 deg (H) = q - 1, deg ( v ) = #q -2. G. The cohomology algebra of (Z, n)
Let 2 be the free cyclic group. Prove that the cohomology algebra H * ( 2 ) over the ring of integers is isomorphic to the exterior algebra generated by an element of degree 1 and the cohomology algebra H*(Z, 2) over the ring of integers is isomorphic to the polynomial algebra generated by an element of degree 2. Next, let K be a field of characteristic zero. By means of the relations 1 and 2 in Ex. F, prove that the cohomology algebra H*(Z, n ; K ) over K is isomorphic to an exterior algebra generated by an element of degree n if rt is odd and is isomorphic to a polynomial algebra generated by an element of degree n if n is even. H. The cohomology algebra of (1(S")
Consider the sphere Sn of dimension n > 2 and study the cohomology algebra of the space S2n = A ( S n ) of loops in Sn with so as basic point. Let T = [Sn; S n , so]. Then T is a contractible fiber space over Sn with projection o : T -+ Sn and fiber Q n = o-l(so). From (13.2), deduce an isomorphism p* : H y Q n ) M Hm-n+l(Qn)
296
IX. T H E S P E C T R A L S E Q U E N C E O F A F I B E R S P A C E
for each m ; p* is a derivation or an anti-derivation according as n is odd or even. Thus, a homogeneous basis { ec }, i = 0, 1,2; * -,of H*(SZn) is given by e, = 1, et = p*-l(ei-l), i = l , 2 ; - . , with dim (et) = i(n- 1). Prove the following Themem. T h e mzlltiplicative structure of the integral cohomology algebra H*(SZn) i s given by the rule epeq = cp,qep+q,where cp,q i s a n integer given as follows: (i) If rt is odd, then (9 + q ) ! cplq =
p!4!
*
(ii) If n i s even, then cp,q = 0 if both 9 and q are odd; otherwise
where [XI denotes the largest integer not exceeding x. Then prove 1. If n is odd, then (el)P = (p!)e,. If n i s even, then (el), = 0, (ez)P = ( p ! ) e 2 p ,and ele2p = e2pel = e2p+1. 2. If n is even, then H*(SZn) is isomorphic to the tensor product of the algebras H*(Sn-l) and H*(Q2n-I). 3. Let K be a field of characteristic zero. If n is odd, then H*(@; K ) is isomorphic to a polynomial algebra generated by an element of degree n - 1. If n is even, then H*(Q,; K ) is isomorphic to the tensor product of an exterior algebra generated by an element of degree n - 1 and a polynominal algebra generated by an element of degree 2(n - 1). 4. Let K be a field of characteristic 9 # 0. If n is odd, then H*(Q,; K ) is isomorphic to a polynomial algebra with infinitely many generators f g , (i = 0 , 1; * *), modulo the ideal generated by (f@, (i = 0 , 1 ; *), f t being of degree 94(n - 1). 5. The cohomology algebra H*(Sn-l x Q2a-l) is isomorphic to the tensor product H*[Sn-l) 18H*(P'n-l). 1. The connective fiber spaces of S3
For each IZ >, 3, let X , denote an n-connective fiber space over S3.Verify the following results on the cohomology algebra H*(X,; 2,) due to [Serre 31 : 1. For the dimensions < 11, H*(X,; 2,) has a homogeneous basis { 1, a , b, c, d } where d i m ( a ) = 4, d i m (b) = 5 , d i m (c) = 8, d i m ( d ) = 9, and where b = Sqla, c = a,, d = ab. 2. For the dimensions < 8, H*(X,; 2,) has a homogeneous basis { 1, e, f , g, h, i 1 where d i m (e) = 5, d i m ( f ) = 6 = d i m (g), d i m (h) = 7 , d i m ( 2 ) =- 8, and where f = Sqle, h = Sqlg = Sq2e, i = Sq2f, Sq2g = 0. 3. For the dimensions < 7, H*(X,; 2,) has a homogeneous basis { 1, i, k } where d i m ( j ) = 6 , d i m ( k ) = 7 , and where Sqlj = 0, Sq2j = 0. 4. H7(X,; 2,) has a basis formed by a single element m with S q l m # 0.
CHAPTER X CLASSES O F A B E L I A N GROUPS
1. Introduction Our principal remaining object is the computation of certain of the groups nm(Sn). To facilitate these computations, a digression on Serre’s class theory
of abelian groups is necessary. It is to this digression that the present chapter is devoted; the study of the groupsnm(Sn)is deferred to the next (and final) chapter . In $$ 2-6, we introduce the formal definitions. In $4 7-10, certain immediate topological consequences are obtained ; the most important of these is the generalized Hurewicz theorem, from which one deduces that the homotopy groups of a simply connected finite polyhedron are finitely generated.
2.The Definition of Classes A collection V of abelian groups is called a class if the following four conditions are satisfied: (CG1) V contains a group which consists of a single element. (CG2)If a group A is isomorphic to some group in V, then A is in V. (CG3)If a group A is a subgroup or a quotient group of some group in %, then A is in V. (CG4)If an abelian group A is an extension of a group in V by a group in %, then A is in V. Examples of classes are listed as follows. Verifications are left to the reader. (1) The class d of all abelian groups. (2) The class 0 of all groups consisting of a single element. (3)The class dfof all finitely generated abelian groups. (4)The c l a s s 9 of all finite abelian groups. (5) The class of all finite abelian groups with order not divisible by any prime number of a given family of prime numbers. In particular, if the given family consists of all prime numbers except p , this reduces to the class alp of all finite abelian groups of order p r , r = 0, 1,2; * * . (6) The class 9 of all torsion groups. An abelian group A is called a torsion group if every element of A is of finite order. Other examples of classes will be given in the next section and in Ex. A at the end of the chapter. 297
298
X. C L A S S E S OF A B E L I A N G R O U P S
I t is an easy exercise to prove that a non-empty collection %' of abelian groups is a class iff, for every exact sequence L -f M --f N of abelian groups, the following condition is satisfied: (CG5) L E%' and N E W imply M E%'. Furthermore, it can also be verified that every class V has the following properties : (CG6) I f A E Y a n d B E W , thenA + B E % ' . (CG7) If A E W and B is finitely generated, then A @ B and Tor(A,B ) are in V. (CG8) If A is finitely generated and A EW,then H m ( A )E%'for every m. See (VI; Ex. G ) . It is, of course, necessary to observe that a class V cannot be a set; thus the usual precautions must be taken to avoid contradictions. See [E-S; p. 1201.
3. The Primary Components of Abelian Groups For any given prime number p, the #-primary component of an abelian group A is defined to be the subgroup of A consisting of the elements of order fir, Y = 0, 1,2, * . If A is a torsion group, then it is the direct sum of its p-primary components for all p. Hence, in order to compute a certain torsion group A , it suffices to compute all of its primary components. See [Ka]. If, for some prime number p, an abelian group A reduces to its p-primary component, that is, if all elements of A have orders which are powers of p, then A is said to be a 9-primary group. A finitely generated abelian group is p-primary iff i t is of order for some r. It is evident that, for a given prime number p, the p-primary groups form a classgp. Let F be a given family of prime numbers. Then the torsion groups with null p-primary components for each prime number p in the family F constitute a class. If F contains all prime numbers, then this class reduces to the class 8;if F consists of all primes except a given prime number p , then it becomes the classPp; if F is empty, then it is the classy.
--
4. The %'-Notions on Abelian Groups For a given homomorphism f : A -f B, let I m ( f ) , Key(/), and Coker(f) denote respectively the image, kernel, and cokernel of f ; this latter is defined by Coker(f) = B/Im(f). The following sequence is obviously exact :
o
-f
K e y ( / )+ A -!- B -f coker(f)-f 0.
4. T H E V-NOTIONS ON ABELIAN
GROUPS
299
Furthermore, any pair of homomorphisms
A ~ . B & C gives rise to a natural exact sequence S(f, g) :
0
-f
Key(/) -f Kerhgf) + Ker(g) + Coker(f) + Coker(gf) + Coker(g) + 0.
In the applications of class theory, the groups in a class V are usually to be neglected in a certain sense. Thus we are led to the following terminology: Let V be a given class. A group A is said to be V-null if A E V. Let f :A -+ B be a homomorphism. Then f is said to be a V-monomorphism if Ker(f ) E %', a V-efiimmphism if Coker(f)EV, and a V-isomorphism if it is both a Vmonomorphism and a V-epimorphism. If a %-isomorphism f : A + B exists, then A is said to be V-isomorphic to B . If V is the class 0,these notions coincide with the corresponding classical notions; and it is more or less obvious that, for an arbitrary class %,' these notions have the same formal properties as the classical notions. The detailed statement of these facts is deferred to Ex. B at the end of the chapter. Two abelian groups A and B are said to be V-equivalent if there exists an abelian group L with two V-isomorphisms f : L + A and g : L + B. Proposition 4.1. Two abelian groufis A and B are V-equivalent if there exists an abelian group M and two V-isomorphisms h : A -+ M and k : B + M .
+
Proof. Suficiency. Let L denote the subgroup of the direct sum A B consisting of the elements (a, b) such that h(a) = k(b). Define homomorphisms f : L + A and g : L + B by / ( a , b) = a and g(a, b) = b. Then one can verify that f and g are V-isomorphisms. Necessity. Let A and B be V-equivalent. Then there exists an abelian group L together with two %'-isomorphismsf : L + A and g : L + B. Let M denote the quotient group of the direct sum A B over the subgroup consisting of the elements (f(l),g(1)) for all 1 E L . Let p : A B + M denote the natural projection. Define homomorphisms h . A + M and k : B + M by h(a) = $(a, 0) and k(b) = fi(0, b ) . Then one can verify that h and k are V-isomorphisms. I The relation of being V-equivalent is obviously reflexive and symmetric. It is also transitive. To verify this, assume that A , B and B , C are both V-equivalent. By definition and (4.1), there are two abelian groups L,M and four %'-isomorphisms f : L + A , g : L + B , h : B + M , k:C+M.
+
+
Since hg : L + M is aV-isomorphism, it followsthat A andC are W-equivalent. This proves the transitivity and the relation of being V-equivalent is an equivalence relation.
300
X. C L A S S E S O F A B E L I A N G R O U P S
5. Perfectness and Completeness The classes of abelian groups with which we will deal in the sequel usually satisfy some further conditions described as follows. A class %? is said to be perfect if A E%?implies that Hm(A)E%? for every m > 0. V is said to be complete if A E%? implies that A @B E 9 and Tor(A,B) E%? for every B. %? is said to be weakly complete if A E %? and B E %? imply that A 8 B E %? and Tor(A,B) E %. %? is said to strongly complete if every finite or infinite direct sum of groups in %? is also in %?. Every complete class is obviously weakly complete, and i t can be verified that every strongly complete class is complete and perfect. See Ex. C a t the end of the chapter. The usefulness of these completeness conditions can be illustrated by the following Proposition 5.1. If %? i s a comfilete class, X a pathwise connected space, X , a subspace of X , and G = { Gx I x E X } a local system of groups in X with each GxE %?, then H m ( X , X,; G) is in %? for each m > 0. Proof. Pick x, E X as in (IX; fj 2) and denote Gxoalso by G. Then Hm(X,Xo; G ) is isomorphic with a quotient group of some subgroup of Cm(X,Xo)@G which is in %?. Hence Hm(X,X,; G) is in V . I As to the examples of classes (1)-(6) in fj 2, one can verify that ( l ) , (2), (6) are strongly complete, and that (3), (4),( 5 ) are prefect and weakly complete but not complete. The classes in § 3 are all strongly complete; in particular, the class P p of the 9-primary groups is strongly complete. For an example of a class which is perfect and complete but not strongly complete, see Ex. A2, at the end of the chapter. In the sequel, we will deal with two different kinds of applications. For the first kind, i t suffices to assume that the class Vinvolved is weakly complete (and, sometimes, perfect). For the second kind, we have to assume that V is complete. However, this difference is not of much practical importance. In fact, the homotopy and homology groups considered in the applications are usually finitely generated; and if V is a given class and Vf is the class consisting of those abelian groups all of whose finitely generated subgroups are in V, then %?f is strongly complete and
w f n d f =undf. 6. Applications of Classes to Fiber Spaces Let us go back to the notation of (IX; 3 3) and assume that n,(B, b,) operates simply on the homology and cohomology groups of the fiber F . Unless otherwise stated, the coefficient group G is the group of integers and hence is omitted from the notations.
6.APPLICATIONS
O F CLASSES TO FIBER SPACES
301
Let V be any given class of abelian groups. Let us consider the spectral homology sequence of (IX; 3 7). Lemma 6.1. I f , for some
n >, 1, Ep”,qE V , then EZq EV.
Proof. Since E;,:’ is the quotient group of a subgroup of Eg,q, it is in V. Then it follows by finite induction that E& is in V. I
+
Lemma 6.2. I f , for a given pair ( 9 , q ) , E T j € V whenever i j =p +q and i Q p , then H p , q ( X X,) , E V. I n particular, if ETj E W whenever i + j = P q, then Hp+q(X,X o ) E V. Proof. The lemma follows from the fact that Hr,j(X, X,) is an extension of H L ~ , ~ + ~X,) ( Xby , ETj and that Ho,m(X, X,) = E&,. I
+
Proposition 6.3. If %? is a weakly complete class and i f , for some integer
> 0 , Hm(B, B,) E V and H m ( F )E V whenever H m ( X ,X,) E V whenever 0 < m < r. 7
0
< m < r, then we have
Proof. According to the above lemmas, it suffices to show that Ei,qE V whenever 0 < p q < r. By (IX; 6.3),
+
+
Ei,q % Hp(B, Bo) 8 Hq(F) Tor(Hp-i(B, Bo),Hq(F)). If p > 1, q > 0, and 0 < p + q < r, the weak completeness of V implies E& E V. If q = 0 and 0 < p < r, we have Ei,o m H p ( B ,B,) E V since H,(F) w 2. To verify the proposition for the cases p = 0 and p = 1, let us first assume B, # o . Then H,(B, B,) = 0 and hence Ei,m= 0 , Ef,m-1 M H , ( B , Bo) 8 Hm-l(F) E V whenever 0 < m Q r. Next, assume that B, = a. Then H,(B, B,) M 2 and hence E& M Hm(F)EV, E:,m-l whenever 0 < m < r . I
M
H , ( B , Bo) 8 Hm-I(F) EV
Proposition 6.4. If V is a complete class and if, for some integer p > 0 and q > 0 , Hm(B, B,) E V whenever m 2 p and Hm(F) E V whenever m > q, then H m ( X , X,) E V whenever m 2 p q.
+
Proof. By the completeness of V, it follows that E t j € V whenever, i j >, fi + q. Hence the proposition is an immediate consequence of the above lemmas. I Throughout the remainder of the section, we assume that B, # and hence H,(B, B,) = 0.
+
Theorem 6.5. If %? i s a weakly complete class and
if, for some integers
p > 0 and q > 0 , we have H 1 ( B ,B,) = 0 , Hm(B, B,) E V whenever 1 < m
+
Hm(B, Bo)
X. C L A S S E S OF A B E L I A N G R O U P S
302
is a V-isomorphism whenever m Q r and is a V-epimorphism whenever m = r + 1, where r = Inf (9,q + 1). Proof. The kernel of u)* is Hm-i,i(X, X,) according to ( I X ; 9 7). Therefore, in order to prove that w* is a V-monomorphism for m Q 7 , it suffices to show that E i j E V whenever i + j Q r and i > 1. Since
E t j M H t ( B ,B,) 8 H A F )
+ Tor(Ht-,(B,B,I, H A F ) ) ,
i t follows that Ei,j = 0, Ef,j = 0, and E&E V whenever i > 1, j and i j Q r. This proves that w* is a V-monomorphism for m < r. It remains t o prove that w, is a V-epimorphism whenever m Q r By (IX; 5 7), the image of w, is E;,". Furthermore, we have
+
E:,o = E mm,O+ l c
*
*
c
En+' c E" c m,O m,O
* * *
1,
+ 1.
c Ef,o= H,(B, B,),
EZ,o/EZ$lw dn(E:,o) c E:-n,n-l,(2 Q n Q m). Since EZ-n,n-l is of total degree m - 1 Q r, it is in % for 2 Q n Q m.This implies that the cokernel Ei,o/EE,:l of 0,is in V. I Note. If p < q + 1, the condition H,(B, B,) = 0 may be replaced by H,(B, B,) E V. In fact, only this is used in the proof of Et,r-l E V.
q 0
Theorem 6.6. If V is a complete class and if, for some integers p > 0 and and Hm(F)E % whenever
> 0 , we have H m ( B ,B,) E %? whenever 0 < m < p < m < q, then the induced homomorphism
: H m ( X , Xo) + Hm(B, Bo)
i s a V-isomorphism whenever m Q r and is a V-epimorphism whenever m = r + 1, where 7 = p + q - 1. The proof of this theorem is analogous to that of (6.5). In the remainder of the section, we are concerned with the important special case where the subspace B, consists of a single point b,. Then we have
H,(B, Bo)
=
0, Hm(B,B,)
Hm(B), (m > 1).
Proposition 6.7. If V i s a weakly complete class, if H m ( X )E V for each m > 0, and if H , ( B ) = 0 and Hm(B) 6 % whenever 1 < m < for some given integer # > 0, then Hm(F) E V whenever 0 < m < p - 1 and the homomorp hisms a Hp(B, bo) w Hp(B) Hp-1 ( F ) +- H p ( X ,F ) >-+
define a V-equivalence of Hp-l(F) and H p ( B ) . Proof. We shall prove this proposition by means of induction. The case
1 is trivial. Assume that p > 1 and the proposition is true for # - 1. Then, by the hypothesis of induction, we have Hm(F) E V whenever 0 < m < p - 2 and H,-,(F) is %-equivalent to Hp-,(B) and hence is in V.
p
=
6.
303
A P P L I C A T I O N S O F C L A S S E S TO F I B E R S P A C E S
Applying (6.5) with q = 9 - 1 and B, = b,, we deduce that H p ( X ,F ) is V-isomorphic to H,(B, b,) under w,. In the exact sequence a
’+H p ( X ) + H p ( X , F ) ---+ H p - l ( F ) -+ H p - l ( X ) we have H p ( X )E V and H p - l ( X ) E %. This implies that morphism. I * *
*
+ a
a
., is a %-iso-
Proposition 6.8. If V is a complete class, Hm(X)E Y for each m > 0 , and Hm(B)E % whenever 0 < m < p for some given integer p > 0 , then Hm(F)E V whenever 0 < m < p - 1 and the homomorphisms
a
Hm(F) Hm+l(X,F ) %-+Hm+l(B,bo) w Hm+l(B) define a %-equivalence of Hm(F)and Hm+l(B)whenever 9 - 1 < m < 2p - 2. The proof of this theorem is analogous to that of (6.7). A space X is said to be %-acyclic if H,,,(X) E % for each m
> 0.
Theorem 6.9. If Y is a weakly complete class, H l ( B ) = 0 , and two of the spaces X, B , F are V-acyclic, then so is the third. Proof. If the two spaces are B and F , then it follows from (6.3)with r = co and B , = 17that X is %-acyclic. If the two spaces are X and B , then it follows from (6.7) with p = co that F is %-acyclic. If the two spaces are X and F , then we shall prove H,(B) E % by induction on p . The case p = 1 is trivial. Assume 9 > 1 and Hm(B)E % if 1 < m < p . By (6.7), H,(B) is %-equivalent to H p - I ( F )and hence H,(B) E % . I As an application of these results, let us consider the special case where X = [ B ;B , b,] and o : X + B is the initial projection. In this case, the fiber F becomes the space A ( B ) = [ B ;b,,b,l
of all loops in B with b, as basic point. Assume that B is simply connected and hence A ( B ) is pathwise connected. Therefore, we may apply the results of this section. Since [ B ;B , b,] is contractible, the following theorem is an immediate consequence of (6.7) and (6.8). Theorem 6.10. If % is a class and Hm(B)E % whenever 0 < m < p , then Hm(A(B))is V-equivalent to Hm+l(B)for the following values of m: ( i )0 < m < p if Y is weakly complete. ( i i ) 0 < m < 2p - 2 if % is complete. Therefore, for a weakly complete class %, A ( B ) is %-acyclic iff B is
V-acyclic. In particular, if B is a space of the homotopy type (n,n) with n > 1, then A(B)is a space of the homotopy type (n,n - 1). Thus, we may apply (6.10) to this special case and obtain Hm(n, n - 1) w Hm+,(n,n ) , O < m < 2n - 2. Furthermore, we have the following obvious
X. C L A S S E S O F A B E L I A N G R O U P S
304
Proposition 6.11. For an abelian group n,an integer n > 1, and a weakly complete class V, the following two statements are equivalent : (i) H m ( n )E 'i$ for each m > 0. (ii) Hm(n, n) E V fm each m > 0.
7. Applications to n-connective f'iber spaces Let B be a pathwise connected space and b, a given point in B. According to (V; 3 8), we may construct inductively a sequence of spaces (B, n), n = 0, 1,2; , and a sequence of maps
--
fin: (B,n) -+ ( B , n -
l), n
=
1,2,3;-.,
as follows. Let (B, 0) = B. For each n > 0, let (B, n) be an n-connective fiber space over (B, n - 1) with projection &. This system { (B, n), pn 1 will be referred to as a connective system of the space B. If B is locally pathwise connected and semi-locally simply connected, then of course we may take (B, 1) to be the universal covering space over B with as the projection. Now let { (B, n),Pn } be any connective system of B. Since ( B ,n) is an n-connective fiber space over an (n- 1)-connected space ( B ,n - 1) with /?, as projection, i t follows that the fiber of this fibering is a space F,, of the homotopy type (nn(B),n - 1). Next, let
wn
=/lj?2...pn:(B,n) -+B,
n
=
1,2;.-,
then it is clear that (B, n) is an n-connective fiber space over B with wn as project ion. Applying (6.9) and (6.11) to the fibering pn, we obtain the following Proposition 7.1. If V is a perfect and weakly complete dass, n n,(B) E %?, then the following two statements are equivalent: (i) (B, n) is W-acyclic. (ii) (B, n - 1) is %-acyclic.
Applying (6.6) with q
=m
> 1, and
to the fibering pn, we obtain the following
Proposition 7.2. If V is a perfect and complete class, n
then
(Bn)* : &(B, 4 is a V-isomorphism for each m > 0.
-+
> 1, and n,,(B) E V,
Hm(B, 92 - 1)
Finally, if B is (n - 1)-connected, then we may take (B, m) = B for every m Q n - 1. The following proposition w ill be used in the sequel. Proposition 7.3. Let V be a perfect and weakly complete class. IfB is (n - 1)connected, nn(B)E V, n > 1, and p > n is an integer such that Hm(B)E %f whenever n < m < p, then the induced homomorphism
(A)*: H m P , n) Hm(B) is a V-isomorphism for m Q p and is V-epimorphism for m -+
=
p
+ 1.
8. T H E
GENERALIZED HUREWICZ THEOREM
305
Proof. Since the fiber F = F n of the fibering /?n is of the homotopy type (n,(B),n- I), it follows from (6.11) that HnL(F)E % for every m > 0. Let B,c B consist of a single point, then we have H , ( B , B,) = 0 and Hm(B,B,) E V for each m < 9. Thus we may apply (6.5) with q = co. Hence the induced homomorphism
(&)# : Hm(X, F ) + Hm(B, Bo), X
=
( B ,4 ,
is a V-isomorphism whenever m < $ and is a V-epimorphism whenever = p + 1. Since Hm(F)E V for each m > 0, it follows from the homology sequence that i : Hm(X) Hm(X, F ) is a V-isomorphism for every m > 0. Finally, using the isomorphisms
m
-+
k : Hm(B)M Hm(B,E,), we obtain (&)*
=
k-'(fin)#j.
m > 0,
I
8. The generalized Hurewicz theorem Theorem 8.1. Let % be a perfect and weakly complete class. If X i s a simply connected space and n > 2 is a n integer such that nm(X)E V whenever 1 < m < n, then the natural homomorphism
hrn :nm(X) i s a %-isomorphism whenever 0 m=n-tl.
+
Hm(X)
< m < n and is a
V-epimorphism whenever
Proof. We are going to prove this theorem by induction on n. If n = 2, then this is implied by the usual Hurewicz theorem since n o ( X ) = 0 and n l ( X ) = 0. See (V; 4.4) and ( V ; Ex. C). Let p > 2 be an integer and assume that the theorem is true for n < p. Let us prove the theorem for n = p. By the inductive hypothesis, i t follows that hm is a V-isomorphism whenever 0 < m < p and is a V-epimorphism for m = p. I t remains to prove that hp is a V-monomorphism and hp+l is a V-epimorphism. Consider a connective system { ( X ,r ) ,/?,.} of the space X as defined in 5 7. Since X is simply connected, we may assume ( X , 1) = X . Then ( X , p - 1) is a ( p - 1)-connective fiber space over X with 0 =
p2ps.
*
.pp-l
: ( X ,p - 1) -+
x
as projection. Thus we obtain a commutative rectangle
nm(X. p - 1) &+ Hm(X, p - 1)
1-*
n m( X )
,*
1-#
-+
Hm(X)
306
X. C L A S S E S O F A B E L I A N G R O U P S
where w*,w# are induced by w , and gm, hm are the natural homomorphisms. By the definition of ( p - 1)-connective fiber space, w* is an isomorphism for m 2 p. Since ( X , p - 1) is (p- 1)-connected, it follows from the usual Hurewicz theorem that g, is an isomorphism and g, is an epimorphism. Hence, it suffices to prove that w# is a V-isomorphism form = p and a V-epimorphism for m = p + 1. Since w = p J S . *j3p-l, it suffices to prove that, for each r = 2,3; * , p - 1, the induced homomorphism,
-
9
: Hm(X, r)
+
H m ( X , 7 - 1)
+
is a W-isomorphism for m = p and is a %‘-epimorphism for m = p 1. For this purpose, let B = (X, r - 1). Then, B is ( r - 1)-connected and we may take a connective system of B with ( B ,r - 1) = B and (23, r) = ( X , Y ) . Since B is simply connected and nm(B)E V whenever 1 < m < p , it follows from the inductive hypothesis that Hm(B) E V whenever 0 < m < p . Therefore, by (7.3), (j3r)# is a %-isomorphism for m = p and is a V-epimorphism for m = p + 1. I Corollary 0.2. Let % be a Perfect and weakly complete class. If X is a simply connected space and n > 2 is an integer such that H m ( X ) E V whenever 1 < m < n, thenn,(X) E V whenever 0 < m < n.
This corollary follows from (8.1) by finite induction on m. Therefore, if X is simply connected and %-acyclic, then X is V-asphericd, i.e. n m ( X )E V for all m > 1. In particular, we have the following Corollary 0.3. The homotopy groups of any simply connected finitely triangulable space are finitely generated.
9. The relative Hurewicz theorem Theorem 9.1. Let V be a perfect and complete class, X a simply connected space, and X , a simply connected subspace of X . If n , ( X , X o ) = 0 and n > 2 is an integer such that n m ( X ,X,) E % whenever 2 < m < n, then the natural homomorphism hm n m ( X , Xo) H m ( X , Xo) is a %-isomorphism whenever 2 < m < n and is a V-epimorphism whenever m-n+l. +
Proof. We prove this theorem by induction on n. If n = 2 , the theorem is true since H , ( X , X,) = 0 and h, is an isomorphism by (V; Ex. C). It follows from the hypothesis of induction that k, is a %-isomorphism whenever 2 Q m < n. Hence H m ( X , X,) E % whenever 0 < m < n. It remains to prove that hn is a %-isomorphism and hn+, is a V-epimorphism. Pick a point x, E X , and consider the space of paths Y = [ X ;X , x,]. Then, Y is a fiber space over X with projection w : Y + X defined by
307
10. T H E W H I T E H E A D T H E O R E M
o(t)= l ( 0 ) for each 5~ Y . Let Y o = [ X ;X,, x,]. Then Yo is pathwise connected and zm(Yo) =nm+l(X, X,), (m > 1). Hence, nl(Yo)= 0 andnm(Yo)E %? for 2 < m < n - 1. By (8.I), the natural homomorphism g, :Z m ( Y 0 ) + H m ( Y o )is a %?-isomorphismfor m = n - 1 and is a V-epimorphism for m = n. Since F = o-l(xo) = A ( X ) is pathwise connected and H m ( X ,X,) E Y whenever 0 < m < 12, we may apply (6.6) with p = n and q = 1. Hence the induced homomorphism o# :Hm(Y, Yo)+. H m ( X ,X,) is a %?-isomorphism for m = n and is a Q-epimorphism for m = n 1. In the diagram
+
w
nm(X, Xo) +---*-nm(Y,
a
Yo)A Z m - l ( Y o )
K n ( X , X,)6 -w# ~ t n ( yyo) , -+a# Hm-i(y0) we obtain hm = w#a#-l gm-lll*o*-'. Therefore, h,, is a %?-isomorphismfor m = n and is a %?-epimorphismfor m = n + 1. I Corollary 9.2. If X and X , are simply connected, n,(X, X,) Hm(X, X,) is in a $erfect and complete class V whenever 2 < m n m ( X , X,) i s also an V whenever 2 < m < 12.
=
0 , and
< n, then
Remark. The theorem (9.1) does not hold if we merely assume that Y is perfect and weakly complete. For example, let X = A x B and X , = A x b, where A and B are simply connected, B is %?-acyclic,and b E B. By making various choices of A , one can see that (9.1) is true for a given class Q iff %? is perfect and complete. Similarly, (8.1) is true iff %? is perfect and weakly complete. 10. The Whitehead theorem Theorem 10.1. Let %? be a perfect and complete class. If X and Y are simply connected spaces, f : X + Y i s a map such that f* : n 2 ( X )-+n,(Y)i s a n epimorphism, and n 2 2 i s a given integer, then the following two statements are equivalent : (1) f* : n m ( X )-+nm(Y) i s a %?-isomorphism for m < n and is a %?-epimorphism for m = n. (2) f# : H m ( X )-+H m ( Y ) i s a %-isomorphism for m < n and i s a Q-epimorphism for m = n. Proof. Consider the mapping cylinder Zfof the map f . By ( I ; 5 12), both X and Y can be naturally imbedded in Zfand Y is then a strong deformation retract of Zf.Thus the map f is decomposed into the composition r i of the inclusion map i : X c Zf and a strong deformation retraction r :2, + Y . Since Y induces isomorphisms on homotopy and homology groups, (1) and (2) are equivalent respectively to the following two statements:
308
X. C L A S S E S O F A B E L I A N G R O U P S
(1’) i, : a ( X )+ n m ( Z f ) is a $9-isomorphism for m < n and is a 55‘-epimorphism for m = n. (2‘) i# : H m ( X ) + H m ( Z f )is a %-isomorphism for m < n and i s a V-epimorphism for m = n. Then i t follows from the homotopy sequence and the homology sequence of ( Z f ,X ) that (1‘) and (2’) are equivalent respectively to the following two statements: (1”) nm(Zf,X ) E V whenever 2 < m < n. (2”) H m ( Z f ,X ) E %whenever 2 Q m f n. By (9.1), we conclude that (1”) and (2”) are equivalent. This entails the equivalence of (1) and (2). I
EXERCISES A. Examples of Classes of Abelian Groups.
In addition to the classes given in 2 and 5 3. we give the following examples : 1. The class of all abelian groups with power not exceeding a given infinite cardinal number N,. In particular, if w, is the power of the set of natural numbers, this reduces to the class -01, of all countable abelian groups. Verify that this class is perfect and weakly complete but not complete. 2. The class of all abelian groups A such that there is an integer N depending on A with Nu = 0 for every a E A . Verify that this class is perfect and complete but not strongly complete. 3. The class of all abelian groups satisfying the descending chain condition. Verify that this class is perfect and weakly complete but not complete. B. Composed Homomorphisms
By using the natural exact sequence S ( f ,g) of two homomorphisms f : A -+ B and g : B + C, prove the following six assertions for a given class V: 1. I f f and g are V-monomorphisms, then so is gf. 2. I f f and g are %?-epimorphisms, then so is gf. 3. If gf is a V-monomorphism, then so is f. 4. If gf is a V-epimorphism, then so is g. 5 . If gf is a V-monomorphism and f is a %‘-epimorphism, then g is a V-monomorphism. 6. If gf is a ‘X-epimorphism and g is a V-monomorphism, then f is a V-epimorphism . C. O n Perfectness and Completeness
Prove the following assertions: 1. For any class V of abelian groups, the following three statements are equivalent :
309
EXERCISES
(a) Q is complete. (b) A E W implies A @ B E V for every abelian group B. (c) For any A E V, every finite or infinite direct sum of groups isomorphic to A is in V. 2. Every strongly complete class is perfect and complete. It is unknown whether there is a class which is not perfect or not weakly complete. D. The C-Generalization of the “Five” Lemma
Prove that the “five” lemma, [E-S; p. 161, remains true modulo a class %‘. Precisely, if we have two exact sequences each with five terms and five homomorphisms of the groups of the first sequence into the corresponding groups of the second sequence with the commutativity relations being satisfied, and if the four extreme homomorphisms are V-isomorphisms, then the middle homomorphism is also a V-isomorphism. E. The C-Inverse Homomorphism Theorem
Consider a class %? and an exact sequence A , >+ / A , f l + A,f”+ A,--+1.
A,
Assume that there exist two homomorphisms g, : A + A , , and g, : A , --t A , such that the endomorphisms f , g , and f4g4 are V-isomorphisms. Define a homomorphism 12: A , A , + A 4 by taking h(x, y ) = f,(x) g,(y). Prove that h is a %-isomorphism.
+
+
F. The Products of C-Equivalent Groups
Let V be a complete class. Prove that, if A and B are %‘-equivalent respectively to A‘ and B‘, then A 8 B and Tor(A,B) are %-equivalent respectively to A‘ @ B’ and Tor(A’,B‘). G. C-Exact Sequences
Let V be a class and A , B two subgroups of an abelian group G. We say that A and B are V-equal if the inclusion homomorphisms A n B + A and A n B + B are V-isomorphisms. Replacing equality by %?-equality, one can define the notion of a V-exact sequence. 1. Establish the elementary properties of 9-exact sequences as in [E-S ; p. 501. 2. Generalize the results in (VIII; 3 8) to obtain various fundamental V-exact sequences. H. O n Induced Homomorphisms
Let X , Y be simply connected spaces, f : X -+ Y a map such that f* : n z ( X )+ n2(Y)is an epimorphism. Assume that the homology groups are finitely generated.
310
X. C L A S S E S O F A B E L I A N G R O U P S
1. Let 9 denote the class of all finite abelian groups, F the class of all torsion groups, and G a field of characteristic zero. Prove that the following four statements are equivalent: (a) f # : H m ( X )-+ H m ( Y ) is an Y-isomorphism for m < n and is an 9-epimorphism for m = n. (b) f# : H m ( X ) + H,(Y) is a 9-isomorphism for m < n and is a Y-epimorphism for m = n. (c) f# : H,(X; G) + Hm(Y; C) is an isomorphism for m < n and is an epimorphism for m = n. (d) f# : H m ( Y , G) + H m ( X ; G ) is an isomorphism for m < n and is a monomorphism for m = n. 2. Let F Pdenote the class of all finite abelian groups of order not divisible by a given prime number p, Y pthe class of all torsion groups with null +-primary component, and G a field of characteristic p. Prove that the following four statements are equivalent : (a) f# : H,(X) + H,(Y) is an gp-isomorphism for m < n and is an Sp-epimorphism for m = n. (b) f# : H m ( X )-+ H m ( Y )is a .Tp-isomorphism for m < n and is a Yp-epimorphism for m = n. (c) f # : H,(X; G) + H m ( Y ;G) is an isomorphism for m < n and is an epimorphism for m = n. (d) f#: H m ( Y ; G) + H m ( X ; G) is an isomorphism for m < n and is a monomorphism for m = n. The usefulness of these two propositions is that, on many occasions, we may replace the calculus mod W by the calculus with coefficients in a field. Secondly, since Y and Y Pare perfect and complete, we may apply (10.1). Finally, the statements (b,), (c), (d) are equivalent even if the homology groups are not finitely generated.
CHAPTER XI H O M O T O P Y GROUPS OF SPHERES
1. Introduction Finally we come to the determination of certain of the homotopy groups of spheres. The calculations are particularly based on the results of the previous two chapters, and, since they are quite technical, we will not attempt to summarize them here. However, in the course of the development, several topics of independent interest appear: Freudenthal's suspension theorem (stated in 3 2 and proved in $9 2-5), pseudo-projective spaces and Stiefel manifolds ($3 10-1 l), and the Hopf invariant of a map f : S2n-l + Sn ( 3 14). We have already seen that if Y < 0 then nn+r(Sn)is the zero group, and, that if Y = 0 then this group is free cyclic; and now in $8 15-17 we shall settle the cases Y = 1,2,3, and 4.A brief report of the cases 5 < Y < 15 is given in the final section.
2. The suspension theorem
+
Let n > 1 and consider the n-sphere Sn as the equator of the ( n 1)sphere Sn+l with u and v denoting respectively the north and south poles of Sn+l. Pick a point so in Sn and consider the space
w = n(sn+l) of loops in Sn+l with so as basic point. There is a natural imbedding i : Sn + W described as follows. For each x E Sn, i ( x ) is the loop in Sn+l joining so to u,u to x , x to v , and v back to so, all by shortest geodesic arcs. That a is a homeomorphism of Sn into W is obvious. Furthermore, the loop i ( s o ) is homotopic to the degenerate loop W,E W which maps I into so by means of a natural homotopy; in other words, the points W , and i(so) of W are connected by a natural path u in W . Hereafter, we shall identify x and i ( x ) for every x E Sn. Thus, Sa becomes the subspace i(Sn) of W and i : Sn -+ W reduces to the inclusion map. For each m > 0, i induces a homomorphism
i, :7cm(Snt so) +nm(W, so), the path u induces an isomorphism u* : nm(W,so) M nm(W,W,). 311
XI. H O M O T O P Y G R O U P S O F S P H E R E S
312
and, according to (IV; 2.2), we have an isomorphism
h* :n m ( W , wo) M
nm+l(Sn+', SO).
Composing i,, a,, and h,, we obtain a homomorphism
C
=
h,o,i,
:nm(Sn,so) +nm+l(S"+l, so)
for each m > 0, called the suspension. One can verify that this definition is equivalent to the more general one given in (V; 4 11) for this special case. Theorem 2.1. (The Suspension Theorem). The suspension I: i s an isomorphism if m < 2n - 1 and is an epimorphism if m = 2n - 1.
Proof. Since a* and h, are isomorphisms, it suffices to prove that i, is an isomorphism if m < 2n - 1 and is an epimorphism if m = 2n - 1. Thus, according to Whitehead theorem, ( X ; lO.l), it suffices to prove that the induced homomorphism i# : Hm(Sn) +. H m ( W )
is an isomorphism if m < 2n - 1 and is an epimorphism if m Since Hm(W) M 2, if m 0 mod(n),
=
2n - 1.
Hm(W)= 0, if m f 0 mod(n), by (IX; 13.4), i t remains to prove the following Lemma 2.2.
i# : Hn(Sn) M Hn(W).
This lemma will be proved in the next three sections; we conclude this section with one immediate consequence of the suspension theorem. Let U and V denote respectively the north and south hemispheres of Sn+l; then, using (V; 1l.l), we have the following Corollary 2.3. The excision homomorphism
e, :nm(U,Sn)+n,(Sn+l, V )
is an isomorphism whenever 2 m
=
2n.
< m < 2n and i s an epimorphism whenever
The identity map on Sn extends to a map f : (En+l,S n ) +. ( U ,9). If we compose this map with the excision e : ( U ,Sn) c (Sn+l, V ) ,we obtain a map g
=
ef : (En+l,Sn) + (Sn+l, V ).
Since V is contractible to the point so, g is homotopic in
(Sn+l,
V ) to a map
h : (En+l,S n ) --f (Sn+l, so) which represents a generator o f n n + l ( S n + l , so). For any element a ofnm(Sn,so), choose a map : S m + S n which represents a. The map has an extension
+
+
y : (Em+l, Sm) + ( E n + l , 9).
3. T H E
313
CANONICAL MAP
Then it can be seen that the composed map hy represents the element X(ct) in ~ ~ + ~ ( Sso). n+l,
3. The canonical map Consider the space of paths X = [Sn+l;so, Sn+l]. According to (111; 13), X is a fiber space over Sn+l with a projection o : X -+Sn+l defined by w ( x ) = x ( 1) for every path x E X and with fiber W = co-l(s,). Let U and V denote the north and the south hemispheres of Sn+1 respectively and let
Q
x, = w-l(S*),
x u = w-l(V),
x, = w-'(V).
We are going to define a map x : (U x
w,S"
x W ) ( X u ,X,) -+
which will be called the canonical map. For each point b E U , let y(b)E X u denote the path joining so to u and then u to b by geodesic arcs. The assignment b -+ y ( b ) defines a cross-section y : U + X u . Then x is defined by x ( b , f ) = f . y ( b ) for each b E U and f E W , where f.y(b) denotes the product of the paths f and y(b). As a consequence of the construction, we have
wx(b, f )
=
b, (6 E U , f E W ) .
Lemma 3.1. The canonical map x is a homotopy eqzlivalence. Proof. Let
A : ( X w ,X,)
-+
( U x W ,Sn x W ) be the map defined by
A(%) = (w(x),X ' [yw(x)l-'),
( X E Xa).
where [yo(x)]-ldenotes the reverse of the path yo(%). Then we have xA(x) = x(w(x),X ' [yw(x)]-l)= ( x . [yw(x)]-1)* y w ( x ) ,
[r(41-1)'
W h f ) = W Y ( b ) )= ( b , [ i * Y ( b ) l *
Hence, x l and ;Ix are both homotopic to the identity maps. I Next, let us define a map
pusnx
w+w
by taking p ( b , f ) = f.i(b) for each b E S and ~ f notes the imbedding in 3 2.
E
W , where i : Sn + W de-
Lemma 3.2. The m a p p is homotopic in X v to the map v : Sn = x I Sn x W .
x W -+ X ,
defined by v
Proof. Intuitively, a homotopy of p to v is accomplished by "unwinding" the path i ( b ) to half its original length. More precisely, define a homotopy Itt : Sn + X,, ( 0 < t < I), by taking
XI. H O M O T O P Y G R O U P S OF S P H E R E S
314
Then Ito = i and k, = y . Define a homotopy Kt : Sn x W + X,, (0 < t by taking K & f ) = f.ht(b), ( b E S " , f E W , t E I ) . Then KO
=p
and k ,
= Y.
< I),
I
4. Wang's isomorphism p* In the present section, we shall construct for each integer q 2 0 an isomorphism p* Hn(Sn)8 Hg(W) M Hn+g(W)-
+
Let m = n q + 1. The construction of p* will be made in six steps as follows. Step 1. Since the south hemisphere V of Sn+l is contractible to the point so, an application of the covering homotopy theorem proves that W is a strong deformation retract of Xu. Hence the inclusion map induces an isomorphism t : H m ( X , W ) W Hm(X, Xu).
Step 2. The inclusion map induces a homomorphism Let
q : Hm(X,, X,) + H m W , Xu)* D , = sn+1\ v , D, = sn+l\ U , D = D ,
n D,; Y , = w - l ( ~ u )Y, , = O - ~ D , ) , Y = yUn Y,.
-
Since Y, and Y , are open sets whose union is X , the excision theorem holds and hence the inclusion map induces an isomorphism
Hm(Yu, Y) Hm(X, Y v ) . Since U , V ,Sn are strong deformation retracts of D,, D,,D respectively, an application of the covering homotopy theorem proves that Xu, Xu, X , are strong deformation retracts of Y,, Yo, Y respectively. Hence 7 is an isomorphism. Step 3. Since the canonical map x is a homotopy equivalence, it induces an isomorphism x* : Hm(U x W ,s n x W ) M Hm(Xu, XO).
Step 4. By the Kiinneth theorem, we get an isomorphism
C : Hn+l ( U ,Sn) 8 Hg(W) w Hm(U x W ,Sn x W ) . Step 5. Since X is contractible, we have an isomorphism
a : H m ( X , W ) w Hm-1(W).
Step 6. Since U is contractible, we have an isomorphism Hn+i(U,Sn) M Hn(Sn). Taking tensor products, we obtain an isomorphism 0 : Hn+l(U,Sn) 8 Hg(W) M Hn(Sn) 8 Hg(W).
5. RELA'I'lON
BETWEEN
p*
AND
i#
315
Composing these steps, we get an isomorphism p*
: Hn(Sn)€3 HQ(W)M Hn+g(W).
= dt-'qx,@-'
This isomorphism p* is the same as the homomorphism p* in Wang's exact sequence (IX; 13.1) for the fibering o : X +Sn+l. See also [Wang 21.
5. Relation between p,, and i# The space W of loops has a continuous multiplication
M:WxW+W defined in (111; 5 11). The total homology group
H(W)=
m
x Hm(W)
m=O
becomes a ring under the Pontrjagin multi~licationdefined as follows. Let U E H ~ ( W and ) BEH*(W).By the Kunneth theorem, u and determine a unique element u x B of Hp+Q(Wx w).The map M induces a homomorphism M , : Hm(W x W ) -Hm(W) for every m. Then the Pontrjagin product of u and /Iis defined to be the element a.B = M,(a x B) E H ~ + Q ( W ) . Proposition 5.1. For every u E H n ( S n ) and @ E HQ(W), we always have
P*(Q €3B)
= B.i#(.),
where i# : Hn(Sn) -+ Hn(W) denotes the homomorphism induced by the imbedding i : S n -+ W of 5 2 . Proof. Consider the diagram
H m (u x
w,S n x
W )5 Hm(X,, XO) A H m ( X , X")
I.
I
HmV, W)
+
I.
where u and z are induced by inclusion maps and the homomorphisms d are boundary operators. The rectangules are all commutative and hence (1)
apqx, = u-ltv,a.
By (3.21, we have
(2)
tv,
= ap,.
By the Kunneth theorem, we have an isomorphism
XI. H O M O T O P Y G R O U P S O F S P H E R E S
316
and a commutative rectangle
Hn(Sn)@ Hg(W)-5Hn+q(Sn x W ) .
Hence we obtain (3)
x
= age-1.
Using ( l ) , (2) and (3), we deduce p*
= at-%x*ce-i
=
a-ltv,age-l
=
cl*age-l
=
cl*x.
Then it follows from the definition of p that
P * b @ B)
= p*%
@
B) = Pi#(.).
I
In particular, if q = 0, then H,(W) is a free cyclic group generated by the element e represented by wo as a 0-cycle of W . For each a E Hn(S"),we have $+(a) = e*i#(a) = p*(a@e).
This proves Lemma 2.2.
6. The triad homotopy groups Consider the space of paths By (IV; 3.1), we have
T
=
[ W ;Sn, so].
nm(T) =nm+i(W,Sn)
for every m. Hence the homotopy sequence of the pair (W,9)gives rise to an exact sequence
*..+%(T) +%(Sn)
T'
+nm+l(Sn+')
+nffl-1(T)+*...
This is essentially the suspension sequence of the triad ( S n + l ; U ,V ) ,ndm(T) being essentially the triad homotopy group ~ + ~ ( S n + U l ;, V ) . See (V; $8 10-11). Because of this exact sequence, it is desirable to determine the triad homotopy groups h ( T ) .The following lemma is an immediate consequence of the suspension theorem (2.1). Lemma 6.1. n m ( T ) = 0 for every m
< 2n -2.
To determine the higher homotopy groups of T , let us study the space of paths Q = [W;Sn, W ] which is of the same homotopy type as Sn. Consider the projection w : Q +W defined by w(a) = a(1) for every a E Q; then Q becomes a fiber space over W with fiber w-'(s0) = T.
7. 0
317
F I N I T E N E S S O F H I G H E R HOMOTOPY G R O U P S
Since Hm(W)= 0 whenever 0 < m < n and Hm(T) = 0 whenever ( I X ; 14.1) an exact sequence
< m < 2n - 1, we have by
H3n-n(T) + *
*
.+Hm+l(Q)+Hm+!(W) +Hm(T)+Hm(Q) + *
*
*+H1(W)+O.
Since Q is of the same homotopy type as Sn, we have Hm+l(Q)= 0 = Hm(Q) for every m > n. This implies that H m ( T ) M Hm+,(W)whenever n < m < 312 - 2. Hence, we deduce the following Lemma 6.2. H2n-1(T)M Z and Hm(T)= 0 whenever 2n - 1 < m < 3n -2 .
Choose a map f : S2n-1+ T which represents a generator of the free cyclic group n2n-l(T)M H2n-I(T). Then f induces an isomorphism
f# : H2n-1(S2n-l) M Han-I(T). Then, by (6.1) and (6.2), it follows that f# : Hm(S2%-l)M H m ( T ) for every m < 3n - 2. An application of Whitehead's theorem proves that the induced homomorphism f* :n m ( ~ 2 n - ~ ) + n m ( ~ ) is an isomorphism if m < 3n - 3 and is an epimorphism if m Hence we have proved the following
to nm(S2n-') for every m is isomorphic to a quotient group of r~,~-,(S~n-l).
Proposition 6.3. nm(T)i s isomorphic
andn,,-,(T)
=
3n - 3.
< 3n - 3
7. Finiteness of higher homotopy groups of odd-dimensional spheres In the present section, we are concerned with an odd-dimensional sphere
Sn. Since the homotopy groups of the 1-sphere S1 are completely computed in (IV; 5 2), we may assume that n > 3. Consider an n-connective fiber space X over Sn with a projection w :X 4%. By definition,
n m ( X ) = 0 , (m G n ) , o,,,: n m ( X )mnm(Sn),
(m-> n).
Then it follows that the fiber F is a space of the homotopy type (2, n - 1). By ( X ; 8.3), nm(Sn) is a finitely generated abelian group for every m. An application of the generalized Hurewicz theorem proves that H m ( X ) is finitely generated for every m. Let K be a field of characteristic zero. Since n - 1 is even, it follows from (IX; Ex. G) that the cohomology algebra H * ( F ; K ) is isomorphic to a polynomial algebra over K generated by an element of degree n - 1. Let a E Hn-'(F; K ) be a generator of H * ( F ;K ) . Consider Wang's cohomology sequence (IX; 13.2):
. - . - t P ( X ; K ) + H " ( F ; K ) -P*+
Hm-n+yF;K)+Hrn+'(X;K) +'
-
*,
318
XI. H O M O T O P Y G R O U P S O F S P H E R E S
where p* is a derivation since n is odd. Since H n ( X ; K ) = 0 = H n - l ( X ; K ) , p* sends Hn-l(F; K ) isomorphically onto H o ( F ;K ) . Therefore, p*(a) is a non-zero element of H o ( F ;K ) w K . Now let us prove that p* :HP(n-l)(F;K ) H(p-9(n-l)(F;K ) for every positive integer p . I n fact, the vector space Hp(r-l)(F;K ) over K admits a p as a basis. Since p* is a derivation, we have
p*(kap) = pkp*(a)ap-l, (k E K ) . Hence p* is an isomorphism for every m > 0. Then an exactness argument proves that H m ( X ; K ) = 0 for every m > 0. Since &(X) is finitely generated and K is of characteristic zero, this implies that H m ( X )is finite for every m > 0. An application of the generalized Hurewicz theorem (X ; 8.1) proves that n m ( X )is finite for every m. Thus, we have proved the following Theorem 7.1. I f Sn is an odd-dimensional sphere and m
i s finite.
> n, then n,,(Sn)
8. The iterated suspension The natural imbedding Sn+l c L I ( S ~ + of~3) 2 induces an imbedding j : A(Sn+l) +.A2(Sn+z). Composing with the natural imbedding i:Sfi-+A(Sn+l), we obtain an imbedding k = ji :Sn +.A2(Sn+z). For each m, k induces a homomorphism k , :nm(Sn, SO) +.nm(A2(Sn+z),SO). As in 3 2, we have a natural isomorphism I , :%(AZ(Sn+z), so) M n m + p + z , so). Proposition 8.1. I,k, is equal to the iterated suspension Proof. By
P.
3 2, there is an isomorphism
a = h,a, :n,(A(Sn+l), so) Similarly, there are isomorphisms
M
~ ~ + ~ ( S so). n+l,
p :nm(AZ(Sfi+3,so)
w nm+l(A(Sn+z),so), y :nm+l(l.i(Sn+2), so) w n m + p + z , so). Then 1, = yp and k , = j*i,. The proposition is a consequence of the commutativity of the diagram:
nm(SB,so) L+ nm(A(Sn+l),so) Jbnm(AZ(Sn+Z),so)
X P
7cm+l(S"+1,so)
1
nm+l(A(s*+z),so)
The following proposition is an immediate consequence of (8.1) and (2.1).
9.
T H E P-PRIMARY C O M P O N E N T S O F Z ~ ( S ~ )
319
Proposition 8.2. The homomorphism k , is an isomorphism if m < 2n - 1 and is an epimorphism if m = 2n - 1. If we study the p-primary components instead of the whole homotopy groups, then we can deduce more detailed information from the iterated suspension X2. Theorem 8.3. Let n > 3 be an odd intcger, p a prime number, and V the class of all finite abelian groups of order prime to p. Then the iterated suspension
X2
nm(Sn) +nm+s(Sn+')
is a %f-isomorphismif m < p(n + 1) - 3 and is a V-epimorphism if m p(n 1) - 3.
+
=
Proof. According to (8.1), it suffices to prove the theorem for the homo-
k , :nm(Sn) +nm(A2(Sn+2)) morphisms induced by the natural imbedding k : Sn + L I ~ ( S ~ + ~ ) . Let K be a field of characteristic 9. Then k induces the homomorphisms k# : H ~ ( L I ~ ( S ~K) ++ ~H ) ;m ( S n ; K).
By Whitehead theorem ( X ; 10.1) and (X; Ex.H2), it suffices to show that k# is an isomorphism for every m Q p(n 1) - 3. By (8.2) k , is an isomorphism if m < 2n- 1 and is an epimorphism if m = 2n- 1. An application of ( X ; 10.1) and (X;Ex.H 2 ) proves that k# is an isomorphism for m < 2n - 1. By ( I X ; Ex.F6 and F7), we have Hm(A*(Sn+2); K ) = 0, ( n < m Q p(n 1) - 3). Since n >3, it follows that k# is an isomorphism for every m < p(n + 1) -3. I
+
+
Corollary 8.4. If n > 3 i s an odd integer, p a prime number, and m < n + 4p-6, then the p-primary components of nm(S") and nm-n+3(S3) are isomorphic.
Proof. We shall prove the corollary by induction on n. When n = 3, there is nothing to prove. Assume that q > 5 is an odd integer and the corollary is true for every odd integer n with 3 Q n < q. By (8.3), the p-primary components of nm(Sq) and nm-,(Sq-2) are isomorphic if m - 2 < p ( q - 1) - 3. Since q > 5, we have ( p - 1) (q - 5) >O and hence 4 + 4 p - - 8 Q P ( q - 1) -3.
This implies m - 2 < p ( q - 1) - 3 whenever m < q + 4p - 6. By the hypothesis of induction, the #-primary components of nm-a(S@-2) and 7 ~ ~ * + ~ ( Sare 3 )isomorphic if m < q + 4p - 6 . Hence, the corollary is also true for n = q. I
9. The p-primary components of nm(S3) The corollary (8.4) reveals the importance of finding the +-primary components of the homotopy groups of the 3-sphere S3.
3 20
XI. H O M O T O P Y G R O U P S O F S P H E R E S
Consider a 3-connective fiber space X over S3 with a projection w : X + S3 and fiber F which is a space of homotopy type ( Z , 2 ) . Lemma 9.1. T h e integral homology groups of X are as foZlows: H m ( X ) = 0 if m i s odd; H,,(X) i s cyclic of order n for every n > 0. T h u s , the first few homology groups are: z,o, o,o, z,, o , z 3 , o,z,, O,Z,,‘
-
Proof. Consider Wang’s cohomology sequence (IX; 13.2) : *
*
+ Hm(X) + Hm(F) -P’+ Hm-Z(F) + Hm+l(X)+*
* *
where p* defines a derivation of H*(F).By (IX; Ex. G), H*(F)is isomorphic to a polynomial algebra over the ring of integers generated by an element of degree 2. Since n m ( X ) = 0 for m < 3, we have Hm(X) = 0 for m < 3. Hence we obtain p* : H 2 ( F )M Ho(F)= 2. Let 0: denote element of H 2 ( F )with p*(u) = 1. Then M generates the algebra H * ( F ) and p*(un) = nc1n-l. Let m = 2 n with n > 2. The sequence becomes 0 + HZ%(X)+ H y F ) p’+ HZfi-Z(F) -+ HZ”+l(X) + 0. Since H2fi(F)is free cyclic with a” as generator and p*(un) = i t follows that p* is a monomorphism and its cokernel is cyclic of order n. Then the exactness implies that H2n(X)= 0, H2n+l(X)M 2,. The lemma follows from this and the duality between homology and cohomology. I Theorem 9.2. If p i s a prime number, then the $-primary component of 4 S 3 )i s 0 if m < 2 p and i s Z, if m = 2p.
Proof. Let %? denote the class of all finite abelian groups of order prime to p. By (9.1), we have H m ( X )E V whenever 0 < m < 2p. An application of the generalized Hurewicz theorem proves that n m ( X )E V whenever 0 < m < 2 p and n z p ( X )is %?-isomorphicto 2,. Since X is a 3-connective fiber space over S3, we have nm(S3)= n m ( X ) for each m > 3. This implies the theorem. I Corollary 9.3. I f n 2 3 i s a n odd integer and p a p r i m number, then the p-primary component of nm(Sn) i s 0 if m < n + 2p - 3 and i s 2, if m = n + 2p-3.
+
+
+
Proof. Since > 2, we have n 2 p - 3 < n 4$ - 6. Then it follows (8.4) that the p-primary components of nm(Sn) and nm-n+3(S3) are isomorphic. I
10. P S E U D O - P R O J E C T I V E S P A C E S
321
10. Pseudo-projective spaces If we adjoin to the n-sphere Sn an (n + 1)-cellEn+l by means of a map
$ : i?En+l = Sn + Sn of degree h as in ( I ; 5 7), we obtain a space P
= e+1
which is called a 9seudo-projective space, [A-H; p. 2661. We shall assume that h > 0; in this case, the homology groups of P are
H,(p) Wz, H n ( p ) W z h ; Hm(P) = 0, m # 0, m # n. Lemma 10.1.
For every m < 2n - 1, we have a n exact sequence
0 -j7Zm(Sn) 8 zh +.nm(P) + TOr(nm-,(Sn),zh)+ 0. Proof. The map $ extends to a map y : En+l+ P in an obvious way. We obtain a commutative diagram: a a * * *+.nm+](P,Sn) -+ nm(Sn)+7zm(P)+n,(P,Sn) -+ nm-](sn) +' * *
iv*
n,+l(En+l,Sn)
- F* a
nm(S@)
F.
nm(En+l,Sn)
a --j
Id*
nm-,(S*)
where the top row is the homotopy sequence of the pair (P,Sn) and
$* nm(Sn) - j n m ( S n ) , y* :nm(E"+', Sn) + n m ( P , Sn) are induced by the maps $ and y. Since q5 is of degree h, we have $*(a) = ha for each u €nm(Sn)whenever m < 2n - 1 ; on the other hand, y* is an isomorphism for every m < 212 - 1. See Ex. A and Ex. B at the end of the chapter. Hence we may replace the homotopy sequence of (P,9) by the exact sequence 4 n*(Sn) L+. nm(S@)-nm(P)+nm-I(S")
nm-l(Sn)
for every m < 2n - 1. Since the kernel and the cokernel of $* :nm(Sn)+ nm(Sn) are isomorphic to Tor(nm(Sn),zh) and nm(Sn)8 zh respectively, the exactness of this sequence implies the lemma. I Let X denote a 3-connective fiber space over S3with projection o : X + S3. Since the p-primary component of n,,(X) is cyclic of order p, there exists a map f :S 2 p +. X which represents a generator [f] of this p-primary component of n,,(X). Consider the pseudo-projective space P = P;p+'. Then S 2 p c P. Since f~[ f ] = 0, f can be extended to a map g : P + X . Composing with o : X +. S3, we get a map X = wg : P + S 3 which induces the homomorphisms X* :nm(P)+nm(S3).
XI. H O M O T O P Y G R O U P S O F S P H E R E S
322
Lemma 10.2. X , is a monomorphism for m < 4p - 1 and sends nm(P) onto the #-primary component of nm(S8)for m Q 4p - 1. Proof.
It suffices to prove the lemma for the induced homomorphisms g* :nm(P)+ n m ( X ) .
It follows from the generalized Hurewicz theorem (X; 8.1) that nm(P) is a +primary group for every m. Hence g, sends nm(P) into the $-primary component of n m ( X ) . Let W denote the class of all finite abelian groups of order prime to $. It remains to prove that g, is a %-isomorphism for m < Irp - 1 and is a W-epimorphism for m = 4p - 1. From the construction of g, one can see that g#: Hzo(P)M H a p ( X ) . Then, by (9.1), g# : Hm(P)+ H m ( X ) is a W-isomorphism for m < 4p. An application of the Whitehead theorem (X; 10.1) completes the proof. I This lemma reveals the importance of finding the homotopy groups of p = PZP+'. P
If p is a $rime number and m Q 4p -2, then -(P) if m is digerent from 2p, 4p - 3, and 4p -2, while: Theorem 10.3.
=0
n,p(P) M 2,; n,-3(P) w zp; nu--,(P)w z, if p > 2. Proof. According to Hurewicz's theorem, we have n m ( P ) = 0 for each
m < 2p and nzp(P)w 2,. Applying (10.1) with n 4p - 3, we obtain an exact sequence
=
2$, h
=
p, and m
<
0 +nm(S'~)8 Z p +nm(P)-+ Tw(nm-i(S'p),Z p ) +O.
By the suspension theorem (2.1), we have % ( S 2 P ) w n++l(Sap-l) for every m < 4$ - 3. Since 2p - 1 2 3, it follows from (9.3) that the $-primary component of nm-I(S'P-') is 0 if m < 4p - 3 and is Z, if m = 4p - 3. Hence we obtain: nm(P) = 0, (2p < m < 4p - 3) ; JC4P-3(P) M Z P
By ( 1 0 4 , the +-primary component of nm(S8) is 0 whenever 2p < m < 4p - 3 and is 2, if m = 4p - 3. Then, by (8.4), we deduce that, if n > 3 is odd, the P-primary component of nm(S") is 0 whenever n + 2p - 3 < m < n 49 - 6. By (2.1), nm(S2P) M nm+l(S'P+') for every m < 9,- 2. Hence, the $-primary component of nm+1(S'P+l) is 0 whenever 4p - 3 < m < 6p - 6. If p > 2, then 4p - 2 < 69 - 6 and hence the @-primarycomponent of Z ~ - ~ ( S is ~ P0.) By (10.1) with n = 2p, h = p, and m = 4p -2, we get n,-,(P) -2,. I
+
323
11. S T I E F E L MANIFOLDS
Corollary 10.4. If p is a #rime number, then the #-primary component of nm(S8)is 0 if 2 p < m < 4 p - 3 and is 2, is m = 4p - 3 . If p > 2 , then the p-primary component of n4p-,(Sa)is 2,. Corollary 10.5. If n Z 3 is an odd integer and p a prime number, then the #-primary component of %(Sn) is 0 if n 2p - 3 < m < n 4 p - 6 and that of nn+ap-n(S") is 0 or Z p .
+
+
11. Stiefel manifolds Let n > 4 be an even integer and consider the Stiefel manifold V = Vn+l,a of all unit tangent vectors on 9,(111; Ex. G). Then, V is simply connected and its homology groups are as follows:
H,(V)
-
2, Hn-1(V)
= z,,
Ha?&-#)
=z
and all other homology groups are zero, [Stiefel 1 , 2 ] and [S; p. 1321. Since V is the tangent bundle of Sn, it is a fiber space over Sn with a projection w : V +.Sn and fibers homeomorphic to 9 - l . According to (V; § 6), this fibering gives an exact sequence * *
d.
* - + n m ( ~5 ) nm(Sn) +n m - I ( ~ n - 1 )
nm-i(V) +.*.
*.
This exact sequence is usually used to deduce properties of the homotopy groups of V . Here, on the contrary, it will be used to study the groups nm(Sn). Let um denote the generator of n m ( S m ) represented by the identity map. Consider the following part of the sequence: * *
.+.nnfV)-+
d
nn(Sn) L+nn-l(Sn-l)
>-+ n?&-l(v) +. 0 .
By Hurewicz's theorem, Z % - ~ ( Vfi:) 2,. Since z, is an epimorphism, the exactness of the sequence implies that the image of d , is a subgroup of nn-l(Sn-l) of index 2 . Hence we deduce that d*(un) = f 2un-1. I n fact, it is , we shall not need this refinement. known that d,(un) = 2 2 ~ , , - ~but The structure of the homomorphism d , with m > n is described by the following Lemma 11.1. If X is a fiber space over Sn with a pathwise connected fiber F , then the homomorphism d , i n the exact hmotopy sequence *
*+.nm(X)+ X m ( S n )
d
A+ nm-l(F) - + n m - l ( X ) +
*
sends the suspension X(a) of any element a ~ n m - l ( S n - l )into the composition d*(un) 0 a.
If a map k : 9
K, :nm-l(Sn-l)
+. F represents d*(un), then k induces a homomorphism nm-,(F) which gives k,(a) = d,(un) 0a for each a in
- 1 +.
XI. HOMOTOPY G R O U P S O F S P H E R E S
324
nmPl( 9 - 1 ) . Hence the relation stated in the lemma means that the following triangle is commutative : nm-i(Sn-l) A %(Sn)
nm-1 ( F ) Proof. The projection w : X + Sn induces an isomorphism w, : n m ( X ,F ) M nm(Sn). By definition, d , is wsl followed by the boundary homomorphism : n m ( X ,F ) -tnm-l(F). A representative map k : Sn-l-+ F of d,(u,) can be constructed as follows. Let h : (En,9 - l ) + (Sn, so) be a map which represents un. It follows from the covering homotopy theorem that there exists a map H : (En, 9 - 1 ) -f ( X ,F ) such that wH = h. Then the restriction k = H 1Sn-l represents d , @n) * Let a ~ n ~ - ~ ( S nIt - l remains ). to prove &(a) = d,X (a). Let : Sm-l-+ Sn-l represent a ; then k,(cr) is represented by A+. Extend to a map y : (Em, 9 - 1 ) (En,9 - l ) . By 3 2, hy represents X ( a ) . Since wHy = hy, H y represents X (a) and hence k+ = H y I Sm-l represents d,X (a).I Next, let us study the homotopy groups of the Stiefel manifold V = Vn+,,z. Using the generalized Hurewicz theorem, we can deduce that: (1) n m ( V ) is finitely generated. (2) nm(V) = 0 if m < n - 1. (3) nn-dV) 2,. (4) nm(V)is a finite 2-primary group whenever n - 1 < m < 2%- 1. (5)n2n-1(V)is isomorphic to the direct sum of 2 and a finite 2-primary group.
a
+
--f
-
Lemma 11.2. If %? denotes the class of all finite 2-primary groups, theiz there exists a map q : S2n-l + V such that the induced homomorphism
:nm(S2n-l) -+nm(V)
i s a %?-isomorphism for every m. Proof. By (5), there is a map q : S2n-l -+V which represents a free element a of nsn-l(V)such that the free cyclic subgroup generated by a is of index some power of 2. Since the natural homomorphism of Z ~ ~ - ~into ( VHzn-l(V) ) is a %-isomorphism, it follows that the induced homomorphism
4% : Hm(S2n-l) -+ Hm(V)
is a %'-isomorphism for m = 2n - 1 and hence for every m.An application of Whitehead's theorem (X; 10.1) proves the lemma. I We list the following additional results on the homotopy groups of V ; these are immediate consequences of (10.2). (6) nm(V)is finite if m > 2%- 1.
13.
THE P-PRIMARY C O M P O N E N T S O F HOMOTOPY G R O U P S
325
(7) If p is an odd prime number, then the +-primary component of h ( V )is isomorphic to that of nm(S2n-l).
12. Finiteness of higher homotopy groups of even-dimensional spheres Theorem 12.1. If S n i s a n even-dimensional sphere and m i s a n integer such that m > n and m # 2 n - 1, then n,(Sn) i s finite and nz,-,(Sn)i s isomorphic to the direct sum of Z and a finite group. Proof. Since nm(S2)mnm(S3) for every m > 2 , the theorem is true for n = 2. Hence we may assume n 2 4 and apply the results of 3 11. If m > n and m # 2 n - 1, then both n,(V) and nm-l(Sn-l) are finite. Therefore, the exactness of the sequence n,(V) -+n,(Sn) +nm-l(Sn-l) implies that n,(Sn) is finite. To study the critical case m = 2 n - 1, let V denote the class of all finite abelian groups. It follows from the exact sequence that
: n z n - i ( V +zdan-I(S") is a %-isomorphism. This implies that n,n-l(Sn) is isomorphic to the direct sum of Z and a finite group. I
13. The p-primary components of homotopy groups of even-dimensional spheres In the present section, we are concerned Cith the p-primary components of the homotopy groups n,(Sn) of an even-dimensional sphere Sn. Since nm(S2)m nm(S3) for every m > 3, we may restrict ourselves to the case n > 4 and apply the results of 8 11. Consider the following part of the exact sequence appearing in 3 11 :
n,+l(Sn)
d
d n,(Sn-1) A+ n,(V) 5-tn,(Sn) >-+ n,-l(Sn-l).
--I-+
Using the homomorphism o* and the suspension X, we define a homomorphism :n m ( ~ + ) z m - l ( ~ n - l )+ n , ( ~ n )
r
by setting r ( a ,p)
= o,(ct)
+ Z(p) for each ct E n,V) and p ~ n , - , ( S ~ - l ) .
Lemma 13.1. If V denotes the class of all finite 2-primary groups, then r i s a
%?-isomorphism. Proof. According to ( 1 1 . l ) , the suspension Z followed by d , is the induced endomorphism k , onnm-I(Sn-l) of a map k : S n - l + Sn-' of degree d = & 2. By Ex. A6 at the end of the chapter, k , is a V-automorphism. Hence the lemma follows as a consequence of ( X ; Ex. E). I Theorem 13.2. If V denotes the class of all finite 2-primary groups, thex the homotopy group n,(Sn) of a n even-dimensional sphere Sn i s %?-isomorphic to the direct s u m of nn(S2n-l) and nrn-,(Sn-').
X I . HOMOTOPY G R O U P S O F S P H E R E S
326
Proof. Since nm(S2) w %(Ss) for every m > 3, the theorem holds for n = 2. If n > 4, then ( 13.2) is a direct consequence of (13.1) and ( 1 1.2). I The importance of (12.1 ) and (13.2)is that the calculation of the homotopy groups of an even-dimensional sphere, except for their 2-primary components, reduces to that of the homotopy groups of odd-dimensional spheres. Precisely, we have the following Corollary 13.3. If n is even and p an odd prime, then the #-primary component of nm(Sn) is isomorphic to the direct sum of those of nm(S2n-l) and n, -1 (9-1). 14. The Hopf invariant
In order to strengthen f13.2), we propose to present Serre's version of the notion of Hopf invariant. Consider the n-sphere Sn and a given point so E Sn. Let Qn = A(Sn)denote the space of loops in Sn with so as basic point. Consider the natural isomorphism j :nsn-s(S2n) w and the natural homomorphism h :nsn-&2") -+ H,-,(@)
w 2.
Let f :9 n - l +Sn be a given map representing an element [f] En,n-l(Sn). The Hop/ invariant of f is the integer H ( f )uniquely determined by where ug denotes the generator ~ * ~ ( ofl ) Hzn-a(@) determined by the homomorphisms p* in (IX; 13.1). For other definitions of Hopf invariant, see Ex. C at the end of the chapter. When n is odd, H ( f ) is always zero; when n is even, there exists a map f with H ( f ) = 2 ; if n = 2, 4 or 8, then there exists a map f with H ( f ) = 1, namely the Hopf maps of (111; Q 5). See [Hopf 21 and [S; p. 1131. Theorem 14.1. Let n be an even integer and f : 9 n - l +Sn a map with HOP/ invariant H ( f) = k # 0. Let V denote the class of all filzite abelian groups of order dividing some power of k. If
Xf :nm-l(Sn-1)
+ nm(S2n-l) +nm(Sn)
is the homomorphism defined by
B) = V a ) + f&),
xf(a,
a Enrn-i(S"-'),
then Xf is a V-isomorphism for every m
B E~c,(S"-'),
> 1.
Proof. The theorem is obvious if n = 2. Hence we assume n > 4. The map f defines a map g : Qzn-1 +S2n in the obvious way and g induces a homomorphism g* :H*(SZn) + H*(BZn-lj
14. T H E
327
HOPF INVARIANT
of the cohomology algebras with integral coefficients. According to (IX;
Ex.H), H*(@) admits a homogeneous basis { ai } with dim (ai) = i(n- 1) for each i = 0, 1, * such that a0 =
1, ( a J z = 0, (as)'
=
(p!)aap,alaap
=
aspal
= aap+1.
Similarly, H*(Bafi-l) admits a homogeneous basis { bi } with dim (bi) = i ( 2 n - 2 ) for each i = 0, 1, * * * such that
bo
=
1, (b1)P = (P!)bp.
Since H ( f ) = k , it follows that g*(a,)
(P!)g*(a,)
= g*(a,p)
=
=
kb,. Then we get
kp(b1)P = k p ( p ! ) b p ;
and therefore g*(am) = Mbp. Since obviously g*(aZp+J= 0, it follows that g* is completely determined. Now let i : Sn-l +Qn denote the natural imbedding in 4 2 and consider the induced homomorphism
i* : H*(@)
+ H*(Sn-1).
By (2.2), e = i*(a,)is a generator of Hn-1(Sn-1). By means of the multiplication in Qn, we may define a map : Sn-l x Sd2n-l+ Bn by setting +(x, w ) = i ( x ).g(w) for each x E Sn-l and w ~ S 2 ~ n - l . By (IX; Ex. H), H*(Sn-l x J2w-l) is naturally isomorphic to the tensor product H*(Sn-l) @ H*(B2n-l). This enables us to determine the induced homomorphism +# : H*(Qn) --f H*(Sn-1 x P - 1 ) as follows: +#(asp) = k p *1 @ b p ,
+
+#(asp+J = +#(a1)+#(aap)
=
@ bp.
Hence, +# : Hm(On) + Hm(Sn-l x L P - l ) is a monomorphism and its cokernel is finite of order equal to a power of k for every dimension m. By duality, this implies that +# : Hm(Sn-l x Qzn-l) + H m ( P ) is a W-isomorphism for every m. An application of the Whitehead theorem shows that the induced homomorphism
+* :nm(Sn-l
x
Qzn-l) +nm(.Rn)
is a V-isomorphism for every m. The group nm(Sn-l x Q2n-l) is isomorphic to the direct sum 7cm(Sm-l) Z , ( Q ~ ~ -;~on ) the other hand, n m ( 8 n ) M nm+l(Sn).Then, the construction of shows that +* reduces to the homomorphism Xf. I
+
+
Corollary 14.2. If H ( f ) = f 1, then X, is an isomorphism and hence the suspension Z :nm-l(Sn-l) +nm(Sn) is a monomorphism for every m > 1.
Because of the existence of a map f with H ( f ) = 2, (14.1) implies (13.2).
328
XI. H O M O T O P Y G R O U P S O F S P H E R E S
15. The groups mn+,(Sn) and J Z , , + ~ ( S ~ ) Since nm(Sl) = 0 for each m we may assume n 2 3.
> 1 and nm(S2)M nm(S3)for every m > 2,
> 3.
Theorem 15.1. i ~ ~ + ~ (i S s cyclic n ) of order 2 for every n
Proof. Let
we have
X denote a 3-connective fiber space over S3. Then, by (gal), n4(S3) w n4(X) w &(X) w 2,.
By the suspension theorem (2.1), we deduce n4(S3)M n5(S4)M M nn+l(SN)M
---
*
-
*.
Hence nn+l(Sn) w 2, for every n > 3. I One constructs the generator of Z ~ + ~ ( as S ~follows. ) Let us consider the Hopf map p : S3 +S2 defined in (111; 3 5 ) . According to (V; 3 6), p represents the generator of n3(S2).Since the suspension X : n3(S2)-+n4(S3) is an epimorphism, the suspended map Zp :S4+ S3 represents the generator of n4(S3). Then the generator of nn+l(Sn) is represented by the (n- 2)-times iterated suspension Zn-2p of the Hopf map p. Theorem 15.2. nn+,(S@)i s cyclic of order 2 for every n
Proof. Applying (10.1) with n = 4, h = 2, m
sequence
=
> 3.
5 , we obtain an exact
0 + 2, 8 2, +n5(Pi)-+ Tor(Z,Z,) + 0.
Since 2,g 2, M 2, and Tor(2,2,) = 0, this implies n5(Pz)M 2,. Then, by (10.2), the 2-primary component of n5(S3)is isomorphic to Z,.Since the $-primary component of x5(S3)is 0 if p > 2 by (9.2), it follows that n5(S3)M 2,. Since the Hopf map S7 + S4 is of Hopf invariant 1, we may apply (14.2) with n = 4 and m = 6. Hence, (2.1) and (14.2) imply that X :n5(S3)M 7z6(S4).Thus, n,(S4) M 2,. Finally, by (2.1). we deduce
n,(S4)
M
n,(S?
M
-
* *
M
nn+,(Sn) M
* * *.
Hence nn+,(Sn) w 2, for every n > 3. I To obtain the generator for nn+a(Sn),let i :S4 + Pz denote the imbedding given by the definition of Pz. Then, by the proof of (lO.l), the generator of n5(Pi)is represented by the composition of i and X2p :Ss+ S4. Composing with the map X : Pi + S 3 in 4 10, we obtain a representative map Xi C2p for the generator of n5(S3).This implies that Xi represents the generator of n4(S3)and hence is homotopic to Zp : S4+S3. Therefore, the generator of n 5 ( S 3 )is represented by = ~p 0~ 2 :p ss + ~ 3 , where p : S3 -+ S2 denotes the Hopf map. Then it follows that the generator of nn+,(Sn) is represented by the ( n - 3)-times iterated suspension Cn-39 of q for every n Z 3.
16. T H E
G R O U P S ntn+3(S")
329
Corollary 15.3. n3(S2)M 2,n,(S2)M Z,, n,(S2) M 2,.
By (V; tj 6 ) , the generators of these cyclic groups are represented by respectively the maps
p
: s3 +s2, p
0cp : s4 +s,, p 0cp 0c2p :ss +SZ. 16. The groups n,,+3(Sn)
Theorem 16.1. n,(S3) M Z,,. Proof. Applying (10.1) with n = 4, h
sequence
=
2, m
= 6,
we obtain an exact
0 + 2, 8 2, -+n,(P5,)-+ Tor(Z,, 2,) -+ 0.
Since 2, 8 2, M 2, and Tor(Z,,2,)M Z,, n,(P;) is isomorphic to an extension of 2, by 2, and hence has 4 elements. Hence, by (10.2), the 2-primary component of n6(S3)has 4 elements. By (9.2), the 3-primary component of n,(S3) is isomorphic to 2, and the $-primary component of n,(S3) is 0 for every prime p > 3. I t follows that n6(S3) has 12 elements and hence is isomorphic to either Z,, or 2, + 2,. Suppose that n,(S3) w 2, 2,. Let X denote a 5-connective fiber space
+
and it follows from the universal coefficient theorem [E-S; p. 1611 that
W ( X ;2,) M Hom ( H , ( X ) ;2,) M 2, + 2,. This contradicts to ( I X ; Ex. I ) ; hence, we conclude that n,(S3) M Zlz. I Examination of the first paragraph of the proof reveals that the composition of the maps S6 -ZSfi + Sb P P S4 s3 ----f
=---f
represents an element of n,(S3) of order 2. A generator of n,(S3) is represented by the characteristic map 5 : S6 + S3 of the fiber bundle Sp(2) over S6 with Sp( 1) as fiber, [Borel and Serre 1 ; p. 4421. For the definition of the characteristic map, see [S; p. 971. Corollary 16.2. n,(S2)w Z,,. A generator of n,(.S2) is represented by the composed map Theorem 16.3. n7(S4)w 2
pt : S6+ S2.
+ Z,,.
Proof. Let us denote by q : S 7 + S 4 the Hopf map in (111; 5 5). Since H ( q ) = 1, we may apply (14.2) with n = 4, m = 7, and f = q. Thus, we obtain an isomorphism
& :n,(S3)+ n7(S7)
n7(S4).
Since n7(S7) M 2 and n,(S3) Z,,, this proves the theorem. I From the preceding proof, it follows that q represents the generator of
330
XI. HOMOTOPY G R O U P S O F S P H E R E S
the free component Z of n7(S4) and the suspended map Xt : S7 + S4 represents an element of order 12 which generates the torsion component Z,, of n7(S4). Theorem 16.4. nfl+3(Sfl) F=Z2, if n 2 5.
Proof. By the suspension theorem (2.1), Z maps n7(S4)onto n,(Sb). According to Ex. D at the end of the chapter, the kernel of X is the free cyclic subgroup of n7(S4)generated by the Whitehead product [e, e l , where e denotes the generator of n4(S4) represented by the identity map on S4. On the other hand, it follows from a theorem on characteristic maps, [S; p. 1211, that
where E = f 1 depends on the conventions of orientation. Hence, in n8(S6),we have P [t]= E2 E [q]. This implies that n8(S6)is isomorphic to 2, with X [q] as a generator. Finally, by the suspension theorem (2.1), we deduce n,(S6)
M
n p )M
* * *
M
nfl+3(sfl) m
- -. *
Hence ~ ~ + ~ ( w S Z,, f l ) for every IZ 2 5. I Obviously, a generator of Z ~ + ~ ( Sn~2) ,5, is represented by the (n-4)times iterated suspension D - 4 q : Sn+3 -+Sn of the Hopf map q : S7 + S4.
17. The groups a,,+&”) Theorem 17.1. n7(S3) w 2,.
Proof. By (9.2), the p-primary component of n7(S3)is 0 for every prime
fi > 3. By (10.4), the 3-primary component of n,(SS)is also 0. Hence n7(S3)
is a 2-primary group. Next, consider a 6-connective fiber space X over S3. Then n7(S3)M n7(X)rn H 7 ( X )and hence we have
Horn (n7(S3), 2,)w H , ( X ; 2,). By ( I X ; Ex. I4), H 7 ( X ;2,) M 2,. This implies that n7(S3)is isomorphic to a cyclic group 2, with q = 2h, h > 1. If h > I , every homomorphism of n7(S3)into 2, can be factored into
n,(S*)+ 2, + 2,. Then it follows from the exact sequence in (VIII; Ex. 17) that Sq’a for every element a E H 7 ( X ;2.J.This contradicts ( I X ; Ex. 14). Hence h andn7(S3)-2,. I According to [Hilton 2; p. 5491, the two maps
6 0c4p : S’
-9,
z p 0q : s7 +s3
=0 =
1
17.THE
GROUPS ~ c ~ + ~ ( s f l )
331
are both essential. Hence they are homotopic and represent the non-zero element of n7(S3). Corollary 17.2. n7(S2)w 2,.
The non-zero element of n7(S2) is represented by the homotopic maps : S7+S2, f~0X P
P 05 0 Theorem 17.3. n,(S4)w 2,
+ 2,.
0q : S7 +S2.
Proof. As in the proof of (16.3),we obtain an isomorphism
%* :n7(S3)
+
ns(S7) W
n,(S4).
Since n7(S3)w 2, and n8(S7) m Z,, the theorem is proved. I The group n,(S4)is generated by two elements a and @ of order 2. a is represented by the homotopic maps
I:
(loI:”) :
S8
I:(I: 0q) : S8 +S4,
+S4,
and @ is represented by q 0ESP :S8 -+ S4. Theorem 17.4. ng(S5)w 2,.
Proof. Consider the following part of the suspension sequence in
n,(T )
*
-+
+
X
X
n,(S4) -+ n,(S5) JLn,(T ) -+ n7(S4)
--+
5 6:
n,(SS)
+ 0.
As mentioned in the proof of (16.4), the kernel of X :n7(S4) +n6(S5) is a free cyclic group. By (6.3),n7(T ) w 3 4 . 9 ) w 2. I t follows from the exactness of the sequence that :n,(T) +n7(S4)is a monomorphism and hence I: :n,(S4)+n,(Ss) is an epimorphism. Since n,(T) M n,(S7) w Z,, the kernel of I: :n6(S4) -+ n,(S6) contains at most two elements. On the other hand, consider the element a of n6(S4) represented by X (6 0X4$) : S8+S4. Inn,(Sy, we have
+
I:b) =
0W P 1 = (&2I::[qI)0 (CYPI) = (&I:hI)0 (21:6[Pl)= 0.
Hence the kernel of Z :n8(S4)-+ n,(Sb) consists of exactly two elements, namely, 0 and a. This implies that n,(SS) w 2,. I The non-zero element of n,(S6)is I:(,!?) represented by the map I:(qOI:Sfi): Sg +Ss. On the other hand, the Whitehead product [e, e] of the generator e of n,(S5) is also non-zero and hence [e, el = X(@), [Serre 3; p. 2301. Theorem 17.5. ~ ~ + ~ ( = S 0f ifl )12 2 6. Proof. By the suspension theorem (2.1), X maps n,(Ss) onto n,,(S6). According to the delicate suspension theorem in Ex. D at the end of the chapter, X [e, el = 0. Hence we obtain n,,(Ss) = 0. Finally, by (2.1), we deduce nl0(S6) n,,(S7)w * * M n,+,(Sn) M * * .
-
Hence ~ ~ + ~ ( = S 0f lfor ) every n Z 6. I
X I . HOMOTOPY GROUPS O F SPHERES
332
18. The groups nn+r(Sn), 5 Q r
< 15
In this final section of the book, we will list the groupsn,,+r(Sn)for the cases r = 5, 6, 7, 8.For more detailed information, see [Serre 5, 61. H. Toda has computed the groups nn+r(S") for 9 < I Q 15. We will not list his results here; the interested reader should refer to poda 1,2] with recent corrections given in poda 31. I = 5.
r = 6.
I=
Y
7.
= 8.
EXERCISES
333
EXERCISES A. The distributive laws
Let u €nm(Sn,so) and /?E ~ , ( X xo). , If u and p are represented by the maps
f : (Ern,sm-1)
--f
(Sn, so), g : (9, so) -+ ( X ,xo)
respectively, that the composed map gf represents an element y of n,(X,xo). Prove: 1. The element y depends only on the elements u,p and will be called the composition /I0u of u and /?. 2 . The right distributive law. For a given Enn(X, xo), the assignment u -+/? 0u defines a homomorphism. In fact, this is the induced homomorphism g,. 3. The left distributive law. If X is an H-space with xo as homotopy unit or if u is the suspension X(6) of some element 6 ~ n ~ - ~ ( Sso), n -then ~ , the assignment B -+/I 0u defines a homomorphism. [S; p. 1221. I n particular, let ( X , xo) = (9, so). Consider a map g : (Sn, so) + (Sn, so) of degree d and study its induced homomorphism
g, :nm(Sn, so) 'nm(Sn, so). Prove : 4. g,(u) = du for every u E n m ( S n , so) if n = 1,3, 7 or if m < 2n - 1. 5 . If m = 3 and n = 2 , then g,(u) = d2ufor every u €n,(S2,so). [Hopf 1 1 . 6 . If V denotes the class of all finite abelian groups of order dividing a power of d, then g, is a %?-automorphism for every m > 0 and n > 3. 6. On relative (n
+ 1)-cells +
Let (X, A ) be a relative (rt 1)-cell obtained by adjoining En+l to A by means of a map g : Sn + A . Then g has an extension f : (@+I, Sn) -+ ( X ,A ) defined by f ( x ) = x for every x E En+l\ Sn = X \ A . This map f is called the characteristic map of ( X ,A ) . If we identify A to a single point, we obtain an ( n + 1)-sphere Sn+l as quotient space with projection h : X -+Sn+l. Choose so E Sn and let xo = f (so) E A . Use so and xo as basic points of the homotopy groups. Verify that the rectangle
nm(En+l,Sn)+ '
+
.
T C-I( ~ Sn)
z
-+
n m ( X ,A )
nm(Sn+')
is commutative, i.e. h,f, = Ed. Prove 1. If 2 is a monomorphism, so is 1,. If 2 is an epimorphism, so is h,.
334
X I . HOMOTOPY G R O U P S O F S P H E R E S
2 . If m < 2n, then f , is a monomorphism, h, is an epimorphism, and
n,(X, A ) decomposes into the direct sum of I m ( f ) *and Ker(h,).
3. If A is r-connected for some Y < n, then f , is an epimorphism whenever < n + Y . See [J. H. C. Whitehead 6 ; p. 141 and [Hilton 1 ; p. 4641. As an application of these results, consider the relative ( p q)-cell ( X ,A ) with X = SP x SQ and A = SP SQ. By 2 and (V; 3.1), we obtain natural isomorphisms
m
+
nm(SP x SQ,SP Z,-~(SP v SQ)M Z,-~(SP)
for every m < 2 p min ( p , q) -2 .
+
SQ)M nm+,(Sp+Q-l)Ker(h,),
+ nm-l(SQ) + nm-l(Sp+Q-l)+ Ker(h,)
+ 2q -2 . Finally, by 3, Ker(h,) = 0 if
m
C. Other definitions of the Hopf invariant
In
9 14, we gave Sene's definition of the Hopf invariant H ( f ) of
map
f : S2n-1 + S n ,
a given
n 2 2.
There are a few different but equivalent definitions given by various authors listed below. Each of these definitions has a natural generalization but the various generalized Hopf invariants are not necessarily equivalent. 1. HoPf's definition. By using a homotopy if necessary, we may assume that f is a simplicial map relative to some triangulations. Choose a pair of distinct interior points, u, v oi n-simplexes of Sn. Then /-1(u),/-1(v) are disjoint ( n - 1)-manifolds in S2n-1; and H ( f ) is the linking number of the (n - 1)-cyclesf-'(u) and f-'(v). See [Hopf 21 and [S; p. 1131. This definition can be generalized to manifolds. 2. Steenrod's definition. Let M denote the mapping cylinder of f with S n c M and S2n-l c M . The integral cohomology algebra H*(M,S2n-l) has a homogeneous basis { a, b } with dim ( a ) = n and dim (b) = 2n. Then, H ( f ) is the integer defined by a2 = H ( f ) b . See [Steenrod 3; p. 9831 and [Serre 2 ;p. 2861. This definition can be generalized to the maps f : Sn+t-l + S n as follows. In this case, H * ( M ,Sn+f-l; 2,) has a homogeneous basis { a, b } with dim (a) = n and dim (b) = n + i . Then, H ( f ) E Z is~defined by Sq'a = H(f)b. 3. Whitehead's definition. Identifying the equator S n - l of Sn to a single point, we obtain a quotient space Sn S n with projection g : S n + S n Sn. The composed map gf represents an element [gf] of nan-l(Sn Sn). Then H ( f )is obtained by projecting [gf] into the direct summand n,n-1(S2n-l) M Z of npn-1 (Sn v 9 ) . Since n,(Sn
Sn)
for every m <4n-4,
M
nm(Sn)
+ nm(Sn) + n,(S2n-l) + Ker(h,)
this definition can be generalized to the maps
335
EXERCISES
f : Sm + Sn for m Q 4rt -4 in an obvious way. See [G. W. Whitehead 21 and [Hilton 11. Thus, for each m < 4n -4, we obtain a Hopf homomorphism
H :fCm(Sn) - + n m ( P - l ) . Among the properties of this generalization of Hopf invariants, verify the following few: (i)If a €nm(Sn),m Q 3n - 3, and B1, B2 E ~ , ( X )then , (Bl
+ 0a = B2)
B1
0 a + Be 0 a +
[Bll
Be1 0 H(a).
(ii)If a ~ n ~ - ~ ( S nwith - ~ )m Q 4n -4, then H X(a) = 0. (iii)If n is odd and m < 4n - 4, then 2H(a) = 0 for every a ~ n ~ ( S n ) . (iv) If a E n m ( S n ) with m < 4 n - 4 and X(a) = 0, then H ( a ) = 0 if n is odd H(a)E 2n,(S2n-l) if n is even. 4. Hilton's definition. If we identify the subset Sn v Sn of the product space Sn x Sn to a single point so, we obtain a 2n-sphere S2n as quotient space with projection h : ( 9 x Sn, S n v Sn) -+ ( P , so). For each m
> 0, define a homomorphism H* :nm(S%)-+nm+l(s2fi)
by taking H* to be composition of the sequence h
n,(S%) &-+ 7c~ ( s ~ V S L " )+ n m + l ( ~ n X S ~ , S ~ V S - L~+ )z ~ + I ( s ~ ~ ) , where g, h, are induced homomorphisms and p denotes the projection of Prove the nm(Sn Sn) onto its direct summand nm+l(S"x Sn, Sn 3). following relations : (i)H* = XH whenever m <4n-4. [Hilton 1; p. 4731 and [Hilton 3 ; p. 1661. (ii)If a Enn(Sq), @ Enm(Sn), then (iii)If n is odd and a ~ h ( S n )then , 2H*(a) = 0. (iv) If n is even, a ~ n ~ ( Sisnof) odd order, and H*(a) = 0, then a for some @ ~ n ~ - ~ ( S n - ~ ) .
=
X(p)
D. The delicate suspension theorem
The suspension theorem (2.1) is the easy part of Freudenthal's result and is usually called the crude suspension theorem. The delicate part of Freudenthal's result has been slightly strengthened to the following form, [G. W. Whitehead 21 : 1. The image of X :nen(Sn) -+Iten+l(Sn+') is the subgroup of nzn+l(Sn+l) consisting of the elements of Hopf invariant zero.
336
XI. H O M O T O P Y G R O U P S O F S P H E R E S
2. The kernel of C :n2n-l(Sn)-+n2n(Sn+1) is the cyclic subgroup of
generated by the Whitehead product [e, el, where e denotes the generator of nn(Sn)represented by the identity map. If n is even, [e, el has Hopf invariant 2 and therefore has infinite order. If n is odd, 2 [ e , e ] = 0, and [e, el = 0 iff there is an element of nan+l(Sn+l) with Hopf invariant 1.
B I B L I O G R A P HY B 0 0 K S and Mimeographed Notes ALEXANDROFF, P. und HOPP, H. : Tapologie I, Springer, Berlin. 1935. BOURBAKI, N. : elements de mathbmatiques. Hermann, Paris, 1939-1948. S. : Homological Algebra. Princeton Univ. Press, CARTAN, H., and EILENBERG, 1956. CHEVALLEY, C.: Theory of Lie Groups I. Princeton Univ. Press, 1946. S., and STEENROD, N. E. : Foundations of Algebraic Topology. EILENBERG, Princeton Univ. Press, 1952. HILTON,P. J. : An Introduction t o Homotopy Theory. Cambridge Univ. Press, 1953. Hu, S. T. : Homotopy Theory I. Dittoed Technical Reports, Tulane University, 1950. HUREWICZ, W., and WALLMAN, H . : Dimension Theory. Princeton Univ. Press, 1941. I. : Infinite Abelian Groups. Univ. of Michigan Press, Ann Arbor, KAPLANSKY, 1954. KELLEY,J. L., General Topology. D. van Nostrand Co. Inc., New York, 1955. S.: Topology. (Amer. Math. SOC.Coll. Publ., Vol. 12). 1930. LEFSCHETZ, S.: Algebraic Topology. (Amer. Math. SOC.Coll. Publ., Vol. 27). LEFSCHETZ, 1942. S.: Topics in Topology. (Annals of Math. Studies, No. lo), 1942. LEFSCHETZ, MORSE,M.: The Calculus of Variations in the Large. (Amer. Math. SOC.Coll. Publ., Vol. 18), 1934. MORSE,M.: Introduction to Analysis in the Large, 2nd ed. Mimeographed, Institute for Advanced Study, Princeton, 195 1. SEIFERT,H., und THRELFALL, W. : Lehrbuch der Topologie. Teubner, Leipzig, 1934. STEENROD, N. E. : The Topology of Fibre Bundles. Princeton Univ. Press, 195I. VEBLEN,0.: Analysis Situs. (Amer. Math. SOC.Coll. Publ., Vol. 5, Part 2). 2nd ed., 1931. PAPERS ARENS,R. 1. Topologies for homeomorphism groups. Amer. J. Math., 68 (1946), 593-610. 2. A topology for spaces of transformations. Ann. of Math., 47 (1946). 480-495. BLAKERS, A. L. and MASSEY,W. S. 1. The homotopy groups ofatriad, I, 11,111. Ann. of Math., 53 (1951), 161-205, 55 (1952). 192-201, 58 (1953), 409-417. 2. Products i n homotopy theory. Ann. of Math., 58 (1953), 295-324. BOREL,A., and SERRE,J.-P. 1. Groupes de Lie et puissances rdduites de Steenrod. Amer. J. Math., 75 (1953). 409-448.
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Index A Absolute neighborhood retract, 26, 29-32, 59, 184, 198 Absolute retract, 26, 29 Admissible transformation, 135-136
B Basic point, 125, 137 Betti number, 28, 277 Bigraded exact couple, 234-236 Bigraded group, 231 Borsuk extension theorem, 29 fibering theorem, 103 homotopy extension theorem, 3 1 Boundary operator, 112, 159 Bridge theorems, 59, 195 Brouwer fixed-point theorem, 4 Bundle property of a map, 65, 99 Bundle space, 65
C %‘-notions, 298 %‘-acyclic,303 %‘+spherical, 306 %‘-epimorphism, 299 %‘-equal,309 %‘-equivalence, 299 %‘-monomorphism,299 %‘-null,299 Characteristic class, 88 Characteristic element of a complex, 193 of a map, 189 of a pair of maps, 186 Classes of abelian groups, 297-310 complete, 300. 308 perfect, 300, 308 strongly complete, 300 weakly complete, 300 Classification problem, 13, 16, 187, 198 Classification theorem of covering spaces, 96 Hopf, 53, 59 primary, 191
343
Closed surface, 28 fundamental group of, 58 Coboundary operator, 208.2 14 Cohomology algebra of a space, 294 of a space of loops, 295 n), 295 of (2, Cohomology group cubical singular, 261-262 of (n,n ) 199 Cohomotopy group, 205-228 Cohomotopy sequence, 2 14-2 16 Cohomotopy set, 205, 213 Complex contractible, 200 CW-, 193, 254 Eilenberg-MacLane, 203 8-, 246 6-,246 semi-simplicial, 140-142 simplicial. 140 singular, 195 Cone over a space, 20, 28 partial, 22 Contraction, 12 Covering homotopy, 62 Covering homotopy extension property, 62 absolute, 62 62, 63 -polyhedral, Covering homotopy property, 24, 37, 62, 63, 70, 99 absolute, 62, 98 polyhedral, 62, 63, 99 Covering homotopy theorem, 62, 66, 314 Covering map property, 43 Covering path property, 36 Covering space, 89-97 generalized, 104 of a torus, 105 regular, 92 universal, 91, 96 Covering theorem, 91 Covering transformation, 93 Cross-section, 70, 73, 99, 313 Curve, 103
INDEX
344 D Deformation, 12 cochain, 179 homeomorph, 103 obstruction theory of, 197 problem, 22 retract, 16, 33 Degree base, 266 complementary, 235 fiber, 266 of a map, 12. 38, 52, 57, 60 of homogeneity, 231 primary, 235 total, 235 Derived triplet. 1 1 1 Difference cochain, 179 Differential group, 229-231 derived group of, 230 filtered. 239, 244 filtered graded, 245-248 Differential operator, 229, 232 Dimension, 266 Direct sum theorems, 150-152
E Eilenberg extension theorem, 180 subcomplex, 45, 174 Eilenberg-MacLane complex, 203 Equivalence theorem for covering spaces, 91 for homotopy systems, 135 Euler characteristic, 277 Evaluation, 74, 76 Exact couple, 232 associated, 263-266 bigraded, 234-236 cohomology, 251 cohomotopy. 254 derived couple of, 232, 26G269 homology, 234, 251 homotopy, 251, 252 of a bundle space, 251 regular, 236-238 Exact sequence, 115 Gysin’s, 280-281 natural, 299 truncated, 284 Wang’s, 282-284 Exactness property, 115 Exactness theorem, 136 Excision theorem, 208
Extension, 1 Extension index, 176 Extension problem, 1, 2, 15, 20, 71, 198 Extension property, 28-29 Extension theorem Borsuk, 29 Eilenberg, 180 Hopf. 53, 59 primary, 190 F
Fiber, 62, 262 Fiber map, 71, Fiber space, 62-106 homotopy groups of, 152-154 n-connective, 156, 304, 320 sliced, 97 spectral sequence of, 259-296, 300-304 Fibering, 62 induced, 72 n-connective, 155 of spheres, 66, 100 Fibering property, I18 Fibering theorem for homotopy groups, 136 for mapping spaces, 83-84 Filtered d-group, 244-245 associated graded group of, 244 exact couple associated with, 245 spectral sequence of, 249 Filtered graded d-group, 245-248 Filtraction. 244-246, 262 Five lemma, 309 Freudenthal’s suspension, 162, 227, 31 1 Fundamental exact sequence, 240-242 Fundamental group, 39-47, 193 P S a group of operators, 130 influence on homology and cohomology, 201-202, 288-290 Fundamental homotopy lemma, 186 C
Gamma functor, 255-256 Graded group, 231. 237, 238-239 associated, 244 Grassmann manifold, 101 Group bigraded, 231 Bruschlinsky, 47-52 differential, 299-231 fundamental, 39-47, 57, 130, 193, 201, 288
INDEX
graded, 231 homology, 44, 261 homotopy, 107-164 7c-, 201 $-primary. 298 torsion, 297, 310 Gysin's exact sequence, 280-281 H Homology group, 44, 261 cubical singular, 261 of (72. n ) , 199 of a group, 199 Homotopy, 1 1 addition theorem, 164-166 class, 13. 16 connecting maps f and g, 11, 15 index, 183 invariant, 18 partial, 13 relative, 15 unit, 81 Homotopy equivalence, 1 17 Homotopy extension property, 13-1 5, 30-34 absolute, 13, 31 covering, 62 neighborhood, 30 Homotopy extension theorem, 30-31 Homotopy group, 107-164 absolute, 107-1 10 abstract, 128, 137 of adjunction spaces, 168 of covering spaces, 154 of fiber spaces, 152-154 of H-spaces, 139 of spheres, 3 1 1-336 relative, 1 10-- 1 12 relations with cohomotopy groups, 224-226 triad, 160, 316 Homotopy property, 1 17 Homotopy sequence of a fibering, 152 of a triad, 160-161 of a triple, 159 of a triplet, 115, 120, 136 Homotopy system, 119 axioms of, 120 equivalence theorem for, 135 equivalent, 12 1 group structures for, 123 inductive construction of, 135
345
properties of, 136 uniqueness theorem for, 121 Homotopy theorem Hopf, 53 primary. 191 Homotopy type, 17. 198 Hopf classification theorem, 53. 59 extension theorem, 53, 59 fiberings of spheres, 66 homotopy theorem, 53 invariant, 326-327, 334-336 map, 66 theorems, 52-56, 59, 255 H-space, 81 Hurewicz theorem, 57, 148, 166, 253 generalized 305 relative, 306 I Induced transformation of a'rnap on cohomotopy groups, 206, 2 I3 on fundamental groups, 42 on homotopy groups, 113, 125 K Kan complex, 141-142 Klein bottle, 28 Kiinneth relations, 202 Kuratowski's imbedding, 27
L Leray. 229, 249 Lifting. 24, 69, 72, 86 Lifting problem, 24, 70, 72 Local system of groups, 129 simple, 131 Loop, 36, 31 1 degenerate, 40 equivalent, 39 product of, 39, 79 representative, 40 reverse, 40 space of, 79-82, 295
M Map, 1 algebraically trivial, 67-69 association, 75 canonical, 3 I3
346
INDEX
cellular. 172 P characteristic, 170, 333 Pair combined, 7 binormal, 208 deformable into a subset, 16, 22,24, 197 n-coconnected, 226 derived, 114 n-simple, 197 essential, 16 Path, 36 exponential, 35 component, 78 fiber, 71 equivalent, 41 homotopic, 11, 15 space of, 78-79, 313, 316 inclusion, 2 Path lifting property, 82-83, 98-99 inessential, 16 Poincarb %-extensible, 175, 194-196 group, 40 n-homotopic, 182, 194-196 polynomial, 277-280 %-normal, 197 Primary component, 298, 319, 325 null-homotopic, 16, 33 of surfaces, 106 R of exact couples, 242-243 Realizability theorem, 169 of filtered d-groups, 244 Regular couple, 236-238 of torus into projective plane, 106 a-, 236 on topological products, 102 6-. 237 partial, 1 graded groups of, 238-239 suspended, 60 two-term condition for, 250 Map excision theorem, 207 Relative homeomorphism, 10 Mapping, 1 Relative homotopy, 15 Mapping cylinder, 18 addition theorem, 166 partial, 2 1, 3 I class, 16 Mapping space, 73-78, 101-102 Relative homotopy group, 1 1 0 - 1 12, 137 evaluation of, 74, 76 Relative n-cell. 1 I , 333 exponential law for, 77 Restriction of a map, 1 fibering theorem for, 83-84 Retract, 5, 25 induced map in, 85 absolute (AR), 26, 29 subbasic sets of, 73 absolute neighborhood (ANR), 26, Maximal cycle theorem, 293 Mobius strip, 27 29-32, 59, 194, 216 deformation, 16, 33 neighborhood, 26 N strong deformation, 16, 32, 33 Retraction, 5 Natural correspondence, 1 19 Retraction problem, 5 Natural equivalence, 122 Natural homomorphism, 44, 48, 287, 305 S Natural projection, 9, 21, 143 Semi-simplicial complex, 140-142 complete, 141 0 degeneracy operators in a, 141 fundamental group of a, 193 Obstruction, 175-204, 227 of a group, 199 cocycle of a map, 177, 195 topological realization of, I70 cohomology class, 180. I84 Singular complex, 45, 196 of a homotopy. 183 admissible subcomplex of, 172 of a map, 177 first Eilenberg subcomplex of, 45, 174 primary, 188- 190 Singular n-cube, 259 set. 181. 185 degenerate, 260 One-point union of two spaces, 145 faces of, 259
INDEX
of degeneracy q, 262 weight of, 263 Singular simplex, 45, 196 Space adjunction, 9-11, 31 base, 61, 65 binormal, 14 bundle, 65, 286 connective system of, 304 contractible, 12 covering, 89-97 director, 65 dominating, 32 fiber, 62 filtered, 234, 248 generalized covering, 104 H-, 81 homogeneous, 99 homotopically equivalent, 17 locally contractible, 32 locally pathwise connected, 43 locally simply connected, 93 mapping, 73-78 n-coconnected. 2 10. 226 n-connected, 57, 148, 166, 210 n-connective, 155 n-simple, 131-134 of curves, 103 of homotopy type (n.n), 168, 198 orbit, 9, 200, 286 pathwise connected, 41, 78 pseudo-projective, 32 1-323 quotient, 9, 99 real projective, 32 1-323 regular covering, 92 semi-locally simply connected, 93 simply connected, 42 solid. 2, 26 topological sum of, 10 total, 61 totally pathwise disconnected, 89 universal covering, 91 Spectral cohomology sequence, 292 Spectral homology sequence. 271 Spectral sequence associated with exact couple, 233 limit group of, 234 of a fiber space, 259-285 of a regular covering space, 285-287 of filtered d-groups, 249 Sphere as homogenous space, 100 connective fiber space over, 296
347
finite groups operating freely 290-291 homotopy groups of, 311-336 Hopf’s fiberings of, 66 Steenrod square, 258 Stiefel manifold, 100-101, 323-325 Suspension, 163, 257. 312, 335 iterated, 318 theorem, 31 1, 312, 335 Suspension sequence of a triad, 163
T Tietze’s extension theorem, 26 Topological identification, 8. 27 Topology admissible, 102 compact-open, 73 identification, 9 of uniform convergence, 102 quotient, 9 weak, 200 Whitehead, 170 Torus, 28 Transgression, 256-258, 284, 293 Triad, 78, 160 generalized, 78 homotopy groups of, 160, 316 Triple, 214-216, 219-222 binormal, 2 14 cohomotopy sequence of, 2 14 Homotopy sequence of, 159 Triplet, 110 derived, 1 1 1 homotopy sequence of, 115, 120, 136 Two-term condition, 241 U
Uniqueness theorem for homotopy, 12 1 Unit n-simplex, 7 Universal coefficient theorem, 202, 329 Universal covering space, 9 1
W Wang exact sequence, 282-284, 320 isomorphism, 3 14 Weight, 244-245 Whitehead exact sequence, 253 product, 138-139, 330, 336 theorem, 167, 307 topology, 170 Wojdyslawski’s theorem, 27
on,
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