Translations of
MATHEMATICAL MONOGRAPHS Volume 199
Geometry of Characteristic Classes Shigeyuki M0r-ita
Editorial Board
Shoshichi Kobayashi (Chair) Masamichi Takesaki
CHARACTERISTIC CLASSES AND GEOMETRY by Shigeyuki Morita Copyright © 1999 by Shigeyuki Morita Originally published in Japanese by Iwanami Shoten, Publishers, Tokyo, 1999 Translated from the Japanese by the author 2000 Mathematics Subject Classification. Primary 57R20, 57R32, 55P62; Secondary 55R40, 20F38, 32G15. Library of Congress Cataloging-in-Publication Data Morita, S. (Shigeyuki), 1946[Tokuseirui to kikagaku. English) Geometry of characteristic classes / Shigeyuki Morita; [translated from the Japanese by the author]. p. em. - (Thanslations of mathematical monographs, ISSN 0065-9282 ; 199) (Iwanami series in modern mathematics) Includes bibliographical references and index. ISBN 0-8218-2139-3 (alk. paper) 1. Characteristic classes. 1. Title. II. Series. III. Series: Iwanami series in modern mathematics.
QA613.618.M67 514'.72-dc21
©
2001 00-054312
2001 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America.
@! The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1
06 05 04 03 02 01
Contents Preface to the English Edition
VB
Preface
ix
Summary and Purpose of This Volume
xi
Chapter 1. De Rham Homotopy Theory 1.1. Postnikov decomposition and rational homotopy type 1.2. Minimal models of differential graded algebras 1.3. The main theorem 1.4. Fundamental groups and de Rham homotopy theory
1 1 18 34 43
Chapter 2. Characteristic Classes of Flat Bundles 2.1. Flat bundles 2.2. Cohomology of Lie algebras 2.3. Characteristic classes of flat bundles 2.4. Gel'fand-Fuks cohomology
49 49 60 66 75
Chapter 3. Characteristic Classes of Foliations 3.1. Foliations 3.2. The Godbillon-Vey class 3.3. Canonical forms on frame bundles of higher orders 3.4. Bott vanishing theorem and characteristic classes of foliations 3.5. Discontinuous invariants 3.6. Characteristic classes of flat bundles II
87 87 91 100
Chapter 4. Characteristic Classes of Surface Bundles 4.1. Mapping class group and classification of surface bundles 4.2. Characteristic classes of surface bundles 4.3. Non-triviality of the characteristic classes (1) 4.4. Non-triviality of the characteristic classes (2)
135
v
113 119 126
135 144 151 160
CONTENTS
vi
Applications of characteristic classes
171
Directions and Problems for Future Research
175
Bibliography
179
Index
183
4.5.
Preface to the English Edition This is a translation of my book originally published in Japanese by Iwanami Shoten, Publishers. Its aim is to introduce the reader to various theories of characteristic classes which have been initiated and developed during the last three decades. They include charateristic classes of flat bundles, characteristic classes of foliations and characteristic classes of surface bundles. I would like to thank the American Mathematical Society for publishing this English edition and their staff for providing excellent support. Shigeyuki Morita February 2001
Preface The purpose of the present volume is to give expositions on the basics of some new theories concerning geometry of characteristic classes which have appeared since the end of the 1960's. Arnong characteristic classes, there are Stiefel-Whitney classes, Euler classes or Pontrjagin classes and Chern classes as typical examples. These classes were introduced during the 1930's and 1940's, and since then they have played fundamental roles in the classification as well as analysis of the structure of manifolds. On the other hand, from the end of the 1960's onward, there arose a few theories which treat finer structures of manifolds than before. These include Gel'fand-Fuks theory, Chern-Simons theory and also the theory of characteristic classes of flat bundles, which are closely related to each other. These new characteristic classes are defined when the above mentioned classical characteristic classes vanish (partially) so that sometimes they are called the secondary classes. Among other things, the characteristic classes of flat bundles have been shown to have intimate relations with not only geometry but also algebraic geometry and number theory. It is plausible that they will play more and more crucial roles in future mathematics. The theory of characteristic classes of fiber bundles, whose structure groups are infinite dimensional groups such as the diffeomorphism groups of manifolds, remain largely unknown. However, in the case of diffeomorphism groups of surfaces, quite detailed studies have been made. This is the theory of characteristic classes of surface bundles which began in the 1980's. As will be mentioned in "Directions and Problems for Future Research" at the end of this book, studies of the above new theories are continuing from various points of view. Although the description of this book is limited to the foundational part, the author would be happy if readers are interested in these theories. ix
x
PREFACE
The period when the subjects of this book were developed overlaps the thirty years since I began mathematical research in graduate school. I would like to express my gratitude to my teachers, superiors and colleagues for their support. Also I would like to express my thanks to the staff of Iwanami Shoten. Shigeyuki Morita May 1999
Summary and Purpose of This Volume As was already mentioned in the preface, we have Stiefel-Whitney classes, Pontrjagin classes and Chern classes as typical examples of characteristic classes. They were introduced one after another during the ten years after the middle of the 1930's, and foundations of the theory of characteristic classes of principal bundles, whose structure groups are finite dimensional Lie groups, were already established in the early 1950's. There are two main methods in the theory of characteristic classes. One is topological and the other is differential geometrical. The former method appeared first. In fact, all the characteristic classes mentioned above were introduced first as the primary obstructions to the existence of sections of certain fiber bundles which are associated to various principal bundles. Next there arose a method to compute the cohomology groups of the classifying spaces, which are described in terms of Grassmannian manifolds, by means of explicit cell decompositions. On the other hand, the latter differential geometrical method describes the way principal bundles are curved by differential forms using the concepts of the connection and the curvature. This is called the Chern-Wei! theory. Roughly speaking, the theory of secondary characteristic classes is a refinement of the latter method. We summarize the contents of this book briefly. In Chapter 1, we give a short exposition of the de Rham homotopy theory. This theory was begun by Sullivan in the 1970's, and it enhances extensively the theorem of de Rham, which describes the cohomology of manifolds in terms of differential forms. Roughly speaking, it is meant to grasp information about homotopy types of spaces by differential forms rather than merely the cohomology groups. This theory is not directly related to characteristic classes. However, it is one of the basic tools for the study of manifolds, and it seems that it will develop further in the future. It is only in §3.5 where we shall use the ingredients of xi
xii
SUMMARY AND PURPOSE OF THIS VOLUME
this chapter. Nevertheless, the general idea behind this theory has meaning throughout all the other chapters. In Chapter 2, we consider characteristic classes of flat bundles. A flat bundle is a principal bundle equipped with a connection whose curvature vanishes identically. Hence, by the Chern-Weil theory, all the characteristic classes (with real coefficients) vanish. The structure of such a bundle can be described completely by a homomorphism from the fundamental group of the base space to the structure group which is called the holonomy group, or according to the context, the monodromy group. However, in general, it is very difficult to study this homomorphism directly. There arises then an idea to describe how the bundle is twisted in terms of cohomology classes. Since the curvature is identically zero, we can construct certain cohomology classes by using the connection form. More precisely, certain characteristic classes, called characteristic classes of flat bundles, can be defined based on cohomology theory of Lie algebras. We also give a quick introduction to the Gel'fand-F\lks cohomology theory, which can serve as characteristic classes of flat bundles whose structure groups are infinite dimensional. In Chapter 3, we treat the theory of characteristic classes of foliations. A foliation is a certain striped pattern on a manifold, and its characteristic classes are certain cohomology classes of the manifold which describe global geometrical properties of the pattern. This theory is closely related to those of Cheeger-Chern-Simons and Gel'fandFuks, and it was established within a rather short period in the first half of the 1970's. It is a typical example of secondary characteristic classes. One characteristic of these classes is the following fact. Namely, in contrast with the classical characteristic classes which are cohomology classes with finite or integral coefficients, characteristic classes of foliations become cohomology classes with essentially real coefficients. This fact appears directly in a celebrated theorem of Thurston which showed that a characteristic class of foliations, called the Godbillon-Vey class, can vary continuously. In §3.5 we shall formulate an open problem which is derived from the above fact. The subject of Chapter 4 is the theory of characteristic classes of surface bundles. The theory of characteristic classes of fiber bundles, whose fiber is a general manifold in a general dimension, is far from being understood because of difficulties in treating diffeomorphism groups which are infinite dimensional. However, in the 2-dimensional case, namely in the cases of surfaces, an exceptional phenomenon
SUMMARY AND PURPOSE OF THIS VOLUME
xiii
occurs which is caused by special properties of geometries on surfaces. More precisely, it turns out that surface bundles, which are objects of topology, have an intimate relationship with the moduli space of compact Riemann surfaces or the Teichmiiller space, which belong primarily to algebraic geometry or complex analysis, so that we can develop a far richer theory than other dimensions. Here we shall give an introductory exposition of this theory.
CHAPTER 1
De Rham Homotopy Theory Let M be a COO manifold. The famous theorem of de Rham claims that the real cohomology group of M can be described entirely in terms of differential forms on it. More precisely, if we denote by A' (M) the de Rham complex of M, then we have a natural isomorphism H*(A*(M)) ~ H*(M;lR). Although the information about the cohomology group is surely important among various topological properties of N1, it is by no means complete. As criteria for topological classifications of manifolds, we have difIeomorphisms, homeomorphisms or homotopy equivalences. Here the last criterion is a rather weak one. However, there is already a big difference between the two conditions regarding two given manifolds: one is that they are homotopy equivalent and the other is that they have isomorphic cohomology groups. The de Rham homotopy theory is the one which aims to get information on the homotopy type, rather than simply the cohomology group, out of the de Rham complex. In this chapter, we summarize the basic part of this theory, which was initiated by Sullivan, following his original papers [SuI] [Su2] [Su3], and the textbook [GMJ, which treats the case of simply connected spaces (see also [MoJ). We refer the reader to the above papers for various applications as well as prospects for further development of this theory. 1.1. Postnikov decomposition and rational homotopy type 1.1.1. Homology theory and homotopy theory. Among the methods of investigating geometrical properties of a given figure or space, we have homology theory and homotopy theory. The former was begun by Poincare around 1900, and the latter was initiated by Hurewicz in the 1930's. Briefly speaking, in homology theory we
1. DE RHAM HOMOTOPY THEORY
2
FIGURE
1.1. Attaching space
decompose a given figure into components like points, segments, triangles and in general k-dimensional simplices (the triangulation), and then we extract a topological invariant called the homology group out of the way they are connected to each other. There is another way of decomposing figures, namely by means of CW complexes which are more flexible than triangulations. Just to make sure, let liS recall the definition of CW complexes. Let Dn = {x E lRn; Ilx:11 :::; l} be the n-dimensional disk and let sn-l = aDn = {x E: lRn; Ilxll = I} be its boundary, namely the (n - 1)-dimensional sphere. DEFINITION
1.1. Let X be a topological space and let f
: sn-l -)
X be a continuous map. We denote by XUjD n
the space obtained from the disjoint union of X and D" by identifying each point x E sn-l with f(x) E X. It is called the space obtained by attaching an n-cell en = D n \ sn-l to X by f or simply the attaching space (see Figure 1.1). The map f is called the attaching map. This operation will exercise its power when we try to kill homotopy groups of topological spaces soon after. A Hausdorff space X is called a cell complex if it is expressed as a disjoint union of cells e). (,\ E A) in such a way that the image of the attaching map of any n-cell is contained in the union of cells whose dimensions are less than or equal to n - 1. A subset Y of X, which is a cell complex itself, is called a subcomplex. DEFINITION 1.2. A cell complex X is called a CW complex if it satisfies the following two conditions.
(i) The closure of X.
e of any cell
e is contained in a finite sub complex
1.1. RATIONAL HOMOTOPY TYPE
(ii) A subset U C X is open if and only if, for any cell e, open m e.
3
en U is
The conditions (i), (ii) above are called closure finite and weak topology, respectively. If X is a CW complex, then the set of all cells of dimension less than or equal to n, denoted by x(n), becomes a subcomplex. It is called the n-skeleton of X. In homology theory of CW complexes, the following fact plays a fundamental role. Namely for any n-cell en of X, we have
k=n k i- n. In contrast, homotopy theory concerns the set of homotopy classes of maps from the sphere sn of each dimension to the given space. Let Po be the base point of sn. DEFINITION 1.3. Let X be a topological space with base point Xo. The set [sn, Xlo of homotopy classes, relative to the base points, of maps from sn to X is denoted by
1rn(X,XO). It can be shown that 1rn(X, xo) has a natural structure of a group and is called the n-th homotopy group of X with respect to Xo. 1r1 (X, xo) is the fundamental group of X and for any n > 1, 1rn(X,XO) is an abelian group. 1f1(X,XO) acts naturally on 1fn(X,XO). The group structure of 1rn(X, xo) does not depend on the choice of base point so that it is frequently denoted by 1rn(X), The definition of homotopy groups is simpler than that of homology groups. But it is balanced by the fact that computation of homotopy groups is much harder than that of homology groups in generaL For the product X x Y of two spaces X, Y, however, it follows immediately from the definition that 1fn (X X Y) ~ 1fn(X) X 1rn (Y), which is simpler than the Kiinneth theorem in homology theory. One of the basics in developing homology theory is the homology exact sequence for pairs of spaces. There exists a similar exact sequence for pairs of spaces in homotopy theory. However, what is more important is the homotopy exact sequence for fibrations.
DEFINITION 1.4. Let E, B be topological spaces. A continuous surjection 1r : E --+ B is called a fibration (or fibering) if 1r has the covering homotopy property with respect to the n-dimensional
1. DE RHAM HOMOTOPY THEORY
4
cube In for all n ~ 0 in the following sense. Namely for any continuous map I : In --7 E and any homotopy It : In X I --7 B of I = "Tr 0 I, there exists a homotopy It : In X I --7 E of f such that "Tr 0 it = It for any tEl = [0,1]. In this case, we call "Tr-l(b) (b E B) the fiber over b. (i) Fiber bundles are important examples of 1.5. fibrations. (ii) Let X be an arcwise connected topological space and let Xo be its base point. Then the set EXAMPLE
PX = {l: [0,1]--7 X;l(O) = xo} equipped with the compact open topology is called the path space of X with initial point Xo. If we define "Tr : PX --7 X by setting "Tr(P) = P(l), then it becomes a fibration. The fiber over Xo is the space consisting of all closed paths based there, namely the loop space of X which is usually denoted by nx. P X is contractible, and it is known that if X has the homotopy type of a CW complex, then so does nx. (iii) Let X, Y be topological spaces and assume that Y is arcwise connected. We show that any continuous map I : X --7 Y can be considered as a fibration in the sense of homotopy. To see this, let Map(I, Y) denote the mapping space consisting of all continuous maps from the unit interval I to Y and set
X= c
{(x,l) E X X Map(I, Y);l(O) = I(x)} X x Map(I, Y).
If we define i : X --7 X by i(.x) = (x,ff(x), then i becomes a homotopy equivalence. Here l f(x) denotes the constant path at f(x). If we define "Tr : X --7 Y by setting "Tr(x,P) = P(l), then it can be shown that it is a fibration. ClearlY"Tr 0 i = .f so that this fibration is homotopy theoretically equivalent to the original map I : X --7 Y. The fiber of this fibration is called the homotopy fiber of I.
THEOREM 1.6 (Homotopy exact sequence for fibrations). For each fibration F --7 E --7 B, there exists a long exact sequence ... --7 "Trn+l (X) --7 7rn(F) --7 "Trn(E) --7 DEFINITION
In case n
7rn
(X) --7 ....
1.7. Let n be a positive integer and let"Tr be a group.
> 1 we assume that "Tr is an abelian group. A topological
1.1. RATIONAL HOMOTOPY TYPE
5
space K (7r, n) is called an Eilenberg-Mac Lane space of type (7r, n) if k=n ki-n. For example S1 is a K(Z,l) and Cpoo is a K(Z,2). In general, a manifold M which is a K(7r, 1), namely 7rn (M) = 0 for any n > 1, is called a K(7r, 1) manifold. Equivalently we may characterize a K(7r, 1) manifold by the property that its universal covering space is contractible. Among such manifolds, closed K (7r, 1) manifolds are particularly important. It is known that any closed Riemannian manifold with negative sectional curvature is an example of such manifolds. THEOREM 1.8. FOT any (7r,n) given in the above definition, theTe exists an EilenbeTg-MacLane space K(7r, n) which is a CW complex. M OTeover it is uniquely defined up to homotopy equivalence.
Sketch of Proof. In case n = 1 we express the group 7r in terms of a system of generators and relations. We fix one O-cell and attach a I-cell to it corresponding to each generator. We then realize each relation by attaching a 2-cell. If we kill higher homotopy groups 7r2, 7r3,'" of the resultant 2-dimensional complex by attaching k-cells for k = 3,4"", then we obtain a K(7r, 1). The cases of n > 1 can be treated basically in the same way. We first express the abelian group 7r in terms of generators and relations. We prepare one copy of the n-sphere corresponding to each generator and attach them to a point. Next we realize the relations by attaching (n + I)-cells. Finally we kill higher homotopy groups 7rn +1, 7rn +2,'" and we are done. Uniqueness can be proved by a simple application of the theorem of J .H.C. Whitehead concerning homotopy equivalences. 0 If n > 1, K(7r, n) is (n-l)-connected by definition so that we have an isomorphism Hn(K(7r, n)) ~ 7rn (K(7r, n)) ~ 7r by the Hurewicz theorem. Hence the universal coefficient theorem implies
sn
Hn(K(7r, n); 7r)
~
Hom(7r, 7r).
Let L E H n (K (7r, n); 7r) be the element which corresponds to id E Hom( 7r, 7r) under the above isomorphism. We call it the fundamental cohomology class of K (7r, n). The following theorem shows that the Eilenberg-MacLane spaces can serve as the classifying spaces for the cohomology theory.
6
1. DE RHAM HOMOTOPY THEORY
THEOREM 1.9. Let X be a CW complex. Assume that either n 1 or n = 1 and 7r is an abelian group. Then the correspondence
>
is a bijection. Sketch of Proof. Let 0: E Hn(Xi 7r) be an arbitrary element and choose a co cycle which represents it. Construct a continuous map I: X --) K(7r, n) as follows. We send the (n - I)-skeleton x(n-l) of X to the base point of K (7r, n), and over each n-cell en of X, we set I to represent the element 0:( en) E 7r = 7rn (K (7r, n)). Since 0: is a cocycle, I can be extended to x(n+l). Moreover since the target of the map I is KC71-, n), we see that it can be extended to the whole of X by an easy application of the obstruction theory- Clearly f* (/_) = 0: by the construction. Thus we have proved that the correspondence of the theorem is surjective. Next suppose that we are given two maps Ii : X --) K(7r, n) (i = 0,1) such that IO(L) = fi(L). We may assume that the restrictions of Ii to x(n-l) are both constant maps. By the assumption, we have lo\X(n) ~ h\X(nJ' Since the target of our map is K (7r, n), again the obstruction theory implies that the homotopy can be extended to the whole of X x I. This completes the proof. 0
1.1.2. Postnikov decomposition. Given a topological space X, if we can construct a triangulation of it or a decomposition as a CW complex, then it is convenient for the study of homological structure_ However, it is not so useful for homotopy theoretical study of the space. The Postnilcov decomposition describes the homotopy type of X in terms of Eilenberg-MacLane spaces as fundamental components. The simplest space whose homotopy groups are the same as those of X would be
K(7rl(X), 1) x K(7r2(X), 2) x ... which is the product of various Eilenberg-MacLane spaces. In general, X is not the product but a certain twisted version of the above space. The way it is twisted is described by what is called the Postnikov invariants.
1.1. RATIONAL HOMOTOPY TYPE
7
THEOREM 1.10. For any connected CW complex X, there exists the following commutative diagram:
1/4
II X
X(3)
-------->
1
II X
13
X(2)
-------->
112
II X
--------> X(I)
=
K(7fI(X),
1)
which is uniquely defined up to homotopy. Here 7fi(X(n») = 0 for all > n and7fi(X) ----> 7fi(X(n») is an isomorphism for any i::; n. Moreover In : X(n) ----> X(n-I) is a fibration whose fiber is K(7f n (X),n).
i
First we set X(I) to be the CW complex which kills (i = 2,3,···) by attaching cells of dimensions 3,4,· .. to X. Next let X(2) J X be the space which kills 7fi (i = 3,4,···) of X. Then it is easy to see that the inclusion X C X(l) can be extended to X(2). We then convert the resultant map X(2) ----> X(l) into a fibration, by the method of Example 1.5 (iii), to obtain a map h : X(2) ----> X(l). By the homotopy exact sequence of this fibration, we see that the fiber of his K(7f2(X), 2). We can continue similar processes to obtain the required tower of maps. 0 PROOF.
7fi(X)
X(k) is sometimes called the k-dimensional coskeleton of X. In the above considerations, there appeared various fibrations whose fibers are Eilenberg-MacLane spaces. We take up a simple type among them whose structure can be analyzed relatively easily. We remark that the fundamental group of the base space of any fibration acts on its fiber by homotopy equivalences because of the covering homotopy property.
DEPINITION 1.11. Let K (7f, n) ----> E ----> B be a fibration. Suppose that the action of 7f1 (B) on the fiber is trivial and in case n = 1 further assume that 7f is abelian. In this case we call it a principal fibration.
8
1. DE RHAM HOMOTOPY THEORY
EXAMPLE 1.12. By definition, any fibration whose base space is simply connected is a principal fibration. In particular, if 7r is an abelian group, then the fibration P K (7r, n + 1) --> K (7r, n + 1) associated to the space of paths of K (7r, n + 1) is a principal fibration and its fiber is OK(7r, n + 1) = K(7r, n). To classify principal fibrations, let us investigate how we can decide whether a given principal fibration K(7r, n) --> E --> B is trivial or not. For that we try to construct a section of it on each i-cell of B for i = 0,1,· . '. For simplicity, assume that this fibration is trivial on each cell of B. In this case, since the fiber is (n -I)-connected, we can construct a section on the n-skeleton B(n) of B by the obstruction theory. The primary obstruction, which we denote by o(E), to extending it on B(n+l) is defined as an element of Hn+l(B; 7r). We call it the characteristic cohomology class of principal fibrations. It is easy to see that the characteristic cohomology class of the principal fibration of Example 1.12 above is nothing but the fundamental cohomology class ~ E Hn+l(K(7r,n+ 1);7r). DEFINITION 1.13. Two fibrations Ei --> B (i = 0,1) over the same base space B are said to be fiber homotopy equivalent if there exist mappings h : Eo --> E 1 , h : El --> Eo which cover the identity of B such that h 0 h, and h 0 h are homotopic, through fiber preserving mappings, to the identities of E l , Eo, respectively. It can be shown that characteristic cohomology classes of two principal fibrations with fiber K(7r, n) which are fiber homotopy equivalent are the same. THEOREM 1.14. Let B be a CW complex. Then the set of fiber homotopy equivalence classes of principal fibrations over B whose fiber'S are K (7r, n) can be naturally identified with the cohomology group Hn+l(B; 7r) by the correspondence E ...... o(E) E Hn+l(B; 7r) where o(E) denotes the characteristic cohomology class of E.
Sketch of Proof. For any element 0 E Hn+l(B; 7r) consider the following commutative diagram of pullback of fibrations o*PK(7r,n+1) - - - - 7 PK(7r,n+1) K(7r,n)
1 B
1
K(7r,n)
~
K(7r,n+1)
where 0 : B --> K (7r, n + 1) denotes the map corresponding to 0 under the bijection of Theorem 1.9. Then we have o(o·PK(7r,n + 1)) =
1.1.
RATIONAL HOMOTOPY TYPE
9
0*(1.) = 0 by the naturality of the characteristic cohomology classes. Here t E Hn+ 1 (K (1r, n + 1); 1r) is the fundamental cohomology class. Hence the correspondence in question is a surjection. Next assume that we are given two fibrations K(1r,n) -+ Ei -+ B (i = 0,1) such I~hat o(Eo) = 0(E1). Since the fiber is (n - I)-connected, both Eo and El are fiber homotopy equivalent to the trivial fibration over the n-skeleton B(n). Then by the assumption, we can conclude that they are fiber homotopy equivalent over B(n+1). Finally since the homotopy groups 1ri of the fiber are trivial for i ;::: n + 1, we see that the above fiber homotopy equivalence can be extended to the whole 0 of B. This completes the proof. Now if X is a simply connected CW complex, its coskeletons X(n) are also simply connected for all n. Hence the fibrations K(1rn(X), n)
---+
X(n)
---+
X(n-l)
which appear in the Postnikov decomposition (cf. Theorem 1.10) of X are principal fibrations. Their characteristic cohomology classes kn+1(x) E H n+1(X(n_l); 1rneX)) (n = 3,4,··· ) are called the Postnikov invariants of X. They are also called the k-invariants. In this terminology, the homotopy type of X can be expressed completely as X ~ K(1r2(X),2)
Xk4
K(1r3(X),3)
Xk'
K(1r4(X),4) x ...
in terms of its homotopy groups and Postnikov invariants. The condition on the fundamental group can be weakened as follows. DEFINITION 1.15. A topological space X is called nilpotent if the action of 1rl(X) on 1ri(X) is nilpotent for any i ;::: 1. Here an action of 1rl (X) on an arbitrary group G is said to be nilpotent if there exists a series
Go
=
G:J G l :J G2 :J ... :J G s = {I}
of subgroups of G which satisfies the following conditions. (i) G k is a subgroup invariant under the action of 1rl(X) for any k. (ii) G k + l is a normal subgroup of G k and the quotient group Gk/Gk+1 is abelian. (iii) 1rl(X) acts on G k /G k +1 trivially. In particular, the fundamental group of a nilpotent space is nilpotent. Clearly any simply connected space is nilpotent. A topological
10
1.
DE RHAM HOMOTOPY THEORY
space whose fundamental group is abelian and acts on higher homotopy groups trivially is called simple. A simple topological space is nilpotent. The following theorem can be proved relatively easily from the definition. THEOREM 1.16. A CW complex X is nilpotent if and only if its Postnikov decomposition has the following refinement. Namely, for any n the fibration X(n) -) X Cn - 1) can be expressed as a composition X(n)
= X(n)
----->
X(\t)
-----> ..• ----->
XC;:) = XCn - 1)
of finitely many fibrations which are all principal fibrations. In this case, by changing the suffix, we can say that a series of principal fibrations K(7r m, d m ) -> Xm -> X m- 1 (m = 1,2,··· ) gives rise to a refinement of the Postnikov decomposition. If we oenote by k m E Hdm.+l(Xm _ 1 ; 7rm ) the characteristic cohomology class of Xm -> X m- 1 , then it can be represented by a certain map km : X m - 1 -> K(1rm, dm + 1). Hence the homotopy type of nilpotent spaces can be described almost in the same way as the case of simply connected spaces. EXAMPLE 1.17. The real projective plane pJR(2 is an example of a topological space with abelian fundamental group which is not nilpotent. In fact, the action of 1rl(p]R2) ~ Zj2Z on 1r2(PJR(2) ~ Z is not nilpotent. 1.1.3. Rational homotopy type. As was mentioned in §1.1.1, it is very difficult to compute homotopy groups in general. for example, even in the case of spheres which are rather simple spaces, the complete determination of their homotopy groups seems to be almost impossible. The problem of determining homotopy types which requires information about the Postnikov invariants is much more difficult. However, if we ignore torsions and consider only 7ri ® Q, then the problem turns out to be much easier. This is what is called the rational homotopy type. Henceforth in this subsection, we work in the category New consisting of nilpotent spaces which are homotopy equivalent to CW complexes and continuous mappings between them.
DEFINITION 1.18. A topological space X E New is called a rational space or Q space if its fundamental group is a Lie group over Q and all homotopy groups 1ri(X) (i ~ 2) are vector spaces over Q. For example, K(Q, n) is a typical example of Q spaces.
l.1. RATIONAL HOMOTOPY TYPE
11
DEFINITION 1.19. Let X E New be a topological space. A continuous mapping e : X ---. Xo is called the localization at 0 if it satisfies the following conditions.
(i) Xo is a Q space. (ii) For any Q space Y and a continuous mapping f : X ---. Y, there exists a continuous mapping fo : Xo ---. Y unique up to homotopy such that f ~ fo 0
e.
By the condition (ii) above, the homotopy type of Xo is uniquely determined depending only on X. Such Xo is called the rational homotopy type of X. Sometimes we write XQ instead of Xo.
e:
THEOREM 1.20 (Sullivan [SuI]). Let X ---. Y be a continuous mapping between nilpotent spaces. Then the following three conditions are all equivalent.
(i)
e is the localization at O.
Hence, in particular Y = Xo. homotopy groups. Namely, e* : 'ffi(X) ---. 1fi(Y) induces an isomorphism 1fi(X) ®z Q ~ 1fi(Y) for any i > O. (iii) localizes homology gro'ups. Namely, Hi(X) ---. Hi(Y) induces an isomorphism Hi(X) ®z Q ~ Hi(Y) for any i > o.
(ii)
e localizes
e
e. :
In the above description, 1f1 (X) ® Q denotes the MaIcev completion of 'ffl(X) (see §1.4.3). If we assume this theorem for the moment, then the rational homotopy type Xo of X can be constructed explicitly as follows. To begin with, for the n-sphere sn we have to convert Hn(sn) ~ Z into Q. To do so, we choose a degree m. map dm : sn ---. S" for each natural number m and let M(m.) = sn x I Ud m sn be its mapping cylinder. Then the homomorphism Hn(sn) ~ Z ---. Hn(M(m)) ~ Z induced by the inclusion sn = sn X {OJ c M(m) is just multiplication by m. Next if we denote by i'v/(m, m') the mapping cylinder of the composition of natural projection M(m) ---. sn with the map dm , : sn ---. sn, then the homomorphism Hn(sn) ~ Z ---. Hn(M(m, m')) ~ Z induced by the inclusion sn C M (m, m') is the multiplication by mm'. We continue this process to obtain an infinite mapping cylinder M(2, 3, ... ) (see Figure 1.2) which we denote by S[). Then it can be shown that the inclusion Srt C So localizes the homology group. Hence by Theorem 1.20, S[) can serve as the rational homotopy type of S". For the general case of a simply connected CW complex X, we can do as follows. First we may assume that the O-cell of X is unique and there is no i-cell. By induction on n, assume that we obtain a
12
1. DE RHAM HOMOTOPY THEORY
FIGURE 1.2.
Localization of the n-sphere
X6
X.
n ) of the n-skeleton x(n) of Observe localization g(n) : x(n) --> that we can use the localization of sn constructed above for the first non-trivial n. Let en + 1 be an (n + I)-cell of X and let f : Sri --> x(n) be its attaching map. In view of Theorem 1.20, there exists n ) making the following diagram a continuous map fa : So ----> commutative up to homotopy.
X6
f sn -------> x(n)
1e(n)
e1
sn -------> x(n) 0 0 fa
We set
X6 n + 1) = X6 n )
U (U
en + 1
Cone(S(;)) ,
{fo} where Cone( So) denotes the cone over SO'. Then the natural inclusion x(n+1) ----->
X6n.+1)
localizes homology so that it is the localization at O. Hence if we set
Xo
=
UX6n), n
we obtain the rational homotopy type of X. The above construction of the localization at 0 of a CW complex X was done by making use of its structure as a cell complex. Alternatively it is also possible to utilize the Postnikov decomposition of X. To begin with, if 7r is an abelian group, the mapping K(7r, n) ----> K(7r Q9 Q, n) induced by the natural homomorphism 7r ----> 7r Q9 Q clearly localizes the homotopy groups. Hence it gives rise to the localization at 0 of K (7r, n). In the general case of simply connected X, we construct Xo by induction on its coskeletons which appear in the Postnikov decomposition (cf. Theorem 1.10). In the first stage, we can use the case of K(7r, n) above. Assume inductively
1.1. RATIONAL H01VIOTOPY TYPE
13
that we have the localization .e(n-l)O : X(n-l) -> X(n-l)O of X(n-l)· The principal fibration X(n) -> X(n-l) is the pullback of the path fibration
PK(7Tn(X), n + 1) -> K(7T n (X), n + 1) by the Postnikov invariant k n+1 E Hn+l (X(n-l); 7Tn(X)). It is enough to define X(n)O to be the pullback of the fibration
by kn +l18l Q EHn+1(X(n_l)O; 7T n (X) 181 Q)
SO!Hn+1(X(n_l); 7Tn (X) 18> Q). Then it is easy to show the existence of the localization map .e(n)O : X(n) -> X(n)O, and the induction is completed. The above construction can be applied also to a general nilpotent space X by using its refined Postnikov decomposition (cf. Theorem 1.16). Thus we obtain the following corollary to Theorem 1.20. COROLLARY 1.21. Any nilpotent space X has the associated rational homotopy type Xo.
Sketch of Proof of Theorem 1.20. For the proof we use the following three facts. (I) (Postnikov decomposition of mappings) Let f : X -> Y be any continuous mapping. Then we can choose refined Postnikov decompositions {Xm -> Xm-d, {Ym ---+ Ym-d of X, Y respectively and maps between them such that f is described by the following homotopy commutative diagram: 1m
Xm
---;
1 X·rn -
1
1 K(7T m , dm + 1)
1m -1
-->
Ym
1 Ym- 1
1
~ K(7T~,dm + 1)
Here the mapping k-m(f) : K(7T m , dm +1) -> K(7T~, dm +1) is induced by the homomorphism f* : 7T.(X) ---+ 7T.(Y).
14
1. DE RHAlvl HOMOTOPY THEORY
(II) Any homomorphism p : 7r -+ 7r' between two abelian groups induces the following homotopy commutative diagram:
----
K(7r', n)
-------;
PK(7r', n)
p.
K(7r,n)
1 PK(7r,n)
1
K(7r, n + 1)
---p.
1 1 K(7r', n
+ 1)
(III) For any homotopy commutative diagram
P
1 E
1 B
f
P'
--->
e
E'
b
B'
-----+
~
1 1
consisting of maps between two principal fibrations, the following two facts hold. (1) If any two of the three mappings {f, e, b} localize homotopy groups, then so does the third one. (2) If any two of the three mappings {f, e, b} localize homology groups, then so does the third one. We prove the theorem by using these facts. First we show that the two conditions (ii) and (iii) concerning the map e: X -+ Y in the statement of our theorem are equivalent. We first consider the case where both X and Yare K(7r, n) spaces. The claim is almost clear for the case of a mapping K (7r, 1) -> K (7r', 1) where 7r and 7r' are abelian groups. This is because each of the two conditions are then equivalent to 7r ® Q ~ 7r'. Next we consider the cases of a mapping e: K (7r, n) -+ K (7r', n) for general n ~ 1. By the theorem of Hurewicz, we have 7r ~ H n (K(7r,n)),7r' ~ H n (K(7r',n)). Hence if e localizes homology groups, then 7r' ~ 7r ® Q. Conversely if e localizes homotopy groups, then 7r' = 7r ® Q. We need to prove that e also localizes homology groups. We show this by induction on n. The case n = 1 was already shown above. If we apply (III) to the above homotopy commutative diagram in (II), under the condition
1.1. RATIONAL HOMOTOPY TYPE
15
e
= w ® Q, and use the induction assumption, then we see that localizes homology groups. Next we prove the equivalence of (ii) and (iii) for general mapping £ : X ---> Y. Assume first that £ localizes homotopy groups. In this case, we apply the Postnikov decomposition in (I) above to £ to obtain a series em : Xm ---> Ym of mappings. Then by using the case of K(w, n), which was already proved, as well as (III), we can see by induction on m that em : Xm, ---> Ym localizes homology groups. Hence so does e. Conversely assume that e localizes homology groups. Let eo : X ---> Xo be the localization at 0, which was already constructed above by using refined Postnikov decomposition (cf. Corollary 1.21). By the construction, £0 localizes homotopy groups. Now by the assumption, we have H* (Y, X; Q) = 0, and the homotopy groups of Xo are vector spaces over Q. If we apply obstruction theory here, we can conclude that there exists a mapping 1 : Y ---> Xo such that eo ::: 1 0 e. It follows from the facts already proved that eo localizes homology groups. Hence we can finally conclude that 1 induces an isomorphism in homology. If we use the theorem of J.H.C. Whitehead extended for nilpotent spaces, then we see that 1 is a homotopy equivalence and so it induces isomorphism in homotopy. Since eo localizes homotopy groups, so also does £. Next we show that the condition (i) implies (iii). We take K(Q, n) as the Q space in the statement of Definition 1.19 and consider any map 1 : X ---> K(Q, n). From the existence of a map 10 : Y ---> K(Q, n) such that 1 ::: 10 0 e we see that f* : H* (Y; Q) ---> H* (X; Q) is surjective and its uniqueness implies that it is also injective. Hence by the universal coefficient theorem, we have £. : H.(X; Q) ~ H*(Y; Q). On the other hand, since Y is a Q space, we have H*(Y; Q) ~ H*(y). Hence e localizes homology groups. Conversely we show that the condition (iii) implies (i); that is we prove that if e : X ---> Y localizes homology groups, it is the localization at O. Let Z be any Q space and let 1 : X ---> Z be a continuous mapping. Since homotopy groups of Z are vector spaces over Q, we see that H*(Y,X;1r*_l(Z)) = 0 by the assumption. Hence by the obstruction theory, there exists a mapping j : Y ---> Z such that 1 ::: joe. Next let l' : Y ---> Z be another map with the same property. Then the obstruction to the existence of a homotopy l' ::: j lies in H*(Y,X;w.(Z)) = O. It follows that j exists uniquely up to homotopy so that we can conclude that e is surely the localization at O. This completes the proof of Theorem 1.20. 0 Wi
16
1. DE RHAM HOMOTOPY THEORY
1.1.4. Examples. In this subsection we take up several examples related to rational homotopy theory. PROPOSITION 1.22. The cohomology groups of the EilenbergMacLane spaces K(Z, n), K(Q, 11.) are given as follows: (i) H*(K(Z,2n);Q) ~ H*(K(Q,2n);Q) ~ Q[L]. (ii) H*(K(Z,2n + l);Q) ~ H*(K(Q, 2n + l);Q) ~ BQ(L). Here L E Hn(K(Q,n);Q) -is the fundamental cohomology class and EQ denotes the exter'iOT algebm over Q. PROOF. We use induction on n. If 11. = 1, we have K(Z, 1) = S1 so that the claim is clear. We assume the case 211. - 1 and deduce the case 211. from it. Consider the fibration
nK(Z,2n) = K(Z,2n -1)
-->
PK(Z,2n)
-->
K(Z,2n)
(see Example 1.12) and let {E~,q, dr,} be the Serre spectral sequence for cohomology with Q coefficients. Since P K(Z, 2n) is contractible, E~q = 0 except for the case (p, q) = (0,0). By the induction assumption, the only E2 terms which might be non-trivial are the cases q = 0,211. - 1, namely E~'o ~ HP(K(Z, 211.); Q),
Hence d2 = ... = d2n-1 us study the differential
E~,2n-1 ~ HP(K(Z, 211.); Q).
= 0 and E2 = ... = E2n, E2n + 1 = Eoo. Let
H P+2n ,o d2n·. EP,2n-l 2n --> 2n . K(Z,2n) is (211. - I)-connected and H2n(K(Z, 211.); Q) ~ Q is generated by the fundamental cohomology class L. It follows that the 2nO correspon d'mg to p = 0"IS an ISO02n - 1 ~ E 2n' ·"" . 1 d2n: E 2;" d luerentm morphism. Using the fact that the Eoo term is trivial as mentioned above, it can be shown by induction on the degrees that the homomorphism Q[L] --> H*(K(Z, 211.); Q) is an isomorphism. Next we deduce the case 2n + 1 from the case 211.. We consider the fibration nK(Z, 211. + 1) = K(Z, 2n)
-->
PK(Z, 2n + 1)
-->
K(Z, 2n + 1).
Then the E2 term of the spectral sequence similar to the above is given by
1.1. RATIONAL HOMOTOPY TYPE
17
Hence d2 = ... = d2n = 0, E2 = ... = E 2n +l. Clearly d 2n +1 : O,2n - t E2n+l,O .Isomorp h·Ism. It f 0 11ows t h at d2n+l: EO,2kn E 2n+l 2n+l . IS an 2n+l-t E~~ti,2(k-l)n is also an isomorphism for any k ::::: 1. The claim now follows from these facts. 0 EXAMPLE 1.23. Let ~ E H 2n - 1 (S2n- \ Q) be the fundamental cohomology class of an odd dimensional sphere s2n-l and let
s2n-l
---->
K(Q, 2n - 1)
be its classifying map. Then it induces an isomorphism in the rational cohomology by Proposition 1.22 above. Hence this map gives the localization at O. Namely S6 n - 1 = K(Q, 2n -1). On the other hand, it is known by Serre that all the homotopy groups of spheres are finitely generated. Hence by Theorem 1.20 we see that 1fi(S2n-l) is a finite group for any i > 2n - 1. EXAMPLE 1.24. For an even dimensional sphere S2n, the pullback of i 2 E H 4 n(K(Q, 2n); Q) by the classifying map s2n - t K(Q, 2n) of its fundamental cohomology class is O. Hence a map
s2n
---->
K(Q,2n)
X.2
K(Q,4n - 1)
is defined. If we use the spectral sequence here, we can deduce that the above map induces an isomorphism in the rational cohomology. Hence it gives the localization at 0 of s2n. Again by the result of Serre above and Theorem 1.20, we see that 1fi(S2n) ®Q is isomorphic to Q for i = 2n,4n - 1 and other homotopy groups are all finite groups. EXAMPLE 1.25. We determine the rational homotopy type of the complex projective space cpn. Let cpn - t K(Q, 2) be the classifying map of the generator of H2(cpn; Q). Then the pullback of i n+1 E H 2n+2(K(Q, 2); Q) by this map is O. Hence a map
cpn
---->
K(Q,2)
X.n+1
K(Q, 2n + 1)
is defined. Again by a spectral sequence argument, we see that the above map induces an isomorphism in the rational cohomology. Hence it gives the localization at 0 of cpn. It follows that k = 2, 2n + 1 otherwise. EXAMPLE 1.26. It is known that the cohomology algebra of the classifying space BGL(n; q ~ BU(n) of n-dimensional complex vector bundles is a polynomial algebra generated by the Chern classes
1. DE RHAM HOMOTOPY THEORY
18 Ci
E H 2i (BGL(n; C); Z) (i = 1,··· , n). Hence BGL(n,C)o EXAMPLE
=
K(Q,2) x .. · x K(Q,2n).
1.27. Consider the fiber bundle U(n - 1)
->
U(n)
-> s2n-l.
It can be shown by induction on n, using spectral sequence, that
H*(U(n);Q) ~ H*(SI x S3 x···
X
s2n-\Q).
Hence U(n)o = K(Q, 1) x K(Q, 3) x ... x K(Q, 2n - 1).
In general, it is known that the rational homotopy type of any simply connected Lie group is the same as that of a product of some numher of odd dimensional spheres. 1.2. Minimal models of differential graded algebras 1.2.1. Differential graded algebras. DEFINITION 1.28. Let K be Q, lR or C. A graded vector space A=EBA(k) k?O
over K which has the following additional structure is called a differential graded algebra over K. (i) An associative product x E A(k), y E A(t) :::} xy E A(k+l') is defined such that yx = (-l)kR xy . (ii) A differential d : A(k) -> A(k+l) such that dod = 0 is defined. (iii) d(xy) = dx y + (l)kxdy. Differential graded algebras are often abbreviated d.g.a., and we use this notation in this book. Similarly, graded algebras are called g.a. for short. The de Rham complex A*(M) = EBkAk(M) of a Coo manifold NI is one of the most jmportant examples of d.g.a. 'so By condition (ii) above, the cohomology group H* (A) of A with respect to d is defined, and it becomes a d.g.a. with trivial differential, namely d == O. DEFINITION 1.29. A d.g.a. A is called co homologically connected if HO(A) = K and connected if A(O) = K. It is called a free d.g.a. if A is free as a graded algebra, that is jf it is expressed as a tensor product of polynomial algebra generated by elements of even degrees and exterior algebra generated by elements of odd degrees.
1.2. MINIMAL MODELS OF DIFFERENTIAL GRADED ALGEBRAS
19
A free g.a. generated by certain generators x,x, namely
P[x,x; degx,x even] ® E(x>.; degx,x odd), is abbreviated to
A(x,x). Here P and E denote polynomial algebra and exterior algebra respectively. Also if V is a finite dimensional vector space, we denote by A(V) the free g.a. generated by elements of V. In this case we always assume that the degrees of elements of V are the same unless otherwise stated. We write A(V)k to emphasize that the degrees are all equal to k. DEFINITION 1.30. A d.g.a. A is called minimal if it is connected, free and d is decomposable; that is
d(A+)
where A+
=
c
A+ A+
EBk>oA(k).
Let B be a d.g.a. A subspace A c B which is closed under the operations of product and differential is called a sub d.g.a. of B. DEFINITION 1.31. Let B be a d.g.a. and let A c B be a connected sub d.g.a. If there exist a finite dimensional vector space V and an isomorphism
B~A®A(V)k
as graded algebras such that dv E A(k+1) for any element v E V, then we call A c B a Hirsch extension or an elementary extension. The above Hirsch extension together with the differential d will be denoted by A®dA(vk In this case, we consider the element o E Hom(V, Hk+1(A)) ~ H k+1(A, V*) which is represented by the correspondence V '3 v
f---t
[dv] E Hk+1(A).
It is called the characteristic cohomology class of the Hirsch extension. We say that two Hirsch extensions A®dA(V), A®d' A(V) are equivalent if there exists an isomorphism as d.g.a.'s which restricts to the identity on A. The proof of the following proposition is ea.c;y, so it is left to the reader.
1. DE RHAM HOMOTOPY THEORY
20
PROPOSITION 1.32. Two Hirsch extensions are equivalent if and only if their characteristic cohomology classes are the same.
Let A be a minimal d.g.a. If we denote by Ak the subalgebra of A generated by all the elements of degrees ::; k, then it becomes a sub d.g.a. of A. DEFINITION 1.33. A minimal d.g.a. A is called nilpotent if there exists a series Ao = K C Al C A2 C ... of sub d.g.a.'s which satisfies the following conditions. (i) A = ukAk. (ii) Ak c Ak+l is a Hirsch extension for any k. (iii) For any k there exists m such that Ak cAm. If only conditions (i) and (ii) are satisfied, A is called generalized nilpotent.
It is easy to see that any minimal d.g.a. which is generated by elements of degree 1 is always nilpotent. EXAMPLE 1.34. A connected, free d.g.a. which is generated by elements of degree 1 is called a dual Lie algebra. It becomes automatically minimal. Let {Xdk be a basis of A(l) and suppose that the differential
is given by dXk
=
z=
afj Xi 1\ Xj'
i,j
Let us consider the dual of this. Namely let V be the dual vector space of A(1) and {x;'h be its dual basis. Then the dual of the differential d is a linear map A 2 V ---> V described by
[x;,xjJ
=
z=at x;'. k
It is easy to see that the equality dod = 0 is equivalent to the Jacobi identity so that V becomes a Lie algebra. In this setting, it can be shown that A is nilpotent in the sense of Definition 1.33 if and only if V is nilpotent as a Lie algebra. 1.2.2. Minimal model. Let A, B be two d.g.a. 'So A linear map
f :A
---> B is called a d.g.a. mapping if it is a homomorphism as graded algebras and commutes with the operation of tal(ing differentials. In this case it induces a homomorphism
f* : H*(A)
~
H*(B)
1.2. MINIMAL MODELS OF DIFFERENTIAL GRADED ALGEBRAS
21
in cohomology. 1.35. Let A be a d.g.a. and let i be a non-negative A d.g.a. mapping
DEFINITION
integer or
00.
p:M----tA
is called an i-minimal model of A if it satisfies the following conditions. (i) M is a minimal, generalized nilpotent d.g.a. and M = Mi; that is M is generated by elements of degrees smaller than or equal to i. (ii) p* : H*(M) -+ H*(A) is an isomorphism for * S i and injective for * = i + 1. If i = 00, it is simply called a minimal model. EXAMPLE 1.36. The minimal model of the de Rham complex A*(sn) of the n-sphere sn is given by p : A(x) -+ A*(sn) for odd nand p : A(x,y) -+ A*(sn) for even n. Here degx = n,degy = 2n -l,dx = O,dy = x 2 and p(x) denotes the volume form of sn.
If we consider the d.g.a. version of the usual homotopy X x I -+ Y of a. continuous ma.pping X -+ Y, then we obtain the following definition. Namely two d.g.a. mappings h : A -+ B (i = 0,1) are said to be homotopic if there exists a d.g.a. mapping H: A
-+
B(t,dt)
such that HII~o = fa, Hlt=l = h- Here B(t, dt) = B Q9 A(t, dt) and degt = 0, degdt = 1, d(t) = dt, d(dt) = O. A(t, dt) serves as a model for the de Rham complex of the unit interval 1= [0,1]. We have also the following alternative definition, due to Sullivan [Su3], of homotopy of d.g.a. maps which corresponds to the adjoint map X -) Y I of the usual homotopy. He obtained this definition by constructing a model for the mapping space Y I = Map( I, Y). In this book we adopt this latter definition because it has some advantage over the former one, especially in its theoretical side. For example the proof that the homotopy becomes an equivalence relation can be given in a much simpler way. We begin by describing the model for yl = Map(I, Y). Let AD be an arbitrary d.g.a. For the moment the reader may assume that AD = K. Let A = Ao(x"J be a d.g.a obtained by adding free generators x" of positive degrees successively to AD such that the differential dx" is expressed as a polynomial (in the sense of graded
22
1. DE RHAM HOMOTOPY THEORY
algebra) of elements which appeared in earlier stages with coefficients in Ao. In other words, A is described as the union of an increasing series Ao C Al C ... c At C ... of sub d.g.a.'s such that the differential of any element XC" which was added to Ae-l to construct At, satisfies dx cx E At-I. In this setting, for each Xcx we prepare a copy Ycx of it and a new generator 8cx whose degree is less than that of Xcx by 1 and set I - -
= A ®Ao A(ocx) = Ao(xcx , Ycx, ocx).
A
PROPOSITION 1.37. In the above setting, it is possible to define a differential d of AI which extends the given one on A ®Ao A such that d8cx has the form d8cx = Ocx - 'TIcx satisfying the following condition. Here Ocx = Xcx - Ycx and'TIcx belongs to ALI where f! is the smallest number such that Xcx E At. Moreover, if we denote by I( ) the ideal generated by the elements in the parenthesis, we have
I(Jcx,ocx) = I(8cx, d8cx ), and it has trivial cohomology. PROOF. We use induction on f!. Let Xcx be any element of Al which was added to Ao· Then dx cx E Ao so that d(x cx - Ycx) = 0 (in case Ao = K we have dx cx = 0). Hence we can set rlcx = 0, and since d8cx = Ocx we have I(Jcc , occ) = I(8cx, d8cc ). Then we have the following natural isomorphisms:
(1.1)
A{ ~ Ao(x cc ) ® A(8cx, dJcc ),
A{/I(8cx , ocx) ~ AI·
0: runs through all the generators Xcx E Al as above. Since A(8cc , dJcc ) clearly has the trivial cohomology, it follows from the first isomorphism that H*(A{) ~ H*(AI). Hence we can conclude the triviality of cohomology of I(Jcx,ocx) from the second isomorphism.
Here
Thus we can start the induction. We assume that the claim has been proved up to Ae-I and consider the case of At· For any generator Xcx E Ai \ At-I, we have clearly
ALI
'3
d(xcx - Ycx) 1-70 E Ae-l/Ie-I ~ A e- I·
A:_
Here Ie-I denotes the corresponding ideal of I . It follows that d(xcx - Ya) E Ie-I· By the induction assumption, the cohomology of Ii-l is trivial so that there exists an element 'TIcx E Ie-I such that
1.2.
MINIMAL MODELS OF DIFFERENTIAL GRADED ALGEBRAS
23
rl1/o< = d(xa - Ya). Hence we can set diSa = 60< - 1/a· Then the equality I(iS a )6a ) = I(iSo<) diS a ) also holds for Ae so that the isomorphisms) which are obtained from (1.1) above by replacing AI, Ao(xa) by At, Ae-I (xa) respectively, hold. Therefore we can conclude that H* (Ie) = by the same argument as above, completing the induction step. D
°
The differential d of AI is not unique. However, we can see from the above proof that any two differentials are interchangeable with each other by an automorphism of AI which is the identity on A Q9Ao A. EXAMPLE 1.38. Clearly AI is not minimal. In order to understand the above definition, we take an example. If A = A(XI, X2), dXI = 0, dX2 = :x:r, then I
-
-
A = A(Xl,YI,61,X2,Y2,62) and the differential is given by diS l = where 1/1 = 0,1/2 = iS1(XI +YI)·
Xl -
YI - 1/1, diS 2 = X2 - Y2 - 1/2
DEFINITION 1.39. Let A, B be two d.g.a.'s and suppose that A is expressed in the form A = Ao(xa). Two d.g.a. maps Ii : A -> B (i = 0,1) which are the same on Ao are called homotopic relative to Ao if there exists a d.g.a. map H : AI -> B which extends the natural map fo Q9 II : A Q9Ao A --> B. In this case we write H : fo ~ h (reI Ao). We see from the remark after the proof of Proposition 1.37 that the above definition of homotopy does not depend on the choice of the differential of AI. The following proposition is almost clear from the definition. PROPOSITION 1.40. Homotopic d.g.a. maps induce the same homomorphism in cohomology. THEOREM 1.41 (Sullivan [Su3]). Let A be a cohomologically connected d.g.a. Then there exists an i-minimal model p:M--->A
for any 0 <::; i :::; 00. It is unique in the following sense. Namely if p' : M' -> A is another i-minimal model, then there is an isomorphism tp : M --> M' such that p ~ p' otp. Moreover the isomorphism tp itself is uniq'ue up to homotopy.
1. DE RHAM HOMOTOPY THEORY
24
The proof of this theorem will be given later: the existence in §L2.3 and the uniqueness in §1.2.4 respectively. Here we make preparation. DEFINITION 1.42. Let f : A ---> B be a d.g.a. map. We define the mapping cone of it, denoted C, as follows. First we set
c(n) = A(n) EB B(n-I)
and then define d : c(n)
--->
d(x, y) ,. (-dx, dy
c(n+l) by
+ f(x))
It is easy to check that dod
(x E A(n), y E B(n-I)).
= O.
Geometrically it is more natural to define d(x, y) = (d:r:, dy
+
(-l)nf(x)) rather than the above formula. Also the above mapping cone C is just a differential graded module and is not a d.g.a. because product is not defined in it. It is possible to define the mapping cone as a d.g.a. However, we use the above definition to avoid unnecessary complication. We leave the proof of the following proposition to the reader. PROPOSITION 1.43. Let f : A ---> B be a d.g.a. map and let C be its mapping cone. Then we have the following Mayer- Vietoris exact sequence ..• ----->
H*(C)
~ H*(A)
L
H*(B)
~ H*+l(C)
-----> ...
where PI and i2 denote the projection to the first factor· and the inclusion to the second factor respectively. Henceforth we write H*(A,B) for H*(C) and call it the relative cohomology group of f : A ---> B. 1.2.3. Proof of the existence of minimal models. In this subsection we prove the part of existence in the statement of Theorem 1.41. We prove the existence of i-minimal model by induction on i. If i = 0, the inclusion K c A of scalars clearly serves as the O-minimal model because A is cohomologically connected by the assumption. Now assume that i ~ 1 and let Po : N ----->
A
1.2. MINIMAL MODELS OF DIFFERENTIAL GRADED ALGEBRAS
25
be an (i - I)-minimal model. We have the following Mayer-Vietoris exact sequence
... -) Hi-l(N)
~
Hi-l(A) -) Hi(N,A) = 0 - ) Hi(N)
~ Hi(A) -) HHl(N,A) - ) Hi+1(N) -) HHl(A)
Hence if we set V sequence:
- > ....
. HHl (N, A), we obtain the following short exact
0-) Coker(Hi(N) --->Hi(A)) -) V -) Ker(H i +1(N) A basis of Coker (Hi (N) {Uj}j such that
--->
U·J
--->
Hi+1(A))
->
O.
Hi(A)) can be described by a certain set
E
A(i) ,
Also a basis of Ker(HHl(N) lows. Fix a generating system
--->
du J· = O. HHl(A)) can be described as fol-
for the kernel and choose Wk E A(i) such that PO(Vk) = dWk. Then {wkh can serve as the required basis. Hence we can take {iij, Wk}j,k for a basis of V where ii j , Wk denote copies of Uj, Wk respectively. Now we consider the Hirsch extension
.Nh ... N The differential d : V
--->
(8Id
A(V);.
N(i+1) is defined by
By the induction assumption, we have N = Ni-I. We can see from this that MI is also a generalized nilpotent minimal d.g.a. If we define a map PI
:M 1 ->A,
which is an extension of Po : N
--->
A, by
1. DE RHAM HOMOTOPY THEORY
26
then clearly PI becomes a d.g.a. map. Now consider the following commutative diagram: ------7
Hi(N)
------7
1
Hi(A)
------+
V
= Hi+1 (N, A)
1
II
------->
Hi(M I )
"" --+
------7
Hi+l(N)
------7
Hi+l(Mr)
Hi(A)
------7
~
Hi+l(A)
------7
------7
II Hi+l(A)
------7
i*l
Hi+l(MI' A)
Since M 1 is obtained from N by adding elements of degree i, we have Hk(N) S:! Hk(Mr) for any k ::; i - l. Also it follows immediately from the definition that
Hi(Mr)
S:!
Hi(N) EB (EBj[UjJ)
S:!
Hi(A)
and Ker(Hi+l(N) --> Hi+l(A)) c Hi+l(N) is sent to 0 by the map Hi+l(N) --> Hi+l(Ml)' Moreover we can see that the map V --+ HHI(Mr,A) is the 0 map. Now if we assume that HI (A) = 0 (geometrically this condition corresponds to the simply connected case), then JV(1) = 0 so that M~i+l) = N(i+l). Hence Hi+l(Md is not bigger than Hi+l(N), and by the construction Ker(Hi+l(N) --+ HHI(A)) vanishes in MI. lt follows that HHI(MI) --> HHI(A) is injective (and hence Hi+l(A11' A) = 0), and so PI : MI --+ A becomes an 'i-minimal model of A. If N is nilpotent, then clearly so is MI' To sum up, if HO(A) = HI(A) = 0, then we can conclude that there exists a minimal model
p:M -->A such that K = MO = MI C M2 C M3 C ... and Mi --+ A is an i-minimal model for each i and J\;(i C MHI is a Hirsch extension. In particular M is nilpotent. In case HI(A) =I- 0, the situation becomes more complicated. Since the mapping Hi+l(M2) --+ Hi+l(A) need not be injective anymore, in order to kill its kernel we have to add elements of degree i to MI to obtain M2. Since Hi+l(M2) --+ Hi+l(A) is not necessarily
1.2. MINIMAL MODELS OF DIFFERENTIAL GRADED ALGEBRAS
27
injective either, we have to add elements of degree i to M2 to obtain = 1,2,··· ). If we set M to
M 3 . Going on similarly, we make Mk (k he the union of all M k and consider
then p* : H* (M) -> H* (A) is bijective for degrees smaller than or equal to i and injective for degree i + 1. Now even if N were nilpotent, M may not be so because starting from N we have to apply possibly infinitely many Hirsch extensions to obtain M. This problem occurs already in the first stage of i = 1, and going on like M o = J( C MO,l C
MO,2 c···
UMO,k
C MI =
k
(1.2) MI = M1,o C MI,I C M 1 ,2 C ... C
M2
= UM1,k k
and as described above, an i-minimal model
M i-
1 = J(
C M
i - 1 "I
C M
i-
I 2 C ... C
Mi
= U M
i-
I ,k
k
is obtained by applying possibly infinitely many Hirsch extensions to the previous (i - I)-minimal model ;\.1 i - 1 . Finally the union
of all these models becomes a generalized nilpotent minimal model of A. The order type which describes the successive Hirsch extensions turns out to be w 2 as it stands. However, the number of generators of previous stages, which appear in the differential of any new generator, is finite. Hence we can reorder them to satisfy the condition of generalized nilpotence (although this operation is rather artificial). Thus we have completed the proof of existence of minimal models.
1.2.4. Proof of the uniqueness of minimal models. In this subsection we prove the latter part of Theorem 1.41, namely the uniqueness of minimal models. Let us recall the precise statement. Let A be a cohomologically connected d.g.a and assume that we are given two minimal models p : J\.Il -> A and p' : M' --+ A. Then our task is to prove that there exists an isomorphism rp : M ~ M' such
1. DE RHAM HOMOTOPY THEORY
28
that the diagram M~A
II
(1.3)
M' -
-+ p'
A
is commutative up to homotopy. Moreover we have to prove also that such map!.p is unique up to homotopy. By definition, M is generalized nilpotent so tha.t it can be expressed as the union of certain increasing series
(1.4)
Mo = K C Ml C M2 C ... C M£ C ...
of Hirsch extensions. It is natural to try to construct the desired map !.p by induction on f. Then there arise certain extension problems of d.g.a. maps. More precisely, we will apply the following proposition where we replace N,A,f,B in the statement by Me,M',p',A respectively. PROPOSITION 1.44. Let No be an arbitrary d.g.a. Let JV = No(x c,) be a d.g.a. obtained fr-om No by adding free generators Xc< of positive degrees successively such that each differential dJ:", is expressed as a polynomial (in the graded sense), with coefficients in No, of generators which appear in earlier stages. Also let N C .if = N Q9d A(V)k be an extension of d.g.a. such that dX(3 E N for any generator X(3 E V. Now assume that, in the following diagram of d.g.a. maps
N
1 --> 9'
B
which is commntative over No, there is given a homotopy H : NI -> 0 9 and g'IN'. Then there exists an element
B, relative to JVo, which connects two maps .f o E Hom(V,H k + 1 (A,B))
satisfying the following conditions. (i) 0 = 0 if and only if there exist a d.g.a. map.q: N Q9d A(V)k -> A which is an extension of 9 and a homotopy if : f 0 .q ':::= g' (reINo) which is an extension of H.
1.2. MINIMAL MODELS OF DIFFERENTIAL GRADED ALGEBRAS
29
(ii) Assume that f is surjective and fog = 9'IN. Then 0 = 0 ~f and only if there exists a d.g.a. map g : N ®d A(Vh ....... A, which 'is an extension of g, such that f 0 ,ij = g'. PROOF. By the definition of homotopy of d.g.a. maps (Definition 1.39), we can write N[ = No(xo"Ya,ba ), and also the differential of 8", has the form dba = Xa - YOt - 'f/Ot· It follows from the cllisumption that dXf3 EN for any Xf3 E V so that there exists all element 'f/fl E I(ba , dba ) C NI such that
d(x(3 - y(3) = d'f/{3, and abo we have H(dx{3) = define the required element spondence
f
0
0 E
g(d:J:{3) , H(dYf3) = g'(dx{3). Now we Hom(V, Hk+1(A, B» by the corre-
where Then dE,(3 = (-dg(dxfj), -dg'(x{3) - dH(17{3)
= (0, H(dx{3 - dYf3 - d'f/(3»)
co
+f
0
g(dx(3))
(0,0)
so that c'{3 is in fact closed under d. We prove (i). Suppose that, as in the claim, there exist g, if which arc extensions of g, H respectively. Then
f 0 g(X(3») (!J(dx(3),if(x{3 - y(3 - 'f/(3) - f 0 g(X(3»)
d( -.c;(x{3),iI(bfJ)) -, (dg(x{3),dif(bf3) =
= (g(dx(3), -g'(X(3) =
- H('f/f3»)
E,(Xfj)
so that 0 = O. Conversely assume that 0 = O. Then for each element x(3 E V there exists an element w(3 •. (a(3, bfl) E A (k) EEl B(k-l), which depeuds linearly on xf3, such that f,f3 = dWfj, namely (1.5)
g(dx(3)
Then we set
(l.6)
=
-da(3,
-g'(X(3) - H('f/f3) = db(3
+ f(afj).
1. DE RHAM HOMOTOPY THEORY
30
The fact that both from
9 and H commute with the
differential d follows
dg(x/3) = -da/3 = g(dx/3) = g(dx/3), dH(8/3)
= db/3 = f
0
g(x/3) - g'(x/3) - H(r]/3)
= H(x/3 - Y/3 - r]/3)
=
H(d8/3).
This completes the proof of (i). Finally we prove (ii). By the assumption, in the above construction, we may take H to be the constant homotopy, namely H(8cx ) = O. Hence H(r]/3) = O. Since f is surjective, there exists an element a~ E A(k-l) such that b/3 = f(a~). If we set wp = (a/3 + da~,O), then dw~ = dW/3 = ~/3. Hence we can use w~ instead of w/3 and so g(X/3) = -a/3 - da~. It follows from this that f o.9(X/3) = g'(X/3), completing the proof. 0 Now let us go back to diagram (1.3) and prove the existence of a map
e
~M'
p
and construct a map
so that the induction works and we can conclude the existence of <po Next, assuming the existence of another map
l.2.
MINIMAL MODELS OF DIFFERENTIAL GRADED ALGEBRAS
31
1/1/.1.11, g. Then the homotopy Telative to No defines an equivalence ,dation in this set.
PROOF. Since it is easy to check the reflexive law as well as the :;Vll1metric law, it is enough to prove that the homotopy satisfies the transitive law. Let x'" be free generators of N over No so that we can write N = No(x o ,). Now suppose that we are given three d,g.a. maps I, : N ---> A (i = 0,1,2) and two homotopies
.II : Nh .II c::= 12 : Nh fo c::=
----->
A
---->
A.
Ilere we may write
NIl = No(xcx, Yeo 15",,),
N!' = No(y"", z"" i5~)
where i5~ is the element corresponding to o~ = Ycx - z"" We have only extend the inclusion
Co
No(xcx, zO') Co
C
No (x", , YO" z"" JO' , i5~)
No(x O' , z"" J~) :) No(xa, z",). Here i5~ is the element corresponding
10 o~
= x'" - Z",. Assuming inductively that the image of di5~ has heen fixed, we must determine the image of i5~. These constructions ('an be done solely in the ideal I(J", , i5~, 0"" o~). On the other hand, it can be shown that this ideal has the trivial cohomology by the same argument as that given right after (1.1). This completes the proof. 0 Let us go back to diagram (l.3) again. Assuming that we are given two isomorphisms 'P, 'P' : M ---> M', both of which make the diagram homotopy commutative, we prove that 'P c::= 'P'. First the above Proposition 1.45 implies that p' 0 'P c::= p' 0 'P'. Hence we have a homotopy H. : MI ---> A connecting them. If we set No = M I8i M here, we can write MI = N o(i5",) because of the definition of MI. Consider the following commutative diagram:
No
----->
1 MI = No (15",)
M'
lp' -----> H
A
32
1.
DE RHAM HOMOTOPY THEORY
If we apply Proposition 1.44 to this diagram, we see that the obstruction to the existence of a d.g.a. map if : MI ---> M', which extends 'P 0 'P', lies in
H*(M',A). Since p' : M' ---> A is a minimal model, the above group is trivial. Hence if exists and this gives the desired homotopy between 'P and 'P'. Finally the fact that 'P : M ---> M' becomes an isomorphism follows from the following proposition. PROPOSITION 1.46. Let M, M' be generalized nilpotent minimal d.g.a. 's wh'ich are generated by elements of degrees smaller- than or equal to i where i = 1,2"" ,00. If there exists a d.g.a. map
'P: M
--->
M'
such that the induced homomorphism 'P' : H*(M) -) H*(M') 'is bijective up to degree i and injective for degree i + 1, then 'P is an isomorphism, PROOF. To begin with, let 171 be the set of all d-closed elements of
M whose degree is positive and the smallest (which we denote by nd, Then we see that there exists a natural isomorphism 171 ~ Hnl (M). We apply similar construction to M' and let VI ~ Hn~ (M') be the corresponding isomorphism, Then from the assumption, we can deduce that n~ = nl and 'P sends 171 bijectively to VI. If we set Ml = A(Vd and M~ = AW{), then they become sub d.g.a.'s of M and M' respectively. Next we set M/icleal MT F1 and choose an isomorphism M ~ M I 0F1 . Regarding M', we can also construct a similar isomorphism M' ~ M~ 0 F{ such that 'P(Fd C F(, Let 172 be the set of all homogeneous elements x E F1 with the smallest positive degree (which we clenote by n2) such that dx E MI' Let V{ be the corresponding set for M' and let n~ be its degree. Then (-dx, x) becomes a relative co cycle with respect to the inclusion MI -) M so that it defines an element of the cohomology group Hn 2 +I(M 1 , M). It can be shown that this correspondence induces an isomorphism V2 ~ Hn2+I(MI,M). By Proposition 1.43, we obtain an exact sequence
0---> H n2(M I ) ---> Hn2(M) --->
172
--->
H n 2+ I (M 1 )
--->
H n 2+ 1 (M)
---> ... ,
1.2. MINIMAL MODELS OF DIFFERENTIAL GRADED ALGEBRAS
33
II' we compare this exact sequence with the corresponding one for .M' and use two isomorphisms Ml 9'. M~, <po : H*(M) 9'. H*(M'), Lhen we can conclude that n~ = n2 and
Mo = [( C Ml C ... C Me C ... of Hirsch extensions. Let Xo: be an element which appears in the first Me with respect. to the inclusions. We write simply x for it. Then since dx is decomposable it can be expressed in terms of finitely many generators Yi. Also it belongs to an earlier stage than Me so that the degrees of y,; are aU strictly less than k. Hence dx E Moo. Now consider the cohomology class [dx] E Hk+l(Moo). It follows from H*(Moo) 9'. H*(M) that [d:z; = O. Therefore there exists a certain element y E Moo such that dx = dy. Now consider [x - y] E Hk(M). By the above isomorphism again, there exist elements z E Moo, W E M sHch that x - Y + dw = z. We then have x = y - dw + z. Since the degree of w is k - 1, we have w E Moo. Hence x E Moo. Thus we obtain a contradiction, and this completes the proof. 0 The proof of existence and uniqueness of minimal models was done by making use of Proposition 1.44. If we examine the proof of it carefully, we see that the following theorem is actually proved. THEOREM 1.47. Let f : A ---> B be a d.g. a. map inducing an isomorphism H*(A) 9'. H*(B) in cohomology. Then for any generalized nilpotent, minimal d.g.a. M, the natural map [M,A]---> [M,B] is a b~jection. Here [M, A] denotes the set of homotopy clas.ses of d.g.a. maps from M to A.
L DE RHAM HOMOTOPY THEORY
34
COROLLARY 1.48. Let A, B be cohomologically connected d.g.a. 's and let f : A -> B be a d.g.a. map. Also let PA : MA -> A and Pl3 : Ml3 -> B be minimal models of A and B re8pectively. Then there exists a d.g.a. map j: MA -> Ml3 which makes the diagram
MA~A
M6~B P13
homotopy commutative. Moreover, such a map is unique up to homotopy. PROOF. It is enough to set A in the statement of Theorem 1.47.
= Ml3,f = Pl3,B = B,M
co
MA 0
1.3. The main theorem 1.3.1. Differential forms on simplicial complexes. The de Rham complex A*(M) of a Coo manifold M is a. d.g.a. over ~ so that in principle we cannot deduce information on the rational homotopy type of M from it. If there a.re given enough cydes of Mover Z, then by investiga.ting values of integrals over them, we can decide whether a given closed form represents a rational cohomology class or not. However, this procedure cannot be considered as an intrinsic structure of the de Rham complex. Then there appeared the de Rham theory for simplicial complexes which turn out to be utilized to obtain information about their structure over Q. To define the de Rham complex of simplicial complexes, first consider the k-dimensional standard simplex
!J.k = {(to,tI'''' ,tk)
E JRk+l;ti
2
O,'L)i = l}.
1.49. (i) The restriction of any Coo differential form on JRHI to 6,k is called a Coo form on it. (ii) The restriction of a form DEFINITION
j 'I(to , ... ,tk)dt· t , 1 1\···l\dt·t..,whose coefficients h(to,'" , tk) E Q[to, tl,'" , tk] are polynomials with rational coefficients to !J.k is called a Q polynomial form.
1.3. THE MAIN THEOREM
35
We denote by A * (6, k) and AQ (6, k) the sets of all Coo forms and Q polynomial forms on 6, k respectively. For any face 6,r c D. k, the restriction of forms induces a natural homomorphism A*(6,k) ----> A*(6,T) and similarly for the Q polynomial forms. DEFINITION 1.50. Let K be a simplicial complex. A Coo q-form on K is a collection
(w,,- ),,-EK of q-forms w,,- E Aq((J) on each simplex (J of K such that if T is a face of (J, w,,-IT = w T . We denote by A*(K) the set of all Coo forms on K and call it the de Rham complex of K. If L c K is a subcomplex, the relative de Rham complex of the pair (K, L) is defined by
A*(K,L) = {w
E
A*(K);w,,- = 0 for any
0'
E
L}.
Similarly the de Rham complex AQ(K) of all Q polynomial forms on K as well as its relative version AQ(K, L) are defined. The exterior differential and the exterior product are defined for
A*(K,L), AQ(K,L) exactly the same way as the case of manifolds. Namely we set
d(w,,-) = (dw,,-),
(w,,-)
1\
(T,,-) - (w"
1\
T,,-).
Thus A*(K, L) and AQ(K, L) become d.g.a.'s over lR and Q respectively. The integration on each simplex induces natural maps
I: A*(K,L)
(w,,-)
f------>
I: A;Q(K,L):OJ (w,,-)
f------>
=:1
1 ~1
{O' ~
w,,}
E
C*(K,L;lR)
{O'
w,,}
E
C*(K,L;Q)
which commute with the differential operators by virtue of the theorem of Stokes. For the second map, we use the fact that the integral of any Q polynomial form on a standard simplex is a rational number. It follows that I induces a homomorphism in cohomology. THEOREM 1.51 (de Rham theorem for simplicial complexes). Let K be a simplicial complex. Then the map I induced by the integration gives isomorphisms
H*(A*(K)) H*(AQ(K))
~ ~
H*(K;lR) H*(K; Q)
of algebras. The same is true for' the relative cohomology groups of any subcomplex L c K.
1. DE RHAM HOMOTOPY THEORY
36
We first show that the claim holds for t:J. k . We assume that the case A*(t:J.k) is already known and consider the case AQ(t:J.k). Since the unique relation between the coordinates and I-forms on t:J.k is to + ... + tk = 1, dto + ... + dtk = 0, PROOF.
we can conclude that there exists a natural isomorphism
A;Q(t:J.k)
=:!
Q[tl,··· ,tkJ 0 E
It is easy to see that the d.g.a. on the right hand side has trivial cohomology. Next we show that for any sub complex L c K the homomorphism
A;Q(K)
----t
A;Q(L)
induced by restriction of forms is surjective. Let W = (w r )rEL E A*(L) be any element. Let 0" E K \ L be a k-simplex and assume that a (k - I)-dimensional face T -< 0" of it belongs to L. Denote by v the vertex of 0" opposite T and choose coordinates to, ... ,tk of 0" such that v corresponds to tk = 1. Then the stereographic projection 7r : 0" \ {v} -+ T from v can be written as 7r(to,··· ,tk) Since
1 1 - tk
= - -(to,···, tk-l).
1
d(l_ t) =
1
0- tk)2 dtk,
if we take N large enough, then WI
= (1 - tk)N 7r*wr
define!:) a Q polynomial form on 0" and clearly wIlr = w r . Next suppose that another (k - I)-dimensional face T' =f. T of 0" also belongs to L. Let v' be the vertex opposite T' and let W2 be the Q polynomial form on (5 obtained by applying the same operation as above to Wr ' - WIlT' (see Figure 1. 3) . It follows from the construction that
where the second equation follows from (wr' - wIlr' )IT()r' = O.
Hence WI
+ W2 is a Q polynomial form on and (WI + w2)lr = Wr , (WI + W2)lr' = (J
Wr'·
1.3. THE MAIN THEOREM
37
v
v
v'~---
tk-,=1
FIGURE
1.3
In this way, if we apply the same operation to all (k -I)-dimensional faces of (J, then we obtain a Q polynomial form on (J whose restriction to (J n L coincides with that of w. By applying this process to all simplices of K \ L, we obtain the desired form. In the case of Coo forms, it is enough to replace (1 - tk)N in the above argument by a Coo function on (J which takes the value 0 near the vertex v and 1 near a neighborhood of T. With the above preparation, we prove the theorem by induction on dim K = n. We treat only the case of Coo forms because the proof for Q rational forms is the same. The claim clearly holds for n = O. We assume that it holds for dim K < n and prove the case dim K = n. Consider the following commutative diagram:
. -->
A * ( .6. n )
_.---+
A * ( 8.6. n )
11
-----------.
0
11
where we omit the coefficient lR in the bottom row. By the fact we proved in the beginning a.s well a.s the induction assumption, the right two 1's in the above diagram induce isomorphisms in cohomology. If we apply the five lemma to the induced long exact sequence in cohomology, we obtain an isomorphism
(1.7)
H*(A*(.6. n , 8.6. n ))
~
H*(.6. n , 8.6. n ).
1. DE RHAM HOMOTOPY THEORY
38
Next we consider the commutative diagram o ----> A*(K,K(n-I)) - - - 4 A*(K)
11 o ---->
C*(K, K(n-I)) -
C*(K)
---4
---->
A*(K(n-I))
---4
C*(K(n-I)) - -
---->
0
... 0
where K(n-I) denotes the (n -I)-skeleton of K. Now A*(K, K(n-I)) is isomorphic to the direct sum of d copies of A * (L\ n, 8L\n) , where d is the number of n-simplices of K. Similarly C*(K, K(n-I)) is isomorphic to the direct sum of d copies of C*(L\",8L\rt). Hence by (1.7), their cohomology groups are isomorphic. By the same argument as above, applying the five lemma to the induced long exact sequence in cohomology, we obtain an isomorphism H*(A*(K))
~
H*(K).
The relative version can be proved similarly. This completes the proof.
o
For the product structure, we refer the reader to [GlVI). 1.3.2. Homotopy groups of minimal d.g.a. 'so
1.52. Let M be a d.g.a. over the ground field K. (i) Let MI C M be the sub d.g.a. generated by elements of degree 1. We define sub d.g.a.'s Ni (i = 0,1,2,· .. ) of MI as follows. DEFINITION
No=K
= sub d.g.a. N2 = sub d.g.a.
Nl
of Ml generated by {x E M\ dx
= O}
of j\1/1 generated by {x E M\ dx ENd
N3 = sub d.g.a. of MI generated by {x E MI; dx E N 2 }
Then each N, C Ni+l is a Hirsch extension and MI If we take the dual of these, we obtain a series . . . ----+
£i
----t . . . ----t
.c 3 ----t £2 ----t £ 1 ----t
°
= UiNi.
1.3. THE MAIN THEOREM
39
of nilpotent Lie algebras £i over K. Here each surjective homomorphism £i+l --; £i is a central extension of Lie algebras. If we set Li to be the Lie group corresponding to £;, the Baker-Campbell-Hausdorff formula implies that the exponential map exp : £i --; L; is bijective and the product in L; can be expressed by explicit polynomial mappings. We define the fundamental group 7rl (M) of M to be the tower . . . ----->
Li
------> . . . ------>
L3
------>
L2
------>
L1
------>
0
of nilpotent Lie groups. These matters will be further explained in §1.4. (ii) Let M+ = EBi>oM(i) be as before and set
I(M) = M+ /M+ M+
= EBi>O I(M)C'i). In other words, I(M)(i) is the set of all indecomposable elements of M. We set
7ri(M) = Hom(I(M)(;), K) and call it the i-th homotopy group of M. The differential d induces a bilinear mapping
7ri(M) ®7rJ (M)
------>
7r·i+J-l(M).
It can be deduced from the fact dod = 0 that 7r. (M) EBi7ri(M) becomes a graded Lie algebra. DEFINITION 1.53. Let (X, xo) be a topological space with a base point. A mapping
7ri(X,XO) x 7rj(X,Xo)
3
(0:,,6)
f------+
[a,,6]
E
7ri+j-l(X,XO),
called t.he Whitehead product, is defined as follows. If two elements a, ,6 are represented by continuous mappings f: (r',ar') --; (X,xo),
g: (Ij,[)Jj) --; (X,xo)
respectively, then [a,,6] is the element represented by
ar'+J
=
Ii
X
uP
U
ali x Ij ~ X.
If i = j = 1, [a,,6] is the usual commutator in the fundamental group. In the cases i > 1 or j > 1, the Whitehead product is distributive with respect to the corresponding factor and also
[,6, 0:]
=
(-1) ij [a, ,6].
40
L DE RHAM HOMOTOPY THEORY
Moreover it is known that the Jacobi identity
(_I)ik[[a,/JJ,"/]
+ (-I)ij[[/J,,,/J,a] + (-1)1k[b,a],!3]
holds where "/ E 7rk(X,XO). With these structures, 7r*(X) = EBk7rk(X) becomes a graded Lie algebra over Z. The next theorem is the main theorem of the de Rham homotopy theory. THEOREM 1.54 (Sullivan [Su2][Su3]). (i) Let K be a simplicial complex which is nilpotent and of finite type; that is the act'ion of Jrl (K) on 7ri (K) is nilpotent and Jri(K) is .finitely generated for any i. Let AQ(K) be the d.g.a. consisting of all Q polynom'ial forms on K and let M f( be its minimal model. Then M J( and the rational homotopy type of K are related as duals to each other 80 that each of them determines the other completely. In particulaT there e:risi;s a natural isomorphism
as graded Lie algebras over Q. (ii) Let f : K -7 L be a simpl'icial map between nilpotent simplicial complexes and let f* : AQ (L) -) AQ (K) be the induced homomorphism. Denote by .1* : M L -7 M f( the d. g. a. map between the m'inimal models induced by f*. Then it is the d'ual of the localization map fa : Ko -7 Lo of f so that each of.f* and fo deteTm'ines the other completely. In particular, under the isomorphism of (i), the mop .1* : Jr * (M K ) -7 Jr. (Md coincides wil;h f. : 1f* (K) ® Q - 7 Jr * (L) 0Q. THEOREM 1.55 (Sullivan [Su3]). Let M be a Coo manifold which is nilpotent and of finite type and let K be a Coo triangnlation of d. If we denote by Ai /1'/ the minimal model of the de Rham comple:r A * (Ai) of Ai, then theTe exists a natural isomorphism MM~MJ(0R
In particular- we have 1f*(Ml'vd ~ 1f.(M) @ JR. A similaT statement as that of the pTev'ioll,s theorem holds fOT Coo mappings between two manifolds.
Sketch of Proof of Theorem 1.54.. The key to the proof is the following fact which exhibits certain bijective correspondence between principal fibrations with fiber an Eilenberg-MacLane space and Hirsch extensions. Let X be a nilpotent simplicial complex and suppose
41
1.3. THE MAIN THEOREM
that there is given a d.g.a. map M --> AQ(X) which induces an isomorphism in cohomology. Let V be a finite dimensional vector space over Q and let
K(V,n)
--4
E
--4
X
be a principal fibration over X with fiber K(V, n). Let
o E Hn+l(x; V) ~ Hom(Hn+ 1 (XiQ), V) be its characteristic cohomology class. Then the dual element can be expressed as 0* E
0*
of
0
Hom(V*,H"+l(X;Q)) ~ Hom(V*,Hn+l(M)).
On the other hand, the isomorphism class of a Hirsch extension
is classified by its characteristic cohomology class 0' E
Hom(V*,HM1(M))
(see Proposition 1.32). Then, as is expected by the fact that the two elements 0*, 0' belong to the same set, we can conclude as follows. Namely, there exists a d.g.a. map
M ®d A(V*)"
--4
AQ(E),
which induces an isomorphism in cohomology and coincides with the given map M -, AQ(X) on M, if and only if the equality 0*
= 0'
holds. This fact is natural and should be easy to understand conceptually. However, we need many technical preparations for a rigorous proof of it. For example, in the above, E is not a simplicial complex as it stands so that we have to construct a model of E --> X in terms of simplicial complex ami simplicial maps. We refer the reader to the book [GM] for details. Assuming the above fact, let us examine the ingredients of the theorem on simply connected simplicial complex K for simplicity. If we set 1f2(K)®Q = V2 , the first stage in the Postnikov decomposition of the rational homotopy type K 0 of K is given by
Ko
--4
K(V2, 2).
On the other hand, there exists a natural isomorphism H2(K; Q) ~ V2* so that we can write MJ{ = A(V;*). Then J(MK )(2) = V2* = Hom(7T'2(K), Q).
1. DE RHAM HOMOTOPY THEORY
42
Next, if we set 11"3 (K) Q9 Ql = V3 , the second stage in the Postnikov decomposition is given by the principal fibration
K(Va,3)
K O(3)
---->
---->
K(V2, 2).
If we denote its characteristic cohomology class, namely the k-invariant of Ko, by
k 4 (Ko) E H4(K(V2' 2); Va) ~ Hom(H4 (K(V2, 2», V3), its dual element can be written as
k4 (Ko)* E Hom(V3*' H 4(K(V2, 2»). Then the claim is that the Hirsch extension
A(Vn Q9d A(Vn which corresponds to the above element is nothing but M 3 , that is the 3-minimal model of AQ(K). Moreover, if we take the dual of the differential where we can identify
A 2 (Vn = {homogeneous polynomial of degree 2 generated by H2 (K; Ql) },
we obtain the dual mapping {homogeneous polynomial of degree 2 generated by H 2 (K; Ql)} ---->
V3 = 7r3(K)
Q9
Ql.
Now the claim is that this corresponds to the Whitehead product
1f2(K) Q9 1I"2(K) ....... 1f3(K). Explicitly, if we choose a basis Xb··· ,Xe of H 2 (K;Ql), we can write M7( = A(X1,··· ,xe). Let xi,··· ,xi be the dual basis of 1I"2(K) Q9 Ql. Let Y1,··· , Ym, Z1>··· , Zn be new generators of MK of degree 3. Here Yl,··· , Ym is a basis of H:3 (K; Ql) and Zl,··· , Zn is a basis of Ker(H 4 (M}) ....... H 4 (K;Ql»). Hence we can write dYi = 0,
Then yi, ... , y:n, zi, and we have
... , z~
dZ k = LatxiXj.
can serve as the dual basis of 1f3(K) Q9 Ql,
* Xj*] [Xi'
k * = "" L...- aijzk·
The situation is similar for higher degrees. Namely, the k-invariant of K 0 at each stage corresponds to the characteristic cohomology class
1.4. FUNDAMENTAL GROUPS AND DE RHAM HOMOTOPY THEORY 43
of the Hirsch extension appearing in the corresponding stage of MK, and quadratic terms of the differential d of the Hirsch extensions correspond to the Whitehead products on 1f*(K) 0 rQ. 1.4. Fundamental groups and de Rham homotopy theory 1.4.1. Lower central series and nilpotent groups. Let r be a group. We put ro = r, and for general k ::::: 0 we inductively define rk+l = [rk, r]. r 1 = [r, r] is the commutator subgroup of r. rk is a normal subgroup of r for any k. The series
r :J r 1 :J ... :J r k :J ... of these normal subgroups is called the lower central series of r. If r k is trivial for some k, r is said to be nilpotent. Any abelian group is nilpotent. The quotient group Nk = r / r k is called the k-th nilpotent quotient of r. Nl = r/[r,r] is the abelianization of r, and Nk is nilpotent for any k. A short exact sequence 1 ~ A ~ G ~ Q ~ 1 consisting of homomorphisms of groups is called an extension of the group Q by A. In the case where A is an abelian group contained in the center of G, it is called a central extension and denoted by
o ----> A
---->
G
---->
Q
---->
1.
If we set Ak = rk-drk, it is easily seen to be an abelian group contained in the center of Nk = r / r k. Hence we obtain a series
(1.8)
0
-----.c,
Ak
---->
Nk
---->
N k- 1 ----> 1 (k
= 1,2,· .. )
of central extensions. By collecting them, we obtain homomorphisms from the given group r to a tower
(1.9)
II
1
r
N3
1
II r
N2
.)
1
II r
----->
N1
=
r/[r,r]
1. DE RHAM HOMOTOPY THEORY
44
of nilpotent groups such that each step tension.
Nk -)
Nk -
1
is a central ex-
1.4.2. Central extensions of groups and the Euler class. Given a group G and an abelian group A, let us consider the problem of constructing central extensions of the form
(1.10)
0
----->
A ~G~Q~1
and also the classification of them. To investigate the structure of such an extension, choose a map s : Q -> G such that 7r 0 s ,idQ. If we can take s to be a homomorphism, then it is easy to sec that G ~ A x Q. Therefore we consider the correspondence
Qx Q
3
(0:,{3) ........ c s (a,{3) = s(a)s({3)8(af3)-1.
E
A.
A simple computation shows that the equality
cs ({3, 'Y) - cs (a{3, 'Y) + cs(a, (3'Y) - cs(o:, (3) = 0 holds for any a, (3, 'Y E Q. It is called the co cycle condition. We denote by Z2(Q; A) the set of all mappings c : Q x Q -> A satis{ying the co cycle condition, and any element of it is called a 2-cocyde of Q with values in A. The above mapping Cs is called the 2-cocycle associated to s. Let C s ' be the 2-cocyde associated to another map 8' : Q -> G. In this case, it can be shown that if we define a map d: Q -> A by d(a) = s'(a)8(a)-1 E A, we have (1.11)
c'(o:, (3) - c(a,f3) = d({3) - d(a[3)
+ d(a).
In general, for any given map d : Q -> A, if we define od : Q x Q -) A by setting lSd(a, (3) to be the right hand side of (1.11), then it becomes a 2-cocycle. Such a 2-cocycle is called a 2-coboundary, and the set of all 2-coboundaries is denoted by B2(Q; A) c Z2(Q; A). The equation (1.11) can now be written as c' - (' = od. Given any element c E Z2(Q; A), define a product on the set A x Q by
(a, a)(b, (3) = (a
+ b + c(a, (3), 0:(3)
(a, bE A, 0:, (3 E Q).
Then it can be shown that A x Q becomes a group which is a central extension of Q by A. We denote this group by A XC Q. In this case, if we define s : Q -> A Xc Q by s(a) - (0, a), then clearly Cs = c. Moreover, for another 2-cocycle c' E Z"2(Q; A) if there exists a map d : Q -> A such that c' - c = lSd, then it can be shown that we can construct a group isomorphism A xc' Q ~ A xcQ which is the identity on A. We then consider the quotient group
H2(Q; A) = Z2(Q; A)jB 2 (Q; A)
1.4. FUNDAMENTAL GROUPS AND DE RHAM HOMOTOPY THEORY 45
;1I1d call it the two dimensional cohomology group of the group Q with coefficients in A. It follows from the definition that the element hl E JI2(Q;A) is well defined independent of the choice of s. We (all it the Euler class of the central extension (1.8). Summing up the above argument, we obtain the following theorem. THEOREM 1.56. Let Q be a group and let A be an abelian group. Then the set of isomorphism classes of central extensions of Q by A can be naturally identified with H 2(Q;A) = Z2(Q;A)/B 2 (Q;A) by associating the Eule'r class.
We identify Sl with the Lie group SO(2). As is well known, the set of isomorphism classes of Sl-bundles Sl ---4 E ---4 X over a topological space X can be identified with H2(X; Z) by associating the Euler class to each Sl-bundle. The above theorem is an analogue of this fact in the context of group theory. In fact, the classification of central extensions of a group Q by Z corresponds exactly to the geometrical classification of Sl-buncHes over K(Q,l). Here the 2dimensional cohomology of a group appeared. We can also consider the cohomology group of any dimension. DEFINITION 1.57. Let r be a group. The cohomology group H*(K(T, 1)) of the Eilenberg-MacLane space K(T, 1) is called the cohomology group of r and is denoted by H* (r).
This is a. geometrical definition of cohomology of groups. There is also a purely algebraic definition (see §3.6.2). The above description of central extemiions in terms of 2-dimensional cohomology of groups is one such example. Also we can consider any r-module for cocHicients of cohomology groups. For a general theory of cohomology of groups, we refer the reader to [Brl. Let M be a manifold, or more generally a topological space, a.nd let M ---4 K (7r1 (M), 1) be the first stage in the Postnikov decomposition of it. This map induces a hOlIlomorphism H*(7rl(M))
--->
H*(M).
This homomorphism already played a key role in the formulation of the Novikov conjecture, which has been one of the fundamental problems in topology of manifolds. It seems that the importance of this kind of homomorphism will increase in the future in not merely geometry but also in algebraic geometry and number theory.
1. DE RHAM HOMOTOPY THEORY
46
r
1.4.3. Malcev completion. Let be a finitely generated group. In this subsection, we define r ® Q by using the results of the preceding subsections §1.4.1 and §1.4.2 (see [M] for details). It is a certain projective system consisting of nilpotent Lie groups over Q and homomorphisms between them. It contains all the nilpotent information over Q of the group r. r ® Q is called the Malcev completion or the rational nilpotent completion of r. Roughly speaking, r ® Q is the diagram
1
II r
(1.12)
N3®Q
-->
1
II r
N2®Q
----?
1
II r
----?
NI®Q=Hl(r;Q)
which is obtained by replacing each nilpotent group Nk by Nk ® Q in the tower (1.9) of nilpotent groups. Nk ®Q can be defined inductively as follows. If k = 1, NI is an abelian group so that NI ® Q is defined and we have a natural homomorphism Nl ---t Nl ® Q. In this case, H*(NI ®Q; Q) ---t H*(NI; Q) is an isomorphism. We now assume the existence of N k - I ®Q and a homomorphism N k - I ---t N k - 1 ®Q such that the induced homomorphism H*(N~'_I ® Q;Q) 2:' H*(Nk-I;Q) in Q cohomology is an isomorphism. Consider the homomorphism H 2 (Nk_ 1 ; A k )
--7
H 2 (Nk _ 1 ; Ak ® Q)
2:'
]-J2(Nk_l ® Q; Ak ® Q).
The Euler class of the central extension (1.8) is an element of the group H 2 (Nk _ 1 ;A k ). We then consider its image, under the above homomorphism, in H2(Nk_1 ® Q; Ak ® Q). If we consider the associated central extension, we obtain a commutative diagram
o -----+
Ak
1
----t
1
1
1
1.4. FUNDAMENTAL GROUPS AND DE RHAM HOMOTOPY THEORY 47
By this, Nk Q9 Q and a natural homomorphism Nk --t Nk Q9 Q Ilave been simultaneously constructed. It can be shown, by an argument using the spectral sequence, that this homomorphism induces ;tn isomorphism H* (Nk Q9 Q; Q) ~ H* (Nk:; Q) in Q cohomology. This completes the induction step. Geometrically, Nk Q9 Q is nothing but [he fundamental group of the rational homotopy type K(Nk , 1)0 (see ~i1.1.3) of the Eilenberg-MacLane space K(Nk, 1). 1.4.4. Fundamental groups and differential forms. The main theorem of the de Rham homotopy theory concerning the fundamental groups can be stated as follows. The Malcev completion of the fundamental group (tcnsored by lR) is equivalent to the I-minimal model of the de Rham complex. In this subsection, we briefly explain this theorem from the viewpoint of constructing both sides explicitly. Let M be a Coo manifold. We denote by the fundamental group Jrl(M) of M and let Ml be the I-minimal model of the de Rham complex A * (M). First we define Jrl (M) Q9 lR as follows. We consider the Malcev completion r Q9 Q and replace each nilpotent Lie group Nk Q9 Q by simply connected Lie group Nk Q9lR over R On the other lland, the I-minimal model Ml was obtained as an increasing series
r
Mo
= lR C Nl C
N2
C ... C Ml =
UNk k
of d.g.a.'s over JR. (see §1, 2, 3 (1.2) and §1.3.2, Definition 1.52 (i)). Now each Nk is a connected free d.g.a. generated by elements of degree 1. Hence, by taking the dual as in §1.2.1, Example 1.34, we obtain a Lie algebra nk over R Moreover, since nl ~ Hl(M;JR.)* is a commutative Lie algebra, it can be shown by induction on k that nk is a nilpotent Lie algebra for any k. In fact, nk can be obtained from nk-l by a central extension
(1.13) of Lie algebras. Here we have the identification Uk
= Hom(Ker(H 2 (Nk _d
--t
H 2 (r; lR)), lR).
Let N~ be the simply connected nilpotent Lie group corresponding to the nilpotent Lie algebra nk. Then the exponential map exp : nk --t is a bijection. Explicitly, it is described in terms of polynomial mappings by the Baker-Campbell-Hausdorff formula. Geometrically, Nk can be naturally identified with the d.g.a. consisting of left invariant differential forms on N~.
N:
48
1.
DE RHAM HOMOTOPY THEORY
The central extension (1.13) of nilpotent Lie algebras induces the associated central extension (1.14)
0 ~ A~ - , Nf
-->
Nf_l - , 1
of nilpotent Lie groups, which is equivalent to the former. Then the key fact, in the de Rham homotopy theory of fundamental groups, is that the above central extension is equivalent to the tensor product with IR of the central extension (1.8) which is associated to the lower central series of = 7fl (M). In particular, we have Nj;· ~ N k @ IR. This can be proved a..'l follows. To begin with, clearly HI (Nk ) ~ HI (r) = Nl for any k. I-Ience, by the long exact sequence of the cohomology of the group extension
r
1
-->
rk - 1 ~ r ~ N k - 1
--t
1
(cE. [BrD, we can deduce
H1(rk_l)r ~ Ker(H 2(Nk _ 1 )
......
H2(r»).
On the other hand, it is easy to see that HI (rk - 1 ; Q)r ~ (rk - 1 / I k)0 Q = Ak ® IQ. It follows that the tensor product with IR of the central extension (1.8) corresponds to
Ker(H 2 (Nk _ 1 ; lR) -, H2(r; IR»). On the other hand, the above central
exten~ion
(1.14) corresponds to
Ker(H 2 (Nk _d ...... H 2 (r;IR»). Then it can be proved by induction on k that there exists a natural isomorphism H 2 (Nk _ 1 ) ~ H 2 (N k _ 1 ;1R) ~ H 2(Nk _ 1 0Q;lR). The claim is now proved. Thus we obtain t.he following theorem. THEOREM 1.58 (Sullivan [Su3D. Let M be a Coo man~fold. Then the tower of nilpotent Lie groups {Nk®lRh, associated to the IR nilpotent completion 1fl (M) ® lR of the fundamental group of M, is canonically isomorphic to the tower of nilpotent Lie groups which is obtained by first d'ualizing the I-minimal model of the de Rham complex A*(M) and then taking the exponential. In the above, if we replace M and its de Rham complex A*(M) by a simplicial complex K and its de Rham complex AQ(K) of Q polynomial forms, then we obtain a similar theorem which holds over Q.
CHAPTER 2
Characteristic Classes of Flat Bundles Let G be a Lie group. One way of investigating global structure of prillcipal bundles with structure group G is by putting connections on them and then studying the associated curvature. This is the ingredient of the Chern-Weil theory. The most important cases are those where G is a general linear group, namely the cases of GL(n, JR) or GL(n, C). These correspond to considering real or complex vector bundles where characteristic classes like Pontrjagin classes or Chern classes play fundamental roles. Now a flat bundle, which we investigate in this chapter, is a principal bundle endoweci with a connection such that its curvature vanishes identically. Any characteristic class with real coefficients of such bundle is zero. Therefore it may appear that it is close to a trivial bundle. However, it will turn out that this is far from being true, depending on the fundamental group of the base space. We need new methods to study the structure of such bundles. The holonomy groups as well as the theory of characteristic classes of flat bundles, which is based on cohomology theory of Lie a.lgebras, can serve as those. In this chapter, we describe basic facts concerning these materials. We also give a brief account of the Gel'fand-F\lks cohomology theory ([GFl] [GF2]) which corresponds to the case where the structure group is infinite dimensionaL 2.1. Flat bundles 2.1.1. Chern-Weil theory. Let G be a Lie group and let 1r : P -) 1\11 be a principal G-bundle over a Coo manifold 1\11. Namely there is given a right action
of the structure group G on the total space P satisfying the following condition. 49
50
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
Local triviality: For any point p EM, there exist an open neighborhood U 3 P and a diffeomorphism r.p : 7r-l(U) S:! U x G such that
7r( ug)
= 7r( u),
r.p( ug)
= r.p('u)g
(u E 7r- l (U), 9 E G).
For example, the tangent frame bundle 7r : P(NJ) ---t M of NJ becomes a principal bundle with structure group GL(n,IR), where dim M = n, and it is a very important principal bundle for the investigation of the structure of M. In fact, for the study of manifolds, it is one of the main tools to consider various bundles over NJ, not merely the tangent frame bundle, and then to examine the structure of them. Now the Chern-Weil theory is the one which can analyze the structure of principal bundles. In this subsection, we review this theory briefly. Given a principal bundle 7r : P ---t NJ, in order to make certain connections between fibers 7r- 1 (p)(p EM), we first introduce what is called a connection on it. More precisely, at each point u E P in the total space, we give a direct sum decomposition (2.1) of the tangent space which is invariant under the action of G. Here Vu denotes the subspace consisting of all tangent vectors at u which are tangent to the fiber 7r- 1 (7r(u)) (these are called vertical vectors). Any tangent vector belonging to Hu is called a horizontal vector (with respect to this connection). The set of all Hu , namely 1-l = {Hu; u E P} becomes a distribution on P. In this terminology, we can say that a connection is nothing but a distribution on the total space which is G invariant and transverse to the fibers.
If we express connections in terms of differential forms, we obtain connection forms. Explicitly, if we denote by 9 the Lie algebra of G, then at each point u E P the projection TuP
--4
Vu S:! 9
induced by the direct sum decomposition (2.1) defines a I-form w E Al(p; g) on P with values in g. The form w thus obtained is called the connection form. Here 11" S:! 9 is the canonical identification. Clearly we have
Hu
= {X
E
TuP;w(X)
= O}.
Conversely, if we are given a I-form w E A1(P; g), then we have the associated distribution 1-l which is defined by the above equality.
51
2.1. FLAT BUNDLES
We can then express the condition that 1i becomes a connection by simple equalities of differential forms. Therefore connection forms are rrequently called simply connections. If there is given a connection W E A1(P; g) on a principal Gbundle, then we can define its curvature n E A 2 (P; g) by d~fferentiat ing it. These two forms satisfy the following fundamental equation: (2.2) which is called the structure equation. Now consider the k-th power
of the curvature form n and compose it with an invariant polynomial f E Jk (G) of G) namely a symmetric multilinear map f:gx···xg--d~
"----v----' k-times
which is invariant under the adjoint action of G. Then we obtain a 2k-form
on P. It can be shown that the form obtained in this way is always closed and moreover it is the pullback of a uniquely defined form on the base space under the projection 7f : P ---> 1\11. This procedure defines a homomorphism w: J(G)
----->
H*(M;JR)
(J(G)
=
Ef7klk(G))
which is called the Weil homomorphism. It can be proved that this homomorphism does not depend on the choice of a connection. Hence, for any element f E J(G), its image w(f) E H*(M;JR) expresses the way a given principal G-bundle is twisted in terms of real cohomology classes of the base space. We call these classes characteristic classes of principal G-bundles. If we unify connection, curvature and invariant polynomials in a single object, we obtain the Weil algebra
W(g)
=
A*g*18i S*g*.
It serves as a model for the de Rham algebra of the total space of
any principal bundle
7f :
P
--->
M endowed with a connection. More
52
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
precisely, the following commutative diagram is defined.
w W(g) - -
~
A*(P)
!~.
i! J(G)
._-..., A*(M) w
Here both i and 7r* are injective, and the homomorphism induced in cohomology by the map w in the bottom row is the Weil homomorphism mentioned above. The above is a brief summary of the Chern-Weil theory. 2.1.2. Definition of flat bundles. DEFINITION 2.1. A connection won a principal G-bundle is called a flat connection if its curvature n is identically O. A principal Gbundle equipped with a fiat connection is called a flat G-bundle. EXAMPLE 2.2. If we put the trivial connection on a product bundle M x G, it is clearly a flat bundle. This is called a trivial flat bundle. The connection form Wo of this bundle is given by Wo = q*O where q : M x G -+ G is the natural projection and () E Al(G; g) denotes the Maurer-Cartan form of G. EXAMPLE 2.3. Let 7r : P -+ M be a fiat G-bundle and let f N -+ M be a Coo map. Then the pullback bundle f* P -+ N by becomes a fiat G-bundle.
f
By virtue of the Chern-Weil theory, which we recalled in the previous subsection, any real characteristic class of a fiat bundle vanishes. However, such bundle is not necessarily a trivial bundle as a principal bundle and furthermore, even if it were so, the fiat connection on it is not necessarily a trivial one. Depending on the base :;pace M, it may happen that there are many fiat G-bllndles on it. In such a situation, it often becomes an important problem to consider all fiat bundles on M and then classify them. Accordingly we first give a criterion of classification of fiat bundles. DEFINITION 2.4. Let 1fi : Pi -+ Mi (i = 1,2) be two fiat bundles and let Wi E A 1 (Pi ; g)(i = 1,2) be their fiat connection forms. A
2.1. FLAT BUNDLES
53
bundle map
from PI to P2 is called a bundle map as fiat bundles if the condition WI is satisfied. This condition can be equally phrased that the differenti8l of .f sends horizontal vectors at any point u in PI to horizontal vectors at .fCu) E P2 . Two Hat G- bundles over the same base space M are said to be isomorphic if there exists a bundle map as fiat bundles between them over the identity map of M.
.f*W2 =
With this terminology, a fundamental problem is this: Determine the set of isomorphism classes of fiat G-bundles over a given manifold NI. 2.1.3. Flat bundles and completely integrable distributions. In this subsection and the next, we shall present a few mutually equivalent conditions which describe fiat G-bundles geometrically. Suppose that there is given a connection w on a principal Gbundle 7f : P ---t M and let ?t = {Hu; u E P} be the corresponding distribution on P. Let us rephrase the condition that this connection is fiat, namely the vanishing of its curvature, in terms of the distribution ?t. For that we recall the theorem of Frobenius. In a general setting, let M be a Coo manifold and suppose that there is given a distribution r on M, which is by definition a sub bundle of the tangent bundle TM. A submanifold N of M is called an integral manifold of T if at each point pEN we have TpN = Tp. A connected integral manifold which is not a proper subset of any connected integral manifold is called a maximal integral manifold. We denote by r( r) the set of all sections of T. Also let 1(r) denote the ideal of A*(M) generated by I-forms ex E AI(M) such that ex(X) = 0 for any X E r(T). Then, if we set
1k(T)
= {17
E Ak(M); 17(XI
,'"
clearly we have 1(T) = €Jk1k(r).
,Xk ) = 0 if Xi
E r(r)
for any i},
54
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
2.5. (i) A distribution T on M is called completely integrable if for any point on M there exists an integral manifold containing it. (ii) T is called involutive if the condition DEFINITION
X, Y
E r(T)
===}
[X, Y]
E r(T)
is satisfied. (iii) If dI(T) C I(T), I(T) is called a differential ideal. THEoREivr 2.6 (Theorem of Frobenius). Let T be a distribution on a Coo manifold M. Then the following three cond'itions are equivalent. (i) T is completely integrable. (ii) T is involntive. (iii) I (T) is a differential ideal.
If we use the theorem of Frobenius above, we can give an answer to the present problem as follows. PROPOSITION 2.7. A connection w on a principal bundle is fiat if and only if the corresponding distribution H is completely integrable. PROOF.
By virtue of the theorem of Frobenius, we have only to
show that w is fiat <===? H is involutive.
If X, Y E r(H), then w(X) = wrY) = O. Hence, by the properties of differential forms, we have (2.3) 1 1 dw(X,Y) = 2{Xw(Y) - Yw(X) -w([X,Y])} = -2w([X,Y]).
On the other hand, the structure equation (2.2) implies (2.4)
dw(X, Y) = n(X, Y).
It follows from (2.3), (2.4) that
(2.5)
n(X, Y)
=
1
-2w([X, YD·
Now if w is fiat, namely n == 0, then (2.5) implies w([X, Y]) = O. Hence [X, Y] E r(H) so that we can conclude that H is involutive. Conversely, if we assume that 7-{ is involutive, then by the above computation we have n(X, Y) = 0 for any X, Y E r(H). Now in general, for any X, Y E TuP(u E P) if we denote by X h , Yh E Hu their horizontal components respectively, then it is well known that
2.1. FLAT BUNDLES
55
H(X, Y) = fl(Xh' Yh ). This is one of the properties of curvature. It LlOW follows that fl == 0, and the proof is completed. 0 Thus we can say that A flat bundle is nothing but a bundle with a connection whose corresponding distribution is completely integrable. 2.1.4. Flat bundles and holonomy homomorphisms. We give another equivalent description of flat bundles. This description is very important for the classification of flat bundles. Let 1r : P - t j\1 be a principal G-bundle with a flat connection wand let H be the corresponding completely integrable distribution on P. Then, for each point u on P if we denote by Lu the maximal integral manifold passing through it, then the projection 1r : Lu - t M becomes a covering map. Now let Po E M be a base point of the base space and choose Uo E 1r- 1 (po). Then a homomorphism p: 1rl(M)
~
G,
called the holonomy homomorphism, is defined as follows. In some cases, it is also called the monodromy homomorphism. For each element 0' E 1rl (IvI) of the fundamental group, we choose a closed curve e with initial point Po which represents 0'. Let l be the lift of e to Luo with initial point uo. Then we can write terminal point of l
= uog
(g E G)
(see Figure 2.1). I
I
I
1l0p(a)-14------..,. p
l
C Luo
M
FIGURE 2.1. Holonomy homomorphism Since 1r : Luo - t IvI is a covering map, we see that the terminal point of l (and hence the element g) is determined uniquely by 0'
56
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
independent of the choice of f. We set
p(o:) PROPOSITION
2.8. p: 1fl(M)
= 09- 1. ~
G is a homomorphism.
PROOF. For any two elements 0:,{3 E 1fl(A1), we choose closed curves fex, ffJ representing them respectively and let in, ffJ be their lifts to Luo with initial points 'lto. Then we have
terminal point of fex = uOp(o:)-l, terminal point of ffJ = uOp((3)-1. Since 1t is invariant under the right action of G, ffJp(o:)-1 is a lift of ffJ to LUOP(Cl)-' = Luo with initial point uOp(o:)-l. Hence the composition la . ffJp(o:)-l of two curves is a lift of fa ·ffJ with initial point uo. On the other hand, the terminal point of this curve is uOp({3)-lp(o:)-l =uo(p(o:)p({3))-l.
Hence we finally obtain p(o:{3) = p(o:)p({3). This completes the proof. C Next we study how the holonomy homomorphism changes if we change the base point uo E 1f-l(pO)' Let Uo = uoh (17, E G) be another base point and let p' be the corresponding holonomy homomorphism. As above, if we denote by .e the lift of a closed curve f, which represents 0: E 1fl (M), with initial point Uo, then lh becomes the lift of the same curve with initial point Since the terminal poiat of ell. is
uo.
uop(u)-lh=u~h-lp(o:)-lh,
we can conclude p'(Ct) = h- 1 p(0:)h. Thus we sec that the two homomorphisms p, p' are conjugate to each other. If two fiat G-bundles over the same base space are isomorphic, then clearly their holonomy homomorphisms coincide (up to conjugacy). Actually, the converse is also true, and we have the following theorem. THEOREM 2.9. Let !vI be a Coo rnan~fold and let G be a Lie group. Then the cor'respondence, 'Which sends each fiat G -bundle over IvI to its holonorny homomorphism, 'induces a b#ection
{isomorphism class of .flat G-bundlcs over M } ~
{conjugacy class of homomorphism p : 1fl (!VI) -, G}.
2.1. FLAT BUNDLES
57
PROOF'. Vife first prove the case where M is simply connected. As above, let Po be the base point of M and let L u () be the maximal integral manifold passing through Uo E 7f-l(pO)' Then, since the projection
7fo : Luo
M
---7
is a covering map and !vI is simply connected by the assumption, this map is a diffeomorphism. We then consider the map
MxG given by Here
~(u)
(see Figure 2.2). I
I
,, I
P tio
7l
I I
,, \
L u, lfo'(1f(U»
\
M 'Po
FIGURE
2.2
Then, for arbitrary elements u E P and 9 E G, we have
58
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
of fiat bundles. Hence the given fl~bundle P can be identified with a quotient of the product bundle lVI x G by a certain free action of 7rl (M) by automorphisms of it. In other words, we can write P = P/7rl(M) = M x G/7rl(M).
We describe this action explicitly. For that, we use the model of the universal covering manifold of M given by the set M = {[eli e ; [0,1]
->
M, e(l) = Po}
of homotopy classes (with endpoints fixed) of paths on !vI with terminal point Po. Here tel denotes the homotopy class of e. The projection L M -> M is given by q([el) = e(O). The (right) action of 7rl(M) on M is given by
Here e· 0: denotes the composition of the two paths hand, we can write
P = {(u, [el)
where
~(u)
(u, [el)
Q.
On the other
P x M;7r(u) = q([f)) = e(O)}.
E
Now the isomorphism (2.6) of correspondence
P3
e,
f->
P as
a flat bundle is given by the
([e],((u, tel))
E
EGis determined by the equality
(2.7)
1l
= v((u,
[il)
(see Figure 2.3).
£
~.e(O)
Po
FIGURE 2.3
M x G
2.1. FLAT BUNDLES
e
59
e
Here denotes the lift of to Luo with terminal point un, and v ,iPllOtes the initial point of Now for an arbitrary element a E 7rl (An of the fundamental !'.roup, we have
e.
('U, [fla) ~ ([f]a,~(u, [f]a)). 'Ii) compute
~(u,
[fla), we consider Figure 2.4.
u
t
a~) FIGURE 2.4
Here
a denotes the lift of a
to Luo with initial point Un. If we let
p(a) act on the whole of this figure from the right, we obtain Figure
2.5.
~(".
FIGURE
lela)
2.5
If we denote by w the initial point of ep(o:), clearly w = vp(o:) On the other hand, by the definition of~, we have u = w~(u, [flo:) Hence we can conclude
(2.8)
u = vp(o:) ~(u, [fla).
From equalities (2.7),(2.8), we obtain ~(u, [e]o:) = p(o:)-l~(U,
[fl).
60
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
Thus, if we define the action of 1l'1 (M) on M x G by M x G 3 ([.e],g) ~ ([.e]a,p(a)-1g) E M x G
(a E 1l'l(M)),
then we obtain an isomorphism P
~
M x G/1l'l(M)
of fiat bundles. Here the right hand side is determined depending only on p. Moreover it is easy to see that, for any homomorphism p : 1l'1 (M) ~ G, the right hand side becomes a fiat bundle. This completes the proof of the theorem. 0 2.2. Cohomology of Lie algebras 2.2.1. Left invariant differential forms on Lie groups. Let G be a Lie group as in the previous section and let 1l' : P ~ M be a flat G-bundle. If we denote by W E A1(M; g) the corresponding connection form, then we can consider that w defines a linear map w : g* -~ A 1 (P).
Explicitly, we have
w(a)(X)
= a(w(X)), (a
E g*,
X
E
TuP).
By taking the exterior powers, w is naturally extended to a linear map (2.9)
w : A*g*
....... A*(P).
Now A*g* can be identified with the set A*(G)G of all left invariant differential forms on G. Since the exterior differentiation of any left invariant form is clearly left invariant, A" g* is a subcomplex of the de Rham complex A*(G) of G. By the assumption, w is a flat connection. Hence the mapping of (2.9) commutes with the operation of taking the exterior differentiation. Hence it induces a homomorphism w* : H*(A*g*;d)
= H*(A*(G)G)
--+ H*(A*(P)) ~ H*(P;~).
It might be natural to expect this homomorphism to have a certain role of invariants for flat G-bundles (cf. §2.3.1). With the above facts as motivation, we define the cohomology group H*(A*g*;d). R.ecall that any element r.p E Akg* defines an alternating multilinear mapping r.p:gx···xg--+R '----v----" k-times
2.2. COHOMOLOGY OF LIE ALGEBRAS
61
In this book, we adopt the convention that
for ai E g*,Xi E 9 (i = 1"" ,k). (There is another convention where one multiplies the above by the constant ~.) To calculate' dtp E Ak+lg*, let Xl,'" ,Xk + 1 E g. Here we consider each Xi to be a (left. invariant) vector field on the Lie group G. Then, by the well-known formula for the ext.erior differentiation, we have
(2.10) k
~1
dtp(Xl,''' ,Xk+l) = 2)-1)i+lXi (tp(X 1 ,· ..
,Xi,'"
,Xk+1 ))
i -- l
+ 2:) -l)i+ J tp ([Xi , Xj], Xl,'" ,Xi,'" ,Xj , ...
,Xk+d·
i<j
Here the symbol Xi means that we omit Xi. Since the function tp(Xt ," . ,Xi,' .. ,Xk+l), which appears in the first term of the right hand side, i;; clearly a constant function on 0, the derivative of it by Xi is O. Hence we have (2.11)
Thus we see that the exterior differentiation
can be described entirely in terms of the Lie algebra g. With these in mind it may be natural to call H*(A*g*; d) the cohomology of the Lie algebra 9 and denote it by H* (g). EXAMPLE 2.10. Let 0 be a connected compact Lie group. Then, by taking the average of differential forms on G by means of the Haar measure, it can be shown that the inclusion A'g* = A*(O)G c A'(G) induces an isomorphism in cohomology. Hence we have
H*(g)
~
HDR(G)
~
H*(G;JR).
2.2.2. Definition of the cohomology of Lie algebras. From the consideration of the previous subsection, it may be natural to define the cohomology group H* (9) of a Lie algebra 9 as follows.
62
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
First we set CO(g) = R For each positive integer k, we set Ck(g) = {ep: 9 x··· x fJ ---; lR;alternating multilinear map}. '----v---' k-times
Here by alternating we mean that for any permutation equality ep(Xa(l), ... , Xa(k))
= sgn (J" ep(X1 , ...
,
(J"
E
610 the
X k ) (Xi E g)
holds. Any element of Ck(g) is called a k-cochain of g. We can identify Ck(g) with Akg*, according to the convention mentioned in the previous subsection. Hence the set C*(g) = ffikCk(g) of all cochains is identified with the exterior algebra A * g*. It induces a product Ck(g)
X
C e(g):1 (ep,'ljJ)
t--->
epA'ljJ E Ck+f(g)
of two cochains which is explicitly given by ep A 'ljJ(XI , ... , Xk+e)
=
1
L
sgn(J" k!e! ep(Xa(I)"" , Xa(k»)'ljJ(Xa(k+l)"" , Xa(k+e»).
aE 6 kH
Now motivated by (2.11), we define a linear map d: Ck(g) - } Ck+l(g)
by setting d
= '""'(-1)i+j1fl([X XlJ' Xl , ... ,X t , ... ' X J ' ... ~
y
t)
, X k.+l )
i<j
for any ep E Ck(g). For example, if k dep(X, Y)
= 1 we
have
= -ep([X, Y]).
LEMMA 2.11. (i) d2 = O. (ii) d(ep A 'ljJ) = dep A 'ljJ + (-l)kep A d'ljJ
(ep E Ck(g)).
If 9 is the Lie algebra of a Lie group G, then the above lemma is clear because we defined d by using the ordinary exterior differentiation of (left invariant) differential forms on G. In the case where 9 is a general Lie algebra, it is easy to prove it by a purely algebraic argument. We leave the details to the reader. We can conclude from (i) above that {C*(g),d} is a cochain complex. We then define the cohomology of Lie algebras as follows.
2.2. COHOMOLOGY OF LIE ALGEBRAS
63
DEFINITION 2.12. Let 9 be a Lie algebra. We call the cohomology group of the cochain complex {C* (g), d} the cohomology of 9 and denote it by H*(g).
2.2.3. Relative cohomology and cohomology with coefficients of Lie algebras. In the cohomology theory of topological spaces, there are notions of relative cohomology group H* (X, A) of a pair (X, A) and also cohomology group H* (X; S) with coefficients in a local system S. They play important roles both theoretically and from the viewpoint of applications. Similarly, in the cohomology theory of Lie algebras, we have relative cohomology with respect to Lie subalgebras and also cohomology with twisted coefficients. We review these briefly. Let 9 be a Lie algebra. For any element X E 9 there is associated a linear map i(X) : Ck(g) - 7 Ck-1(g),
which is called the interior product and is defined by
= rp(X, Xl,·· . , X k- 1 ).
(i(X)rp)(Xb·· . , Xk-d
Clearly 'i(X)2 = 0, and it can be shown that
i(X)(rp 1\ '!jJ) = (i(X)rp)
1\
'!jJ
+ (_l)krp 1\ (i(X)'!jJ) (rp E Ck(g)).
Namely i(X) is an anti-derivation of degree -1. Now let subalgebra of g. If we put
~
be a Lie
Ck(g,~) = {rp E Ck(g); i(X) rp = i(X) drp = 0 for any X E ~},
then it can be shown that
C*(g,~) =
EB Ck(g,~) k
becomes a sub complex of C*(g). DEFINITION 2.13. H*(g,~) = H*(C*(g, ~)) is called the relative cohomology of the Lie algebra 9 with respect to the Lie subalgebra
~.
For each element X E g, we define the Lie derivative
Lx : Ck(g)
-7
Ck(g)
with respect to X by Lx = i(X)d+di(X). Then it is a derivation of degree o. In terms of this, we can also write C*(g,~) =
{rp E C*(g); i(X) rp = Lx rp = 0 for any X E ~}.
64
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
There is another version of relative cohomology which generalizes the above one slightly. Let G be a Lie group and let K be a Lie subgroup of G. Let g, e be Lie algebras of G, K respectively. By restricting the adjoint representation Ad : G --t GL(g) to K, K acts on 9 by automorphisms. If we set Ck(g, K)
= {cp E Ck(g); i(X)cp = 0 for any X cp(Ad(g)X 1 ,··· , Ael(g)Xk )
E
= cp(X1 ,···
e, ,
X k ) for any X E e},
then it can be checked that
EB Ck(fI, K)
C*(g, K) =
k
becomes a sub complex of C*(g). In other words, the set of all elements whose interior product with respect to any element of eis 0 and which are invariant under the action of K makes a subcomplex. DEFINITION 2.14. H*(g,K) = H*(C*(g,K)) is calleel the relative cohomology of the Lie algebra 9 with respect to a Lie subgroup
K. If K is connected, it is easy to see that C*(g,K)
= c*(g,e).
Therefore Definition 2.14 is a generalization of Definition 2.13. Also if K is a closed subgroup of G, then the set A * (G / K)G of all Ginvariant differential forms on the homogeneous space G / K becomes an important sub complex of the de Rham complex A * (G / K) of G / K. In this case, it is easy to see that there exists a natural identification C*(g,K) 9" A*(G/K)G.
Hence we have H*(g,I<) 9" H*(A*(G/K)G).
Next we consider the cohomology of Lie algebras with arbitrary twisted coefficients. Let 9 be a Lie algebra and let V be a fI-module; that is to say, there is given a homomorphism 9 --t gl(V). Then we set C k (g; V) = {cp : 9
X ...
x 9
--->
V; alternating multilinear}
'--v-----' k-times
and define the coboundary operator
d: Ck(g; V)
--t
Ck+I (g; V)
2.2.
COHOMOLOGY OF LIE ALGEBRAS
65
by k+J
dcp(XJ,··· ,Xk+l) = 2)-1),+lXi CP(X I ,···
,Xi,···
,Xk+d
i=l
+ ""(-l)i+jcp([X ~
't)
Xl J ' Xl , ...
X ... 'J" X ... x k + J ) .
,tl
i<j
It can be checked that d 2 = O. Observe here a similarity between the above definition of d ancl the equality (2.10). DEFINITION 2.15. Let 9 be a Lie algebra and let V be a g-module. We denote by H* (g; V) the cohomology of the cochain complex C*(g; V) and call it the cohomology of 9 with coefficients in V.
2.2.4. Cohomology of .5[(2, IR). As an example of cohomology of Lie algebras, here we compute the cohomology of the Lie algebra
s[(2,IR)
=
{X E M(2, IR); TrX = O}
consisting of all 2 x 2 real matrices with Tr = O. We choose
Xo
=
G ~l)'
Xl
(~ ~),
=
X2
=
(~ ~)
for a basis of s[(2,IR) and let
CPo, CPI, CP2 E C I (£1[(2, IR))
=
£1[(2, IR)*
be its dual basis. Since
we have
Hence {
IR
k = 0,3
° ki
0,3,
and we see that [cpoCPjCP2l is a generator of H 3 (s[(2,IR)). Next we consider the maximal compact subgroup of 5L(2,IR), which is 50(2), and compute the relative cohomology H* (s[(2, IR), 50(2)) - H* (s[(2, IR), so(2))
with respect to it. We can take X = -Xl Then we have
i(X)cpo
=
0, i(X)CPI
=
+ X2
-1, i(X)CP2
as a basis of so(2). = l.
66
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
We can deduce from these facts that any non-trivial element of C*(sf(2,1R),'s0(2)) is a multiple of CPo /\ (CPI + CP2). Hence we have Hk(sf(2 IR) so(2)) = {IR
"
k = 0,2 k # 0,2.
()
2.3. Characteristic classes of flat bundles 2.3.1. Characteristic classes of flat product bundles. Let P ~ M be a flat G-bundle and let wE Al(P; g) be the corresponding flat connection. As we saw in §2.2.1, W induces a homomorphism which commutes with the operation of taking exterior differentiation, namely a d.g.a. map 7f:
w: A*g*
----t
A*(P).
Hence, in view of the motivation for the definition of cohomology of Lie algebras mentioned there, we have a linear map (2.13)
w*: H*(g)
= H*(A*g*;d)
- \ H*(P;IR)
which can be regarded as giving characteristic classes of Hat bundles. However, characteristic classes are generally defined as cohomology classes of base spaces. Nevertheless, in the above we have cohomology classes of the total space rather than those of the base space. If we are given a section s : 1.1 ---> P, then we obtain cohomology classes of the base space by composing the induced homomorphism s* : H* (P; IR) ---> H* (M; IR) with (2.13). On the other hand, it is the same thing to give a section s to a principal G-bllndlc and to give a trivialization P '" 1\11 x G as principal G-bundlcs. This can be seen by the correspondence M x G 3 (p,g) t--+ s(p)g E P. With these facts in mind, we make the following definition. DEFINITION 2.16. Let 7f : P ~ M be a Hat G-bundle. If there is given a trivialization P ~ A1 x G as principal G-bundles, it is called a flat G-product bundle. Let 7ri : Pi ~ Mi (i = 1,2) be two flat G-product bundles. A bundle map
PI
(2.14)
j ----------t
1n2
nIl Ml
P2
------t
f
M2
2.3. CHARACTERISTIC CLASSES OF FLAT BUNDLES
67
as flat G-bundles is called a bundle map as flat G-product bundles if it satisfies the condition .f 0 S1 = S2 0 f. In particular, two flat Gproduct bundles 1ri : Pi -> M (i = 1,2) over the same base space are called isomorphic as flat G-product bundles if there exists a bundle map PI -> P 2 as flat G-product bundles over the identity map of M. With this preparation, we define characteristic classes of flat product bundles as follows. 2.17. Let 1r : P -> M be a flat G-product bundle and P he the corresponding section. Then the homomorphism
DEFINITION
let.') : Jv!
->
'W:
H*(g)
w'"
s*
---t
H*(P;JR.)"':""'" H*(M;JR)
obtained above is called the characteristic homomorphism. For each element a E Hk(g), the cohomology class w(a) E Hk(M; JR) is called the characteristic class of flat G-product bundles corresponding to a. It should be clear from the definition that the following proposition holds. PROPOSITION 2.18. Suppose that the-re is given a bundle map (2.14) as flat G-product bundles between two flat G-product bundles 1ri : Pi -> Mi. (i = 1,2). Then the diagram
H*(g)
w
---->
H*(Nh; JR)
ir
II H*(g)
---->
H*(NhlR)
w
is commutative. In particular, characteristic classes of mutually iso-
morphic flat G-product bundles coincide. The above property is called the naturality of characteristic classes of flat G-product bundles. 2.3.2. Definition of characteristic classes of flat bundles. By improving the consideration in the previous subsection, we define characteristic classes of any general flat G-bundle 1r : P -> M. In this case, a given G-bundle may not be a trivial bundle as a principal Gbundle. Hence we cannot assume the existence of a section s : M -> P, and so it is not possible to obtain cohomology classes of the base space as in Definition 2.17.
68
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
We therefore take a maximal compact subgroup K of G and consider the quotient space G / K. Then the projection 'if : P -> M is the composition of two maps as shown by
P ---) P/K ---) M.
(2.15) Clearly the first map
P
(2.16)
-~
P/K
has the structure of a principal J(-bundle. Here we recall an important general fact concerning principal bundles. Let e be the Lie algebra of K. Then a differential form on P is the pullback of a form on P / J( if and only if the following two conditions are satisfied. (i) The interior product with respect to a fundamental vector field on P induced by any element of e is O. (ii) It is invariant under the action of K. By combining this fact with the considerations of §2.2.3, we obtain the following commutative diagram. w
C*(g) = A*g*
A*(P)
--}
r
(2.17)
r
C*(g, K) = A*(G/K)G
-
----->
A*(P/K)
w
Here both C' (g, K) -> C* (g) and A * (P / K) injections. Next we consider the second ma.p
->
A * (P) denote natural
P / J( ---) Jv!
(2.18)
appearing in (2.15). It is easy to see that this has the structure of a fiber bundle with fiber G / K. As is well known, G / K is diffeomorphic to a Euclidean space of suitable dimension, and in particular it is contractible. Hence the map (2.18) becomes a homotopy equivalence, and we obtain an isomorphism H*(P/J(;~) ~ H*(M;~)
(2.19)
in cohomology. From (2.17), (2.19), we obtain a linear map w: H*(g,K) ~ H'(P/K;~) ~ H*(M;JR). DEFINITION
2.19. Let
'if:
P
->
M be a flat G-bundle. The map-
ping w : H* (fj, K) ---) H* (M; 1R)
2.3. CHARACTERISTIC CLASSES OF FLAT BUNDLES
69
defined above is called the characteristic homomorphism. For any a E Hk(g,K), the cohomology class w(a) E Hk(Nf;JR) is called the characteristic class of fiat G-bundles corresponding to a. Similar to the case of fiat G-product bundles, the following result holds which shows that. characteristic classes of fiat G-bundles are natural with respect to bundle maps. PROPOSITION 2.20. Let 7fi : Pi - t Mi (i = 1,2) be two flat Gbundles. Suppose that there is given a bundle map as flat G-bundles as in Definition 2.4. Then the diagram
H*(g,K)
w
~
ir
II H*(g,K)
H*(M1; JR)
--->
w
H*(M2; JR)
is commutative. In particular, chamcteristic classes of mutually iso-
morphic flat G-bundles coincide. 2.3.3. Classifying spaces of flat bundles and characteristic classes. Here we briefiy mention the relation between characteristic classes of fiat bundles, which we considered in the previous two subsections §2.3.1 and §2.3.2, and the classifying spaces of flat bundles. For a Lie group G, the classifying space of principal G-bundles is usually denoted by BG. As is well known, in the case where G is a general linear group such as GL(n, JR) or GL(n, C), its classifying space can be explicitly described by (limits of) Grassmannian manifolds. Then how can we describe classifying spaces of fiat G-bundles? According to Theorem 2.9, to give a fiat G-bundle over a manifold M is the same thing as giving a representation 7f1 (M) - t G. If we express this fact from the topological point of view, we can say that the space BG Ii = K(G,l) serves as the classifying space of flat G-bundles. Here G li denotes the group G equipped with the discrete topology. Hence characteristic classes of flat G-bundles given in Definition 2.19 induce a homomorphism w: H*(g,K) ----> H* (BG Ii ; JR) = H*(GIi;JR). Next we consider flat G-product bundles. Recall that a flat Gproduct bundle is a principal G-bundle equipped with two structures:
70
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
one is a flat connection, and the other is a trivialization as a principal G-bundle. Hence the classifying space of flat G-product bundles is played by the homotopy fiber (see Example 1.5(iii), §1.1.1) of the natural map BG6 ~ BG. This space is usually denoted by BG, and we obtain a fibration BG
----T
BGo
----T
BG.
lf G is connected, G becomes a topological group explicitly described as G
=
{(g,e) E GO x JVIap(J,G);e(O)
= g,e(l) = e}
C G6
x Map(I,G).
In this case, Definition 2.17 gives rise to a homomorphism w: H*(g)
----T
H*(BG;IR).
Summarizing the above, we can say that characteristic classes of various G-bundles are given by the following commutative diagram. H*(g)
w
--------->
H*(BG;IR)
--------->
H*(BG 6 ;1R)
r
T (2.20)
H*(g,K)
w
r
T J*(K)
--------->
H*(BK;IR) ~ H*(BG;IR)
w
Here in the bottom row, we have used the well-known facts that G is diffeomorphic to the product of its maximal compact subgroup K and some Euclidean space and also its corollary that BG is homotopy equivalent to BK.
2.3.4. Chern-Simons forms and Chern-Simons invariants. In this subsection, we mention very briefly the Chern-Simons theory, which can be considered as a certain refinement of the Chern-Weil theory. For details, we refer the reader to the original paper [ChS] of Chern and Simons and also a closely related paper [CS] of Cheeger and Simons. Let f E Jk(G) be an invariant polynomial of a Lie group G. As was already mentioned in §2.1.1, to any principal G-bundle 7r : P ~ !VI with a connection w E A 1 (P; g), there corresponds a certain closed form f(D,k) E A 2k (p). Moreover, this form is the pullback of a closed form f(nk) E A2k(M) of the base space by 7r, and the de Rham cohomology class [f(D,k)] E H2k(M; IR) represents the characteristic
2.3. CHARACTERISTIC CLASSES OF FLAT BUNDLES
71
class corresponding to f. Now as will be easily checked, the form f(0, k ) E A 2k (p) itself is an exact form because of the following reason. By the naturality of characteristic classes with respect to bundle maps, the above form represents the characteristic class of the pullback bundle on the total space by the projection 7f. On the other hand, this bundle is clearly trivial. In the above cited paper, Chern and Simons constructed an explicit (2k -I)-form Tf(w) E A 2k - 1 (p) such that
This differential form is called the Chern-Simons form. In the context of the Weil algebra W(g), the Chern-Simons form is nothing but an element T f E W 2k - 1 (g) such that dT f = f E Ik(G) C W2k(g). From the triviality of the cohomology of W(g), we can conclude that such a form certainly exists and the difference of any two such forms is an exact form. It can be said that Chern and Simons specified one natural element among them. Here we briefly mention only the result. Let VI, ... ,Vm be a basis of 9 and let wI, ... ,w ffi be the dual basis of g*. Also let 0,1, ... ,0,7n be the corresponding basis of Slg*. Then the two elements Tn
w
=
L wi ® V., E g* ® 9 c W(g) ® 9 ;.=1 m
D =
L ni ®
Vi
E Blg* ® 9
c
W(g) ® 9
i=l
are the universal connection form and the universal curvature form, respectively, which are defined at the level of the Weil algebra. We can define natural maps 1\ :
(W(g) ® g01.:) ® (W(g) ® gOt')
[ , 1: (W(g) ® g) ® (W(g) ® g)
----+
----+
W(!J) ® g0(k+£)
W(g) ® g,
and the following two important equations: 1 dw = -2'[w,w] d0, =
[0"w]
+ 0,
(structure equation) (Bianchi identity)
hold in W(g) ® g. Now the Chern-Simons form Tf E W 2k - 1 (g) corresponding to an invariant polynomial f E Ik (G) c Hom(g0 k , lR)
72
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
is given by k-l
Tf =
L
Ai f(w /\ [w, w]i /\ nk -
i - 1)
i=O
where A _ ()i k! (k - I)! i -1 2i(k + i)!(k -1 - i)'· If T f (w) becomes a closed form for some reason, we can define the integral
1
Tf(w)
associated to any (2k - I)-dimensional cycle c in P. The invariants obtained in this way are called Chern-Simons invariants. For example, let M be a closed oriented 3-dimensional Riemannian manifold. As is well known, the tangent bundle T j\lJ is trivial so that the orthonormal frame bundle P(M) has a section s. Hence, by considering the Levi-Civita connection on P(l11) and the first Pontrjagin class, a Chern-Simons invariant is defined by CS(M)
=
r
./s(M)
~ TPI
E
IR/Z .
2
Here we set the value of the integral to be in IR/Z because if we change the section, the value changes by a certain integer. (However, later a close relationship with the 1J-invariant of Atiyah-Patodi-Singer [APS] was found and this indeterminacy disappears now.) As another example, let !vJ be a closed oriented 3-dimensional manifold and let AM denote the set of all connections on the trivial SU (2)-bundle over M. Then the Chern-Simons invariant corresponding to the second Chern class C2 induces a function CS:AM
-~R
After the celebrated work of Witten [W], this function has been playing a fundamental role in the recent development of low dimensional topology using the gauge theory (cf. e.g. [Koh]). Now if w is a flat connection, the Chern-Simons form Tf(w) becomes a closed form. Hence we can consider its de Rham cohomology class [Tf(w)] E H 2k - 1 (P; IR). This is called the Chern-Simons class. Also for a flat G-product bundle, if we pull back the ChernSimons class by the section s : !vI ---t P, we obtain a cohomology class
2.3. CHARACTERISTIC CLASSES OF FLAT BUNDLES
73
of the base space. More explicitly in the context of the classifying space, there exists a natural closed form Tf E A2k - 1 fl* such that
H 2k - 1 (fl)
[Tf]
3
~
[Tf]
E H2k-l(BG;~).
Here l' f is nothing but the projection of l' f, which was defined as an element of the Weil algebra, to A" fl". Furthermore, if the characteristic class [f] E H2k(BG;~) corresponding to f is the image of an integral class, then the Chern-Simons class is defined as a cohomology class
[Tf] E H 2k - 1 (BGo; ~/Z) with coefficients in
~/Z.
For example, we have a cohomology class
TCk E H 2k - 1 (BGL('I7" C)0; C/Z) which is defined for flat complex vector bundles. EXAMPLE 2.21. As an example, let us consider the case G = G L( 1, C) = C*. Although this case is almost trivial, it is suggestive of higher dimensional generalizations. In this case, it is easy to see that BG = K(Z,2),BG o = K(C*,l) and BG = K(C,l). Also we have H* (BG; Z) = Z[Cl]' Then it can be shown that the classes TCI E Hl(BGo;C/Z) ~ Homz(C*,C/Z) and TCl E Hl(BG;C) ~ Homz(C, C) are given by the following commutative diagram:
C (2.21)
cxp
X ..!~; ---->
1
C*
C
1
modZ
-----> 2~{log
C/Z
2.3.5. Nontriviality of characteristic classes of flat bundles. We consider the nontriviality of characteristic classes of fiat bundles (see Definition 2.19). Let G be a connected semisimple Lie group. Then, based on a deep theory due to Borel and HarishChandra, Borel [Bor] proved that there exists a discrete subgroup reG such that r\G becomes compact. By a well-known lemma of Selberg, we may assllme that r is torsion free. Hence, jf we denote by K a maximal compact subgroup of G, M = r\ G I K = K(r,l) becomes an oriented closed manifold. Then we define a fiat G-bundle over M by 11" : r\ « G I K) x G) -} M, where the action of on (G I K) x G is given by ,([g], h) = ([,g], ,h) h E G, [g] E GIK,h E G). Then the associated flat GIK-bundle r\(GIK x G I K) --> r\ G / K has a section induced by the natural diagonal
r
74
2.
CHARACTERISTIC CLASSES OF FLAT BUNDLES
embedding G / KeG / K x G / K. Clearly this section is a homotopy equivalence. It follows that the characteristic homomorphism H* (g, K) - H* (M; lR) of the bundle 7r is induced by the natural map A*(G/Kf ------> A'(M).
In the top degree which is equal to the dimension of M, this map is an isomorphism because both are generated by volume form::;. On the other hand, it is known by Koszul [Kos] that the cohomology H*(g, K) satisfies the Poincare duality theorem. Sillce the above map is a homomorphism as algebras which preserves products, we can deduce that the induced homomorphism in cohomology is injective. Thus we obtain the following theorem. THEOREM 2.22 (Borel, Harish-Chandra, Selberg, Koszul). Let G be a connected semisimple Lie group and let K be a ma:t"imal compact subgroup of it. Then the homomorphism
H*(g,K) _ H*(BG 6 ;lR) is injective. The following example is very important because it is related to various concepts such as characteristic classes of flat bundles, ChernSimons invariants, 1]-invariant, characterisitc classes of foliations (see §3.4) and geometry of negatively curved manifolds. EXAMPLE 2.23. Consider the Lie group PSL(2,C) which is defined as SL(2, q/{±l}. As is well known, it serves a..<; the orientation preserving isometry group of the 3-dimensional hyperbolic space .IflI3. More precisely, the quotient space PSL(2, C)jSO(3) of PSL(2, C) by the maximal compact subgroup SO(3) can be identjfied with lHf3, and moreover the principal SO(3)-bundle
SO(3)
------>
PSL(2,q
------>
PSL(2,C)/SO(3)
is identified with the orthonormal frame bundle P(lHf3) of JH[3. Now the Lie algebra of PSL(2, q is s[(2, C), and explicit computation shows that H*(s[(2,q) ~ H*(S3 x S3;lR),
H*(sl(2,Q,SO(3)) ~ H*(S3;lR).
We can check that H3(S((2,Q,SO(3)) ~ lR is generated by the volume form v of JH[3. Furthermore, under the natural isomorphism
2.4. GEL'FAND-FUKS COHOMOLOGY
75
the real part is generated by the Chern-Simons form TPI on P(JH[3) corre::;ponding to the first Pontrjagin class, while the imaginary part is generated by v (d. [Y] for example). Now let r c PSL(2, C) be a torsion free discrete subgroup such that the quotient space r\p S L(2, C) is compact. It was proved by Thurston, in his theory of hyperbolic 3-manifold::;, that there are plenty of such groups. In fact, in the above ::;ituation M: r\PSL(2,C)/SO(3) becomes what is called a compact hyperbolic 3-manifold, and r\PSL(2, q can be identified with the orthonormal frame bundle P(Iv!) of Iv!. Since it is known that Iv! is parallclizable, P(M) is diffeomorphic to M x S3. Thus we obtain a homomorphism f/*(sl(2,C)) - .., H*(P(M);JR.) ~ H*(M x S3;JR.) which turns ont to be injective if the Chern-Simons invariant of M is nontrivial. Moreover, if we denote by PSO(2) C PSL(2, q the set of all diagonal matrices, then the corresponding quotient P(M)/ PSO(2) -> 111 becomes a Cpl-bundle over M where Cpl = SO(3)/ PSO(2). It can be shown that it has the structure of a foliated Cpl.-bundle, which will be defined in §3.1.1, and its structure group i::; P SL(2, C) acting holomorphically on Cpl. We refer the reader to [Mor2] fo), details. 2.4. Gel'fand-Fuks cohomology 2.4.1. Characteristic classes of flat bundles, continued. We recol1::;ider the construction of characteristic classes of :flat bundles given in §2.3.1 from a more geometrical point of view. Let 1r : P -> IvI be a flat G-prodllct bundle; namely there is given a trivialization P ~ M x G together with the corresponding section s : IvI -> P and a flat connection w. The composition
* A* 9 * ----; W A*(P) ----) .,. A*(M) sow: of two d.g.a. maps w, s* induces a linear map s* ow* :
H*(g) --> H*(M;JR.),
and this gives rise to the characteristic classes of flat G-product bundles. We examine the above map in detail. Any element tp E Ck(g) ~ Akg*; in other words any alternating multilinear map tp:gx .. ·xg-->JR. '-v----' k-tirnes
76
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
determines a certain element s*w(cp)
E
Ak(M). More ex.plicitly, if
Xl"",XkETpM
(pEM),
then we have
s'w(cp)(X1 ,··· , Xk) =w(cp)(S.Xl,··· , s.Xk )
(2.22)
=cp(w(s*Xd,'" ,w(s*Xk ))·
On the other hand, let Xi E T(p,e) (M x G) (e E G denotes the identity element) be the lift of Xi which is horizontal with respect to wand define E TeG by the equality
xl
-
Xi
=
-j
S.Xi + Xi .
xl
Then we see that is nothing but the projection of Xi to the fiber direction with respect to the product structure P = M x G (cf. Figure 2.6). (pIx G
MXG
M
)
p
FIGURE
Xi
2.6
Then, since _
-
-I _
-I
W(S.Xi) - W(Xi - Xi) - -w(Xi ), by substituting it in (2.22), we obtain (2.23)
xf
can be done at any point (p, g) on the Now the construction of fiber {p} x Gover p E M. Hence, by varying 9 in G, we can regard
2.4. GEL'FAND-FUKS COHOMOLOGY
xl
as a vector field on {p} X G invariant. Hence we can write
= G.
77
Clearly this vector field is left
xl E TeG = :£(G)G. With these facts in mind, let us denote (2.23) by ~(XI'''' ,Xk)'
It may appear that this makes the matter unnecessarily complicated. However, it will play an essential role in the next subsection where we shall consider general flat product bundles with infinite dimensional structure groups. Thus we have defi.ned a correspondence
Ck(g)
3 cp
f-.
~ E
Ak(M).
It can be easily checked by tracing the above argument that this correspondence is defined even if w is not flat. An important point here is the fact that if w is flat, then the equality
holds. Let us recall one more important fact that, as was mfmtioned in (2.11) of §2.2.1, the exterior derivative dcp E Ck+l(g) is defined purely algebraically by the equality dcp(Xl"" ,Xk+l) = ~(-l)i+j,n([X ~ ..,... 1.,
X·]J' Xl , ... ,1.., j(. ...
x· ...
'J"
X k +l ).
2.4.2. Flat bundles whose fibers are general manifolds. All the flat bundles which we considered up to the previous subsection have (fi.nite dimensional) Lie groups as their structure groups. We shall try to extend those considerations to the case of fiber bundles whose structure groups are infinite dimensional such as the diffeomorphism groups of manifolds. For any Coo manifold F, let Diff F denote the diffeomorphism group of F. We can regard Diff F as the structure group of general differentiable fiber bundle 1f : E - 7 M whose fiber is F. Henceforth we call such bundle simply an F-bundle. let
DEFINITION 2.24. Let /VI be an n-dimensional Coo manifold and E - 7 1M be an F-bundle. An n-dimensional distribution 'H =
1f :
78
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
{Hu;u E E} on E which is transverse to any fiber, or equivalently which satisfies the condition
Eu = Vu u H u
(11,
E
E),
is called a connection. Here Vu C TILE denotes the subspace consisting of all tangent vectors at u which are tangent to the fiber. If F is a closed manifold, for any smooth curve £ : [a, b] can define the parallel translation
--->
JI/l we
he : Ee(n) ~ Ee(b)
along £ in the same way as the case of principal bundles. It gives a diffeomorphism from the fiber Ee(a) over £(a) to the fiber Ee(l;) over £(b). However, if F is not compact, it is not always possible to define parallel translations. Then we define as follows. DEFINITION 2.25. A connection on a fiber bundle is called a strict connection if parallel translation is always defined along any smooth curve on the base space.
If there is given a connection on a principal bundle, then we can "differentiate" it to obtain the curvature. Then the condition that the curvature is identically zero is equivalent to the one that the distribution induced by the connection is completely integrable. With this fact in mind, we make the following definition. DEFINITION 2.26. A connection on a fiber bundle is called a Hat connection if it is complE:tcly integrable, An F-bnndle equipped with a flat connection is called a flat F-bundle, If in addition the connection is a strict connection, it is called a strictly flat F -bundle.
As was already remarked, in case F is a closed manifold, any flat F-bundle is always a strict fiat F-bundle. DEFINITION 2.27. Two flat F-bundles 1fi : Ei ---> M (i = 1,2) over the same base space are called isomorphic as flat F-bulldlcs if there exists an isomorphism El ~ E2 as F-bundles such that it sends the connection on El to that on E 2 .
Now let 1f : E --7111 be a strict flat F-bundle. Then, by applying the same argument as that of §2.1.4, we obtain a homomorphism
p: 1fl(M)
~
DiffF
2.4. GEL'FAND-FUKS COHOMOLOGY
79
such that it induces an isomorphism
E
~
M x F/IT1(M)
as flat F-bundles. This homomorphism is called the holonomy homomorphism or equivalently the monodromy homomorphism depending on the contexts. Here IT1 (M) acts on M as the universal covering transformation group and also on F through the homomorphism p. The following theorem can be proved in the same way as Theorem 2.9. THEOREM 2.28. Let M and F be Coo manifolds. Then the correspondence, 'Which sends any .str-ict fiat F -b'undle over !VI to its holonomy homornorph'lsm, induces a bijection
{isomorphism class of stTict fiat F -bundle over M} ~
{conj1Lgacy class of homomorphism ITdM)
---+
DiffF}.
2.4.3. Definition ofthe Gel'fand-Fuks cohomology. In this subsection, we define the Gel'fancl-F'uks cohomology group HCF(F) for any Coo manifold F. As is well known, the set l:(F) of all Coo vector fields on F has a structure of a Lie algebra with respect to the bracket. Hence its cohomology group as a Lie algebra is defined (cf, §2.2). However, it is too huge an object for the computation of cohomology group, and also it is difficult to give geometrical meaning to it. On the other hanel, it turns out that l:(F) has a natural topology, called the Coo topology, and it becomes a topological Lie algebra, Therefore we may cOIlHider only continuous cochains of l:(F) with ret:>pect to this topology and then take the cohomology of the subcomplex of C*(l:(F)) cOIlsisting of continuous co chains , We call this cohomology the continuous cohomology of l:(F) and denote it by H:(l:(F)). The Gel'fand-Fllks cohomology HCF(F) of F is nothing but this cohomology group. Before Htating the precise definition, we continue to consider flat bundles. DEFINITION 2.29. Let is given an isomorphism E F-product bundle.
IT :
~
E ---+ M be a flat F-bundle. If there !VI x F as F-bundles, we call it a flat
In other words, a flat F-product bundle is a trivial F-bundle M x F together with a completely integrable distribution on it whose
80
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
dimension is the same as that of M and which is transverse to the fibers. Then for each point p E ]\,1/ on the base space, we can define a linear map TpM 3 X f-----> Xf E X(F) as follows. For each point (p, u) E M x F on the fiber over p, we consider the lift X E T(p,u) of X which is horizontal with respect to the given connection and then set Xf (71.) E T"F to be its orthogonal projection to F (d. Figure 2.7). (p)XF
MxF
M
)
p
FIGURE
x 2.7
If we move the point u in the above construction, then we obtain a vector field on F, and this is the definition of Xf. Now suppose that there is given an element of Ck(X(F)), namely an alternating multilinear map (2.24)
r; : X(F) x ... x X(F) ---; R ,
v'---""'-'
k-times
Then, if we set
(2.25) for Xl,'" , X k E TpM, we obtain an alternating multilinear map ij : TpM x ... x TpM ---; R
,
" k-times
,
If ij defined in this way is of class Coo with respect to p, then we obtain a k-form ij E Ak(M) on M. This condition will be satisfied if r; is continuous with respect to the topology of X(F).
2.4. GEL'FAND-FUKS COHOMOLOGY
81
With these considerations in mind, we now define the Gel'fandFuks cohomology. First we define the C= topology of X(F). Let X E X(F) be a vector field on F. For any coordinate neighborhood U of F and its local coordinates Xl, ... ,X n , let n
X
=
a ax,
LJi(X)-. i=1
be the local expression of X on U. Let K be a compact subset of U, r a non-negative integer and E > 0 a positive number. For any vector field Y E X(F), we consider its local expression n
Y = Lgi(X)
a ax
i=1
'
on U and define
N(X;U,K,T,E) to be the set of all Y such that the inequality
alalJi I IaX~'alalgi ... ax;;:n (x) - aX~l ... ax~n (x) < E holds any point x E K for all
lal = a1 + ... + an
~ T.
DEFINITION 2.30. Let F be a Coo manifold. We define a topology on X( F) by requiring that the set of all N (X; U, K, r, E) as above with arbitrary U, K, I, E forms a subbase for open sets. It is called the C= topology.
It is easy to see that the bracket operation is continuous with respect to this topology. Therefore if we denote by A~(X(F)) the set of all alternating multilinear maps
X(F) x ... x X(F) -----.., '-
~
,/
k-times
which are continuous with respect to the Coo topology and set A~(X(F))
= EBkA~(X(F)),
then it becomes a sub complex of C*(X(F)). Now we make the following definition. DEFINITION 2.31. Let F be a Coo manifold and let X(F) be the Lie algebra of vector fields equipped with the Coo topology. Its continuous cohomology H* (A~ (X(F)) is called the Gel'fand-Fuks cohomology of F, denoted by HCF(F).
82
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
2.4.4. Characteristic classes of fiat F-product bundles. We can define bundle maps for flat F-product bundles similarly as in §2.3.1. PROPOSITION 2.32. Let 7r : E -> !vI be a flat F -p-rvduct bundle. Then for any", E A~(:r(F)) the element ij defined by (2.25) is of class COO . Hence it is an element of A k (Af). FurtheTmoTe we have
dTJ
=
dr,.
As a corollary to this proposition, we obtain the following theorem. THEOREM 2.33. Let 7r : E the cOTTespondence
->
A~(X(F)) 3
TJ
M be a fiat F -product bundle. Then ~ ij E
A'(M)
'is a d.g,a. map. Hence it induces a homomorphism
HCF(F)
---'>
H' (M; 1R).
This homomorphism is natural with respect to bundle maps. Hence any element of the Gelfand-Fub cohomology [JTOUp HCF(F) serves as a chamcteristic class of flat F -product b7mdlc8. If we generalize the considerations in §2.3.3 to the present situation, we can see that the homotopy fiber of the natural map BDiffli F -> BDiff F plays the role of the clatitiifying tipace of flat F-prodllct bllndIes. If we denote this space by BDiff F, then we obtain a fibration
BDiff F - • BDiff'" F -) BDifi F. Then the above theorem claims that there is a llOIllomorphisIll HCF(F) - , H*(BDiffF).
Proof of Proposition 2.32. We first show that r, becomes a Coo form on M. It is clear from the definition that r, is linear with reBpect to functions on M; namely for any f E coo(.M) and Xl,' ,. ) Xx: E X(M), we have 1)(1Xl,'"
,fX k )
= f7i(X 1,'"
,Xk ).
Hence it is enough to prove that ij is of cla..'ls Coo, Let X be a vector field defined on a neighborhood U of p E M. Then for each point q E U a vector field
2.4. GEL'FAND-FUKS COHOMOLOGY
83
on F is determined which clearly varies in a Coo fashion if we move q. It follows from this fact together with the definition of ij that we have only to show the following. Namely for any Coo family, with respect to the parameter t, of k vector fields Xi(t) E X(F) (i = 1" .. ,k), the function
(2.26) becomes a Coo function of t. This condition is guaranteed by the cont.inuity with respect to the Coo topology of vector fields which is contained in the very definition of the Gel'fand-Ful<s cohomology. Let us check this point hriefly. For any element 1] E A~(X(F)), the contiulli ty iIll plieH that
HellCC we call conclude first that (2.26) is of class Co. Next from the equality
I. T/(Xl(t),"', Xk(t)) - 1](X1 (O),··· ,XdO)) t-U t
In1~~~~~~~~--~~~~~~~~
k
= L 1](X1 (O),···
,X;(O),··· ,XJ.:{O)),
i=1
we s(~e that (2.26) is of dass C l . If we continue with a similar argument, we Hee eventually that (2.26) is of class Coo. Next we prove the equality (2.27)
d1]
• dij.
For each vector field X E- X(J\!I) on the base space, we denote by }i E X(E) the vector field 011 the total space E = 111 x F which is obtained hy cOllsidering horizontal lift at each point on it. Then, by virtue of the flatness of the connection together with the theorem of :Frobenimi, we have
(2.28)
[X,Y]
= [X,Y]
for any X, Y E X(M). Now to prove (2.27), it suffices to show that the equality
(2.29) holds for any Xl,'" ,Xk+l E X(J\!I). Since it is enough to show this locally, we consider the arbitrary local coordinate system (U; Yl,'" ,Ym)
84
2. CHARACTERISTIC CLASSES OF FLAT BUNDLES
We know already the linearity with respect to functions. Hence we have only to prove (2.27) assuming that
Xi = -
[}
(i = 1,,,, ,k + 1).
[}Yi
Now set
Xi
+ ~i
= Xi
(here we write ~i instead of X/ to simplify the notation). ~i is a vertical vector field on the product manifold E = jVf x F so that for each point p E NI, we may regard ~i(P) E X(F). Since we have clearly [Xi, X j ] = 0, it follows from (2.28) that [Xi,Xj]
= O.
On the other hand, we have
Hence we obtain [~i,~j]
(2.30)
[}
=
a~j Yi
[}
- 8~i' Yj
If we write A for the left hand side of (2.29), then we have A =dT](X1 ,'" ,Xk+d
(2.31 )
=dr) (6 , ... '~k+l)
=L(_l)'i+jT]([~i'~JJ,6,'"
,t;.,"· ,t
j , .. ·
,~k+l)'
i<J
On the other hand, if we write B for the right hand side of (2.29), we can use the fact [Xi, X J ] = 0 to see that B =di/(X 1 ,'" ,Xk+d
(2.32)
_- 'L....-( " -1) i+l XiT](X ' 1 , " ' , Xi,"
. ,Xk + 1 )
Both A and B are functions on U, and we have to show A = B. Let U be any point and for simplicity assume that the local coordinate
p E
2.4. GEL'FAND-FUKS COHOMOLOGY
of that point corresponds to the origin Yi in (2.31), we obtain
= O.
85
If we substitute (2.30)
(2.33) A(P)
=L7)(~~'(O)- ~~j(0),6(0), ... i<j
'€i(O),.·· ,tj(O),· ..
y,
YJ
'~k+l(O»).
On the other hand, it follows from (2.32) that B(p)
= L(-1)Hl~7)(6,··· ,€i,···
,.
(2.34) =
0Yi
'~k+dl Yi=O
L( _l)i+l L 7)(6(0), ... , ~~J (0),··· '~k+l(O»). i
j"ii
Y.
If we compare A(p) with B(p) carefully, we see that they coincide.
This completes the proof.
0
CHAPTER 3
Characteristic Classes of Foliations In this chapter we treat the theory of characteristic classes of foliations. Foliations are certain patterns on manifolds. Locally they look thc same everywhere. However, globally we can draw various patterns. Characteristic classes of foliations represent their global behaviour in terms of cohomology. This theory was established during a short period from the end of the 1960's to the beginning of the 1970's. Here we first consider the Godbillon-Vey class, which is the representative among characteristic classes of foliations. We give the definition for the case of codimension 1 and describe Thurston's examples showing continuous variation of the class. VYe then give a survey of the general theory of characteristic classes of foliations from the viewpoints of the Gel'fand-Fuks theory mentioned in Chapter 2. For details, we refer the reader to [Bot2] [BH] [HI [KT]. In §3.5 we formulate a problem which arises inevitably from a significant feature of characteristic cla~ses of foliations, that is the continuous variation. 3.1. Foliations 3.1.1. Definition of foliations. Roughly speaking, a foliation on a Coo manifold means that we cover the whole manifold evenly by a. family of submanifolds whose dimensions are the same. Here by evenly we mean that locally the situation is the same as the decomposition lR n =
U lR
n- q
x {x}
x<..!~q
of the n-dimensional Euclidean space lR n into parallel translations of the (n - q )-dimensional subspace lR n - q . In this chapter, we use the word "submanifold" in the broadest sense. Namely we call images of injective immersions submanifolds. 87
88
3. CHARACTERISTIC CLASSES OF FOLIATIONS
DEFINITION 3.1. Let M be an n-dimensional Coo manifold. A foliation on M of co dimension q is a family F = {LOl}OlEA of (n-q)dimensional connected submanifolds La which satisfy the following conditions. (i) La's are disjoint from each other, and they cover the whole M. (ii) For any point pin M, there exist an open neighborhood U :3 P and a diffeomorphism 'P : U -> jRn = jRn-q X jRq such that for any a E A, the image under 'P of any connected component of Un La can be written as {x = (Xl,'" , Xn) E jRn; X n - q +l = Cn-q+l,'" ,Xn = Cn} (see Figure 3.1). Here each Cj is a constant. Each La is called a leaf of F.
FIGURE
3.1
EXAMPLE 3.2. Any n-dimensional Coo manifold M admits two trivial foliations. One is the co dimension n foliation on l\!J whose leaves are points on M, and the other is the codimension 0 foliation with a single leaf which is M itself. In general, any Coo fiber bundle 'Jr : E -> l\!J determines a foliation on the total space E whose leaves are fibers of 'Jr. The co dimension of this foliation is equal to the dimension of the base space M. EXAMPLE 3.3. As a simple example of foliations, we define linear foliations on a 2-dimensional torus T2 = jR2 I'll}. Fix a constant
3.1. FOLIATIONS
89
C E JR U {oo}. Then the set of all parallel translations of the line through the origin in JR2 whose slope is c defines a foliation on JR2 of co dimension one. Since this foliation is clearly invariant under parallel translations, it induces a foliation Fe on the quotient space T2. If cEQ U {oo}, all leaves of Fe are diffeomorphic to 8 1 . On the other hand, if c 1. Q U {oo}, then all leaves of Fe are diffeomorphic to JR and moreover they are dense in T2. Thus, although all Fe look the same locally, they have distinct global features according to the value of c. EXAMPLE 3.4. A codimension one foliation on JR x I is depicted in Figure 3.2 (i). If we rotate this foliated manifold around the axis JR x H}, then we obtain a foliation on JR x D2 as shown in (ii). This foliation is invariant under the translation by 1 in the direction of JR. Hence if we take the quotient by this, we obtain a codimension one foliation on 8 1 x D2 as shown in (iii). This is called the Reeb component. If we glue two Reeb components according to the wellknown decomposition 8 3 = Sl X D2 U D2 X S1 of the 3-sphere, we obtain a codimension one foliation on S:3. This is called the Reeb foliation. EXAMPLE 3.5. Flat bundles, which were defined in §2.1.2, serve as important examples of foliations. Let G be a Lie group and let 7r : P ---+ M be a flat G-bundle. Then the set F of all maximal integral manifolds in P defines a foliation whose codimension is equal to the dimension of G. Similarly, the total space of any flat F-bundle 7r: E ---+ B, defined in §2.4, admits a foliation whose leaves are maximal integral manifolds. Therefore such bundle is sometimes called a foliated F-bundle.
Let F be a foliation on M of codimension q. Then the set
r (F)
=
{X
E
T M; X is tangent to a leaf}
becomes a subbundle of the tangent bundle TNI of M. This is called the tangent bundle of:F. The quotient bundle
v(F) = TM/r(F) is called the normal bundle of:F. Clearly r(F) is an involutive subbundle of T M. Namely, if we denote by r(r(F)) the set of sections of r(F), we have
90
3. CHARACTERISTIC CLASSES OF FOLIATIONS
ii)CCCCCC (ii)
(iii)
FIGUR.E
3.2. Reeb component
Conversely, by the theorem of F'robenius (Theorem 2.6 ill !i2.1.3), any involutive subbulldle T of T!vI becomes the tangent bundle of a foliation. This is because snch subbundle is c:ompletcly integrable and the set of all maximal integral manifolds de tines a foliatioll, as is easily seen. Next we describe foliations in terms of differential forms. Let 7 C T ll,iI be a snbbundle and set v ~ T IYI/ 7. If we denote by v* the dual bundle of v, it becomes a subbundle of the cotangent bundle T* M of M. Explicitly, we can write
T*M:J v* = {w E T*MiW(X) = 0 for any X E T}. Then, as was already described in §2.1.3, we have another formulation of the F'robenius theorem as follows. Namely, T is completely integrable if and only if the ideal [(7) of the de Rham complex A*(!vI) of !vI generated by sections of r(v*) of v* is a differential ideal. This
3.2. THE GODBILLON-VEY CLASS
91
condition can be described locally as follows. Let U be a coordinate neighborhood of Nf and let wI, ... ,w q be elements of r( v*) which are linearly independent at any point of U. Then there exists a system of I-forms w} such that q
(3.1)
dJ..ui =
Lwj /\w j . j=l
Thi::; condition is called the integrability condition. Let F be a foliation on M of co dimension q. A Coo map f N --> M is called transverse to F if the composition 7r 0 f* : TpN - 7 Tf(p)M --> Vf(p)(F) is ~urjective at any point pEN. In this case, if we set f,
= {X
E TN;
f*(X)
E
T(F)},
it becomes a subbundle of TN of codimension q, and it is easy to see that it satisfies the integrability condition above. The corresponding foliation is denoted /*(F), and we call it the foliation induced from F by f. The leaves of /*(F) take the form of connected components of the inverse images of those of F by f. In particular, any submersion f : N - 7 M determines a foliation on N whose leaves are connected components of f-l(p) (p E A1). In the above, we described a minimum of materials which will be nece~sary henceforth in this book. For more details of foliations, see e.g. [L][T][To]. 3.2. The Godbillon-Vey class 3.2.1. Definition of the Godbillon-Vey class. If there is given a foliation F on a Coo manifold M, we simply write (M, F). DEFINITION 3.6. Suppose that for any given foliation (Nf, F) of codimen::;ion q, there is defined a cohomology class
Q(F) E H*(M; JR) which satisfies the following naturality condition. Namely, for any Coo map f : N --> 111 which is transverse to F, the equality
Q(f*(F)) holds. Then we say that of codimension q.
Q
=
J*(Q(F)) E H*(N; JR)
is a characteristic class of foliations
92
3. CHARACTERISTIC CLASSES OF FOLIATIONS
We can also consider coefficients other than lR. for characteristic classes of foliations. However, the case of real coefficients is the most important. The first example of characteristic classes of foliations is the one given in [GV], and it is now called the Godbillon-Vey class. Let F be a codimension 1 foliation on M and assume that the normal bundle I/(F) is trivial for simplicity. Then there exists a I-form W E r(//* (F)) such that it does not vanish at any point on M. It follows from the integrability condition (3.1) that there exists a I-form Tj such that
dw = Tj /\ w.
(3.2)
LEMMA 3.7. The 3-form H = Tj /\ dTj on M is closed, and its de Rham cohomology class [H] E H 3 (M;lR) does not depend on the choice of wand Tj. PROOF. We first show that 0 is closed. If we differentiate (3.2) and use the fact that." /\ dw = 0, we obtain dTj /\ w = O. Hence there exists a certain I-form ~ such that d." = ~ /\ w. Then we have
dO
= d." /\ d." = 0
so that 0 is closed. Next if we have dw = Tj' /\ w for another I-form .,,', then (Tj' - .,,) /\ w = 0 so that there exists a function f such that Tj' -." = fw. Hence we can concl ude
H' = .,,' 1\ d.,,' = (." + f w) 1\ (dTj + df /\ w = ." /\ drl + ." 1\ df /\ w = 0 - d(fdw).
+ f dw)
Thus we obtain [0'] = [0). If w' is another I-form which defines F, then there exists a nowhere vanishing function 9 sHch that w' = gw. Then dw' = dg 1\ w
+ gdw
) , ( dg =-+.,,/\w.
9
Hence we can set Tj'
= ~ + Tj,
0' = .,,'
1\
and we have
ciTj' =
(d g +.,,) g
1\
dTj
= 0 _ d (d g 1\.,,). 9
3.2. THE GODBlLLON- VEY CLASS
Thus we obtain [0'] DEFINITION
=
93
[0], completing the proof.
D
3.8. We denote by gv(F) the cohomology class [0]
E
H 3 (M;IR) obtained above and call it the Godbillon-Vey class. In the above definition, if we change W to -W, the form 0 remains unchanged. Hence we can conclude that the Godbillon-Vey class is defined even if the normal bundle v(F) is non-trivia.!. Let f : N --> M be a map transverse to F. If w is a 1-form which defines F, then clearly we can take j*w for a defining I-form of j*(F). It follows that the Godbillon-Vey clas::; is a characteristic class of co dimension 1 foliations. EXAMPLE
3.9 (Roussarie). Let M be the 3-dimensional Lie group
PSL(2,IR) =
{(~ ~)
;a,b,c,d E' IR,ad- bc= 1}/{±1}.
We take a basis of the corresponding Lie algebra 5((2, IR) to be
Xo =
G~1)'
Xl =
(~ ~),
X2 =
(~ ~).
Then we have
Hence, if we consider Xi as left invariant vector fields on M, the 2-clirnellsional subbulldlc T C TM spanned by Xo and Xl becomes completely integrable. Therefore it defines a foliation F on M of codirnension 1. To compute the Godbillon-Vey class of F, let Wi be the dual basis of Xi and consider them as left invariant I-forms on lvI. We can take W2 as a 1-form which defines F, and we have dw 2 = -2woAw2. From this we obtain n = -2wo A d( -2wo) = 4wo A WI A W2 which is nothing bllt a volume form of lvI. Unfortunately, lvI is diffeomorphic to Sl x ]R2 so that H3(M; IR) = 0 and hence the Godbillon-Vey class must be zero. However, as will be mentioned below, it is known that there exist plenty of torsion-free discrete subgroups r c lvI such that the qnotients r\M become 3-dimensional closed manifolds. Since F is clearly left invariant, we obtain a codimension 1 foliation r\F on r\M. Then r\O is a volume form of a closed manifold so that its d(~ Rham cohomology class is non-trivial. Hence we can conclude that gv(r\F) i 0 E H 3 (r\M; IR) ~ R Thus we obtain a foliation whose Godbillon-Vey class is non-trivial.
94
3. CHARACTERISTIC CLASSES OF FOLIATIONS
We add a few more facts concerning the above example. The group PSL(2,1R) acts on the upper half plane lHI = {z
= x + iy E C; Y > O}
by orientation preserving i!:iometries with respect to the Poincare metric. On the other hand, if I: denotes any closed surface of genus 9 2:: 2 equipped with a metric of constant negative cllrvature, then its universa.l covering manifold is isometric to 1HI. Hence a homomorphi!:im p: 1fl (I:) --> PSL(2, IR) is defined up to conjugacy. We can take !:iuch 1m p corresponding to any I: as which we used in the above exa.mple. In this case, r\M can be identified with the unit tangent bundle 11 I: = {X E TI:; II X II = I} of I:, and the foliation r\F 011 it is called the Anosov foliation, which is an important subject of the theory of dynamical systems.
r
3.2.2. Continuous variation of the Godbillon-Vey class. If there is given a codimension 1 foliation F on a dosed oriented 3dimensional manifold M, we can evaluate its Godbillon-Vey class on the fundamental cycle of M to obtain a number gv(F)[M]. This is called the Godbillon-Vey number or Godbillon-Vey invariant. In [Thl], Thurston proved the following theorem by analy;-;ing the above example 3.9 ill detail. THEOREM 3.10 (Thurston [Thl]). There exists a family F t (t E IR) of codirnen8'ion 1 foliations on S3 s·u.eh that the Godbillo7l,- Vey rmrnber' of F t is eqnal to t.
This theorem showed decisively that the Godbillon-Vey class is a cohomology class with essentially real coefficients. Here we shall briefly explain his ideas. The foliation all PSL(2, IR) described in the previous subsection can be expressed more geometrically as follows. As wa..':> already mentioned, PSL(2, IR) acts on the upper half plane IHI by isometries. Therefore it also acts on the unit tangent bundle TIlHI = {v E TIHI; Ilv II ---' I} of 1HI. If we fix a tangent vector v() E TIlHI at the point i E 1HI of length 1, the correspondence PSL(2, IR) 3 A r-> f(A) =
3.2. THE GODBILLON-VEY CLASS
Ava
E
95
TIIHI becomes a bijection, and we obtain a commutative dia-
gram PSL(2, JR)
11 1HI 1HI Here .f(A) = Ai E 1HI. It follows that we can identify the mctp f : PSL(2,lR) ----> IHI with the unit tangent bundle ?T : TIIHI ----> 1HI which is a principal SO(2)-bundle over 1HI. Under this identification, we can describe the foliation on PSL(2, JR), given in Example 3.9, in the context of geometry of TllHI as follows. Let va E TllHI be an arbitrary point and set. ?T( vo) = Zo E: JlI. As is well known, geodesics of IHI are nothing but the intersect.ions with 1HI of circles (or lines) which are perpendicular to the real line. Hence there exists a unique geodesic pa.%ing through Zo which is tangent to Vo E TIIHI at zoo Let 90 denote it and let :.co, Xoo be the intersection points of its closure 90 with lR U { 00 }. Here we assume that the direction from :1:0 to Xoo coincides with that of Va. Then the leaf LV(l passing through va can be descri bed as
LVI) . : {positive unit tangent vectors on all geodesics through xo} (d. Figure 3.3). Vo
Xo
FIGURE 3.3. The leaf Lvo passing through Vo For each v E TllHI with 71'( v) = Z E IHI, let 8 denote the angle at z which is measured counterclockwise from the perpendicular to the real line to v (cf. Figure 3.3). Then 8 is defined globally on the whole T1IHI, and the correspondence TIIHI 3 v f-> (7I'(v), 8(v)) E IHIxlRj271'Z becomes as an orthonormal frame a diffeomorphism. If we take -y y , y
t tx
3. CHARACTERISTIC CLASSES OF FOLIATIONS
96
field, then the corresponding dual l-forms are 6 = -~dY,6 = ldx. y y Hence the canonical I-forms (see §3.3.I) of TllHI are given by Wl
W2
= cos e 6
=-
+ sin e 6
sin e 6
1
= -
Y
(sin e dx - cos e dy)
+ cos e 6 = ~ (cos e dx + sin e dy). Y
If we denote by Wo the Riemannian connection form, then we see from
that the first structure equation is given by Wo = de
1
+ -dx. y
Then we have dw o = Wl 1\ W2 = y\ dx 1\ dy. Hence if Gaussian curvature, the second structure equation dw o
=
J(
denotes the
-J(Wl 1\ W2
implies that K == -1. Thus we have verified that lHI certainly has a constant negative curvature -l. Coming back to our original situation, for each v E Lvo, the angle between the line, joining Xo and z, and the positive direction of the real axis is ~. Therefore Lvo is represented by the equation (}
y
2
x - Xo
tan- = - - - .
(3.3)
If we differentiate (3.3) and arrange it, we obtain
de =
sin e dy y
_ 1-
cos e dx. y
Hence, as a I-form which defines the foliation on T1IHf in question, we can take W=
e d 1 - cos e. dx de - sin - - y+--
= Wo Since
y
- W2·
y
3.2. THE GODBILLON-VEY CLASS
97
if we set ry = -WI, we have dw = ry 1\ w. Hence the Godbillon-Vey form is give by
n = 'I] 1\ dry = WI
1\ Wo 1\ W2
1 = - 2 dO 1\ dx 1\ dy. Y
The results obtained above are invariant under the action of P SL(2, JR) from the left. Hence we can use them in the computation of the Godbillon-Vey number of the Anosov foliation F on the unit tangent bundle TILg of any surface Lg equipped with a metric of constant negative curvature. Since dx 1\ dy is the volume form of lHl, we obtain
;2
gv(F)[T1L g ] = {
n=
-
lT1Eg
=
-27rvol(Lg)
=
47r 2 (2 - 29).
~dO 1\ dx I\. dy
( lTlEg Y
Here vol denotes the volume (or rather areal) form, and we have used the fact vol(Lg) = 27r(29 - 2) which is a consequence of the theorem of Gauss-Bonnet. Thus we have seen the non-triviality of the Godbillon-Vey numbers quantitatively. However, the values obtained above are discrete. To show that they vary continuously, Thurston considered the following. Instead of the upper half plane 1HI we consider the unit disk ]])) = {z E C; Izl < I} which is isometric to 1HI, and inside it we take a "hyperbolic square" ABCD as depicted in Figure 3.4 where each edge is a geodesic. We denote it by K r , Here 0 < r < 1 is a suitable parameter such that the four vertices approach the boundary of IDl when r --+ 1. Let a( r) be the angle at the four vertices of Kr which have the same value. We have limr-->l a(r) = O. Now let Isom+1Dl be the orientation preserving isometry group of IDl which is the same as the group of all holomorphic transformations of 1Dl. Hence we have Isom+1Dl ~ Isom+lHI = PSL(2, JR). Let Ir E Isom+1Dl be the element which sends the edge AD to BC, and similarly let 9r E Isom+1Dl be the element which sends AB to DC. Then it can be shown that the composition
A~B~CSD~A is a rotation around the fixed point A by the angle 4 a( r). Then, for a sufficiently small c > 0, we remove from Kr the c-neighborhood of
98
3. CHARACTERISTIC CLASSES OF FOLIATIONS /\
})
FIGURE
3.4. "Hyperbolic square" Kr
each vertex and let Kr(c) be the remaining part. If we paste the four edges of Kr(c) by the two maps j,.,gr, then we obtain a figure To which is a torus with an embedded disk removed. On the other hand, since the unit tangent bundle 7r : T1Jl)) -, Jl)) of Jl)) is isometric to TllHI, the Anosov foliation is defined on it. We consider the restriction of the Anosov foliation to n-1(Kr(c)). Since this foliation is invariant under the action of Isom+Jl)), if we paste the boundary of n-1(Kr(C:)) by iT> gT> we obtain a codimension 1 foliation on To x S1. We denote it by:Fr . Now the boundary of To x Sl is a torus T2, and the restriction of:Fr to it is a linear foliation on T2 (cf. Example 3.3) corresponding to the angle 4 a( r). Let To X S1 :) (z,w) ~ (z,w n ) E To X S1 be the n-fold covering along Sl and let :F,~1I) be the pullback of :F,. by this map. Then the restriction of :F;n) to the boundary 8To x SJ = T2 is a linear foliation of angle ~4a(r). a(r) is a monotone decreasing function of rand lim"_d a(r) = O. Hence there exists a certain r.l with 1 > r' > r such that
a(r-')
= ~a(r-). n
Then the restrictions of the two foliations :F,., , :F;n) on To x Sl to the boundary are linear foliations with the same angle. Hence we can paste them along the boundaries. If we paste two copies of To along their boundaries, we obtain a genus 2 closed surface ~2' We
3.2. THE GODBILLON- VEY CLASS
99
thus obtain a foliation F(r, n) on E2 x 51. It is expected that the corresponding Godbillon-Vey number is given by gv(F(r, n))[E2 x 51]
=
-21f(vol(Kr') - n vol(K.,.))
= -21f{(21f - 4a(r,l)) - n(21f - 4a(r))}
1) - 81f(n 2 - l)a(r').
= 41f2(n -
If this is true, then we can conclude that the Godbillon-Vey numbers move continuously by choosing r, n appropriately. However, the I-forms w on each of the two copies of To x 51 which define the given foliation and which were used in the above computation of the Godbillon-Vcy number do not connect smoothly to each other. This disadvantage can be overcome as follows. First we denote by Fe(r, n) the above foliation becauxe F(r, n) depends on the choice of c as it stands. However, actually the foliation-preserving diffeomorphism class of F£(r, n) does not depend on E because the place where we paste the two copies of foliated manifolds is a torus with a linear foliation. Next it can be shown that, for some co > 0, we can take a I-form w;' on To x 51 which defines Fr for any c E (0, co) satisfying the following conditions. (i) (ii) (iii) (iv) (v) (vi)
w;' = wr near the boundary of 1(l x 51. w;' = w away from the boundary of 1() x 51.
w;.
d.w; = T};' 1\ r,; = 0 near the boundary of To x 51. 1/; = -WI away from the boundary of To x 51. lime~() ./~)XSI T}; 1\ dT}; WI 1\ dw 1 == -21fvol(Kr ).
IK"
Here in (i), w" is a closed I-form on a neighborhood of the boundary of To x 51 which definex the linear foliation there. From the above, we can conclude that gv(F(r,n))[E2 x 51] =
r
lToxsl
T}.~, 1\ dT};'
= lim ( /" 17;' £-.0 lToxsl
1\
- n
r
lToxsl
T}~ 1\ dT};
dT};' - n /"
lToxsl
T}; 1\ dT};)
= - 21fvol(Kr ,) - n(-21fvol(Kr )).
Therefore we see that the above rough computation was correct. We thus have proved the following theorem.
100
3. CHARACTERISTIC CLASSES OF FOLIATIONS
THEOREM 3.11 (Thurston). For any t E JR., there exists a codimension 1 real analytic foliation F t on ~2 x Sl such that qv(Ft )[I: 2 x S1] = t.
3.3. Canonical forms on frame bundles of higher orders 3.3.1. Canonical forms and connection. Here we recall some well-known facts about the tangent frame bundle 7r : peN) -> N of an n-dimensional Coo manifold N. First there is defined on the total space peN) of the tangent frame bundle certain I-form () E Al(p(N);JR.n) with values in JRn as follows. It is called the canonical form or canonical I-form. Each point u E peN) defines a linear isomorphism
TuP(N) 3 X
f->
O(X)
=
More explicitly, if Vl, ... , Vn denotes an ordered basis of TpN determined by p, then the components 0 = (()l, ... , ()n) of 0 with respect to the standard basis of JRn are given by
TuP(N) 3 X
f->
7r.(X) = ()l(X)Vl
+ ... + On(X)vn .
The canonical I-form is invariant under diffeomorphisms as described in the following proposition. The proof is easy and is left to the reader. 3.12. Let f : M -> N be a diffeomorphism and peN) be the diffeomorphism induced by f. Then j'()N = ()M where OM and ()N denote the canonical forms of M and N respectively. PROPOSITION
let
j : P(M)
->
Now suppose that an (affine) connection w E Al(p(N);gf(n;JR.» is given on peN). Ifn E A2(p(N)jgf(njJR» denotes the corresponding curvature form, the following second structure equation (3.4)
holds. Then how can we express dO ? Let X, Y E TuP(N) be tangent vectors at an arbitrary point U E peN). If we denote by X h , Y h the horizontal components of them, then we know that n(Xh' Y h) = dw(Xh' Y h ). We define 8 E A 2 (P(N)j JRn) by setting
8(X, Y) = dO(Xh, Y h )
3.3. FRAME BUNDLES OF HIGHER ORDERS
and call it the torsion form. equation
101
Then the following first structure
(3.5) holds. If we denote by ei , e i , WJ, OJ the components of e, e, w, 0 with respect to the standard basis of IR n and gf(n; IR), then the above two structure equations (3.4);(3.5) can be written as n
dwj
= -
LW~
/\ w; + OJ
k=l
n
de i
= -
L wj /\ e + e j
i.
j=l
A connection W with e = 0 is called a torsion free connection. If we have a connection w, it determines a closed form Pi(O) which expresses the i-th Pontrjagin class [Pi (N)] E H4i(N; IR). This differential form depends on the choice of connection. Nevertheless it may be natural to ask whether it may be possible to determine it more canonically. As preparation for giving an answer to this question, in the next subsection we introduce a concept of jet bundles due to Ehresmann. 3.3.2. Tangent frame bundles of higher orders. The tangent frame bundle P( N) of an n-dimensional Coo manifold N consists of all frames at all points of N. Here a frame at a point pEN is an ordered basis of 1~N and so it represents a first order approximation of local coordinates at p. If we consider higher order approximations, we obtain the concept of tangent frame bundles of higher orders or simply frame bundles of higher orders. 3.13. Let x be a point in IR n and let f : U ----7 IR n be a Coo function defined on some neighborhood (which may depend on f) of x. The k-jet of f at x, denoted by j;(f), is an equivalence class of all such functions divided by the following equivalence relation. Here two functions f and 9 are said to be equivalent (denoted by f ",k g) if they satisfy the following conditions. If k = 0, the condition is simply f(x) = g(x). If k ~ 1, DEFINITION
f '" k
9
olal f oxO'.
{==? - -
I Ix
olal 9 I oxO< x
=--
3. CHARACTERISTIC CLASSES OF FOLIATIONS
102
for any multi-index a = (al,··· , (Xn) with I(XI = (Xl + ... + an k. Here Xl,· .. , Xn are the standard coordinates of rn;n and ox'" oxr" ... O.T;;''' .
<
Now if we set
Gk(n)
=
{j~(f); f is a local diffeomorphism of rn;n with f(o) = o}
where 0 denotes the origin of rn;n, it becomes a Lie group with respect to the natural topology and the product induced by compositions of mappings. Clearly Gl(n) can be identified with GL(17"rn;), and for any k, GL(n,rn;) is a subgroup of Gk(n) consisting of all linear isomorphisms of rn;n. Furthermore there existti a natural projection Gk(17,) -; Gk-l(n), and we obtain a series
... - , Gk(n)
-----7
Gk-l(17,)
-----7 • . . .-
.,
G1(n) - , {id}
of Lie groups. DEFINITION 3.14. Let N be an n-dimensional Coo manifold and let .Ik(N) denote the set of all k-jets j~(rp), at thc origin 0 E rn;n, of local diffeomorphisms rp from neighborhoods of 0 to N. Then .Ik (N) has a natural structure of a Coo manifold, and the canonical projection .Ik(N) :3 j~(rp) I--> rp(o) E N becometi a principal bllndle with structure group G k (17,). In particular Jl (N) coincides with the tangent frame bundle P(N) . .Ik:(N) is called the tangent frame bundle of order k.
The projection Gk(n) -; Gk-1(n) induces a natllfal projection .Ik(N) -; .Ik-l(N), and we obtain a series • . • -----7
.Ik(N) - , J',-l(N) - , ... - , Jl(N) - , N
of principal bundles. It is easy to see that the quotient space G k (n)/G k - 1 (N) becometi an affine space for any k :::: 2. Hence all Gk(n) are homotopy equiva.lent to each other, namely Gk(n) c::= GlCn) c::= O(n). Similarly we have .Ik(N) c::= .Il(N). Also the subgroup O(n) C Gk(n) consisting of all k-jets at the origin of all orthogonal trantiformations of rn;n becomes a maximal compact subgroup of Gk(n) for any k. Hence Jk(N)/O(n) c::= N. If N is oriented, we can consider only orientation preserving local c!iffeomorphisms so that we can define the oriented tangent frame bundle 7k(N) --7 N of order k. It is a principal bundle whose structure group is the corresponding subgroup SGk(n) C Gk(n) of index 2. Further we have 7k(N)/SO(n) ~ N.
3.3. FRAME BUNDLES OF HIGHER ORDERS
103
3.3.3. Canonical forms on the frame bundle of second order. The definition of canonical forms on the tangent frame bundle P(N) Jl(N) given in §3.3.I was generalized to the cases of frame bundles of higher orders by Kobayashi [Kol]. In this subsection, we treat the case of the frame bundle 7f : J2(N) -> N of second order following [Ko2]. Let U E J2(N) be an arbitrary point and set 7f(u) = pEN. Then u, can be represented by the 2-jet f;('Pu) of some local diffeomorphism 'Pu : U -> N defined on a neighborhood U of the origin 0 in IR'n such that 'Pu(o) = p. 'Pu induces a natural linear isomorphism Jl('Pu) : TidJ1(IRTI) ~ TpCu)Jl(N). Here p : J2(N) -> Jl(N) is the canonical pro.iection and id denotes the .iet of the identity map, Then we define a I-form e E Al(J2(N);IJdJ1(lRn)) with values in TidJl(IR'n) by
T u J 2 (N) :;. X
f--7
B(X) . J1('Pu)-1(p.(X))
E
TidJ1(IRn),
The I-form () on J2(N) obtained in this way is called the canonical form of second order. Similar to the canonical form described in §3.3.1, this I-form is abo invariant under diffeomorphisms as the following proposition shows. We leave the proof which is not difficult to the reader. PROPOSITION 3.15. Let f : M -> N be a d~ffeomorphism and let J2(f) : J2(M) -> J2(N) be the diffeomorphism induced by .t. Then J2 (f)' BN = BM where eM and eN denote the canonical forms of second order of All and N respectively.
In particular, eN is a 1-form invariant nnder the natural action of the diffeomorphism group Diff N of N. Now let A(n) denote the Lie group consisting of all affine transformations of IRn. It has a structure of a semi-direct product 1 ---} IRn ---} A(n) ---} GL(n, IR) ---} 1 of GL(n, 1R) with the normal subgroup IRn consisting of all parallel translations. We can describe it explicitly as
A(n)
= {
(~
n;
A
E
GL(n, IR), x
E
IRn} C GL(n
+ 1, IR).
We can see that the natural projection 7f : A(n) -> IRn has a structure of a principal GL(n, IR)-bundle over IR n and the correspondence
104
3. CHARACTERISTIC CLASSES OF FOLIATIONS
is an isomorphism as principal GL(n, JR)-bundles. Under this correspondence, id goes to the unit element of A(71). Hence we obtain an identification TidJ1(JRn) ~ o(n). Here o(n) denotes the Lie algebra of A(n). By this identification, we can regard the canonical form f) of second order as a I-form with values in o(n). Let ei, Ej be the standard basis of o(n) = JRnE£)gl(n,lR) and let gi, g; (i,j = 1,··· ,n) be the components of on J2(N).
e with respect to them.
PROPOSITION 3.16. Let p : J2(N) jection. Then we have p' gi = f)i.
-->
Then they are I-forms
Jl(N) be the natural pro-
It is enough to use the fact that 71". : Ti d Jl(JR 2 ) ~ o(n) = JR'" E£) 9[(n, IR) --> To(JRn) = IR n is the projection. 0 PROOF.
J2(N) --> N is a principal G2 (n)-bundle. Hence if we denote by the Lie algebra of G 2 (n), then each element A E 0 2 (71,) defines the associated fundamental vector field A* E X(J2(N». Also each element 9 E G 2 (n) acts on J2(N) from the right, which we denote by Rg : J2(N) --> J2(N). If we set 9 = j~(J), the correspondence 71" :
92 (n)
Jl(IR"') J U:') j~(h) ~ f/)(J
0
h 0 rl) E Jl(JRn),
which is defined on a neighborhood U of id E .Jl(lRn), becomes a local diffeomorphism. We denote by Ad(g) : TidJ1(JRn) ~ o(n) - ; TidJ1(JRn) ~ o(n) the differential of this map at. id. PROPOSITION 3.17. The canonical form g E Al(.J2(N); 0(71,)) of second order has the following two properties. (i) For each element A E g2(71,), we have f)(A*) = p(A), wher-e p: g2(n) --> g[(n, IR) C 0(71) is the natural projection. (ii) For each elem.ent 9 E G 2(n), we have R~f) = Ad(g-l)B. PROOF. It is enough to compute according to the definition of the canonical form. 0
The following proposit.ion gives the structme equation for the canonical form of second order.
3.3.
FRAME BUNDLES OF HIGHER ORDERS
105
PROPOSITION 3.18. Let () = «()i, ()j) be the canonical form of second order on J2(N). Then we have the equality n
d()i
=-
L oj 1\
()j.
j=1
Since 0 is invariant under diffeomorphisms (see Proposition 3.15), it is enough to prove the above proposition for the case N = ~n. In [Ko2J, the claim for this case is proved by a direct computation with respect to explicit coordinates in J2(~n). However, in this book we shall prove the more general structure equation for canonical forms of higher orders (see Proposition 3.23 in §3.3.5) following Bott [Bot2]. Here we proceed further assuming Proposition 3.18 for the moment. Now suppose that there is given a section s : Jl(N)
----+
J2(N)
which is cquivariant with respect to the right action of GL(n, lR). Namely we assume that s(ug) = s(u)g
('I.l E Jl(N),g E GL(n,lR)).
Then if we set Ws
= (s*()j),
it becomes a 1-form on peN) with values in g[(n;~). Moreover, Proposition 3.17 implies that it has the following three properties. (i) For any A E gr(n,lR), we have ws(A*) = A. (ii) For any g E GL(n,~), we have g*ws = Ad(g-l)ws . "') d()i ( III ,
- -
L..Jj=l S *()ij
",n
1\ ()j .
In other words, Ws is a torsion free connection form on peN). Conversely, it is known that any torsion free connection form w can be obtained in this way. More precisely, the following proposition is known to hold. PROPOSITION 3.19 (Kobayashi [Ko2J, Proposition 7.1). There are canonical bijections among the following thTee sets.
(i) The set of all torsion free connection forms on Jl (N). (ii) The set of all GL(n,~)-equivaTiant sections to p : J2(N)
peN). (iii) The set of all sections of J2(N)/GL(n,lR)
->
N.
->
106
3. CHARACTERISTIC CLASSES OF FOLIATIONS
Now put a torsion free connection w in Jl (N) and let n be its curvature form. For example, we put a Riemannian metric on N and consider the associated Levi-Civita connection. Then the i-th Pontrjagin form
Pi(n) is defined which is a closed form on Jl(N) (or actually on N). This differential form depends on the choice of a connection. However, we shall see below that the Pontrjagin forms are determined canonically on J2(N). The structure equation of the connection w is given by n
On the other hand, by Proposition 3.19, if we consider the section s : Jl(N) -+ J2(N) which corresponds to the torsion free conection w, then we have i *()i Wj = S j'
Now we define a 2-form R by the equality
=
(Rj) on .I2(N) with values in gl(n,R) n
(3.6)
d();
=-
L ()~ /\ ()j + R; . 1<=1
Then clearly we have ni. Hj
=
S
'Rij'
If we differentiate both sides of (3.6), we obtain the Bianchi's identity n
·JRij =
(j,
'~ \ ' (R'i,,/\
()kj
-
f}i /\ rlk
Rk) j .
k=l
From this fact, we sec that the i-th Pontrjagin form
is defined as a closed form on .I2(N), and we have Pi(n) = s*pi(R). The differential form Pi(R) is determined independent of the choices of metrics on N or connections on .II (N) and is invariant under diffeomorphisms. Therefore we may say that it is the universal Pontrjagin form.
:U. FRAME BUNDLES OF HIGHER ORDERS
107
3.3.4. Canonical forms and formal vector fields. We consider the frame bundle of k- th order 1r : Jk (N) ~ N. Suppose that a point 'U. E Jk (N) is represented by the k-jet j~ (tp), at the origin, of a local diffeomorphism tp from a neighborhood of the origin of]Rn to N. Then a tangent vector X E T,.Jk(N) can be represented by a family tpt : ]Rn ~ N of local diffeomorphisms such that 'Po = 'P, because 'Ut = j~(tpd E Jk(N) can be regarded as a curve passing through 'U. Now tp~l 0 tpl. is a local diffeomorphism defined in a neighborhood of the origin of ]Rn. Hence it can be described by n functions.
Yi=Yi(t,Xl,"',X n )
(i=l,···,n).
vVe define a vector field Zx, which is defined in a neighborhood of the origin of JR'fl, by
Zx If we flet
then we have (3.7) Actually, only the (k -I)-jet at the origin of each coefficient Zi of Zx ifl defined. Based on this fact, we make the following definition. DEF'INI1'JON
3.20. A formal sum
a
n
Z = ~ .f.lTl,··· , .7;,,) aXi where each coefficient !i(Xj,'" , .T n ) E JR[[Xl,'" , xnll is a formal power series is calleel a formal vector field on JR n . The set of all formal vector ficldfl has a natural structure of a Lie algebra with reflpect to the usual bracket operation. vVe denote it by an. vVe also denote by a11.[k - 1] the quotient space of an obtained by neglecting those terms with degrees ~ k of each coefficient. In other words, an [k - 1] is the set of all (k - 1)-jets of formal vector fields on ]Rn.
Zx
In the terminology of the above definition, we ca.n now write an[k - 1]. We define I-forms
E
ei ,
e}I"-je
(f::; k - 1)
108
3. CHARACTERISTIC CLASSES OF FOLIATIONS
on Jk(N) as follows. Any tangent vector X E TuJk(N) determines a formal vector field Zx as above. Then using the expression (3.7) of Zx, we set
()i(X)
= Zi(O)
()i
(X) _
j1···je
-
Of zi OX . ... OX' J1
(0).
Je
We call these I-forms the canonical forms on Jk(N). Actually there exists a natural isomorphism
such that the correspondence
TuJk(N)
:;I
X
f-----+
Zx
on[k - 1]
E
9:" TidJk-l(l~n)
realizes the composition
TuJk(N) ~ Tp(u)Jk-1(N) Jk-~)-1 lidJk-1(JR n ). In case k = 2, let us check that the above definition coincides with that given in the previous subsection §3.3.3. Formerly, we considered the composition
Tu J2 (N) ~ Tp(u)J1(N)
J1i::2.-t
TidJ1(JRn)
9:"
a(n)
=
JRn EB gl(n,JR)
and for any tangent vector X E TuJ2(N), we defined (()i(X),()j(X)) to be the image of X under the above map. On the other hand, if X is represented by the 2-jet j;( i.pt) of a family i.pt of local diffeomorphisms defined in a neighborhood of the origin, then its image in li d .J1 (JR n ) is represented by j~(i.p-l 0 i.pt). Hence we have
OJ(X) =
~ (!l~-(i-th component of i.p-l 0 i.pt)(O)) I uXj t=o
ut
o
=!:l (OYi ~(t, ut
uXj
0) )
I t=O
= ~Zi (0). c)xj
Since a similar result holds for ()i(X), we see that the two definitions are the same. We leave the proof of the following proposition to the reader because it is not difficult.
3.3. FRAME BUNDLES OF HIGHER ORDERS
3.21. (i) Let p : Jk(N) the natural projection. Then we have
PROPOSITION
= OiJ1···Je.
P*OiJ1·"Je.
-4
109
Jm(N) (m < k) be
(e < m).
(ii) Let f E DiffN be any element and let Jk(f) : Jk(N) be the din:eomorphism induced by f. Then we have
Jk(f)*ot ... je =
O;l ... je
Jk(N)
-4
(e < k).
3.3.5. Structure equations for canonical forms. We pre-pare a few facts in order to obtain the structure equation for canonical forms (see [Bot2] for more details). Suppose that a point U E Jk(N) is expressed as 'U = j~(cp) where cP is a local diffeomorphism defined in a neighborhood of the origin in ~n. Then as was mentioned earlier, a tangent vector X E TuJk(N) at u is represented by a family CPt (CPo ~ cp) of local diffeomorphisms. We can write CPt
= cP 0 (cp-1 0 cpd
where
X(v) = the tangent vector Then we can see that ZX(v)
E
TvJk(N) determined by
(
0
CPt).
= Zx for any v. Hence the function OiJ1·· ·Je. (X)
becomes a constant function on Jk(N) for any e < k. Actually X(v) depends on the choice of 'if}, and rigorously speaking we need an argument using a section Jk(N) -) Jk+l(N). However, if p: Jk(N) -4 Jk-1(N) denotes the projection, then we can see that p*(X(v)) depends only on v so that the above computation is valid. In a general situation, Hl.lppOSe that a Lie group G acts on a Coo manifold M from the right a..s M x G -41\11. For any element A E g, we define a vector field A * on 111 by A*(p) = the tangent vector E TpM determined by t ~ P exptA
where p EM. Then as is well known the correspondence g '3 A
J-----+
A* E X(M)
becomes a homomorphism of Lie algebras. (In the case of a left action, a similar definition implies [A*,B*] = -[A,B]* so that we have to set A * to be the vector field determined by the correspondence t ~ P exp - tA.) In our situation, the infinite dimensional group
3. CHARACTERISTIC CLASSES OF FOLIATIONS
llO
DiffIRn acts on Jk(N) from the right infinitesimally by composition of mappings:
Jk(N) x DiffIR n
3 (j~('P),
f) ... --) j~'('P 0 f)
E
Jk(N).
The differential of this action at the identity (actually only this is defined) is given as follows. Suppose that X E X(JR n ) is expressed by it E DiffJRn (fo id). Then we define a vector field X* on Jk(N) by the correspondence
Jk(N) 3 u = j~('P)'
• X*(u) = the tangent vector E TuJk(N) determined by 'P 0 it-
Here X* is clearly linear with respect to X. Similar to the case of X, actually X* (u) depends on the choice of 'P. However, its image under the natural projection to Jk-l(N) depends only on X and u. LEMMA
3.22. In the above setting, 'We have the equaltiy [X*, Y*] = -[X, Y]*
(X, Y EOn).
PROOF. Suppose that X and Yare represented by families it and gt of local diffeomorphisms in a neighborhood of the origin respectively. Then, in the equality
[X, Y] = lim Y - (ft.)*Y, t-,Q
t
(ft)* Y corresponds to the family ft 0 gt 0 f - t of local diffeomorphisms. On the other hand, in Y*]( 'lj, ) [.v* /\.,
_ I'un ?,*(u) - ((ft).y*)(u) ,
-
t-,{)
t
((ft) *Y*) (u) corresponds to 'P 0 f t 0 gt 0 it. Since the operation of taking conjugates by ft is opposite to each other, the claim is [: proved,
The minus sign which appears in the above lemma is simply due to a technical reason. It shows that the correspondence X --) X* is essentially a homomorphism of Lie algebra.'). Going back to the earlier situation, for each X, Y E T"Jk(N), let X, Y be the correilponding vector fields on Jk(N) respectively, Then the ahove lemma implies that (3.8) The following proposition gives the structure equation for general canonical forms of higher orders,
3.3. FRAME BUNDLES OF HIGHER ORDERS
111
PROPOSITION 3.23. Let O~h ... jk_l be the canonical form of order k on .Jk(N). Then on Jk+l(N), we have the equality
Here S = {Sl,'" , sp} T'ltnS through all subsets of {I,··· , k -I}, and we denote by = {h,'" , t q } (p + q = k - 1) its complement. In pa:rt'tc,ltlar, for k = 1,2,3 we have
se
L 0; 1\ oj dO; = - L O;k 1\ L ()11\ ()] dO i
=-
j
()k -
k
k
dO;k
.-
L ojk€ e
1\
()e -
L O;e e
1\
O~ -
L e
()k€ 1\
OJ -
L ()~
1\
()]k'
e
PROOF, The proof can be done almost the same way for all k. Here we prove the case k = 2. In view of Proposition 3,21 and also since the d,g,a. map p* : A*(y-l(N)) --> A*(Jf(N)), which is induced by the projection p : Je(N) --> Je-l(N), is injective, it is enough to prove that the equality holds on Je(N) for a sufficiently large e. Then for any two tangent vectors X, Y E TuJf(N), let
3. CHARA CTERIS TIC CLASSE S OF FOLIAT IONS
112
X, Y
and also conside r obtain
dej(X, Y)
Then by equalit y (3.8), we
E ?£(Je(N)).
dej(X, Y) = Xej(Y) - yej(JY) - (1j([X, YJ)
=
(the i-th = -~ OXj
- ) (0) compon ent of Z[X,YI
(the i-th compon ent of [Zx, Zyl) (0) ~ OXj
=
OWi n a (l)Zk ~-
= -,-
OXj
UXk
k=l
OZ;) Wk ::.,.) UXk
= I:
0 2 Zi (02Wi - 'Wk Zk OXjOX k; OXjOXk k=l n
(0) OZi )
O'Wk OZk OWi --- + ~ax" OXj
ax"
(0)
OXj
n
I: (e"(X)ejk(Y) - (1"(Y)tijk(X)
=
k=l
+ ej(x)ek(Y) - e;(Y)(1k(X)) n
= -
:L(ejk
!\
ek + ek !\ ej)(x, Y).
k=l
o
This comple tes the proof. Now if we set JOO(N)
Ii!!! Jk(N),
=
e=
then the totality
w, (1;,
ej k> ... )
of canonic al forms can be regarde d as an element of n Then Propos ition 3.23 can be considered as the structu re equatio we that For which expresses de. We put it in a more tractab le form. choose a basis of an as follows. We set
a
e -.t
-
OXi"
=
e~
a
-:rJ'~,. , uXi
and more generally we define e j, t
.. jk -
~"" . '" (_l)k 2k~xJl syrn
. ~
x JkOX ' ~
3.4. CHARACTERlSTIC CLASSES OF FOLIATIONS
I I \
Here sym means that we take the symmetric sum with respecL I.., I I,,· indices. In other words, we set
(-I)k
Ok
(the coefficient of e:l-j,,) = l.
aXjl ... aXjk
If we denote the canonical form by
o= L i
Oi ei
+L
OJ eI
+L
O} k e{ k
+ . .. ,
i,j,k
i,j
then it becomes a I-form on Joo(N) with values in express the structure equation simply by
Un.
We C11.11
1111\\
1 dO = - -2 [0 ' OJ .
(3.9)
3.4. Bott vanishing theorem and characteristic claS8(~8 of foliations 3.4.1. Bott vanishing theorem. Let M be an n-dimeJll'jlllllll Coo manifold and let :F be a foliation on M of co dimension (/. 11,1' Definition 3.1 in §3.1.1, for any point p E M there exist all '1111'11 neighborhood U '3 P and a diffeomorphism IR". (iii) For any p E U", n Uj3, there exists a local diffeomorpliisill 'Y(3", :
a neighborhood of fa(P)
~
a neighborhood of f(-l(II)
of IRq such that the equality f(3 =
'Y{3", 0
f", holds ncar 71.
It is easy to check that we can adopt the above descriptioll a~: /111 alternative definition of foliations. Now consider the fiber llllllclll' f~(Jk(lRq))
-->
U",
which is the pullback of the tangent frame bundle Jk (IRq) of 01'(1101' h: (Ii IRq by the map f",. Over the intersection Ua n U{3, these bundl('s I'all be naturally identified by an isomorphism induced by 'Y{3a' H(~IIl'" WI'
3. CHARACTERISTIC CLASSES OF FOLIATIONS
114
obtain a principal bundle over M with structure group Gk(q) which we denote by Jk(F). If we consider Jk(F) for all k, we obtain a series
.J
,
OO.
of principal bundles. In particular, Jl(F) is the principal GL(q, lR)bundle associated to the normal bundle //(F) of:F. Now recall that the canonical forms () on tangent frame bundles of higher orders are invariant under diffeomorphisms (see Proposition 3.21). Hence we obtain a I-form
ejko"')
()(F) = (e"', ()j, on JOO(F) with values in
Oq.
In particular we have a system of I-forms ()i
()i
,
J
on .J2(F) with values in o(q). THEOREM 3.24 (Batt vanishing theorem [BoU]). Lei; 111 be a Coo manifold and let F be a foliation on NI of codimension q. Then the s'Ubalgebm of H* (NI; Q) genemted by the mtional Pontrjagin classes pi(l/(F)) of the normal bundle v(F) ofF is trivial in riegTees Namely we have
>
2q.
(4('i 1 -I- ... -I- ie) > 2q).
Pi, (v(F))··· Pic (//(F)): 0
PROOF. If we generalize the consideration in §3.3.3 to the case of .1 00 (F), we see that the following facts hold. N amdy, if we define 2-fo1'm8 Rj on J2(F) by setting q
Rj ... d()j +
L ()~; 1\ e.~ , k=l
then we can define canonical Pontrjagin forms
in terms of them. Moreover, if we choose a GL(q, lR)-eqllivariant section s : Jl(F) ~ J2(F), then (s*()j) becomes a connection form of Jl(F) and s*(p.;(R)) becomes a Pontrjagin form representing the Pontrjagil1 class Pi(l/(F)). Now recall that we have canonical forms
3.4. CHARACTERISTIC CLASSES OF FOLIATIONS
115
on .J3(F) and they satisfy the equality d(); = -
q
q
k=l
k=l
L ()1 !\ ()~' - L
Ojk !\ ()k
(see Proposition 3.23). If we choose a section follows from S*()iJ = (JiJ that
8 :
.J2(F)
-->
.J3(F), it
q
R; = d(); + L
()1 !\ ()J
k=l q
'C.o
+ L ()1 !\ ()j)
s*(d();
k~l
q
-8*
(L Ojk !\ ()k). 1.:-1
Hence if
e> q, we can conclude
(3.10) The claim of the theorem now follows from this.
o
3.4.2. Definition of characteristic classes of foliations. We defi.ne characteristic classes of foliations by making use of the equality (3.10) which played a fundamental role in the proof of the Bott vanishing theorem (Theorem 3.24). First of all, t.he Weil algebra (cf. 32.1.1) of g[(q, JR) can be decribed as W(g[(q, JR)), A*g[(q, IR)* ® S*g[(q, IR)* = E(w;) ® p[n;l.
As was mentioned earlier, here E and P denote the exterior and polynomial algebras respectively. Also and nj denote the components of the universal connection and the curvature forms with respect to the standard basis of g[(q, IR), respectively. By the definition of Rj, we sec that the correspondence
w;
w'i )
f-----4
Oi
J'
nJi
f-----4
Ri J
induces a d.g.a. map
(3.11)
~
A*(.J2 (F)).
The equality (3.10) show::> that this map
116
3. CHARACTERISTIC CLASSES OF FOLIATIONS
PROPOSITION 3.25. Let I denote the ideal of the Weil algebra W(g[(q, JR» generated by the elements of the forms nil ... ni."+1 . JI
}'I+I
Then I becomes a differential ideal. Namely we have dI c I. PROOF. If we differentiate both sides of the structure equation i
dw j
=-
q "i
k
LWk I\Wj
i
+OJ'
k=l
then we obtain q
dn;
=
2:(0; 1\ wJ - w~ 1\ nj) k=l
(Bianchi's identity). The claim follows immediately from this.
0
COROLLARY 3.26. If we set W(g[(q,JR» " W(gl(q,JR»/I, then it becomes a d.g.a. and the map (3.11) induces a d.g.a. map :
W(g[(q,JR»
---->
A*(.P(F)).
Hence we obtain a homomorphism :
H* (W(g[(q, JR»)
---->
H* (J 2(F); JR).
As is well known, the cohomology of the Weil algebra is trivial. However, we can see that the cohomology of the quotient of the Wei! algebra divided by the ideal I as above has many nontrivial cohomology classes as follows. We define invariant polynomials Ci E Ik(GL(q, lR)) (i = 1,··· ,q) by det(I +
1 21f
-n)
= 1 + Cl
+ ... + cq .
Here I denotes the identity matrix and n = (nj). Also it is known that C2i is the invariant polynomial corresponding to the i-th Pontrjagin class. By the triviality of the cohomology of the Wei I algebra, there exists an element Ui E W 2i -l(g[(q,JR» such that dUi = Ci. We choose one such element. For example, we could choose the ChernSimons form TCi corresponding to Ci (cf. §2.3.4). We use the same symbols for their projected images in W(g[(q, lR». Then, if we denote by Wq the subalgebra of W(g[(q, JR» generated by Ui, Ci, we can write Wq=E(Ul'··· ,uq)0Pq [Cl,··· ,cq].
3.4. CHARACTERISTIC CLASSES OF FOLIATIONS
Here Pq denotes the polynomial algebra on generated by elements of degrees > 2q.
Ci
117
truncated by the ideal
PROPOSITION 3.27. The inclusion map Wq C W(fJ!(q,JR)) induces an isomorphism
H*(Wq) ~ H*(W(fJ((q, JR)))
in cohomology. PROOF.
We define a filtration {Fkh in W(fJ!(q,JR)) by
Fk .~ {a E W(g!(q, JR)); the degree of a
with respect to curvature
~
k}.
Then we can see that the E2 term of the associated spectral sequence is equal to H*(g!(q,JR)) 0 P'l[Cl,'" ,cq ]. On the other hand, if we put similar filtration in W q , then the corresponding E2 term in this case is clearly equal to Wq itself. As is well known, the projection
E(u},··· ,uq )
--+
A*ll!(q,JR.)*
induces an isomorphism in cohomology. Hence the above two E2 terms turn out to be isomorphic to each other by the natural homomorphism induced by the inclusion. Then the claim follows from the comparison theorem of spectral sequences. 0 We denote the element
of Wq simply by U[CJ. Also we set PROPOSITION
IJI = jl + ... + jt.
3.28 (Vey). The elements
{U[cJ;
PI
~ q, i l ~ jl, i l
+ IJI > q}
form a basis of H*(Wq) ~ H*(W(g!(q,JR))).
It is clear from (3.11) that all the above elements are closed under d. If the normal bundle v(F) of a foliation is trivialized, then by using the corresponding section s : M -> Jl(F), we obtain a homomorphism
3. CHARACTERISTIC CLASSES OF FOLIATIONS
l18
We can see that the image o:(F) E H*(M; 1R) of an arbitrary element 0: E H* CWq) under the above map becomes a characteristic class of foliations whose normal bundles are trivialized. In the case of general foliations whose normal bundles arc not necessarily trivial, we can construct characteristic classes as follows. The action of the orthogonal group O(q) on the Weil algebra W(gl(q,IR» through interior product and adjoint representation induces the action on the truncated Weil algebra W(gl(q, JR». Hence, if we denote by W(gl( q, 1R) )o(q) the subalgebra of W(g((IJ, 1R» consisting of all elemeut.s which are killed by thc int.erior product of any clement of O(q) anci which are also invariant under the adjoint action of it, then we obtain a homomorphism H* (W(gl(q, 1R»O(q»
-------+
H*(J 2 (F)jO(q); JR).
If i is odd, it can be shown that we can choose W(g((q, IR))O(q)' Then we set
Ui
from the set
WO q = E(Ul, U3,"') ® Pq[Cl,'" ,cq] C W(gl(q, 1R»O(q)· As for this, similar to Proposition 3.27 and Proposition 3.28, the following two propositions hold. PROPOSITION
3.29. Thf. above inclusion map induccs an isom.or-
phism in cohomology. PROPOSITION
3.30 (Vcy). The elements
{UTC.!; ·tk is odd [or any k:, 8
=j:. 0
=>
PI :::; q,
8
= 0 ~. all.ie are + 1.11 > q}
cven,
i 1 < the smallest odd .ie, i 1
[onn a IJO.o9is of H*(WO q ) ~ H*(W(gl(q, 1R»O(q)· Thlit; we obtain a homomorphism H*(WOq)
-------+
H*(J 2 (F)jO(q);JR) ~ H*(M;IR),
and we can see that the image o:(F) E H* (M; 1R) of an arbitrary element 0: E H*(WOq) under the above map becomes a characteristic class of foliations. In particular, we call the characteristic class represented by the element Ul ci the Godbillon-Vey dass of codimension q.
3.5. DISCONTINUOUS INVARIANTS
119
In the above definition of characteristic classes of foliations, we llse only information about J2 (F). It might be natural to expect that we cOllld obtain more characteristic classes other than the above by considering jet bundles Jk(F) of higher orders. However, the following theorem shows that unfortunately this is not the case. TUEOREM
3.31 (Gel'fand-Fuks [GF2]). The inclusion map
Wq C W(g[(q,JR)) c A~(uq) induces an isomorphism in cohomology. Here A;: (u,,) denotes the co chain complex consisting of continuous cochaills of the Lie algebra uq with respect to certain natural topology on it. 3.5. Discontinuous invariants In this section, we consider the sit.uation where we are given a real cohomology class Q E HI) (X; lR) of a topological space X or more generally, a system of real cohomology classes of it. We discuss how we can obtain infinitely many higher dimensional cohomology classes from O~ by twisting it using discontinuous endomorphisms of lR and thcn takillg cup products of them. In this book, we call the values of thelll on variolls integral cycles of X discontinuous invariants. As is well known, classical characteristic classes snch as Chern classes or Pontrjagin classes are integral cohomology cla.':;ses. In contrast with this, characteristic cla.<;scs of fiat bundles or those of foliations, which we treated in §2 and 33, are essentially real cohomology classes. Thurston's reslllt concerning continuous variation of the GodbillonVey class (cf. 33.2.2, Theorem 3.10, Theorem 3.11) shows this fact distinctly. Then there arises a natural question whether discontinuous invariants induced by these characteristic classes are non-trivial or not. Here we formulate this problem (see [Mor2] for details). This is an extremely difficult and deep problem. 3.5.1. Discontinuous invariants induced by a real cohomology class. First we consider the ca.se where we are given a single real cohomology class Q
E
HCJ(X;JR)
(q: odd)
of a topological space X of an odd degree. Then the correspondence Hq(X; /l) 3 u r-4 (n, u) E lR given by the Kronecker product induces
120
3. CHARACTERISTIC CLASSES OF FOLIATIONS
a natural homomorphism (3.12) which we denote by the same symbol o. Now for any natural number k, we define a homomorphism (3.13) which generalizes (3.12) as follows. Here A~(lR;) denotes the k-th exterior power of lR which is regarded as a vector space over Q. Also it is easy to see that there exists a natural isomorphism A~(lR) ~ A~(lR). Now let be a basis of HCJ(X; Q) and let
be the corresponding expression of o. DEFINITION 3.32. Let 0 E Hq(X; lR) be an odd dimem;ional real cohomology class. For any natural number k, we define a homomorphism Ok : Hkq(X; Z) --+ A~(lR) by
L
Ok(U)=
(Xil···X·i.k,u)aill\···l\a·'kEA~(lR)
il <···
where
U
E
Hkq(X; Z).
PROPOSITION 3.33. The above map Ok is determined independently of the choice of a basis of Hq (X; Q). PROOF.
Let
YI,'" ,Ym be another basis of Hq(X; Q). Then there exists a regular matrix C = (Cij) E GL(m,Q) such that Tn
y.,
=
LCijXj. j=l
If we denote by C-1
=
(Cij)
the inverse matrix of C, then we have m
Xi =
L CijYj' j=l
For any two multi-indices I = (i 1 ,'" ) ik) (1 ::; i1 < ... < ik ::; m) and J = (j1,'" ,jk) (1 ::; 11 < ... < jk ::; m), we denote by c(I, J)
121
3.5. DISCONTINUOUS INVARIANTS
and c(J, J) the minors of order k, corresponding to (I, J), of C and C- 1 respectively. Now if we set = b1 Yl
Q
+ ... + bmYm,
then
m
ai
=L
Cjibj.
j=l
In this case, we have
c(1 , J)yJ1. ... y'Jk and also
C(J' ' J)b·, J1 J'=(j~ ,
Hence for any element ( X·1.] ···X·~k'
U
E
!\ ... !\ b, . Jk
.. ,j£)
Hkq(X; Z), we have
u)a It !\···!\a 1/..:
L {(L c(1, J)Yh ... u) L c(J', I)b !\ ... !\ b = L {L L e(J',I)c(I, J)(Yj, ... Yj"" u)bj~ !\ ... !\ b = L (Yii ... Yjk' u)b !\ ... !\ b =
j;
Yjk'
I
J
J
J'
j ;.}
J'
j ;.}
I
j1
ik ·
J
Here we used the Laplace expansion theorem c(J',J)c(J,J) = This completes the proof.
8J'J.
0
It is clear from the definition that if k > m, then Qk = 0 and also that if a is a rational cohomology class, then Qk = 0 for any k > 1.
EXAMPLE 3.34. We give a simple example where discontinuous invariants can describe discontinuous variation of geometric structures. Let T2 = ]R2/Z2 be a 2-dimensional torus and consider linear foliations Fe (e E ]Ru {oo}) on it. Here Fe denotes the foliation whose leaves are parallel displacements of the line Y = ex. As was already mentioned in §3.1.1, Example 3.3, if C E QU {oo}, then all leaves are compact (in fact diffeomorphic to 8 1 ), while otherwise all leaves are dense in T2. Thus global properties of Fe vary discontinuously with respect to the parameter c. Now if we differentiate y = cx, we obtain dy = cdx. Hence Fe is defined by a closed I-form We = -cdx + dy
122
3. CHARACTERISTIC CLASSES OF FOLIATIONS
on T2. The de Rham cohomology class represented by this 1-form is [we] = -c[dx] + [dy] E H l (T 2 ;1R). Hence the value of the discontinuous invariant [weh : H2(T2;Z) ----> A~(IR.) induced by [we] is
[w c h([T2])
= -c 1\ 1 E A~(IR).
vVe can now conclude that All leaves of :Fe arc compact
{==}
[w ch([T2])
=
o.
Similar results as above with appropriate modification hold for the case where we are giveIl an even dimensional real cohomology class Ct E Hq(X;IR) (q: even). vVe describe only different points from the odd case and leave the details to the reader as an exercise. In the case where q is eveIl, Ct also induces a homomorphism (3.14) For any k > 1 we define a homomorphism (3.15) as follows. Here S~(IR) denotes the k-fold symmetric power of IR which is regarded as a vector space over Q. In this case also, it is easy to see that there exists a natural isomorphism S~(IR) ~ S(t~(IR). Given k real numbers al,··· ,aA; E IR, we express their product in Sk(lR) by
(h ... t.LA;
E
Sic (IR)
to distinguish it from the 1\s1Ial product in IR. Now let
be a basis of Hq(X; Q) and let Ct
= alXl
+ ... +
be the corresponding expression of
am·T m
(Y.
DCI"INITION 3.35. Let (t E Hq(X;IR) be an even dimensional real cohomology class. For any natural number k, we define a homomorphism Ctk : Hkq(X;Z) ----7 A~(IR) by
Ctk·(U) .
=
~ k!
"
(x ... X 11) a, .... a, E S:'(IR) 'k' " '·k '"
.~."'
where 11 E Hkq(X; 7l).
3.5.
DISCONTINUOUS INVARIANTS
123
PROPOSITION 3.36. The above map Cl'.k is determined independently of the choice of a basis of Hq(X; Q). We omit the proof because it can be done almost the same way as the odd dimensional case.
We can interpret the above results in terms of the homology group of J«lR, q) as follows. PROPOSITION 3.37. The homology group of J«IR, q) is given as follows.
(i) If q is odd, then we have *=0
* = kq otherwise.
(ii) rt q is even, then we have
H.(K(lII., q); Z)
~ {~~(lII.)
*=0
* =-
kq otherwise.
PR.OOF. vVe choose a basis {a-x; A E A} of IR as a vector space over Q. Then we can write lR = fun FQF where F runs through all finite :mbsets of A and QF denotes the finite dimensional subspace of lR generated bya-x (A E F). It follows that
The claim now follows by applying Proposition 1.22 in §1.1.4 and the Kunncth theorem on the right hand side of the above equality. 0 It i1; the same thing to give a real cohomology class and to give a homotopy class of maps Cl'.:
X
----+
Cl'.
E Hq (X; lR)
J«lR,q).
The ind1lced homomorphism
is nothing but the discontinuous invariants induced by
Cl'..
124
3. CHARACTERISTIC CLASSES OF FOLIATIONS
3.5.2. Discontinuous invariants. In the previous subsection, we introduced discontinuous invariants arising from a single real cohomology class. Here we generalize it. Let A be a d.g.a. over lR (see Chapter 1). Let M be a Coo manifold (or a simplicial complex) and let A*(M) be its de Rham complex (or the d.g.a. consisting of all Coo forms on M defined in §1.3.1). An A differential system on M is, by definition, a d.g.a. map I: A The map
->
A*(M).
f induces a homomorphism f* : H*(A) - > H*(MilR),
and we obtain a system of real cohomology classes of M. Two A differential systems Ii : A -> A * (M) (i = 0, 1) on M are called homotopic to each other if there exists an A differential system F : A -> A*(M x 1) on M x I such that FIMX{O} = Io,FIMX{l} = k According to Sullivan [Su2l, the classifying space, denoted by BA, which classifies homotopy classes of A differentia.l systems can be constructed as follows. Namely BA is the geometric realization of the simplicial set whose k-simplex is an A differential system on the standard k-simplex b.. k. Here we use natural operations induced from the usual ones on differential forms as the restriction to each face of a simplex as well as the degeneracy operation of the simplicial set. Then by a general theory, we have a. bijective correspondence {homotopy class of A differential system on M} = [M, BAl. Now if we denote the classifying map of an A differential system f : A -> A * (M) on M by I:M->BA by abuse of notation, then it induces a homomorphism I*: H.(M;'Z)
->
H.(BAi'Z).
We call this the discontinuolls invariants induced by f. Here the crucial point is that the ground field K is lR. In case K = Q, if we define similarly as above, we obtain nothing interesting. This is because of the following reason. On the classifying space BA, there is defined the tautological A differential system A
->
A*(BA).
The induced homomorphism H*(A)
->
H*(BA; K)
3.5. DISCONTINUOUS INVARIANTS
125
in cohomology turns out to be an isomorphism provided K = Q. In contrast with this, if K = JR, it is far from being an isomorphism. We shall show this by an example. EXAMPLE 3.38. Let K be a simplicial complex and let AQ(K) and A*(K) be the de Rham complex consisting of all Q polynomial forms and all Coo forms respectively (see §1.3.1). Let M~ and M~ be their minimal models (cf. §1,2). Then BM~ becomes the rational homotopy type Ko of K (cf. §1.1.3). Hence we have H*(BM~; Q) ~ H*(K;Q). However, though KJR = BM~ is, so to speak, the real homotopy type of K, it turns out that H*(KJRi JR) is much bigger than H* (K j JR). For example, in case K is an odd dimensional sphere s2n+1, we have Ko = K(Q,2n+1) and KJR. = K(.JR,2n+1). We know from the result in §3.5.1 that the latter space has a huge cohomology group. In general, it is a very difficult problem to compute H*(K!Ri Z).
Let G be a Lie group and let 9 be its Lie algebra. Let Bg denote the classifying space of the d.g.a. A" g* consisting of all left invariant differential forms on G. For example, we have
Bsl(2, JR) = K(JR, 3),
Bs[(2, C) = K(C, 3)
(cf. §2.2.4 and §2.3.5, Example 2.23). Then there arises a natural question of how many geometric cycles are in H. (Bgj Z). Here by geometric cycles we mean those cycles which come from flat Gproduct bundles over manifolds. This problem is unsolved even for the very first case where we do not take cup products. For example, in the case of .&[(2, C), the real part of
H3(Bs[(2, C)j Z)
~
C
corresponds to the values of the Chern-Simons class associated to the first Pontrjagin class, while its imaginary part corresponds to the values of the cohomology class induced by the volume form of the hyperbolic 3-space JHI3. It is unknown which values they can take on geometric cycles. This problem is closely related to the shape of the image of the map
(17, i vol) : {isometry class of closed hyperbolic 3-manifold}
-->
C
which is induced by the 17-invariant and the volume of closed hyperbolic 3-rnanifolds. In regard to foliations, we first consider the Godbillon-Vey class of codimension 1. Recall that for any foliation (M, F) of codimension 1, its Godbillon-Vey class gv(F) E H3(M; JR) is defined and Thurston's
126
3. CHARACTERISTIC CLASSES OF FOLIATIONS
result, mentioned at the beginning of this section, shows that it can take any real number on geometric 3-cycles. Then a natural question . arises as to the values of higher discontinuous invariants
gVk(!VI,F): H3dlVJ;7/,)
---7
A~(IR)
(k = 2,3,···)
which are induced by the Goclbillon-Vey class. Concerning this problem, even the non-trivialities of them are completely unknown. The situation is the same for the case of foliatiOJls of general co dimensions. For simplicity, we consider a foliation (lVI, F) of codirnensioll q whose normal bundle is triviali:;,ed. Then there is defined a system
8(F) = (8\ 11;, 8jk' 8jke,"') of canonical forms on JOO(F) (see §3, 4,1). If we pull back these forms by the section .s : !VI --; Jl(F) which is induced by the trivializatioll of the normal bundle together with the homotopy equivalence ,]1(F) ~ Jk(F) (k ~ 2), then we obtain a certain system of differential forms on !VI. In view of the structure equation (3.9) in §3.3.5 for 8(F), it is reasonable to denote the dassifying space of this differcntial system by Bu q . Then the problem is to determirH~ geometric cycles in H* (Bu q ; 7/,). If we use the terminology of Haefliger's classifying space Bfq (cf. [H]), then the above problem is equivalent to asking whether the canonical map Bfq ---7 BU q is a homotopy equivalence or not, as already mentioned in Sullivan's paper [Su2). This is also an extremely difficult question.
3.6. Characteristic classes of flat bundles II 3.6.1. Classifying space of foliated F-bundles. Let G be a Lie group. The relations among characteristic classes of various Gbundles were described ill the commutative diagram (2.20) in §2.3.3. It is an important problem to generalize this commutative diagram to the case of various F-bundles for a givcn closed Coo manifold F. As was already mentioned in §2.4.4, there exists a fibering (3.16)
BDiffF
---7
BDifF F --; BDiffF
consisting of classifying spaces of varions F -bundles. Here Diff<> F denotes the group Diff F equipped with the discrete topology. The fiber BDitf F is the classifying space of foliateel F-product bundles, that is foliated F-bundles together with trivializations as differentiable Fbundles. It was described in §2.4 that the Gel'fand-F'uks cohomology
3.6. CHARACTERISTIC CLASSES OF FLAT BUNDLES II
127
group HCF(F) of F plays the role of characteristic classes of fiat F-product bundles. Next BDiffo F serves as the classifying space of fiat F-bundles (see §2.4.2, Definition 2.26) or equivalently foliated F-bundles (cf. §3.1.1, Example 3.5), for, by Theorem 2.28 in §2.4.2, the isomorphism class of any flat F-bundle 7r : E -+ M is determined completely by its holonomy (or monodromy) homomorphism p: 7r1(M) ----. DiffF.
The third classifying space BDiff F in (3.16) classifies usual differentiable F-bundles. Then the problem mentioned above is to construct two unknown theories in the following commutative diagram: H*GF (F)
(3.17)
----+
H*(BDiffFiJR.)
r r
unknown theory
----. H* (BDiffO Fi JR.)
unknown theory
-; H*(BDiff Fi JR.)
In the case of commutative diagram (2.20) corresponding to a finite dimensional Lie group G, it is essential that G admits a maximal compact subgroup K. More precisely, the relative cohomology H*(g, K) of the Lie algebra of G with respect to K and the algebra J(K) of invariant polynomials of K occupy the two places corresponding to the blanks in the above diagram. Unfortunately, however, the diffeomorphism group Diff F in general does not have subgroups which can play the role of maximal compact subgroups except for a few cases like F = S1, S2, . . .. Therefore we cannot generalize the Chern-Weil theory, which concerns finite dimensional Lie groups, as it stand:::;. The theory of characteristic classes for differentiable fiber bundle:::; with general fibers is a vast unknown domain. We can make use of characteristic classes of foliations, which we considered in §3.4, to construct characteristic classes of foliated Fbundles, though only partially, as follows. Let 7r : E -+ M be a foliated F-bundle and let F be the corresponding foliation on E. We set dimF = q and let et E H*(WO q ) be a characteristic class of codimension q. Then et(F) E H*(Ei JR.) is defined. Here we consider the Gysin homomorphism (or integration along the fiber) 7r* :
H*(EiJR.) ----. H*-q(M;lR)
128
3. CHARACTERISTIC CLASSES OF FOLIATIONS
(see §4.2.3). Then it is easy to check that 7r*(a(F)) E H*-
It is unknown how large a part the cohomology classes of BDiffc5 F obtained in this way can occupy in the whole. 3.6.2. Cohomology of groups. We can cOllsider any characteristic class of flat bundles as an element of the cohomology of the corresponding structure group (either a Lie group G or a diffeomorphism group Difi'F) equipped with the discrete topology. In §1.4, we defined the cohomology H* (T) of a general group T as the cohomology of the Eilcnberg-MacLane space K(T, 1). Here we give a more algebraic definition. As was already mentioned, we refer the reader to the book [Br] for details. For each non-negative integer k, we consider the free abelian group generated by all the elements (gl , . .. , gk)
(gi E T)
of Tk and denote it by
Ck(r). Any element of CdT) is called a k-chain of T. If k Co = Z. We define the boundary operator
8: Ck(r)
~
0, we set
C k- 1 (T)
by
8(g1,··· ,gk) =(92,··· ,9k)-(9192,93,'" ,gk)+(gl,g293,94,··· ,9k)
- ... + (-1)k-l(9l,· ..
, gk-2, gk-lgk)
+ (_l)k(gl'· ..
,9k-l).
If k 1, we define 8(g) = O. Then it is easy to see that 808 = Hence C.(r) = $kCt,;{T) becomes a chain complex. We set
H.(T) = H* (C. (T)) and call it the homology group of T. Next any element of
Ck(T) = Hom(Ck(T), Z)
o.
3.6. CHARACTERISTIC CLASSES OF FLAT BUNDLES II
is called a k-cochain of
r.
129
The coboundary operator
0: Ck-1(r)
--+
Ck(r)
is defined by af(e) = f(8e) (f E Ck-1(r), C E Cdr)). c*(r) = EBkCk(r) becomes a cochain complex. We set
Then
H*(r) = H*(C*(r)) and call it the cohomology group of r. The above definition may appear to be artificial at first. However, one may understand its geometrical meaning by the following interpretation. Suppose that a fiat bundle is given on a manifold 1\1/ and let p: 7rl(M) --+
r
be its holonomy homomorphism. As K(7rl(M), 1) is the 1-coskeleton of M (see §1.1.2), a continuous mapping M ---t K(7rl(M),1) inducing an isomorphism on the fundamental group is defined. On the other hand, the homomorphism p induces a continuous mapping K(7rl(M), 1) ---t K(r, 1). Then the induced homomorphism
H*(r)
= H*(K(r, 1»
--+
H*(K(7r1(M), 1)
--+
H*(M)
in cohomology gives characteristic classes of the given fiat bundle. Clearly they are determined completely by the homomorphism
p* : H*(r)
--+
H*(7rl(M))
between cohomology of groups. Now suppose for simplicity that M = K(7rl (M), 1) and also that there is given a triangulation K on M. Choose a maximal tree T of the I-skeleton K(l) of K and collapse it to a single point. Then we obtain a cell decomposition K of M which has only one O-cell. Here by a tree we mean a simply connected 1dimensional simplicial complex. Now if we denote by Po the above unique O-cell, then each oriented I-cell of K determines an element a E 7'11 (M) of the fundamental group of M with respect to the base point Po. We denote this I-cell by (a). Next we observe that the three edges of an oriented 2-cell can be described as in Figure 3.5. Therefore it may be reasonable to denote such 2-cell by (a,{3). The boundary of this cell is given by
8(a,{3)
= ({3) - (a,B) + (a).
The situation is similar for the cases of higher dimensions. Thus the reader may understand that the above definition of cohomology
3. CHARACTERISTIC CLASSES OF FOLIATIONS
130
Po
(a./J)
(a)
(fJ)
Po
L_--------
Po
(a/J)
FIGURE
3.5. 2-ceU
of groups comes from the usual definition of homology of cell complexes. The above description concerns cohomology of groups with constant coefficients. We can abo consider cohomology with twisted coefficients. Geometrically this corresponds to considering the cohomology of K(r,l) with respect to a local coefficient system on it. Since the importance of cohomology with twisted coefficients will grow more and more in the future, we state its definition here. Let r be a group and let V be a r-modlile. In other words, V is an abelian group, and there is given a homomorphism
r ---+ Aut(V). We set
Cdr; V) Ck(r) ® V and call any element of it a k-chain of r with coefficients in V. The boundary operator a: Cdr; V)
---+
Cf.:-l(r; V)
is defined by a((gl,··· ,gk)
Q9
v)
=(g2,··· ,gk)®.QIIv-(glg2,93,··· ,gk)®V+···
+ (-l)k-l(gl,···
, .Qk-2, gk-lgk) ® v
+ (-l)k(gl,·
.. , 9k-d ® v.
If k = 1, we set a((g) ® v) = g-lv - v E V. Then it is easy to check that aoa = O. Hence c*(r; V): c)kCdr; V) becomes a chain complex. Now we set
H*(r; V) = H*(C*(r; V)) and call it the homology of of
r with coefficients in V.
Next any element
:3,6, CHARACTERISTIC CLASSES OF FLAT BUNDLES II
is called a k-cochain of operator
r
131
with coefficients in V, The coboundary
is defined by (Of)(.1)l,'·· ,9k) =9d(92,'" ,9k) - f(glg2,93,'" ,9k) +(-1)k- 1f(91,'" ,9k-2,9k-19k)
+, ..
+ (-l)kf(gl,'" ,9k-I). E V we set of (g) = 9V -
If k: = 0, for any f = 1) v. Then c*(r; V) = EBh,Ck(r; V) becomes a cochain complex. Now we set H*(r; V)
and call it the cohomology of if k: = 0, then Ho(r; V) = HO(r;V)
= H*(C*(r; V))
r
with coefficients in V. In particular,
vir
= VT
=
{v E V;gv =
1)
for any 9 E r}.
Here V / r denotes the quotient module of V induced by identifying v with gv for any v E V and 9 E
r.
3.6.3. Characteristic classes of foliated 5 1-bundles. If we lise the unit circle in the plane as a model for 51, then 50(2) becomes a subgroup of Difi'+5 1 . As is well known, the inclusion 50(2) C Diff+5 1 is a homotopy equivalence. Hence BDiff+51 = K(Z, 2), and DitLj-5'l becomes homotopy equivalent to the universal covering space DifLj-5 1 of Diff+5'l. It follows from these facts that the theory of characteristic classes of va.rious 5 1 -bundles is rela.tively simple. Here we summarize known results concerning it. The conclusion is that the three homomorphisms from left to right in the following commutative diagram are all injective.
HCF(5'l) ~ JR[(X] 0 E(f3)
r
(3.18) H* (1:(5'1),5'0(2)) ~ ~[(X,
xl/( (XX)
~
H*(BDiff+5 1 ; ~)
--)
H* (BDifft5'\ JR)
~
H*(BDiff+5'l; ~)
l'
~[xl
r r
3. CHARACTERISTIC CLASSES OF FOLIATIONS
132
First, the determination of HCp(S1) in the top row is due to Gel'fand and Fuks (see [GFl]). Here a and ,B are 2 and 3-cocycles respectively and can be described as follows. If we express three given vector fields X, Y, Z on S1 as X = f Y = gft, Z = hft, then we have
gt'
ex(X, Y) =
r If; .J g Sl
fill dt,
gil
f3(X, Y, Z)
=
r .J
SI
f
f' f"
g h
g'
gil dt.
11,'
h"
These co cycles are closely related to the Godbillon-Vey class. More precisely, a is equal to the 2-dimensional cohomology class obtained by integrating the Godbillon-Vey class of foliated Sl-bundles along the fiber, and f3 coincides with the 3-dimensional cohomology class obtained by pulling back the Godbillon-Vey class of foliated S1-product bundles by sections induced by their trivializations. In view of these facts, we see from Thurston's result mentioned in §3.2 that ex varies continuollsly (in the real analytic category). Thurston further showed that any power ex k (k 2': 2) of ex varies continuously but at the moment only in the Ceo category. It is easy to deduce from this result and the relation of f3 with the Godbillon-Vey class mentioned above that the classes a k f3 also vary continuously for all k. The class X in the second and third rows in (3.18) denotes the Euler class of oriented S1-bundles. Concerning it, we can show the non-triviality of any power Xk of the Euler class in H2k(BDiff~S\ JR) by an argument using the results of Mather [Ma] and Thurston [Th2] and also Sullivan's model for the free loop space Map(Sl, Jill) of any topological space M (see [MorJ.]). However, it is an important open problem to determine whether the powers of ex and X of degrees 2': 2 are non-trivial for the real analytic foliated S1-bundles or not. On the other hand, we can use Thun;ton's construction to show that the discontinuous invariant
induced by a is surjective for any k. However, a similar problem for
f3 is equivalent to the problem of non-triviality of the discontinuous invariants arising from the Godbillon-Vey class mentioned in §3.5.2 and is open. PROBLEM
map
3.39. Determine the kernel as 'Well as the image of the
3.6. CHARACTERISTIC CLASSES OF FLAT BUNDLES If
which is induced by the characteristic classes Puks.
0'.,
133
{3 of Gel 'fand and
3.6.4. Thurston cocyle representing the Godbillon-Vey class. Here, at the end of this chapter, we describe what is called the Thurston co cycle which represents the Godbillon-Vey class of foliated F-bundles. If F is a q-dimensional oriented closed manifold, then the classifying space of oriented foliated F-bundles is BDiff~F. Hence, by integrating their Godbillon-Vey class (see §3.4.2) along the fiber, we obtain a certain cohomology class gv E Hq+1 (Diff~F; JR). THEOREM 3.40 (Thurston). Let v be a volume form of F. For any element f E D~ff+F, define a function p,(f) > 0 by the equality f*v = p,(f)v. Then, ~f we define gv E Gq+1 (Diff+F; JR) by
gv(h, ... ,fq+d =
llogP,(fQ+l) dlogp,(fqfQ+d··· d log p,(!I '" f q +1) ,
it becomes a group cocycle representing the Godbillon- Vey class gv E HQ+l(Diff+F; JR) mentioned above.
In the case F = S1, the Thurston co cycle can be described more explicitly as follows. Express any element f E Diff+S1 by a periodic diffeomorphism f E Diff+JR and choose dt for the volume form. Then we have ,Af) = Df· Here D f denotes the derivative of .f. Hence gv(j, g) =
r log
lSI
Dg D log D(fg) dt
becomes the cocycle which represents gv. Now transform the above integral as g
v(f
r
)- ~ /IOgD9 ,g - 21sl DlogDg
logD(fg) / dt D log D(fg) .
Then we see that the Thurston cocycle is a globalization of the co cycle 0'. of Gel'fand-Fuks described in §3.6.3, and conversely the latter is an infinitesimal version of the former.
CHAPTER 4
Characteristic Classes of Surface Bundles In t.his chapter, we describe basics of the theory of characteristic classes of surface buncHes. By surface bundles we mean differentiable fiber buudles whose fibers are closed orientable surfaces. Roughly speaking, the theory Il...
bundles 4.1.1. Characteristic classes of differentiable fiber bundles. Before going into the details of the subject, we mention a few general facts concerning characteristic classes of fiber bundles. Let F be a Coo manifold. We call differentiable fiber bundles whose fibers are diffeomorphic to F simply F-bundles. Then, it is a fundamental problem to determine the set of all isomorphism classes of F-bundles
over a given manifold !vI. The structure group of such bundles is Diff F which is the diffeomorphism group of F equipped with the Coo 135
136
4.
CHARACTERISTIC CLASSES
OF
SURFACE BUNDLES
topology. Hence, if BDiff F denotes its classifying space, then the general theory implies that there is a natural bijection {isomorphism class of F-bundles over M}
~
[M, BDiff F].
Here the right hand side stands for the set of homotopy classes of continuous mappings from lVI to BDiff F. For example, in the case where M is the n-dimensional sphere sn, we have canonical identifications [sn,BDiffF]
~ 7rn(BDiffF)/7r1 (BDiffF) ~
7rn- l (Diff F)/7ro(Diff F)
so that we need to know the homotopy groups of BDiff F or Diff F. In particular, if n = 1, then [SI, BDiff F] can be identified with the set of all conjugacy classes of 7ro(Diff F) which is the group of path components of Diff F. Unfortunately, however, it is almost impossible to compute these groups for a general manifold F. The problem to determine the set [M, BDiff F] should be even more difficult. In such a situation, it is natural to seek methods of determining whether two F-bundles given over the same manifold are isomorphic to each other or not. One such method is obtained by applying characteristic classes of F-bundles. DEFINITION 4.1. Let A be an abelian group and let k be a nonnegative integer. Suppose that, to any F-bundle 7r : E ----) M, there is associated a certain cohomology class 0:(7r) E Hk(M; A) of the base space in such a way that it is natural with respect to any bundle map. Then we say that 0:(7r) is a characteristic class of F-bundles of degree k with coefficients in A. Here by natural we mean that for any bundle map
E1
j
-----;
E2
1~2
~11 lVh
--->
f
M2
between two F-bundles 7ri : Ei ----) Mi (i
=
1,2), we have the equality
In the terminology of the classifying space, we can write 0: E Hk(BDiffF; A) and if f : X ----) BDiffF is the classifying map of the given bundle 7r : E ----) M, then we have 0:(7r) = 1*(0:). Namely
4.l. MAPPING CLASS GROUP
137
characteristic classes of F-bundles are nothing but elements of the cohomology group of BDiff F. It follows immediately from the definition that two F-bundles over the same base space which have a different characteristic class are not isomorphic to each other. Thus it is desirable to define as many characteristic classes as possible. 4.1.2. Surface bundles. A 2-dimensional Coo manifold, which is compact, connected and without boundary, will simply be called a closed surface. The classification of closed surfaces was done already in the beginning of the twentieth century. As is well known, the Euler number together with the property of being orientable or not can serve as a complete set of invariants. In particular, the set of all the diffeomorphism classes of closed orient able surfaces can be described by the series 52,
T2,
2:09 (g
= 2,3",,),
Here 52 and T2 denote the 2-dimensional sphere and torus, respectively, and 2: g stands for a closed orient able surface of genus g. Of course we have 2:0 = 5 2,2: 1 = T2. Henceforth we assume that an orientation is fixed on each 2: 09 , DEFINITION 4.2. A differentiable fiber bundle with fiber 2: g is called a surface bundle or a 2: g -bundle. Let 7r : E -} !vI be a 2: g -bundle. Then the set of all tangent vectors on the total space E which are tangent to the fibers, namely the set ~
= {X E TE; 7r*(X) = O},
becomes a 2-dirnensional vector bundle over E. '''Ie call ~ the tangent bundle along the fiber of the given 2: g -bundle. Sometimes the notation T7r will also be used for~. This concept is defined not only for surface bundles but also for general fiber bundles. DEFINITION 4.3. A surface bundle 7r : E --+ 111 is said to be orient able if its tangent bundle along the fiber T7r is orientable. If a specific orientation is given on T7r, then it is called an oriented surface bundle. Henceforth in this book, all surface bundles are assumed to be oriented and all bundle maps between them are assumed to preserve the orientation on each fiber.
138
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
DEFINITION 4.4. Two L:g-bundles 1Ti : Ei --+ M (i = 1,2) over the same manifold NI are said to be isomorphic to each other if there exists a bundle isomorphism El ~ E2 which preserves the orientation on each fiber. Our principal problem can now be stated as follows. Determine the i::iet of isomorphism clasiie:;; of L:g-bundles over a given manifold. Let Diff+L:g denote the group of all the orientation preserving diffeomorphiiims of L: g equipped with the Coo topology. It serves as the structure group of oriented L:g-bundles. In the case where 9 = 0, namely for the sphere, it was proved by Smale [8] that the natural inclm;ion
SO(3) C Diff+S2 is a homotopy equivalence. It follows from this fact that any S2_ bundle is isomorphic to the sphere bundle of somc uniquely defined 3-dimensional oriented vector bundle. Hence the classification of S2_ bundles over a given manifold NI is equivalent to that of 3-dimensional oriented vector bundles over M. Since the homotopy typcof the classifying space BSO(3) is known, wc may say that this problem is solved. Also for the 3-dimensioual sphere S3, Hatcher [Hat] proved that the inclusion SO(4) C Diff+S3 is a homotopy equivalence. Hence the problem is solved also in this case. However, it wa..<; proved by Farell and Hsiang [FH] that a similar property does llot hold for odd dimensional spheres of sufficiently high dimensions. 4.1.3. Mapping class groups of surfaces. Let F be a COG manifold and let Diff F be its diffeomorphism group. The group of path components of Diff F, namely 7fo (Diff F),
is called the diffeotopy group of F. In this book, we denote this group by V(F). If we write DiffoF for the identity component of Diff F, then we have
V(F)
=
DiffF/DiffoF.
We can also define this group as follows.
4.1.
MAPPING CLASS GROUP
139
DEFINITION 4.5. Two diffeomorphisms (, 1) = '1/;.
E
[0,1]
It can be shown that the isotopy is an equivalence relation. In fact, two difFeomorphisms are isotopic if and only if they belong to the same connccted component of DiflF. Hence we can say that D(F) is the group consisting of all the isotopy classes of diffeomorphisms of F. It follows that we have a natural identification
{isomorphism class of F-bundles over Sl} ~
{conjugacy class of V(F)}.
Now we denote the orientation preserving diffeotopy group D+CEg) of Eg by Mg and call it the mapping class group of Eg. Namely we set Mg = 7fo(Diff+Eg). As will be mentioned later in §4.2.2, the mapping class group Mg plays an important role also in the Teichmiiller theory regarding complex structures on Eg. For this reason, Mg is also called the Teichmiiller modular group. Is is clear from the definition that isotopy is a much stronger condition than homotopy. However, in the case of 2-dimensions, namely for surfaces, it is classically known that they are equivalent. Hence we can say that M 9 is the group of all homotopy classes of orientation preserving diffeomorphisms of E g . Moreover it is also known that M 9 is canonically isomorphic to the group of all homotopy classes of orientation preserving homotopy equivalences of E g . In fact, from the time of Nielsen, who flourished in the first half of the twentieth century, it has been known that there exists a natural isomorphism
My
~
Out+7fl(Eg)
=
Aut+7f1 (Eg)/Inn7f1(Eg).
Here Aut+ 7f1 (Eg) denotes the normal subgroup of the automorphism group of 7f1 (Eg), with index two, consisting of those elements which act on H2(Eg; Z) = H2(7f1 (E g); Z) trivially, and Inn 7f1(Eg) denotes the normal subgroup of all the inner automorphisms. We refer the reader to the book [Bi] for basic facts about the mapping class group.
140
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
4.1.4. The action of Mg on the homology group of surfaces. The mapping class group M 9 of ~g is a natural quotient group of the diffeomorphism group Diff+~g so that it acts on spaces of various structures on the surface. Here we consider the simplest as well as the most important case among them, namely the action on the homology group of surfaces. As is well known, the first homology group of ~g is a free abelian group of rank 2g so that Hl(~g;Z) ~ Z29.
If we are given two homology classes u, v E HI (~g; Z), then their intersection number u· v E Z is defined. Suppose that these homology classes are represented by oriented closed curves on ~g which are in a general position. Then u . v E Z is equal to the sum of signs at each intersection point so that it is bilinear with respect to both u and v. Also we have v . U = -1£. v. Thus the intersection number defines a skew symmetric bilinear form {l:
HI (~g; Z) 0 HI (~g; Z)
->
z.
Moreover this bilinear form is nondegenerate. Namely if 1£. v = 0 for all v, then we can deduce that 1£ = o. In fact, by the Poincare duality theorem, the natural mapping
HI (~g; z)
31£1-> 1£* E
HI(~g; z) ~ Hom(HI (~g; Z), Z)
given by 1£* (v) = 1£ . v is an isomorphism. If we choose a basis XI,X2,··· ,Xg,YI,Y2,··· ,Yg of HI(~9;Z) as shown in Figure 4.1, then we have the following equalities concerning the intersection numbers Xi· Xj = Y'i . Yj = 0,
J;i·
Yj =
OJ.j.
Any basis which has the above property is called a symplectic basis. There are infinitely many symplectic bases. For example, if we replace Xl with Xl + YI in the above basis, then it remains symplectic. Now since Mg contains only those mapping classes which are represented by orientation preserving diffeomorphisms of ~ g, its action on HI(~g; Z) keeps the intersection number unchanged, Hence a natural homomorphism p: Mg
---4
Aut(HI(~g;Z);{l)
is defined. To describe this homomorphism explicitly in terms of matrices, we fix a symplectic basis Xl, X2, ... , Xg, Yl, Y2, ... , Yg of HI (~g; Z). Then the image p(cp), under p, of any element cp E Mg
4 .1. MAPPING CLASS GROUP
FIGURE 4.1.
141
Symplectic basis
can be represented by a 2g x 2g integral matrix. Let us write down the condition of preserving the intersection number in the context of these matrices. We first represent two homology classes u) v E HI CEg; 1.) as
'/1, =u1xl V ='IJtXI
+ ... + 7L g X g + 'Ug+IYl + ... + U2gYg + ... + VgXg + Vg+IYl + ... + V2gYg
with respect to the above basis. Then we have 7J,.
v = UIVg+l
+ ... + U g V2g
-
Ug+IVI _ ... - U2gVg.
We define a matrix J E GL(2g) 1.) by setting
J=(~I
b)
where 0) I are the 9 x 9 zero matrix and the unit matrix) respectively. Then the intersection number can be simply expressed as
u'v=(u)Jv). Here ( ) ) denotes the usual inner product of the Euclidean space R 2g. Now if we denote by A E GL(2g) Z) the matrix corresponding to p(
Au· Av =
U·
v
holds for any two homology classes u, v E HI ('Llg; 1.) ~ 1.29 . On the other hand, we have
A'/1,' Av u .v
=
(Au,JA'u)
= (-tAJAu,v)
= (u, Jv) = (- J u, v).
Here we used the fact that t J = -J. From this we obtain the condition tAJA = A.
142
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
Hence if we define a subgroup Sp(2g, Z) of G L(2g, Z) by setting
Sp(2g,Z) = {A E GL(2g,Z); tAJA = A}, then the action of My on HI (E y; Z) can be represented by the following homomorphism
p: My
->
Sp(2g, Z).
This homomorphism was essentially obtained around the end of the nineteenth century and is a classically important one. Also it was already known then that p is surjective. The group Sp(2g, Z) is called the integral symplectic group or the Siegel modular group as it played a crucial role in the theory of automorphic functions due to Siegel. If 9 = 1, then
Sp(2,Z) = SL(2,Z) = {A E GL(2,Z);detA = I} and the representation p: Ml phism. Hence we have
Ml
->
S:!
Sp(2, Z) is known to be an isomorSL(2, Z).
However, in the cases where 9 > 1, it is known that p has a big kernel. In fact, the group Ig
= Ker p = {cp E
Mg; p(cp)
= id}
is called the Torelli group, which turns out to be an extremely important normal subgroup of the mapping class group. Recently active research has been done concerning this group. However, its structure remains largely uncovered so that we must say that it is still a mysterious group. We end this section by the following fundamental short exact sequence of groups 1
->
Ig - ) My ~ Sp(2g, Z)
->
1.
4.1.5. Classification of surface bundles. As was mentioned in §4.1.2, in the case 9 = 0 Smale showed that the inclusion SO(3) C Diff+S2 is a homotopy equivalence so that BDiff+S2 ::: BSO(3).
Next we consider the case 9 = 1, namely surface bundles whose fibers are diffeomorphic to the torus T2. If we identify T2 with ]R2 /Z2, then T2 acts on itself by diffeomorphisms. Hence T2 can be naturally
4.1. MAPPING CLASS GROUP
143
considered as a subgroup of DiffoT2 which is the identity component of Diff+T2. Moreover it is known by Earle-Eells [EE] that the inclusion
T2
C
DiffoT2
is a homotopy equivalence. On the other hand, we have an isomorphism Diff+T2 jDiffoT2 = M1 ~ SL(2, Z) which was already mentioned in the previous subsection. From these facts we obtain a fibration
BT2
-->
BDiff+T2
-->
BSL(2, Z) = K(SL(2, Z), 1).
The structure of the group SL(2, Z) is classically well known, and we have a homotopy equivalence BT2 c:: ClP'oo x ClP'oo. Based on these facts we can compute the cohomology of BDiff+ T2 which serve as the characteristic classes of T2-bundles. But here we omit the details. In the cases where g ;::: 2, the situation changes drastically. More precisely, Earle and Eells proved in the above cited paper [EE] that DiffoEg is contractible so that BDifl'+Eg = K(M g , 1).
It follows immediately from this that PROPOSITION 4.6.
Let g;::: 2. Then for any Coo
man~fold
M, we
have a natural bijection {isomorphism class of Eg-bundle over M} ~
{conjugacy class of homomorphism 7f1(M) ---) Mg}.
In particular if M is simply connected, then any Eg-bundle over it is trivial. However, in general, it is almost impossible to determine the set of all conjugacy classes of homomorphisms from a given group to Mg. It may be better to understand the above proposition as a starting point for the construction of a classification theory rather than a direct role. Now let a be a characteristic class of Eg-bundles of degree k with coefficients in an abelian group A. Then we can write 0:
E Hk(BDiffEg; A) = Hk(K(M g, 1); A) = Hk(Mg; A)
(cf. §l.4.2, Definition l.57). In other words, characteristic classes of surface bundles of genus 9 ;::: 2 are nothing but cohomology classes of the mapping class group Mg.
144
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
4.2. Characteristic classes of surface bundles 4.2.1. Definition of characteristic classes. Let I: g be an oriented closed surface of genus 9 and let 7r:E--7M be an oriented I:g-bundle. If we denote by ~ the tangent bundle along the fiber of 7r, then by Definition 4.3 ~ has a structure of an oriented 2-dimensional real vector bundle. Hence its Euler class
is defined. For each non-negative integer i, we consider the power ei + 1 E H 2 (i+l) (E; Z) of the Euler clast; e. We then apply the Gysin homomorphism (cf. §4.2.3) 7r* : H 2 (i+1)(E; Z) - - 7 H 2i (M; Z) to ei+1 and obtain a cohomology class of the base space M which we denote by ei(7r) = 7r*(ei +l) E H 2i (M;Z). DEFINITION 4.7. The cohomology class ei(1r) E H 2i (M; Z) which is defined for any I:g-bundle 7r : E ~ M as above is called the i-th characteristic class of surface bundles. The fact that ei in fact defines a characteristic class of surface bundles, namely that it is natural under the bundle maps, can be checked as follows. Let 7ri : Ei ~ Mi (i = 1,2) be two I:g-bundles and let
E1
~l
j
---->
1
M1
E2
1~2 ---->
f
M2
be a bundle map between them. Then by the definition of bundle maps, the restriction of .f to each fiber is an orientation preserving diffeomorphism. Hence if ~i denotes the tangent bundle along the fibers of 7ri, then we have
4.2. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
145
The naturality of the Gysin homomorphism (cf. Proposition 4.8(iii)) implies that
ei( 1f 1) = J*(ei( 1f2)) which shows that ei is indeed a characteristic class. Hence from the description of §4.L5, we can write ei E H2i(BDiff+~g; Z)
= H2i(Mg; Z)
for 9 ;:: 2. In other words, ei can be considered as a cohomology class of Mg of degree 2i. 4.2.2. Characteristic classes of surface bundles and the moduli space of Riemann surfaces. The mapping class group of surfaces has an intimate relationship with an important space called the moduli space of Riemann surfaces. In particular, over the rationals, any characteristic class of surface bundles can be considered as an element of the cohomology group of the moduli space. In this section, we briefly mention this matter. The concept Riemann surface was introduced by Riemann himself as the denomination indicates. It is a surface obtained by suitably modifying the domain of a many-valued analytic function in such a way that it becomes single valued. However, here we call a (real 2dimensional) surface which is given a fixed complex structure a Riemann surface. In particular, if a complex structure on ~g is specified, then it is called a genus 9 compact Riemann surface. We denote by Mg the space of all biholomorphism classes of genus 9 compact Riemann surfaces and call it the moduli space of Riemann surfaces of genus g. We can also define Mg as follows. Let Tg denote the space of isotopy classes of complex structures on ~9 whose induced orientations coincide with the fixed one. It is called the Teichmiiller space of genus g. The mapping class group Mg acts on Tg naturally, and there is a canonical identification Mg =
Tg/M g .
The Teichmi.iller space was introduced by Teichmi.iller in the 1930's. As was already mentioned in §4.L3, the mapping class group is also called the Teichmiiller modular group because of the above fact. Many things are known concerning Tg (see e.g. [IT]), but here we only mention the following two facts. One is that in the cases 9 2': 2, Tg is known to be homeomorphic to IR6g-6 and the other is that the
146
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
action of M 9 on Tg is properly discontinuous. It follows from these facts that there exists a continuous map BDiff+2::g
~
Mg
which is well defined up to homotopy such that it induces an isomorphism H*(Mg; Q) ~ H*(BDiff+2::g; Q) = H*(Mg; Q) in the rational cohomology. In this way we find that the two spaces, namely the moduli space of compact Riemann surfaces and the classifying space of surface bundles which play fundamental roles in the theory of Riemann surfaces and in topology respectively, are very close to each other. Let us reconsider the above matters from a somewhat different point of view. First of all, for a general Coo manifold F let R(F) denote the space of all the Riemannian structures on it. Recall here that a Riemannian structure is given by assigning an inner product, that is a positive definite symmetric bilinear form, on the tangent space TpF of each point p which varies smoothly with respect to p. By the way, any two points in the space of inner products on a finite dimensional vector space over JR: can be joined by a segment. This is because any linear combination (l-t)/tO+t/-l1 (t E [0,1]), where /-lo, /-l1 are two inner products, is again an inner product. It follows easily from this fact that R(F) is contractible. In the case where F = 2::g, we can consider the subspace Ro(2::g) of R(2::g) which consists of all the metrics of constant Gaussian curvature I{ == 1,0, -1 accordingly as 9 = 0, I, :::: 2. In the cases g :::: 2, it can be shown that Ro (2:: g) is naturally a strong deformation retract of R(2::g) as follows. To begin with, any metric on 2::g gives rise to a complex structure via the isothermal coordinates. Secondly, it is a classical result that any complex structure on 2::09 (g :::: 2) is obtained as a quotient space of the upper half plane JH[ with respect to some discrete representation 1[1(2:: 09 ) ---> PSL(2,JR). On the other hand, the group P SL(2, JR:) of biholomorphisms of JH[ can also serve as the group of orientation preserving isometries of JH[ with respect to the Poincare metric. Hence any complex structure on 2::09 is equivalent to the associated metric of constant negative curvature. Thus we find that any point in R(2::g), namely any metric on 2:: g, will uniquely determine an associated point in Ro(2::g), that is a metric of constant negative curvature, which is conformally equivalent to the original one.
4.2. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
147
Let 'if : E ---> M be a Eg-bundle and let us give a fiber metric on it. Then each fiber Ep (p E E) becomes an oriented closed surface equipped with a Riemannian structure. The above fact implies that each metric on Ep defines the associated structure of a Riemann surface. Keeping in mind these structures as a whole, we can consider 'if : E ---> 1\1/ as a family of Riemann surfaces over M. Byassociating the Riemann surface Ep E Mg to each point p EM, we obtain a continuous mapping f: M --+ Mg. It is easy to see that the homotopy class of this mapping does not
depend on the choice of the initial fiber metric. If we apply this operation to the universal Ey-bundle 'if : EDiff+Eg ---> BDiff+Eg, then we obtain a map BDiff+Eg --+ Mg. The moduli space Mg of compact Riemann surfaces is a very important space in algebraic geometry, and many deep results have been obtained by many mathematicians, notably by Mumford (cf. [HL][HM]). Among other things, Mumford began the theory of the Chow algebra A*(Mg) of Mg ([MuD. In pa.rticular, he defined a series of canonical elements
It follows easily from the definitions that, under the natural isomor-
phism H*(Mg;Q) ~ H*(Mg;Q), the cohomology class of Ki corresponds to the characteristic class ei E H2i(Mg; Q) of surface bundles introduced in §4.2.1 multiplied by the factor (_1)'i+1. The class Ki is in fact defined as an element of the Chow algebra A * (My) of the Deligne-Mumford compactification Mg. The classes Ki are called the tautological classes of the moduli space and their cohomology classes, namely the classes ei are called the Mumford-Morita-Miller classes. 4.2.3. The Gysin homomorphism. In §4.2.1 we defined characteristic classes of surface bundles where we made essential use of the Gysin homomorphism. This homomorphism is very important for the study of surface bundles as well as general manifolds. In this subsection we briefly summarize basic facts concerning it. Let F be an oriented closed manifold and let 1f:E--+M
148
4. CHARACTERlSTIC CLASSES OF SURFACE BUNDLES
be an F-bundle over NI. We assume that this bundle is oriented; that is the tangent bundle along the fiber of 7r, denoted by ~ = {X E TE; 7r*X = a}, is orientable and is given a specific orientation. Although we are only concerned with the case F = ~9' the Gysin homomorphism is defined for general F-bundles. If we denote by {E~,q} the spectral sequence for the cohomology of the above F-bundle, then its E2 term is given by
Eg,q
~
HP(M;7-i5).
Here Hq stands for the local coefficient system associated to the qdimensional cohomology Hq (7r- 1 (p); Z) (p E M) of the fibers. If F is n-dimensional, then clearly Hq = 0 for q > n so that
Eg,q
=a
(q> n).
By the assumption, 11.71 is isomorphic to the constant local system Z. Hence E~,n ~ HP(M; Z). On the other hand, the homomorphism dr : E;?-r,n+r-l -+ E~,71 is trivial for any p and r 2: 2 so that we obtain a series of monomorph isms Ef,;,n c ... c Ef,71 C E~,71 ~ HP(M; Z). Now we denote the homomorphism
HP(E; Z)
-----+ g~-n,n C E~-n,n ~
HP-n(M; Z)
which is the composition of the natural projection HP(E; Z) with the above monomorphism (with shifted degree) by
-+
E"g;;n,n
(4.1)
and call it the Gysin homomorphism of the F-bundle 7r : E -} M. Sometimes the symbol 7r! is used instead of 7r *. Note that this homomorphism goes in the opposite direction to the usual one which is induced by the projection 7r and also that it decreases the degree by n, namely the dimension of the fiber. Similarly we have the Gysin homomorphism (4.2)
in homology. The above method of defining the Gysin homomorphism using the spectral sequence is valid over Z, and we may say that it is theoretically the best one. However, it might not be easy to see its geometrical meaning. To cover this point, let us examine the Gysin
4.2. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
149
homomorphism in the context of de Rham cohomology, although the coefficients must be reduced to R Here it can be explained by means of an operation (4.3)
defined in the de Rham complex which is called the integration along the fiber. Any differential p-form on E can be expressed locally as a sum of the forms like w
=
L hj(x, y) dXi l 1\ ... 1\ dXis 1\ dYjl 1\ ... 1\ dYj,· i,j
Here Xl, ... , Xm (m = dim M) and Yl, ... ,Yn are local coordinates of M, F respectively and dYl 1\ ... 1\ dYn is assumed to coincide with the given orientation of F. Also the summation is taken over the multiindices i = (iI,'" ,is), i l < ... < is, j = (jl,'" ,jt), jl < ... < jt s + t = P with s + t = p. We now set 7r * (w)
=
L
i,j(t=n)
(1
iij (x, y)dYI 1\ ... 1\ dYn) dXi l 1\ ... 1\ dX"p_n'
F
It can be easily shown that 7r * is in fact uniquely defined by the above. The integration along the fiber commutes with the exterior differential d, namely d
0 7r
*
= 7r *
0
d.
Hence it induces a homomorphism
and it can be shown that this coincides with the Gysin homomorphism (4.1) which was defined by using the spectral sequence. In the cases where the base space 1\11 is an oriented closed manifold, there is one more interpretation of the Gysin homomorphism. Here we describe it simply. Suppose that there is given a continuous mapping i : N -4 N' between two oriented closed manifolds N, N'. Then a homomorphism
i* : HP(N; Z)
----+
HP-d(N'; Z)
which is also called the Gysin homomorphism is defined to be the composition
150
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
Here n = dim N, d = dim N - dim N', and D and D' denote the Poincare duality maps of N, N' respectively. Similarly, the Gysin homomorphism
j* : Hp(N'; Z)
----4
Hp+d(N; Z)
in homology is defined by setting 1* = D-l 01* 0 D'. Now let us go back to the original situation where we are given an F-bundle 7r : E --+ M and assume that M is an oriented closed manifold. Then the total space E is also a closed manifold with the induced orientation which is locally equal to the one on thc product M x F. In this case, it can be shown that the Gysin homomorphisms 7r*
:HP(E; Z)
7r*
:Hp(M;Z)
----4 ----4
HP-n(M; Z) Hp+n(E;,£)
associated to the projection 7r defined above coincide with the former definitions (4.1), (4.2). The following proposition concerns basic properties of the Gysin homomorphism. We leave the proof of it to the reader because it is relatively easy.
(i) Let F be an oriented closed manifold M be an oriented F -bundle. Then for any ex E HP (M; '£) and {3 E Hq (E; '£), the equality
PROPOSITION 4.8.
and let
7r :
E
--+
7f* (7f* ex U (3) = a
U
7f * ((3)
holds. (ii) For any u E Hp(M; Z) and'Y E HP+n(E; Z), we have
("(, 7f*(u)) = (7f*("(), u). In particular, in the situation of (i), if we further assume that M is an oriented closed manifold and p + q = dim E, then
(7f* ex U {3, [ED
=
(ex U 7r * ((3), [M]).
(iii) The Gysin homomorphism is natural with respect to b'undle maps between oriented F -bundles. More precisely, the composition of the Gysin homomorphism followed by the induced homomorphism in cohomology of the base spaces is equal to the composition of the induced homomorphism in cohomology of the total spaces followed by the Gysin homomorphism, The following lemma treats a special case of the property of Gysin homomorphism of covering maps. We will use it in the next section.
4.3. NON-TRIVIALITY OF THE CHARACTERISTIC CLASSES (1)
151
LEMMA 4.9. Let M be an n-dimensional oriented closed manifold and let!!.-- : M ---> M be a finite covering. We put the orientation on M induced from M. Suppose that the Poincare dual [M] n a E Hn-k(M; Z) of a cohomology class a E Hk(M; Z) is represented by an (n - k)-dimensional oriented submanifold B of M. Then the Poincare dual of 1T*(a) E Hk(M; Z) can be represented by the oriented submanifold jj = 1T- 1(B) of M.
PROOF. Let N(B) be a closed tubular neighborhood of B in M. If we denote by v the normal bundle of B, then we can identify N(B) with the disk bundle D(v) with respect to a suitable metric. We set W M\Int N(B) and consider the following natural homomorphism:
Hk(D(//), aD(I/); Z) ~ Hk(N(B), aN(B); Z) ~
Hk(M, W;Z)
-7
Hk(M;Z).
If we denote by U E Hk(D(v), aD(v); Z) the Thorn class, then as is well known, the image of U under the above homomorphism is nothing but the Poincare dual of [B], namely a E Hk(N£; Z). On the other hand, 1T-1(N(B)) can serve as a closed tubular neighborhood N(B) of B, and moreover under the homomorphism
induced by the projection..'... the above Thorn class U clearly goes to that of the normal bundle of B. The claim follows from this immediately.
o 4.3. Non-triviality of the characteristic classes (1) 4.3.1. Ramified coverings. In this section as well as in the next, we prove the non-triviality of the characteristic classes of surface bundles defined in §4.2.1. The proof will be given by explicitly constructing surface bundles with non-zero characteristic classes. In this subsection, we briefly discuss ramified coverings, which are essential for us to use in our construction. The concept of ramified covering (or branched covering) is obtained by generalizing that of covering spaces, and there are various formulations in the framework of algebraic varieties, complex manifolds or differentiable manifolds. Roughly speaking, a submanifold, called the ramification locus or branch locus, is given in the base manifold, and away from there it is a usual covering space. Suitable
152
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
conditions are required on the ramification locus according to each framework mentioned above. Here we consider only the most simple type of ramified coverings, namely cyclic ramified coverings. Let m be a positive integer. An m-fold cyclic ramified covering is defined by taking the map (4.4) as a model. This map is the identity at the origin, and outside of there it is a usual covering map. From a slightly different viewpoint we can also interpret it as follows. The cyclic group 'LIm of order m acts on C naturally by
C '3
27ri
Z f---+
exp- z E C m
where ( denotes the generator of 'LIm. This action is free outside of the origin, and the quotient space can be canonically identified with
C
--->
C/('Llm)
~
C
is equivalent to the above map (4.4). Now a ramified covering is defined, locally, by taking a direct product of this model with other manifolds. More concretely, assume that the cyclic group 'LIm acts on an oriented Coo manifold N by orientation preserving diffeomorphisms satisfying the following condition. Namely the fixed point set F
=:
{p EN; ((p) '-- p}
is a submanifold of N of co dimension 2, and the action is free outside of F. Then, it can be checked that the quotient space N =-= NI(Zlm) has a natural structure of an oriented Coo manifold by investigating the action of Zim on the normal bundle of each connected component of F. If we denote by 7r:N---+N the natural projection to the quotient space, then F = 7r(F) becomes a submanifold of N of codimension 2. Moreover the restrictioIl7r : F --> F is a diffeomorphism and 7r : N \ F --> N \ P is a covering map in the usual sense. Finally it is easy to see that the map 7r : N --> IV is equivalent to the above model F x C '3 (p, z) ~ (p, zm) E P x C near F.
4.3. NON-TRIVIALITY OF THE CHARACTERISTIC CLASSES (1)
153
E(v)
N(8)
-HH--t-iH--t-i-t-+-t-t-+ IJ
FIGURE
4.2
In such a situation, we call 7r : N -> N an m-fold cyclic ramified covering ramified along F. It is also called simply an m-fold ramified covering. 4.3.2. Construction of ramified coverings. Let M be an oriented closed COO manifold. Assume that there is given an oriented submanifold B c M of codimension 2. Following Atiyah [Aj and Hirzebruch [Hi], let us recall a sufficient condition to the existence of an m-fold ramified covering of M ramified along B. Let a E H2(M; Z) be the Poincare dual of the fundamental homology class [Bj E H n - 2 (M; Z) (n = dim M) of B. The first Chern class of complex line bundles induces a canonical bijection {isomorphism class of line bundles over M} ~ H2(M; Z). Hence there exists a complex line bundle 'f/ over M which corresponds to Q. This bundle can be constructed explicitly as follows. Let v be the normal bundle of B in M and denote by E(v) its total space. v is a 2-dimensional real vector bundle over B, and it has a natural orientation induced by those of .M, B. Hence we can consider vasa complex line bundle. Let N(B) be a closed tubular neighborhood of B. Then as is well known, by choosing a Hermitian metric on v, we can construct a diffeomorphism
N(B)
~
{v E E(v);
Ilvll < E} (6) 0)
such that B c N(B) is sent to the O-section of v (d. Figure 4.2). Let 7r : E(v) -> B be the projection and set 7r'
= 7r 0
N(B) ---+ B.
The total space of the pullback bundle 'f/o = (7r')*(v) of v by the map 7r', which is a complex line bundle over N(B), can be described as
E('f/o) = {(p,v);p E N(B),v E w- 1 (7r'(p))}.
154
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
fR ,-
M
,....
f-
I-
1
I- l-
8
f- .....
FIGURE
Then a natural section s : N(B)
--7
4.3
E(7)o) is defined by
N(B) :7 P f---+ s(p) = (p,
w.
It is clear from the construction that 7)IB = v. Moreover the zero locus of s is precisely equal to B, and the image 1m s of s intersects the O-section M c E(ry) transversely (cf. Figure 4.3). In the above argument, we first assumed that M is a closed manifold. However, it can be shown that this assumption is unnecessary for the construction of the complex line bundle 7). In case !VI is a complex manifold and B is a complex submanifold of codimension 1, the above construction can be done entirely in the complex analytic category. Namely we can construct 17 as a holomorphic line bundle over M, and the section s can be chosen to be holomorphic. More concretely, choose a family Ui (i ~ 1) of coordinate neighborhoods of M with the property that B C UiUi and there is a coordinate function fi : Ui --7 C such that
We also set Uo = M \ B and consider the constant function fa = 1 on Uo· If we put iij = fdj-l, then {iij} becomes a I-co cycle associated to the open covering {Udi20 of M with values in C*. We can now
4.3.
NON-TRIVIALITY OF THE CHARACTERISTIC CLASSES
(1)
155
define "1 to be the holomorphic line bundle determined by this 1cocycle. Also the section s which is induced by Ii is holomorphic and clearly satisfies the above condition. In [Hi] there is a description of a more general construction including the case where B is expressed as the difference of two complex submanifolds. With the above preparation in mind, we prove the following proposition. PROPOSITION 4.10 ([Hi]). Let M be a closed oriented C= manand let B c M be an oriented submanifold of codimension 2. Suppose that, for some positive integer m, the homology class [B] E H n - 2 (M; Z) determined by B is divisible by m~n H n - 2 (M; Z). Then there exists an m-fold cyclic mmified covering M ----> M mmified along B. ~fold
PROOF. Let ex E H2(iVJ; Z) denote the Poincare dual of [B] E H n - 2(M; Z). By the assumption, there is an element (3 E H2(M; Z) such that m(3 = ex. If we denote by "1 the complex line bundle corresponding to ex, namely Cl ("1) = ex, then there exists a section s : M -) E("1) satisfying the following three conditions: (i) (ii) (iii) Let r/
("1')0 m
s = 0 on B. s -I- 0 on M\B. 1m s meets with M C E("1) transversely.
be the complex line bundle with Cl "1. Hence we can define a mapping
(3.
Then we have
=
f : E("1') - -~ E(rJ) by settin.!Lf(v) : v ® ... ® v (v E E("1')). If we set M = then f : lVI ----> 1m s = M is the desired ramified covering.
r
l(Im s),
D
4.3.3. Non-triviality of the first characteristic class 1',1' In this subsection, we prove the non-triviality of the first characteristic class el of surface bundles. The proof is given by slightly modifying Atiyah's argument in [A] in such a way that it can be adapted to the cases of higher dimensions later. The first characteristic class 1',1 is closely related to the signature which is an important invariant of closed oriented 4-manifolds. Just to mal
H2k(M; Q) ® H2k(M; Q)
---7
H4k(M; Q) ~ Q
156
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
which is induced by the cup product, namely the number of positive eigenvalues minus that of negative eigenvalues. Now to see the above fact, let 7r:E~M
be an oriented Eg-bundle and assume that the base space M is also an oriented closed surface. Then the total space E becomes a c~ed 4-manifold equipped with a natural orientation so that its signature sign E is defined. PROPOSITION 4.11. Let 7r : E --+ M be an oriented Eg-bundle over a closed oriented surface M. Then we have the equality
(el' [MJ) = 3 signE. PROOF. Let ~ be the tangent bundle along the fiber of the given Eg-bundle and let TE,TM be the tangent bundles of E,M respectively. Then we have
TE
~ ~
EB 7r*(TM).
It follows that
PI(E) =PI(~ EB 7r*(TM)) =PI(~) +7r*(pI(M))
=X(~)2 ,,~ e2
because PI = X2 for any 2-dimensional oriented real vector bundle and clearly PI (M) = O. We can conclude from this that
(el' [MJ)
c
(e 2, [E])
=(pl(E), [E]) =3signE
where we have used proposition 4.8(ii) of §4.2.3 for the first equation and the Hirzebruch signature theorem for the last equation. 0 In view of the above proposition, to prove the non-triviality of e1 it is enough to construct a surface bundle over a surface which has non-zero signature. Such a surface bundle was first constructed by Kodaira [Kod] in the framework of'the theory of complex surfaces. Slightly later, but independently, Atiyah [A] gave similar surface bundles. First we set MI = E9t (g1 ~ 2). Choose an m-fold cyclic covering PI : M2 --+ Nh of Ml and let (J' : M2 --+ M2 be a generator of its
4.3. NON-TRIVIALITY OF THE CHARACTERISTIC CLASSES (1)
157
covering transformation group. If we denote by 92 the genus of M 2 , then from the equality 2-292 = m(2-291) we obtain 92 = m9l-m+1. Next let P2 : M3 ---t M2 be the finite covering induced by the surjective homomorphism 7rl(M2)
----+
H l (M2;Z)
----+
Hl(1\Ih;Zjm) ~ (Zjm)2gz.
In the product manifold M3 x M 2 , let map U i P2 (i = 1,··· ,m) and set D
r r:;'P2
denote the graph of the
= r r:;P2 + ... + r r:;m P2·
Here we have r r:;'" P2 = r P2 because urn = id. If we fix a complex structure, namely a structure of Riemann surface, on M l , then M 2 ,1\13 also have the induced structure of Riemann surfaces and D becomes a non-singular divisor of the complex surface M3 x M 2 . However, topologically it is enough to understand D as a topological sum of m submanifolds r r:;iP2 (i = 1, ... ,m) of co dimension 2. Now we will show below that the homology class [D] of D is divisible by m in H 2 (M3 x M 2 ; Z). In view of Proposition 4.10, it will then follow that there exists an m-fold cyclic ramified covering f : E ---t 1\Ih X M2 ramified along D, and we obtain the following commutative diagram consisting of bundle maps between surface bundles (4.5)
D
D = f2l(D2)
D2 = rll(D l )
Dl
n
n
n
n
E
f
------+
1\113
X
rrl
pl
M3
1\Ih
M2
fz
------+
l\Ih
X
M2
It
------+
pl P2
------+
M2
Ml
X
Ml
pl PI
------+
1\Ih
Here h = (Pl, PI), h (P2, idMJ and p denotes the projection to the first factor. Also Dl C Ml X Ml is the diagonal set and D2 = fll(Dd· The point here is the fact that D = f21(D2). Now to prove that [D] is divisible by m, it is enough to show that the Poincare dual of [DJ, denoted by [D]* E H2(M3 x M 2; Z), is divisible by m. For tha.t we have only to prove that the mod m reduction of [D]*, which we denote by [Dl:n E H2(M3 x 1\Ih; Z/m), vanishes. If we use here Lemma 4.9 of §4.2.3, we obtain
158
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
On the other hand, the group H2(M1 x Nh; Z) can be expressed as a direct sum H2(M1 x {pt}; Z) EEl (Hl(Nh; Z)
Q9
Hl(M1; Z)) EEl H2( {pt} x M 1 ; Z)
by the theorem of Kiinneth. As an element of this group, we write
~an
for some Cti,(3i E H1(Nh; Z). Clearly both pi : H2(Ml; Z/m) --t H 2 (M2 ;Z/m) and P2 : Hl(NhZ/m) --t Hl(M3;Z/m) are trivial homomorphisms. Hence we obtain [D];'" = f2' fi([D 1 ];"') = O. The mapping 'iT: E ---+ M3 in diagram (4.5) which is constructed above is a surface bundle over M 3 . Its fiber is a covering of M2 ramified along m points on it. This construction of surface bundles is obtained by generalizing Atiyah's argument in [A], which treats the case m = 2, for arbitrary m. Hirzebruch [Hi] also develops another method which yields surface bundles with smaller genera. However, our method above is more suitable for generalizations in higher dimensions. Let us prove that the characteristic class el of this surface bundle is non-zero. For that, we first prove the following general proposition which will be frequently used in the next section. PROPOSITION 4.12. Let'iT: E --t NI, ir : E --t NI be two surface bundles over the same base space M. Suppose that there is given a mapping f : if; --t E, between the total spaces, which is an mfold cyclic ramified covering ramified along an oriented submanifold DeE of codimension 2. Suppose further that D intersects each fiber of 'iT transversely at exactly m points and the following diagram is commutative where i5 = f-1(D).
D
D
n
n
E~E
M=M Then we have the following two equalities:
4.3. NON-TRIVIALITY OF THE CHARACTERISTIC CLASSES (1)
159
(i) J*(v) = mv (ii) e = J*(e) - (m -l)v = J*(e - (1- ~)1/) where v E H2(E; Z), v E H2(E; Z) denote the Poincare duals of D, D, respectively, and e E H2(E; Z), e E H2(i{ Z) denote the Euler classes of 7[', Jr, respectively. PROOF. (i) is clear from the assumption (cf. the proof of Lemma 4.9). Vie prove (ii). Let N(D), N(D) be closed tubular neighborhoods of D, D, respectively. Then in the commutative diagram
f'
----->
H2(E \ IntN(D))
----->
H2(E \ IntN(D))
the images of the two Euler classes e E H2(E;Z),
e E H2(E;Z) in
~:~:~~D~)\::~::;:;n:':, ~;) t~o:~;;::~:;:);h:xact togethzwith the isomorphism
/H2(e, E \ IntN(D)) ~ H2(N(D), 8N(D)) ~ H n - 2(D), we can conclude that there exists an integer a E Z such that
e=
(4.6)
J*(e)
+ avo
Now if we denote by g, g' the genera of well known (4.7)
7[',
Jr, respectively, then as is
2 - 2g' = m(2 - 2g - m)
+ m.
On the other hand, if we restrict both sides of (4.6) to the fiber of Jr, then we obtain . (4.8)
2 - 2g' = m(2 - 2g)
+ am.
From (4.7), (4.8) we can now conclude a = 1-m,
completing the proof.
o
Let us go back to the commutative diagram (4.5) and compute the first characteristic class el of the surface bundle 7[' : E --> Jvh. If
160
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
we denote by have
e the Euler class of 7r, then by the above proposition we
e=
1*((2 - 292) [Nh]* - (1-
~)[D]*). m,
Hence el
=(e2 , [EJ)
=m(((2 - 292)[M2]* - (1-
~ )[D]*)2, [M3 m,
=m2g2 + 1 (((2 - 292)[M2 ]* - (1 =m2g2 + 1 (-2(1-
~ )(2 -
m
x M2])
~ )[D2]*)2, [M2
m
292)m + (1 -
x M 2])
~ )2m(2 m,
292))
=(292 - 2)m2g2(m2 - 1). Since this number is clearly positive we have finished the proof of non-triviality of el. 4.4. Non-triviality of the characteristic classes (2) 4.4.1. A method of constructing surface bundles. We generalize the construction of surface bundles due to Kodaira and Atiyah which was used to prove the non-triviality of e1 in §4.3.3. For that we summarize this construction simply as follows (d. diagram (4.5)). We set Ml = Eg and consider the trivial Eg-bundle Ml x Ml ........, MI' Then the image of the section Nh '3 P f-> (p, p) E Ml X M 1 , namely the diagonal set D 1 , is a submanifold of the total space of codimension 2 and intersects each fiber transversely. However, for any integer m > 1, its homology class [D 1 ] is not divisible by m in H 2 (M1 x M 1 ; Z). We then take a suitable finite covering of the above trivial bundle along the fiber as well as the base space so that the homology class of the inverse image of Dl will be divisible by a given number m. If we consider the associated ramified covering, applying Proposition 4.10, then the resultant surface bundle satisfies the required conditions. To generalize this procedure in higher dimensions, it is necessa.ry to apply a similar construction as above to surface bundles which are already twisted rather than the trivial ones. More precisely, we can overcome this difficulty by combining the following three operations. The first operation is to begin with a given Eg-bundle 7r : E ........, M and construct a new Eg-bundle out of it which has a section and whose dimension is higher by 2. For that it is enough to consider
4.4. NON-TRIVIALITY OF THE CHARACTERISTIC CLASSES (2)
161
the pullback bundle 7r' : E* ---+ E which is induced from 7r by the projection 7r itself. To be more precise, we set E*
= {(u,u')
E E X
E;7r(u)
= 7r(u')}
and 7r' (u, u') = u. Then we can define a bundle map q : E* ---+ E, which covers 7r : E ---+ M, by setting q(u,u') = u', and the following diagram is commutative:
E~M
The diagonal set D = {(u, u); u E E} of Ex E is a sub manifold of codimension 2 and intersects each fiber at a single point. In fact it is the image of the section E 3 u t--+ (u, 'u) E E*. In case the base space M is oriented, E has the induced orientation so that D becomes also an oriented submanifold. For simplicity, we assume that M, and hence E'~' also are closed manifolds. We denote by vEH2(E*;'?l)
the Poincar dual of the homology class [D] E Hn-2(E*; '?l) (n dim E*) of D (notice that this cohomology class is defined without the abo.ife assumption) and let Vm
E H2(E*; '?lIm)
be the mod m reduction of v. The second operation is, for each positive integer m, to take an rn-fold covering along the fibers. Although this is a trivial operation for trivial bundles, we need some work in the general situation. LEMMA 4.13. Let 7r : E ---+ M be an oriented 'Eg-bundle and assume that g ~ 1. Then for any positive integer m, there exists a finite covering lVh ---+ M such that the p:::llback 'Eg-bundle E1 ---+ M1 by this map admits an m-fold covering E1 ---+ E1 along the fibers. Namely E1 ---+ E1 is an m-fold covering map and the composition E\ ---+ Ml has the structure of a surface bundle whose fiber is 'E g, which is an m-fold covering surface of 'Eg (and hence g' = m(g - 1) + 1). PROOF. For the proof we use the following two facts regarding the mapping class group. The first one is the well-known isomorphism
Mg
~
Out+7rl('Eg),
162
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
due to Nielsen, which was already mentioned in §4.1.3, and the second one is that Mg has the property of being virtually torsion free. Here a group G is called virtually torsion free if there exists a normal subgroup H c G of finite index which is torsion free; namely it has no non-trivial element of finite oreler. In fact, it is known that the subgroup of Mg consisting of all the elements which act on HI ('£g; '1,/3) trivially has this property. First let us choose an rn-folel regular covering ~gl -> ~g." In other words, we fix a normal subgroup 1f1 (~g/) C 7rl (~g) of inci~l m. Since 1f1 (~g) has only finitely many normal subgroups of indlx- m, we can choose a normal subgroup Tl c Aut+7r1(~g) of finite index such that any element of it preserves 1fl (~gl ). Consider the natural homomorphism 1':
Tl
---+ Allt+1fl(~91) ---+
Mgl
where Aut+1fl(~gl) -> Mgl is the projection. It is easy to see that for any element 'Y E Inn1fl(~g) n T I , the oreler of r("() is finite. Now choose a normal subgroup T2 c Mgl of finite index which is torsion free. Clearly we have r-1(n) nInn1fI(~g) c KerT. If 7r : Aut+1f1(~g) -> Mg denotes the projection, then 1f(T- 1 (T2 )) is a subgroup of Mg of finite index. Therefore, if we set T3 to be the intersection of all the subgroups which are conjugate to this subgroup, then it becomes a normal subgroup of'M.9 of finite index. From the construction, there is a natural homomorphism T3 -) Mgl. Now let h : 1fl (M) -> Mg be the holonomy homomorphism of the given ~g-bundle 7r : E -> Iv! anel let Ml -> Iv! be the finite covering induced by the kernel of the composition 1f1 (M) -> M 9 -> M g / T3 . Then it can be shown that the ~gl-bundle, which is defined by the homomorphism 1fl (IvIt) -> T3 -> Mgl, satisfies the required conditions. D The third operation, associated to any cohomology dass u E
H2(M; Z/m) of the base space of a given ~g-bundle 1f : E
->
M,
is to construct an appropriate finite covering p : Iv! -> M such that p* (u) = O. This is not always possible as can be seen easily by considering the case where 111 = S2 for example. We need to impose certain conditions on the base space. DEFINITION 4.14. For each non-negative integer n, we define a class Cn consisting of 2n-dimensional connected Coo manifolds recursively as follows. Elements of Co are O-dimensional manifolds each of
4.4. NON-TRIVIALITY OF THE CHARACTERISTIC CLASSES (2)
163
which is a single point, and C1 is the class consisting of closed orientable surfaces of genus g 2: 2. In general, Cn + 1 is defined to be the class consisting of any finite covering of the total space of an oriented I:g-bundle with g 2: 2 whose base space belongs to Cn- We denote by C the disjoint union of all Cn and call members of it iterated surface bundles. PROPOSITION 4.15. Let E be a manifold belonging to the class Cn and let m be a positive integer. Then for any cohomology class u E H2(E; '!LIm) there exists a finite covering p : E ---+ E such that p*('u) o. PROOF. We use induction on n. The statement is clear for n = 0,1. So assume that n> 1 and let E be a manifold belonging to By definition, there exists a manifold Mo E Cn - 1 and a I:g-bundle Eo ---+ NIa over Ma such that E is a finite covering of Eo. It is easy to see that the composition E ---+ Eo ---+ Mo is a fiber bundle. Its fiber, denoted bJ'\I:, is not necessarily connected but consists of some pieces of finite coverings of :E g . 7rl(Mo) acts on the finite set 7ro(:E) of connected comnonents of :E. We take a suitable finite covering NIa ---+ Mo of Mo/which kills this action and let E' ---+ Ma be the pullback of E ---->Mo by this mapping. Clearly we have Ma E Cn - 1 · If we let Eb be a connected component of E', then from the construction the fiber of the map Eb ---+ Ma is connected so that it is a :Eg-bundle for some g 2: 2. On the other hand, since Eb is a finite covering of E, the pullback of u defines a cohomology class u' E H2(Ea; '!LIm). If we apply Lemma 4.13 here, we can conclude that there is a finite covering 1Vh ----> MIl such that the pullback :Eg-bundle El ----> Ml has an m-fold covering Ei ----> 1Vh along the fibers. Then Ei ---+ Ml is an :E g ,bundle for some :E g , which is a certain m-fold covering of :E g . 7rl(M1) acts on the finite group HI (:E g ,; '!LIm). We take a finite covering NIi ----> Nh which kills this action, and we further take another finite covering M2 ---+ Mi such that the homomorphism Hl(Mi;'!Llm) ----> H 1 (Nfz;'!Llm) is trivial. Let E2 ---+ M2 be the pullback :Eg,-bundle by the map Adz ----> MI. Summarizing our construction so far, we obtain the following commutative diagram:
en.
E2 (4.9)
1f
---->
1
Eo'
M2
---->
E'1 - .. -- '.... -+ El
1
Eg'
---->
1
1
Eg
M1=Nh
E'0 Eo
---->
M'0
164
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
Let us write p : E2 ~ Eo for the composition of the three mappings in the upper row of the above diagram. We show that there is an element v E H 2(M2;'Ljm) such that p*(u') = 7f*(v). For that we consider the Serre spectral sequence {E;.,q, dr } for 'Ljm-cohomology of the fiber bundle 7f : E2 ~ M 2. The local system over N/2 defined by the 'Ljm-cohomology of the fibers is clearly trivial. Hence the E 2 term is given as E~,(j = HP(Nh; Hq(L,gl; Zjm)). p*(u') is an element of H2(E2; Zjm), and we have the following two short exact seqllences:
o ---+ K
---+
o --> EC;;}
H2(E2; Zjm)
---+
Er;;}
K --> E;~/
---+
0
-->
.-->
0
where K = Ker(H2(E2; Zjm) ~ H2(L,g'; Zjm)). Now since L,gl ~ L,g is an m-fold covering, the image of p*( u') in E~2 is O. Hence p*(u') belongs to K. On the other hand, by the construction of two finite coverings M{ ~ Nh and M2 ~ M{, the image of p* (u') in E~1 is also O. Hence from the above short exact sequence we see that p*(u') E E~2. But since E~2 = Im(H 2(M2;Zjm) ~ H 2 (E2;'Ljm)), there exists an element v E H2(Nh;Zjm) such that p*(u') = 7f*(v). Now since M2 is a finite covering of M~ E Cn - 1 , M2 also belongs to the class Cn - 1 . Therefore by the induction assumption, there is a finite covering p : M3 ~ M2 such that p* (v) = O. If we let E ~ M3 be the pullback L,gl-bundle by the map M3 ~ M 2, then the composition E - t E2 - t E is a finite covering. We can now conclude that this bundle satisfies the required condition; namely the pullback of the cohomology class u E:: H2(E; 'Ljm) to E is O. This completes the ~~ 0 With the above preparation, out of any L,g-bundle 7r : E ~ M whose total space belongs to the class C, we construct a new surface bundle as follows. To begin with, we apply the first operation; namely we consider the pullback bundle 7r:
E*
-->
E
by the projection 7r. This bundle is equipped with a canonical section s : E --+ E*, and its image D c E* is an oriented sllbmanifold of codimension 2. We denoted by Vm E H2(E*; 'Ljm) the mod m reduction of the Poincare dual of its homology class. Now we consider the
4.4. NON-TRIVIALITY OF THE CHARACTERISTIC CLASSES (2)
165
following commutative diagram consisting of various surface bundles:
E*
--7
E3
1
1
E ---
E
'B.q "
--7
'B., --7
E2
--7
1
'By'
E2
--7
'E~
1
--7
'B.,
Ei
--7
E*
1
'B.
E1 - - - E1
-----..;
E
Here the four columns from the right are the same as those of Diagram (4.9) in the proof of Proposition 4.15 where we replace Eb ~ M6 by "E* --t E and u' E H2(Eb; Z/m) by /.1m E H2(E*; Z/m), respec'tively. In particular, 'Ei -) Ei is an m-fold covering along the fibers (cf. Lemma 4.13). The image in H2(E2; Z/m) of the cohomology class /.1 m is equal to the pullback to E2 of some element v E H2(E2; Z/m). Now since E2 is a finite covering of E, it belongs to the class C. 'Hence, again by Proposition 4.15 there is a finite covering p : E --.:;\ E2 such that p* (v) = O. Therefore if we let E3 ~ E be the pullpack ~g,-bundle, then the image of /.1m in H2(E3;Z/m) is O. It y>llows that, if we denote by D* C E3 the inverse image of D by..the map E3 ~ E*, then the pair (E 3 , D*) satisfies the conditions ~f Proposition 4.10. Hence there exists an m-fold cyclic ramified covering E* --t E3 ramified along D*. The projection E* --t E has a structure of a Eg,,-bundle where E g" is an m-fold cyclic covering of E g , ramified at m points on it. In particular we have g" = mg' + ~(m2 - 3m) + 1 = m 2 g - ~m(m + 1) + 1. We call t.he above process by which we obtain a Eg,,-bundle fro!?, a given Eg-bundle E ~ M an m-construction. The total space E* of the new surface bundle clearly belongs to the class C. Hence for any positive integer m' we can apply the m'-construction on E* ~ E. We have thus proved that, starting with any surface bundle whose total space belongs to the class C, we can construct various surface bundles by applying mj-constructions successively for j = 1,2, .... The construction due to Kodaira and Atiyah mentioned in §4.3.3 is nothing but the 2-construction on the trivial Eg-bundle Eg ~ pt over a single point.
4.4.2. Non-triviality of ei. We compute the characteristic classes of surface bundles constructed in the preceding subsection. Let 7r : E --t M be a Eg-bundle such that E belongs to the class C
166
4.
CHARACTERISTIC CLASSES OF SURFACE BUNDLES
and let E* ~E* ~ E 7r
1E~ E
7r'
-->
r
1
Eg
E
7r
----->
1
Eg
M
7r
be the m-construction on it. We denote by e E H2(E; 71.,) and e E H2(E*; 71.,) the Euler classes of 7f : E ---> M and if : E* ---> E, respectively, and let D be the image of the section s : E ---> E*. Furthermore we set D = r- 1 (D) C E* and write v E H2(E*;7I.,), v E H2(E*;7I.,) for the Poincare duals of D, D, respectively. PROPOSITION 4.16. We have the following equalities. (i) r*(v) = mv and hence v = ~ r*(v). (ii) v 2 = q*(e) v = (7f')*(e) v. (iii)
e = r* ( q* (e) - (1 - ~) v).
PROOF. The Euler class of 7f' : E* ---> E is clearly q* (e). Hence (i) and (iii) follow from Proposition 4.12. On the other hand, if i : D ---> E* is the inclusion, it is easy to see that i*q*(e) = i*(7f')*(E) is equal to the Euler class of the normal bundle of D in E*. If we use the Thom isomorphism theorem, (ii) follows. D PROPOSITION 4.17. If we denote by ek, ek the k-th characteristic classes of 7f : E ---> M and if : E* ---> E, respectively, then we have
ek = m 2 f* (7f*(e k ) -
(1 - m-(k+1l)e k ).
PROOF. By Proposition 4.16, we have (4.10)
ek+1
= r*(q*(e) _ (1- ~) v)k+1 =
r*(q*(e k+1) - (1- m,-(k+1l)(7f')*(e k )v).
By applying the Gysin homomorphism
7f*: H 2 (k+1l(E*;Q)
--->
fJ2k(E;Q)
to (4.10), we obtain
ek =
m,2f*(7f*(ek) -
This completes the proof.
(1- m-(k+l l )ek ). D
NON-TRIVIALITY OF THE CHARACTERISTIC CLASSES
4.4.
(2)
167
If the base space M of a I:;g-bundle 7r : E --> M is an oriented 2n-dimensional closed manifold, then by evaluating any polynomial in ei of degree 2n on the fundamental cycle of M, we obtain various numbers. More precisely, associated to any partition I = {iI, ... ,ir } of n, the corresponding number
is defined. These numbers are called characteristic numbers of surface bundles. Let I = {i1' ... ,i r } be a partition of some natural number. For any subset J = {jl,'" ,js} of I, we describe its complement JC = I \ I as JC = {k 1 , ... ,kt } (s + t = r). PROPOSITION 4.18. Let 7r : E --> M be an oriented I:;g-b~ndle over an oriented 2n-dimensional closed manifold 111 and let if : E* --> E be an m-construction on it.\ Then for any" partit~on I = {iI, ... ,i r } of n + 1 the characteristic n)U1nber of 7T : E* --> E associated to I is given by eI[E] = dm 2r
L( _l)t (1 - m~(kl+1)) ... j
Here d is the mapping degree of E of I.
-->
E and J runs through all subsets
PROOF. By proposition 4.17, we have
eI[E] = =
eij ...
edE]
m 2 f* (7r*(eiJ -
(1 -
m~(il+!))eil) ...
m 2 f*Cn'*(e;J -
(1- m~(ir+I))eir)[E]
= dm 2 (7r*(eil) - (1- m~(il+l))eil) ... (7r*(eiJ - (1 - m-(tr+l))eir)[El 1'
= dm?1'L(-1)t(1-m-(k 1 +l)) ... j
(1 -
m-(k'+!))ejek1+.+k,_1 [M].
The last equality follows from §4.2.3, Proposition 4.8 (ii).
0
In particular, it follows from the above proposition that if en[Ml is non-zero, then en+! [E] is also non-zero for any m > 1. Hence by
168
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
induction we can conclude that all the characteristic classes non-trivial.
ei
are
4.4.3. Algebraic independence of characteristic classes of surface bundles. In view of the description of §4.1.5, the characteristic classes ei of surface bundles can be regarded as cohomology classes of the mapping class group. Namely we can write ei
E H2i(Mg; Z)
(i = 1,2,··· ).
In this subsection, we give an outline of the proof of the following theorem which shows the algebraic independence of these characteristic classes. This theorem was proved by Miller [Mi] and the author [Mor3] independently. However, the range of injectivity below is due to Harer's improved stability theorem [Har3] which was obtained later. THEOREM 4.19. Let n be any positive integer. Then, for any g with 9 :2 3n, the homomorphism
Q[el' e2,···]
-~
H*(Mg; Q)
is injective up to degree 2n. Sketch of Proof. If we examine the formula of Proposition 4.18, it is almost clear that there is no algebraic relations between the characteristic classes of surface bundles which are obtained by successively applying mrconstructions for appropriate mj (j = 1,2, ... ). However, the theorem does not follow from this because it is not obvious whether algebraic independence for one specific genus implies the same property for all larger genera or not. To solve this problem, we introduce mapping class groups of surfaces with base points or boundaries. First assume that a base point * E :E g is given on :Eg . In this case we denote by Diff+ (:E g , *) the group of orientation and base point preserving diffeomorphisms of :E g and set
M
g ,.
= 7fo(Diff+(:Eg, *)).
We call it the mapping class group of :E g relative to the base point. By forgetting the base point, we obtain a projection 7f : M g ,. --+ M g " Furthermore, by the assumption that g ;::: 2, it can be easily shown that there is a canonical isomorphism Ker 7f ~ 7f1 (:Eg). Thus we obtain the following short exact sequence: 1 ---7 7fl(:Eg) ~ M g,. ~ Mg
--+
1.
4.4. NON-TRIVIALITY OF THE CHARACTERISTIC CLASSES (2)
169
Here we can regard any 'Y E 1f1 (~g) to be an element of M g,. as follows. Choose a closed curve e based at the base point which represents 'Y- Then i(ry) is the isotopy class, keeping the base point fixed, of a diffeomorphism defined by pulling the base point along e in the reverse direction and coming back to the initial point. Next let D2 C ~g be an embedded disk and let Diff (~g, D2) be the subgroup of Diff+~g consisting of all elements which are the identity on D"2:-We set
M g ,1
= 1fo(Diff(~g,D2))
and call it the mapping class group of ~g relative to D2. It is also called the mapping class woup of the surface ~g \ IntD2 relative to the boundary. By taking t\he base p¢nt * on D2, a natural projection 1f : M g ,1 --> M g ,. is definel:LJt.cin be shown that the kernel of this projection is isomorphic to an infinite cyclic group generated by the isotopy cla.ss, keeping D2 fixed, of a diffeomorphism which rotates D2 by 360 degrees. Thus we obtain the following short exact sequence:
o ----+ Z ----+ M g, 1
----+
M g, *
----+
L
We can deduce from the result of Earle-Eells [EE] that BDiff+(~g, *)
= K(M g,., 1),
BDiff(~g, D2)
= K(Mg,l, 1).
BDiff+ (~g, *) is the classifying space of surface bundles equipped with a section, and BDiff(~g, D2) serves as the classifying space of surface bundles with a section together with a trivialization of the normal bundle of its image. Now if we regard Diff (~g, D2) as the group of diffeomorphisrns of 1: y \IntD2 which restrict to the identity on the boundary, then any element of it defines an associated diffeomorphism of ~g+l which is the identity on the last handle. This operation induces a homomorphism
M g ,1
----+
M g +1,1.
Hence if we are given a surface bundle 1f : E --> M whose holonomy group (or monodromy group) can be lifted to Mg,l, we can modify 1f to obtain another surface bundle the genus of whose fiber is greater than the original by one (and hence any number). More precisely, we can perform this operation as follows. By the assumption, a section s : lvI -, E and a diffeomorphism between a closed tubular neighborhood N(Ims) of its image and M x D2 are given. We now set To = T2 \ Int D2 and E = (E\IntN(Ims)) uM x To.
170
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
Here the right hand side represents the manifold which is obtained by pasting the two pieces along their boundaries by the identification 8N(Im s) ~ M X 8 1 = 8(M x To). Then the natural projection 'IT : E -+ 1\1/ becomes a surface bundle whose genus goes up by one and whose holonomy group is contained in M g + 1 ,1. This operation can be generalized as follows. Suppose that there are given k genera 91, ... ,9k with L 9j ::; g. In this case, once we fix an embedding Uj E.g] \ Int D2 C E.g of disjoint union of E gj \ Iut D2 into Eg, we obtain a homomorphism
(4.11)
L : A1 yl ,1
X ... X
Mgk,l
----->
Mg,l.
Under such a situation, it is not hard to see that the characteristic classes behave as follows. Namely, with respect to the homomorphism
L* : H*(M g,l;Z) induced by
L
----->
H*(Mg1,1 x ...
X
Mgk,l;Z)
in cohomology, we have k
(4.12)
L*(ei)
= 2:Pj(e'i)
(i
= 1,2",,),
j=l
Here Pj : M gl ,l X ... X Mg"l --+ Mgj,l is the projection to the ,i-th factor. With the above preparation, we can prove the theorem as follows. In the preceding subsection, we showed ei =f. 0 for any i by computing the characteristic classes of iterated surface bundles which are obtained by applying the m-constructions successively. We can enhance this construction to obtain surface bundles which have sections with trivial normal bundles such that Ci =f. 0 in the rational cohomology group. The holonomy group of such bundles lies in Mg,l for some g. Hence we can conclude that Ci =f. 0 E H 2i (M g ,1;Q). It then follows from equation (4.12) that the same fact holds for any genus ~ g. We can now prove that the homomorphism in the statement of the theorem is injective up to degree 2n as follows. For each Cj U = 1, ... ,n) choose gj U = 1,,,' ,n) such that ej is non-zero in H*(M gj ,l; Q). Next, for each j choose d j such that jdj ~ n and set 9 =~ L j djg j . Then we obtain a homomorphism
L: (M g1 ,1)d1 x ... x (M gn ,l)d"
----->
Mg,l
similarly to (4.11). If we use (4.12) here again and the theorem of Kiinneth, we see that the characteristic classes are algebraically
4.5. APPLICATIONS OF CHARACTERISTIC CLASSES
171
independent in H*(Mg,liQ) up to degree 2n. This completes the ~~
0
The above is an outline of the proof of Theorem 4.19 given in the author's paper ~or31. The proof in [Mi] is basically the same except for the fact tnatit uses the following fundamental result of Harer [Har2] to show the existence of 9 with ei # 0 E H 2i (M g ,li Q) without giving explicit surface bundles whose holonomy groups lie in Mg,l. It is called Harer's stability theorem which claims that the natural homomorphisms
H*(Mg,liZ)
-+
H*(MgiZ),
H*(Mg+1,liZ)
-+
H*(Mg,liZ)
are isomorphisms in a certain stable range (up to degree approximately t g). It follows that the stable cohomology algebra of the mapping class group lim H*(Mg; Q) g->oo
is defined. Later it was ~j?)YHarer [Har3] that the stable range can be doubled for the rational cohomology. As mentioned already, the range of injectivity in the theorem is due to this improved result. Using the theorem of Milnor-Moore regarding the structure of Hopf algebras, Miller [Mi] further proved that the above stable cohomology algebra is isomorphic to the tensor product of polynomial algebra on primitive elements of even degrees and exterior algebra on primitive elements of odd degrees. 4.5. Applications of characteristic classes 4.5.1. The Nielsen realization problem. The mapping class group Mg has many finite subgroups. For any fixed g, it is an interesting problem to classify conjugacy classes of finite subgroups of Mg and also to give upper bounds of orders of them. By the way, there was a classical problem concerning finite subgroups of mapping class groups called the Nielsen realization problem which can be described as follows. Let 7r : Diff+Eg -+ Mg be the natural projection. A finite subgroup G C Mg is called realizable as a transformation group of Eg if it can be lifted with respect to the projection 7r, namely if there exists a subgroup G C Diff+Eg such that the restriction of 7r to G gives rise to an isomorphism 7r : G ~ G. Now the above problem is the question Is it true that any finite subgroup of Mg is realizable?
172
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
and it originates from the works of Nielsen in the 1940's. Nielsen himself gave an affirmative answer to this problem for the cases where G is cyclic, and there have been obtained many results for various other cases. Finally Kerckhoff [Ke] solved it for general cases affirmatively. The above problem is meaningful also for infinite subgroups of Mg. In the case where 9 = 1, namely for the torus, if we take T2 = ]R2/'l} as its model, then it is easy to construct a right inverse homomorphism to the projection 7T:
Diff+T2
----+
M1 ~ 8L(2,Z).
More precisely, by letting 8L(2, Z) act on T2 linearly, we obtain a homomorphism s : 8L(2, Z) --> Diff+T2 which clearly satisfies 7T08 = id. It follows that any subgroup of M1 is realizable as a diffeomorphism group of T2. There arises naturally the question whether the same thing is true for the cases g ~ 2 or not. In the next subsection, we give a negative answer to this problem by using the characteristic classes of surface bundles. 4.5.2. The Nielsen realization problem for infinite subgroups. We prove the following theorem. THEOREM 4.20 ([Mor3]). Let 7T : DifJ'+L.g --> Mg be the natural projection and let 7T* : H*(Mg;Q) --> H*(Diff!L.g;Q) be the induced homomorphism where Diff!L.g denotes the group DifJ+"Eg equipped with the discrete topology. Then for any i ~ 3, we have 7T*(ei)
= O.
On the other hand, we know from Harer's improved stability theorem [Har3] together with Faber's work [F2] that e3 i- 0 in H6(M g ; Q) for all g ~ 5. Hence as a corollary to the above Theorem 4.20, we obtain the following result. THEOREM 4.21. The natuml projection 7T : Diff+L.g not have a right inverse homomorphism for all 9 ~ 5.
-->
Mg does
This can be shown as follows. Assume that there is a homomorphism s : M.9 --> Diff+"E g such that 7T 0 8 = id. Then we have s*1r*(e3) = e3 i- O. But this contradicts the conclusion 7T*(e3) = 0 of Theorem 4.20.
4.5. APPLICATIONS OF CHARACTERISTIC CLASSES
173
Proof of Theorem 4.20. Suppose that there is a homomorphism s : Mg ---. Diff+Eg which is a right inverse to the projection 7r. For any Eg-bundle 7r: E ---+ M, consider the composition of s and the holonomy homomorphism p : 7rl(M) ---. Mg sop: 11"1 (M) ---+ Diff+Eg. Then we can conclude that 11" : E ~ !vI has a structure of a foliated bundle with fiber E.g (cf. §3.1, Example 3.5 and §2.4, Definition 2.26). In other words, on the total space E there is a foliation F of codirnension 2 such that any leaf of F is transverse to the fibers of 11". Clearly the normal bundle u(F) of:F can be naturally identified with the tangent bundle ~ alor(g'the fibers of 11". If we apply here the Bott vanishing theorem (§3.4.\Theorem 3.24), then we have
pi(//(F)) == ptr6 = o. On the other hand, we have Pl(~)
= X2(~) = e2 so
that
e4 = 0 E H 8 (E; «:)1). It follows that ei = 0 for all i ~ 3. Since this holds for any foliated Eg-bundle, we can conclude 11"* (ei) = O. This completes the proof. 0 It is well known that the mapping class group is also isomorphic to the group 1I"0(Homeo+Eg) of connected components where Homeo+E g denotes the group of orientation preserving homeomorphisms of E g • Henee we have a projection
7r : Homeo+Eg
---+
Mg.
Similar to the above ease of diffeomorphisms, we can also ask whether there exists a right inverse homomorphism to this projection or not. However, since the Bott vanishing theorem does not hold for CO foliations, our proof above is not applicable anymore. III Fad, Thurston [Th2] proved that the projection 11" induces an isoUlorphism 11"* :
H*(HomeotE g ) ~ H*(M y)
in homology. Hence there is no cohomologicnJ oh:4rllctions to the existence of a right inverse homomorphism. Nevert.lu,jpss I.he following conjecture seems to be reasonable. CONJECTURE 4.22. The natural projection "lr : /lomeo+Eg /V/ 9 does not have a right inverse homomorphism..
---+
174
4. CHARACTERISTIC CLASSES OF SURFACE BUNDLES
In other words, we expect that M 9 would not act naturally on by homeomorphisms. In contrast with this, Cheeger and Gromov constructed a natural action of Mg on the unit tangent bundle T1~g of ~9 by homeomorphisms such that the following diagram is commutative: ~g
Mg C O\lt7fl(~g)
+ - - Out7fl(T1~g)
Here the homomorphism in the lower row is induced by the isomorphism 7f1 (Tl ~g) /Z ~ 7fl (~g) where Z C 7f1 (Tl ~9) is the center of 7f1(Tl~9)'
Directions and Problems for Future Research Here we summarize th, material in each chapter treated in this book, and based on them ws:ention a few open problems which are left for future research. In Chapter 1, we describe t_@J>asic part of the de Rham homotopy theory following Sullivan's original paper [Su3] and the book [GM] of Griffiths and Morgan. The fundamental theorem of this theory is that the minimal model of the de Rham complex of a simply connected manifold is equivalent to the real homotopy type of it. Also in the case of non-simply connected manifolds, the I-minimal model of the de Rham complex is naturally isomorphic to the Malcev completion of the fundamental group tensored with lR. This theory is closely related to Quillen's rational homotopy theory [Q] as well as Chen's theory [eh] of iterated integrals which were introduced in nearly the same decade. Since the beginning, these theories have been developed from various points of view until the present time. Of particular importance is the refinement of them to the cases of Kahler manifolds or algebraic varieties. Already in [DGSM], there was given an important application of this theory to the topology of compact Kahler manifolds. Through the works of Morgan [Mol and Hain [Hal] the refined rational homotopy theory is still making progress. On the other hand, some of the problems presented in the original paper [Su3] remain open_ In particular, it is expected that one can construct a theory which could obtain more information about manifolds than the nilpotent ones using cohomology of local systems on them. In Chapter 2, we described characteristic classes of fiat bundles with finite dimensional Lie groups as fibers in terms of cohomology of the corresponding Lie algebras. We then introduced the theory 175
176
DIRECTIONS AND PROBLEMS FOR FUTURE RESEARCH
of Gel'fand and Fuks from the viewpoint of characteristic classes of flat bundles whose structure group is the diffeomorphism group Diff F of a given manifold F, namely foliated F-bundles. Dupont's lecture note [Dul] contains compact and clear descriptions starting from the Chern-Weil theory reaching to characteristic classes of flat bundles. The theory of characteristic classes of flat bundles has been developed continuously from various points of view. In particular, several deep results have been obtained in connection with algebraic geometry or number theory, and it is very likely that this situation will continue in the future. We refer the reader to papers [BI], [Du2], [R], [BE], [DHZ] and references therein. In Chapter 3, we described the general theory of characteristic classes of foliations. Roughly speaking, there are two approaches to this theory: one is from the viewpoint of the Chern-Weil theory and the other is from that of the Gel'fand-Fuks cohomology theory. In this book, we adopted the latter following [Bot2], [BH], [H]. In the 1970's, many results were obtained concerning non-triviality of characteristic classes of foliations. However, there are still many problems that remain unsolved. Most of the results obtained so far depend on classical theory concerning Lie groups and geometry of homogeneous spaces. It is expected that one could construct essentially new examples or discover new phenomena that go beyond the known ones. The problem of determining whether the discontinuous invariants induced by characteristic classes with real coefficients, described in §3.5, is in general a completely open problem not only for characteristic classes of foliations but also for characteristic classes of flat bundles treated in Chapter 3. The only known results are those given in [Mor2] and Tsuboi's result in [Ts], which solves this problem for the case of piecewise linear foliations of codimension 1. The most interesting cases include the Godbillon-Vey class of Coo or real analytic foliations of codirnension 1 and also the Cheeger-Chern-Simons classes of flat GL(n, C)-bundles. However, we n
DIRECTIONS AND PROBLEMS FOR FUTURE RESEARCH
177
[F2], Looijenga [Lo] and others. On the other hand, from the viewpoint of topology, a series of fundamental results [Harl], [Har2], [Har3], [HZ] due to Harer concerning the mapping class groups of surfaces and also Johnson's pioneering deep works on the structure of the Torelli group (see [J]), both in the 1980's, paved the way to a modern study. Shortly afterward, the theory of characteristic classes of surface bundles treated in thiH book was begun. We gave only the definition of the Torelli group in the text. However, it will surely play a central role in future research of the mapping class group as well as the moduli space of algebraic ,curves. There the representation theory of the symplectic group Sp(2g, Q) and Uw de Rham homotopy theory will serve as main techniqu~s. Recent; n~slllt.s along these lines include Kawazumi's work [K] whic~ gerwraliz(~s the characteristic classes of surface bundles to the case~Lh f.wisL<~d coefficients, Hain's work [Ha2] which determines the striJ(:tl1l'(~ of the Malcev completion of the Torelli group completely, and 0111" paper [KM] in which we investigated group cocycles represcut.ill).!; characteristic classes of surface bundles and also the continuolls coholllology of the mapping class group relative to some natural filtratioll on it. In this way, vivid progress is })(:illg made from the viewpoints of both topology and algebraic !-';c()lIldry. We refer the reader to [HL], [Mor4] for more details on n:cmll. d(:vdopments.
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Index A differential system, 124
Connected (d.g.a.), 18 Connection, 78 Continuous cohomology, 79 Coskeleton, 7 Covering homotopy property, 3 CW complex, 2 Cyclic ramified covering, 152
Adjoint map, 21 Anosov foliation, 94 Attaching map, 2 Attaching space, 2 Branch locus, 151 Branched covering, 151
Decomposable, 19 Dense leaf, 121 d.g.a., 18 Diffeotopy group, 138 Differential graded algebra, 18 Differential ideal, 54 Discontinuous invariants, 119 Divisor, 157 Dual Lie algebra, 20
Canonical form of frame bundles of higher orders, 108 of second order, 103 of tangent frame bundle, 100 Cell complex, 2 Central extension, 43 Characteristic class of F -bundles, 1:36 of foliations, 91 of surface bundles, 144 Characteristic cohomology class of Hirsch extensions, 19 of principal fibrations, 8 Characteristic homomorphism of flat G-bundles, 69 of flat G-product bundles, 67 Chern-Simons class, 72 Chern-Simons form, 71 Chern-Simons invariant, 72 Closure finite, .3 Co cycle condition, 44 Codimension, 88 Cohomologically connected (d.g.a.), 18 Cohomology of groups, 45, 128 Completely integrable, 54 Conjugate (homomorphisms), 56
Eilenberg-MacLane space, 5 Elementary extension, 19 Extension of groups, 43 Exterior power, 120 Family of Riemann surfaces, 147 Fiber homotopy equivalence, 8 Fibering,3 Fibration, 3 Flat F-bundle, 78 Flat F-product bundle, 79 Flat G-bundle, 52 Flat G-product bundle, 66 Flat connection, 52 Foliated bundle, 89 Foliation, 88 Formal vector field, 107 Free (d.g.a.), 18 Fundamental cohomology class, 5 183
184
INDEX
Gel'fand-Fuks cohomology, 81 Generalized nilpotent (minimal d.g.a.), 20 Godbillon-Vey class, 92, 93 Godbillon-Vey invariant, 94 Godbillon-Vey number, 94 Gysin homomorphism, 148 Harer's stability theorem, 171 Hirsch extension, 19 Holonomy homomorphism of flat F-bundles, 79 of flat G-bundles, 55 Homotopy (of d.g.a. maps), 21 Homotopy fiber, 4 Homotopy group, 3 i-minimal model, 21 Integrability condition, 91 Integral manifold, 53 Integral symplectic group, 142 Integration along the fiber, 149 Interior product, 63 Intersection number, 140 Involutive (distribution), 54 Isotopy, 139 Iterated surface bundle, 163 Leaf,88 Linear foliation, 88 Localization at 0, 11 Loop space, 4 Lower central series (of groups), 43 MaJcev completion, 46 Mapping class group, 139 Mapping cone (of d.g.a. maps), 24 Maximal integral manifold, 53 Maximal tree, 129 rn-construction, 165 Minimal (d.g.a.), 19 Minimal model, 21 Moduli space of Riemann surfaces, 145 Monodromy homomorphism of flat F-bundles, 79 of flat G-bundles, 55 Mumford-Morita-Miller classes, 147 Naturality of characteristic classes
flat G-product bundles, 67 of F-bundJes, 136 of foliations, 91 Nielsen realization problem, 171 Nilpotent group, 43 minimal d.g.a., 20 topological space, 9 Nilpotent quotient, 43 Normal bundle (of foliations), 89 Oriented surface bundle, 137 Path space, 4 Postnikov invariant, 9 Principal fibration, 7 Properly discontinuous, 146 Ramification locus, 151 Ramified covering, 151 Rational homotopy type, 10, 11 Rational nilpotent completion, 46 Rational space, 10 Reeb foliation, 89 Relative cohomology group of d.g.a., 24 of Lie algebras, 63 Relative homotopy, 23 Siegel modular group, 142 Signature, 155 Simple (space), 10 Simplicial set, 124 Skeleton, 3 Strict connection, 78 Strictly flat F-bundle, 78 Subcomplex, 2 Submersion, 91 Surface bundle, 137 Symmetric power, 122 Symplectic basis, 140 Tangent bundle (of foliations), 89 Tangent bundle along the fiber, 137 Tangent frame bundle of order k, 102 Tangent frame bundles of higher orders, 101 Tautological classes, 147 Teichmiiller modular group, 139, 145
INDEX
Teichmiiller space, 145 Torelli group, 142 Torsion form, 101 Trivial flat bundle, 52 Unit tangent bundle, 174 Weak topology, 3
185
