Supersymmetry and equivariant de Rham theory

Victor W. Guillemin Shlomo Sternberg

Supersymmetry and Equivariant de Rham Theory

Preface This is the second volume of the Springer collection Mathematics Past and Present. In the first volume, we republished Hormander's fundamental papers Fourier integral operntors together with a brief introduction written from the perspective of 1991. The composition of the second volume is somewhat different: the two papers of Cartan which are reproduced here have a total length of less than thlrty pages, and the 220 page introduction which precedes them is intended not only as a commentary on these papers but as a textbook of its own, on a fascinating area of mathematics in which a lot of exciting innovatiops have occurred in the last few years. Thus, in this second volume the roles of the reprinted text and its commentary are reversed. The seminal ideas outlined in Cartan's two papers are taken as the point of departure for a full modern treatment of equivariant de Rham theory which does not yet . exist in the literature. We envisage that future volumes in this collection will represent both variants of the interplay between past and present mathematics: we will publish classical texts, still of vital interest, either reinterpreted against the background of fully developed theories or taken as the inspiration for original developments.

Contents Introduction 1 Equivariant Cohomology in Topology 1.1 Equivariant Cohomology via Classifying Bundles . . . . . . 1.2 Existence of Classifying Spaces . . . . . . . . . . . . . . . . 1.3 Bibliogaphical Notes for Chapter 1 . . . . . . . . . . . . . .

xiii

1 1 5 6

2 GY Modules

2.1 2.2 2.3

2.4 2.5 2.6 3 The 3.1 3.2 3.3 3.4 3.5 4

Differential-GeometricIdentities . . . . . . . . . . . . . . . . The Language of Superdgebra . . . . . . . . . . . . . . . . . From Geometry to Algebra . . . . . . . . . . . . . . . . . . . 2.3.1 Cohomology . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Acyclicity . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Chain Homotopies . . . . . . . . . . . . . . . . . . . 2.3.4 Free Actions and the Condition (C) . . . . . . . . . . 2.3.5 The Basic Subcomplex . . . . . . . . . . . . . . . . . Equivariant Cohomology of G* Algebras . . . . . . . . . . . The Equivariant de Rham Theorem . . . . . . . . . . . . . . Bibliographicd Notes for Chapter 2 . . . . . . . . . . . . . .

33 Weil Algebra The Koszul Complex . . . . . . . . . . . . . . . . . . . . . . 33 The Weil Algebra . . . . . . . . . . . . . . . . . . . . . . . . 34 Classifymg Maps . . . . . . . . . . . . . . . . . . . . . . . . 37 W* Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Bibliographicd Notes for Chapter 3 . . . . . . . . . . . . . . 40

The Weil Model and t h e Cartan Model 4.1 The Mathai-Quillen Isomorphism . . . . . . . . . . . . . . . 4.2 4.3 4.4

4.5 4.6

The Cartan Model . . . . . . . . . . . . . . . . . . . . . . . Equivariant Cohomology of W' Modules . . . . . . . . . . . H ((A @ E)b) does not gepend on E . . . . . . . . . . . . . The Characteristic Homomorphism . . . . . . . . . . . . . . Commuting Actions . . . . . . . . . . . . . . . . . . . . . . .

41 41 44 46 48 48 49

x

contents 4.7 4.8 4.9

contents

The Equivariant Cohomology of Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . . . Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . . Bibliographical Notes for Chapter 4 . . . . . . . . . . . . . .

5 Cartan's Formula 5.1 The Cartan Model for W *Modules 5.2 Cartan's Formula . . . . . . . . . . 5.3 Bibliographical Notes for Chapter 5

8.4

8.5

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

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

6 Spectral Sequences 6.1 Spectral Sequences of Do-yble Complexes . . . . . . . . . . . 6.2. The First Term . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The Long Exact Sequence . . . . . . . . . . . . . . . . . . . 6.4 Useful Facts for Doing Computations . . . . . . . . . . . . . 6.4.1 Functorial Behavior . . . . . . . . . . . . . . . . . . . 6.4.2 Gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Switching Rows and Columns . . . . . . . . . . . . . 6.5 The Cartan Model as a Double Complex . . . . . . . . . . . 6.6 HG(A) as an S(g*)G-Module . . . . . . . . . . . . . . . . . . 6.7 Morphisms of G* Modules . . . . . . . . . . . . . . . . . . . 6.8 Restricting the Group . . . . . . . . . . . . . . . . . . . . . . 6.9 Bibliographical Notes for Chapter 6 . . . . . . . . . . . . . .

61 61 66 67 68 68 68 69 69 71 71 72 75

7 Fermionic Integration 7.1 Definition and Elementary Propertie . . . . . . . . . . . . . 7.1.1 Integration by Parts . . . . . . . . . . . . . . . . . . 7.1.2 Change of Variables . . . . . . . . . . . . . . . . . . . 7.1.3 Gaussian Integrals . . . . . . . . . . . . . . . . . . . 7.1.4 Iterated Integrals . . . . . . . . . . . . . . . . . . . . 7.1.5 The Fourier Transform . . . . . . . . . . . . . . . . . 7.2 The Mathai-Quillen Construction . . . . . . . . . . . . . . . 7.3 The Fourier Transform of the Koszul Complex . . . . . . . . 7.4 Bibliographical Notes for Chapter 7 . . . . . . . . . . . . . .

77 77 78 78 79 80 81 85 88 92

8 Characteristic Classes 8.1 Vector Bundles . . . . . . . . . . . . . . . . . . . . : . . . . 8.2 The Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 G = C r ( n ) . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 G = O ( n ) . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 G = S 0 ( 2 n ) . . . . . . . . . . . . . . . . . . . . . . . 8.3 Relations Between the Invariants . . . . . . . . . . . . . . . 8.3.1 Restriction from U(n) to O(n) . . . . . . . . . . . . . . 8.3.2 Restriction from SO(2n) to U ( n ) . . . . . . . . . . . 8.3.3 Restriction from U(n) to U ( k ) x U(!) . . . . . . . . .

8.6 8.7

xi

Symplectic Vector Bundles . . . . . . . . . . . . . . . . . . . 101 8.4.1 Consistent Complex Structures . . . . . . . . . . . . 101 8.4.2 Characteristic Classes of Symplectic Vector Bundles . 103 Equivariant Characteristic Classes . . . . . . . . . . . . . . . 104 8.5.1 Equivariant Chern classes . . . . . . . . . . . . . . . 104 8.5.2 Equivariant Characteristic Classes of a Vector Bundle Over a Point . . . . . . . . . . . . . . 104 8.5.3 Equivariant Characteristic Classes as Fixed Point Data105 The Splitting Principle in Topology . . . . . . . . . . . . . . 106 Bibliographical Notes for Chapter 8 . . . . . . . . . . . . . . 108

9 Equivariant Symplectic Forms

111

Equivariantly Closed Two-Forms . . . . . . . . . . . . . . . The Case M = G . . . . . . . . . . . . . . . . . . . . . . . . Equivariantly Closed Two-Forms on Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . . . 9.4 The Compact Case . . . . . . . . . . . . . . . . . . . . . . . 9.5 Minimal Coupling . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Syrnplectic Reduction . . . . . . . . . . . . . . . . . . . . . . 9.7 . The Duistermaat-Heckman Theorem . . . . . . . . . . . . . 9.8 The Cohomology Ring of Reduced Spaces . . . . . . . . . . 9.8.1 Flag Manifolds . . . . . . . . . . . . . . . . . . . . . 9.8.2 Delzant Spaces . . . . . . . . . . . . . . . . . . . . . 9.8.3 Reduction: The Linear Case . . . . . . . . . . . . . . 9.9 Equivariant Duistermaat-Heckman . . . . . . . . . . . . . . 9.10 Group Valued Moment Maps . . . . . . . . . . . . . . . . . . 9.10.1 The Canonical Equivariant Closed Three-Form on G 9.10.2 The Exponential Map . . . . . . . . . . . . . . . . . 9.10.3 G-Valued Moment Maps on Hamiltonian G-Manifolds . . . . . . . . . . . . . . . . 9.10.4 Conjugacy Classes . . . . . . . . . . . . . . . . . . . 9.11 Bibliographical Notes for Chapter 9 . . . . . . . . . . . . . .

141 143 145

10 T h e Thorn Class a n d Localization 10.1 Fiber Integration of Equivariant Forms . . . . . . . . . . . . 10.2 The Equivariant Normal.Bundle . . . . . . . . . . . . . . . . 10.3 Modify~ngu . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Verifying that r is a Thom Form . . . . . . . . . . . . . . . . 10.5 The Thom Class and the Euler Class . . . . . . . . . . . . . 10.6 The Fiber Integral on Cohomology . . . . . . . . . . . . . . 10.7 Push-Forward in General . . . . . . . . . . . . . . . . . . . . 10.8 Loc&ation ........................... 10.9 The Localization for Torus Actions . . . . . . . . . . . . . . 10.10 Bibliographical Notes for Chapter 10 . . . . . . . . . . . . .

149 150 154 156 156 158 159 159 160 163 168

9.1 9.2 9.3

111 112 114 115 116 117 120 121 124 126 130 132 134 135 138

Contents

xii

11 The Abstract Localization Theorem 11.1 Relative Equivariant de Rham Theory . . . . . . . . . . . . 11.2 Mayer-Vietoris . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 S(g*)-Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 The Abstract Localization Theorem . . . . . . . . . . . . . . 11.5 The Chang-Skjelbred Theorem . . . . . . . . . . . . . . . . . 11.6 Some Consequences of Eguivariant Formality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Two Dimensional G-Manifolds . . . . . . . . . . . . . . . . . 11.8 A Theorem of Goresky-Kottwitz-MacPherson . . . . . . . . 11.9 Bibliographical Notes for Chapter 11 . . . . . . . . . . . . .

Introduction

Appendix 189 Notions d'algebre differentide; application aux groupes de Lie et aux variBtb oh opkre un groupe de Lie Henri Cartan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 La transgression dans un groupe de Lie et dans un espace fibr6 principal Henri Cartan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Bibliography

221

Index

227

The year 2000 will be the fiftieth anniversary of the publication of Hemi Cartan's two fundamental papers on equivariant De Rham theory "Notions d'algebre diffbrentielle; applications aux groupes de Lie et aux variettb oh o g r e un groupc?de Lie" and "La trangression dans un groupe de Lie et dans un espace fibr6 principal." The aim of this monograph is to give an updated account of the material contained in these papers and to describe a few of the more exciting developments that have occurred in this area in the five decades since their appearance. This "updating" is the work of many people: of Cartan himself, of Leray, Serre, Borel, Atiyah-Bott, Beriie-Vergne, Kirwan, ~athai-Quillen'andothers (in particular, as far as the contents of this manuscript are concerned, Hans Duistermaat, from whom we've borrowed our treatment of the Cartan isomorphism in Chapter 4, and Jaap Kallunan, whose Ph.D. thesis made us aware of the important role played by supersyrnmetry in this subject). As for these papers themselves, our efforts to Gpdate them have left us with a renewed admiration for the simplicity and elegance of Cartan's original exposition of this material. We predict they will be as timely in 2050 as they were fifty years ago and as they are today.

Throughout this monograph G will be a compact Lie group and g its Lie algebra. For the topologists, the equivariant cohomology of a G-space, M , is defined to be the ordinary cohomology of the space

the "E" in (0.1) being any contractible topological space on which G acts freely. We will review this definition in Chapter 1 and show that the cohcmology of the space (0.1) does not depend on the choice of E. If M is a finite-dimensional differentiable manifold there is an alternative way of defining the equivariant cohomology groups of M involving de Rham theory, and one of our goals in Chapters 2 - 4 will be to prove an equivariant

'

Contents

xii

11T h e 11.1 11.a 11.3 11.4 11.5 11.6

Abstract Localization Theorem hlative E q u i ~ i a n de t Rham Theory . . . . . . . . . . . . Mayer-Vietoris . . . . . . . . . . . . . . . . . . . . . . . . . . S(g*)-Modules . . . . . . . . . . . . . . . . . . . . . . . . . . The Abstract Localization Theorem . . . . . . . . . . . . . . The Chang-Skjelbred Theorem . . . . . . . . . . . . . . . . . Some Consequences of Equivariant ' Formality.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Two Dimensional G-Manifolds . . . . . . . . . . . . . . . . . 11.8 A Theorem of Goresky-Kottwitz-MacPherson . . . . . . . . 11.9 Bibliographical Notes for Chapter 11 . . . . . . . . . . . . .

173 173 175 175 176 179

a

Introduction

180 180 183 185

Appendix 189 Notions d'algkbre diffkrentielle; application aux groupes de Lie et aux va.riBt& ou o&re un groupe de Lie Henri Cartan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 La transgression dans un groupe de Lie et dans un espace fibr6 principal Henri Cartan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Bibliography

Index .

The year 2000 will be the fiftieth anniversary of the publication of Henri Cartan's two fundamental papers on equivariant De Rham theory "Notions d7alg&brediffbrentielle; applications aux groupes de Lie et aux variktk oh opkre un groupe de Lie" and "La trangression dans un groupe de Lie et dans un espace fibr6 principal." The aim of this monograph is to give an updated account of the material contained in these papers and to describe a few of the more exciting developments that have occurred in this area in the five decades since their appearance. This "updating" is the work of many people: of Cartan himself, of Leray, Serre, Borel, Atiyah-Bott, Berline-Vergne, Kirwan, Mathai-Quillen.and others (in particular, as far as the contents of this manuscript are concerned, Hans Duistermaat, from whom we've borrowed our treatment of the Cartan isomorphism in Chapter 4, and Jaap Kalkman, whose Ph.D. thesis made us aware of the important role played by supersymmetry in this subject). As for these papers themselves, our efforts to update them have left us with a renewed admiration for the simplicity and elegance of Cartan's original exposition of this material. We predict they will be as timely in 2050 a s they were fifty years ago and as they are today.

Throughout this monograph G will be a compact Lie group and g its Lie algebra. For the topologists, the equivariant cohomology of a G-space, M, is defined to be the ordinary cohomology of the space

(Mx E ) / G

(0.1)

the "E' in (0.1) being any contractible topological space on which G acts freely. We will review this definition in Chapter 1 and show that the cohomology of the space (0.1) does not depend on the choice of E. If M is a finite-dimensional differentigblemanifold there is an alternative way of defining the equivariant cohomology groups of M involving de Rham theory, and one of our goals in Chapters 2 - 4 will be to prove an equivariqt

xiv

Introduction

Introduction

version of the de Rham theorem, which asserts that these two definitions give the same answer. We will give a rough idea of how the proof of this goes:

en

1. Let ,tl, ... ,

be a basis of g. If M is a differentiable manifold and the action of G on M is a differentiable action, then to each 5, corresponds a vector field on M and this vector field acts on the de Rham complex, R(M), by an "interior product" operation, L,, and by,a ''Lie differentiation" operation, L,. These operations fit together to give a representation of the Lie superalgebra

,

xv

One has to check that it is independent of A, and one has to check that it gives the right answer: that the cohomology groups of the complex (0.5) are identical with the cohomology groups of the space (0.1). At the end of Chapter 2 we will show that the second statement is true provided that A is chosen appropriately: More explicitly, assume G is contained in U ( n ) and, for k > n let Ek be the set of orthonormal n-tuples, (vl, . . . ,v,), with v, E Ck. One has a sequence of inclusions:

-

and a sequence of pull-back maps R(Ek-l)

R(Ek) + R(Ek+l) + . . .

g-1 having L,,, a = 1,. . . ,n as basis, go having L,, a = 1,. . . ,n as basis and gl having the de Rham coboundary operator, d, as basis. The action of G on Q(M) plus the representation of j gives us an action on R(M) of the Lie supergroup, G*, whose underlying manifold is G and underlying algebra is J.

and we will show that if A is the inverse limit of this sequence, it satisfies the conditions (0.3), and with

Consider now the de Rham theoretic analogue of the product, M x E . One would like this to be the tensor product

the cohomology groups of the complex (0.5) are identical with the cohomology groups of the space (0.1).

however, it is unclear how to define R(E) since E has to be a contractible space on which G acts freely, and one can show such a space can not be a finite-dimensional manifold. We will show that a reasonable substitute for R(E) is a commutative graded superalgebra, A, equipped with a representation of G* and having the following properties: a It is acyclic with respect to d . b. There exist .elements 0* E A' satisfying L , B ~ = 6;.

(0.3)

(The first property is the de Rham theoretic substitute for the property "E is contractible" and the second for the property "G acts on E in a locally-free fashion".) Assuming such an A exists (about which we will have more to say below) we can take as our substitute for (0.2) the algebra R(M) @ A (0.4) As for the space (0.1), a suitable de Rham theoretic replacement is the complex (R(M) @ A)bas (0.5) of the basic elements of R(M) @ A, "basic" meaning G-invariant and annihilated by the L,'s. .Thus one is led to define the equivariant de Rharn cohomology, of M as the cohomology of the complex (0.5). There are, of course, two things that have to be checked about this definition.

0

.

-4.

(0.6)

E = lim Ek &

2. To show that the cohomology of the complex (0.5) is independent of A 'we will &st show that there is a much simpler candidate for A than the "A" defined by the inverse limit of (0.6). This is the Weil algebra

and in Chapter 3 we will show how to equip this algebra with a representation of G*,and show that this representation has properties (0.3), (a) and (b). Recall that the second of these two properties is the de Rham theoretic version of the property "G acts in locally kee fashion on a space E". We will show that there is a nice way to formulate this property in terms of W, and this will lead us to the important notion of W* module. Definition 0.0.1 A gmded vector space, A, is a W* module if it is both a W module and a G* module and the map

is a G module rnorphism.

3. Finally in Chapter 4 we will conclude our proof that the cohomology of the complex (0.5) is independent of A by deducing this from the following much stronger result. (See Theorem 4.3.1.)

Theorem 0.0.1 If A is a W' module and E an acyclic W* algebm the G* . modules A and A @ E have the same basic cohomology. (We will come back to another important implication of this theorem in $4 below.)

xvi

Introduction

Introduction

0.3 Since the cohomology of the complex (0.5) is independent of the choice of A, we can take A to be the algebra (0.7). This will give us the Wed model for computing the equivalent de Rham cohomology of M. In Chapter 4 we will show that this is equivalent to another model which, for computational purposes, is a lot more useful. For any ' G module, R, consider the tensor product @ (0.9) equipped with the operation

xa,a = 1,. . . ,n, being the basis of g* dual to Q, a = 1,. . . ,n. One can show that d2 = 0 on the set of invariant elements

making the space (0.11) into a cochain complex, and Cartan's theorem says that the cohomology of this complex is identical with the cohomology of the Weil model. In Chapter 4 we will give a proof of this fact based on ideas of Mathai-Quillen (with some refinements by K a h a n and ourselves). If 0 = R(M) the complex, (0.10) - (O.11), is called the Cartan model; and many authors nowadays take the cohomology groups of this complex to be, by definition, the equivariant cohomology groups of M. Ram this model one can deduce (sometimes with very little effort!) lots of interesting facts about the equivariant cohomology groups of manifolds. We'll content ourselves for the moment with mentioning one: the computation of the equivariant cohomology groups of a homogeneous space. Let K be a closed subgroup of G. Then HG(G/K) Z S(k*)K. (0.12) (Proof: Rom the Cartan model it is easy to read off the identifications

and it is also easy to see that the space on the far right is just S(k*)K.)

A fundamental observation of Bore1 [Bo] is that there exists an isomorphism Hc(M) g H(M/G)

(0.14)

provided G acts freely on M. In equivariant de Rham theory this iesult can easily be deduced from the theorem that we cited in Section 2 (Theorem 4.3.1

.

: L

i

xvii

in Chapter 4). However, there is an alternative proof of this result, due to Cartan, which involves a very beautiful generalization of Chern-Weil theory: If G acts freely on M one can think of M as being a principal Gbundle with base X = M/G (0.15)

and fiber mapping

5 C

Put a connection on this bundle and consider the map

which maps w @ xfl . ..x$ towho, @ p? . . .p? the p,s being the components of the curvature form with respect to the basis, &, . . .,5,, of g and uhor being the horizontal component of w . R(X) can be thought of as a subspace of R(M) via the embedding: R(X) -+ n*R(X); and one can show that the map (0.17) maps the Cartan complex (0.11) onto R(X). In f&t one can show that this map is a cochain map and that it induces an isomorphism on cohomology. Moreover, the restriction of this map to S(g*)G is, by definition, the ChernWeil homomorphism. (We will prove the assertions above in Chapter 5 and will show, in fact, that they are true with R(M) replaced by an arbitrary W* module.)

One important property of the .Cartan complex is that it can be regarded as a bi-complex with bigradation

and the coboundary operators

This means that one can use spectral sequence techniques to compute HG(M) (or, in fact, to compute HG(A), for any G* module, A). To avoid making "spectral sequences" a prerequisite for reading this monograph, we have included a brief review of this subject in §§ 6.1-6.4. (For simplicity we've coniined ourselves to discussing the theory of spectral sequences for bicomplexes since this is the only type of spectral sequence we'll encounter.) Applying this theory to the Cartan complex, we will show that there is a spectral sequence whose El term is H ( M ) @ S(g*)G and whose E, term is HG(M). Fkequently this spectral sequence collapses and when it does the (additive) equivariant cohomology of M is just

xviii

Introduction

We will also use spectral sequence techniques to deduce a number of other important facts about equivariant cohomology. For instance we will show that for any G* module, A, HG(A) Z H T ( A ) ~ (0.21) T being the Cartan subgroup of G and W the corresponding Weyl group. We will also describe one nice topological application of (0.21): the "splitting principlen for complex vector bundles. (See [BT] page 275.)

Introduction

xix

Let A be a commutative G algebra containing C. From the inclusion of C into A one gets a map on cohomology

'

and hence, since HG(C) = S(g*)G,a generalized Chern-Weil map:

The elements in the image of this map are defined to be the "generalized characteristic classes" of A. If K is a closed subgroup of G there is a natural restriction mapping HG(A) HK(A) (0.26) and under this mapping, G-characteristic classes go into K-characteristic classes. In Chapter 8 we will describe these maps in detail for the classical compact groups U ( n ) ,O(n) and SO(n) and certain of their subgroups. Of particular importance for us will be the characteristic class associated with the element, "Pfaff', in S(g')G for G = SO(2n). (This will play a .pivotal role in the localization theorem which we'll describe below.) Specializing to vector bundles we will describe how to define the Pontryagin classes of an oriented manifold and the Chern classes of an almost complex (or symplectic) manifold, and, if M is a G-manifold, the equivariant counterparts of these classes.

The first half of this monograph (consisting of the sections we've just described) is basically an exegesis of Cartan's two seminal papers from 1950 on equivariant de Rharn theory. In the second half we'll discuss a few of the post-1950 developments in this area. The first of these will be the MathaiQuillen construction of a "universal" equivariant Thom form: Let V be a d-dimensional vector space and p a representation of G on V. We assume that p leaves fixed a volume form, vol, and a positive definite quadratic form l l ~ 1 1 ~Let . S# be the space of functions on V of the form, e-ll"la/2p(v), p(v) being a polynomial. In Chapter 7 we will compute the equivariant cohomology groups of the de Rham complex

+

and will show that H;(R(V),) is a free S(g*)-modulewith a single generator of degree d. We will also exhibit an example of an equivariantly closed dform, u, with [Y]# 0. (This is the universal Thom form that we referred to above.) The basic ingredient in our computation is the Fermionic Fourier transform. This transform maps A(V) into A(V*) and is defined, l i e the ordinary Fourier transform, by the formula

Let M be a G-manifold and w E R2(M) a G-invariant symplectic form. A moment map is a G-equivariant map.

.tD1,.. . ,$d being a basis of A'(V), TI,.. . , r k the dual basis of A'(V*),

with the property that for all [ E g

being an element of A(V), i.e., a "function" of the anti-commuting variables +I, . . . ,.tDd, and the integral being the "Berezin integral": the pairing of the integrand with the d-form vol E A ~ ( V * ) .Combining this with the usual Bosonic Fourier transform one gets a super-Fourier transform which transforms R(V), into the Koszul complex, S(V) @ A(V), and the Mathai-Quillen form into the standard generator of H$ (Koszul). The inverse Fourier transform then gives one an explicit formula for the Mathai-Quillen form itself. Using the super-analogue of the fact that the restriction of the Fourier transform of a function to the origin is the integral of the funytion, we will get from this computation an explicit expression for the lLuniversal"Euler class: the restriction of the universal Thom form to the origin.

qjc being the ( component of 4. Let k.

Let

& = &(")

Proposition 2.5.3 R(E) is acyclic. Proof. By Proposition 2.5.2, (and using, say singular homology and cohe mology) H ~ ( E= ~ 0, ) e >o

denote the set of all orthonormal n-tuples v . = (vl, ... ,vn) with vi E Coo. For k > n let Ek be the set of all orthonormal n-tuples with v, E Ck. From the inclusion

if k >> e is sufficiently large. Thus, by the usual de Rham theorem, if p E R4(E) is closed, then pk := j;p = dvk

one gets inclusions

for k sufficiently large. We claim that we can choose the vk consistently, i.e. such that Vk = i;vk+1.

ik : Ek -' Ek+1 and, by composing them, inclusions

which compose consistently and give rise to inclusions

We use these inclusions to put the "final topology" on & (using the terminology of Bourbaki, [Bour] 1-2: A set U C E is declared to be open if and only if each of the subsets jcl(U) is open.) As a consequence, a series of points converges if and only if there exists some k such that all the points lie in Ek and the sequence converges thkre. In particular, a continuous map, f of a compact space X into E satisfies f (X) C Ek for some k.

Indeed for any choice of vk+l we have

Choose an (C - 2)-form 0 on Ek+1 such that iZ.0 = A. Replacing vk+l by vk+, - dp gives us a consistent choice, and proceeding inductively we get a consistent choice for all large k. Hence we can h d a v E R(E) with j;v = vk and dv = p. U(n) acts freely on E by the action (1.9), and this induces an action of U ( n )on R(E). So we can apply (1.3) to conclude that HE(M) = H* ((M x E)/G). Proposition 2.5.4 O(E) satisfies property (C).

(2.49)

30

Chapter 2. G* Modules 2.6 Bibliographical Notes for Chapter 2

Proof. Let zij be the functions defined on E by setting z,,(v) = ith coordinate of the vector vj where v = (vl, . .. ,v,) and let Z be the matrix with entries zij. So Z has only finitely many non-zero entries when evaluated on any Ek. We may thus form the matrix

which is an element of R(E) from whose components we get 0's with the property (C). Let R(M x E) be the inverse limit of the sequence

We claim that

H*((M x E)/G) = Hbas (R(M x E ) ) .

(2.50)

Proof. This follows from the fact that for each i the two sequences

and 0

.

.

+ H"(R(M

X

&)bas) + H ((R(M x Ek+l)bas)

+

...

are termwise isomorphic.

2.6

31

Bibliographical Notes for Chapter 2

1. For a more detailed exposition of the super ideas discussed in section 2.2 see Berezin [Be], Kostant [Kol] or Quillen [Qu].

2. The term "Gmodule" is due to us; however the notion of G' module is due to Cartan. (See 'LNotionsd'alg6bre diffkrentielles, ..." page 20, lines 15-20.) 3. In this monograph the two most important examples which we will encounter of G* modules are the de Rham complex, R(M), and the equivariant de Rham complex, RK(M) (which we'll encounter in Chapter 4. If M is a (G x K)-manifold this complex is a G' module). From these two examples one gets many refinements: e.g., the complex of compactly supported de Rham forms, the complex of de Rham currents, the relative de Rham complex associated with a G-mapping f : X + Y (see [BT] page 78), inverse and direct limits of de Rham complexes (an example of which is the complex R(E), in §2.5), the Weil complex (see Chapter 3), the Mathai-Quillen complex (see Chapter 7), the universal enveloping algebra of the Lie superalgebra, g,. . .. 4. The subalgebra g-1 CBgo of 3 is the tensor product of the Lie algebra g and the commutative superalgebra, C[x], generated by an element, I, of degree -1. The representation theory of the Lie superalgebra

Proposition 2.5.5 The inclusion map with generators XI,.. . ,x, of arbitrary degree, has been studied in detail by Cheng, (See [Ch]). induces an isomorphism on whomology:

Proof. By a spectral sequence argument (see Theorem 6.7.1) it is enough to see that the inclusion induces an isomorphism

H ( Q ( M )8 R(E))

-+

5. Another interesting representation of a Lie superalgebra on R(M) occurs in some recent work of Olivier Mathieu: Let M be a compact sympiectic manifold of dimension 2n with symplectic form w. Since w is a non-degenerate bilinear form on the tangent bundle of M, it can be used to define a Hodge star operator

H(R(M x E))

on ordinary cohomology. But the acyclicity of R(E) and the contractibility of E imply that this map is just the identity map of H*(M) into itself. Since a ( & ) is a G* algebra which is acyclic and has property (C), we conclude that

Let be the operator, E p = w A p. Let

HG(Q(M)) = H (R(M x &)bas)

and hence that

be the operator - * E*. Let

1

H G ( ~ ( M )=) HE(M).

32

Chapter 2. G* Modules

be the operator ( - I ) ~* d* and let

be the operator (n-k). identity. These operators define a representation on R(M) of the simple five-dimensional Lie superalgebra

Chapter 3 d E g' and E € g2, with generators, F E g-', 6 E g-l, H E and relations: [E,F] = H, [H,F] = -2F,[H, El = 2E, [F,dl = 6 and [E,b] = -d. R o m the existence of this representation Mathieu [Mat] deduces some fascinating facts about symplectic Hodge theory: For instance M is said to satisfy the Brylinski condition if every cohomology class admits a harmonic representative. Mathieu proves that M satisfies this condition if and only if the strong Lefshetz theorem holds: i.e., iff the map E~ : E F k ( M ) P + & ( M )

-

is bijective. See [Mat]. 6. The fact that, in equivariant de Rham theory, there is no way to differentiate between free G-actions and actions which are only locally free has positive, as well as negative, implications: The class of manifolds for which R(M) satisfies condition (C) includes not only principal Gbundles but many other interesting examples besides (for instance, in symplectic geometry, the non-critical level sets of moment mappings!)

7. Atiyah and Bott sketch an alternative proof of the equivariant de Ftham theorem in Section 4 of [AB]. One of the basic ingredients in their proof is the "Weil modeln of which we will have much more to say in the next two chapters.

The Weil Algebra 3.1

The Koszul Complex

Let V be an n-dimensional vector space, and let A = A(V) be the exterior algebra of V considered as a commutative superalgebra, and let S = S(V) be the symmetric algebra considered as an algebra all of whose elements are even: So we assign to each element of AV its exterior degree, but each element of Sk(V) is assigned the degree 2k. The Koszul algebra is the tensor product A 8 S. The elements x 8 1 E A' 8 So and 18 x E A' 8 S1 generate A 8 S. The . Koszul operator d ~ is defined as the derivation extending the operator on generators given by

Clearly d$ = 0 on generators, and hence everywhere, since dZK is a derivation. We can also use this same argument, and the fact that the commutator of two odd derivations is an even derivation, to prove that the Koszul operator is acyclic. Indeed, let Q be the derivation defined on generators by

So Q' = 0 and [ Q , ~ K=] id on generators. But since [ Q , d K ] is an even derivation, we conclude that

Thus the only cohomology of d ~ lies in A' 8 9,which is the field of scalars. In fact, the cohomology is the field of scalars, since d K 1 = 0. It will be convenient for us to write all of this in terms of a basis. Let x ' , ..., xn be a basis of V and define

32

Chapter 2. G' Modules

be the operator (-l)k

* d*

and let

be the operator (n-k) identity. These operators define a representation on R(M) of the simple five-dimensional Lie superalgebra

*

Chapter 3 d E g1 and E E g2, with generators, F E g-2, b E g-l, H E and relations: [E, F] = H , [H,F] = -2F, [H,E] = 2E, [F, d] = 6 and [E,bj = -d. From the existence of this representation Mathieu [Mat] deduces some fascinating facts about symplectic Hodge theory: For instance M is said to satisfy the Brylinski andition if every cohomology class admits a harmonic representative. Mathieu proves that M satisfies this condition if and only if the strong Lefshetz theorem holds: i.e., iff the map E~ : P - ~ ( M )+ H"+~(M) is bijective. See [Mat].

6. The fact that, in equivariant de Rharn theory, there is no way to differentiate between free G-actions and actions which are only locally free has positive, as well as negative, implications: The class of manifolds for which R(M) satisfies condition (C) includes not only principal Gbundles but many other interesting examples besides (for instance, in symplectic geometry, the non-critical level sets of moment mappings!)

7. Atiyah and Bott sketch an alternative proof of the equivariant de Rham theorem in Section 4 of [AB]. One of the basic ingredients in their proof is the "Weil model" of which we will have much more to say in the next two chapters.

The Weil Algebra 3.1

The Koszul Complex

Let V be an wdimensional vector space, and let A = A(V) be the exterior algebra of V considered as a commutative superalgebra, and let S = S(V) be the symmetric algebra considered as an algebra all of whose elements are even: So we assign to each element of AV its exterior degree, but each element of Sk(V) is assigned the degree 2k. The Koszul algebra is the tensor product A 8 S. The elements x 8 1 f A' 8 So and 18 x E A' 8 S' generate A 8 S. The Koszul operator dK is defined as the derivation extending the operator on generators given by

Clearly d$ = 0 on generators, and hence everywhere, since d2K is a derivation. We can also use this same argument, and the fact that the commutator of two odd derivations is an even derivation, to prove that the Koszul operator is acyclic. Indeed, let Q be the derivation detined on generators by

So Q2 = 0 and [ Q , ~ K= ] id on generators. But since [Q,dK] is an even derivation, we conclude that

which is the field of scalars. Thus the only cohomology of d K lies in AO 8 9, In fact, the cohomology is the field of scalars, since dK1 = 0. It will be convenient for us to write all of this in terms of a basis. Let x', .. . ,xn be a basis of V and define

34

Chapter 3. The Weil Algebra

3.2 The Weil Algebra

and 2'

35

Theorem 3.2.1 W is an acyclic G' algebm satisfying condition (C).

:= 1 g x t .

Then the Koszul operator d = d~ is expressed in terms of these generators as dB' = z' (3.3)

We recall that Whnris defined to be the set of all elements in W satisfying (2.39). We claim that W = ~ ( g *8) Whor. (3.9)

and

Proof of (3.9). Define dr' = 0.

and the operator Q is given by Then

3.2

The Weil Algebra

The Weil algebra is just the Koszul algebra of g*: by (3.8). So the pb are horizontal elements of W. Moreover, The group G acts on g via the adjoint representation, hence acts on g* via the contragredient to the adjoint representation (the coadjoint representation) and hence acts as superalgebra a u t o m o r p h i i on W. A choice of basis, El,. . .,En of g induces a dual basis of g* and hence generators, e l , . . . ,P ,z i , . . .zn which satisfy

so we can use the Ba and pb as generators of W . So W is the tensor product of the exterior algebr.a in the 0 and the polynomial ring C [ p l , . . . ,pn], and it is clear that an element in this decomposition is horizontal if and only if it lies in C [ p l , . . . ,pnl, i.e.

and Lazb = - c ; ~ z ~ ;

(3.7)

The Koszul diierential d = dK is clearly G-equivariant. This means that we have an action of the go e g l part of j on W. We would like to define the action of g-,, i.e. prescribe the operators L,, SO as t o get a G* action which is acyclic and has property (C) with the .Qb as connection elements. Recall that if G is connected, as we shall assume, property (C) means that, in addition to (3.6), the elements 8' satisfy (2.38). So we define the action of L, on the eb by (2.38), i.e. L,O~

=.5;'

This completes the proof of (3.9) Identifying Cb',.. .,p"] with the polynomial ring S(g*) and recalling that Wb, are the G-invariant elements of Whorwe obtain:

We obtained the element pb by adding the term ac$$jSk to zb. fiom the definition of the structure constants, the element

Since d ~ + , ~ , d = La we are forced to have is precisely the map of g 8 g -. g given by Lie bracket. Hence the (oldfashioned) Jacobi identity for g can be expressed as So we use (2.38) and (3.8) to define the action of L, of generators, and extend as derivations to all of W. To check that we get an action of j on W, we need only check that the conditions (2.17)-(2.22) hold on generators, which we have arranged to be true. We have proved:

, Therefore Lapb= -ctkPk.

36

Chapter 3. The Weil Algebra

3.3 Classifying Maps

We will now show that the operator d acts trivially on Wb-. For this purpose, we first compute dpa: We have, by (3.3) and (3.10),

If we apply d we obtain

where the remaining terms cancel by Jacobi's identity. Thus

37

and pa (and extended so as to be a derivation). One must do some work to then prove that d2 = 0 and that the Weil algebra is acyclic. The advantage of using supersymmetric methods such as the "change of variables" (3.10) is that these facts are immediate consequences of the existence and acyclicity of the Koszul algebra. We will see another illustration of the power of this technique when we come to the Mathai-Quillen isomorphism in Chapter 4. There is an important interpretation of the operator d which is natural from the point of view of the standard treatment. (We will not use any of the following discussion in the rest of the book): We may think of Whor= S(gW)as a g-algebra, that is as an algebra which is a 9-module with g acting as derivations. Then we may use the Chevalley-Eilenberg prescription for computing the Lie algebra cohomology of S(g*) where the complex is taken ~ on generators by to be ~ ( g *8) S(g*) with differential operator d c given

Combining (3.15) with (3.13) we can rewrite (3.15) as Then (3.14) and (3.15) say that

Now a derivation followed by a multiplication is again a derivation, so the operator ebLbis (an odd) derivation as is d. Since the pa generate Who,we conclude that dw = e b l b w v w E who,. In particular, if w E Wbas=

Law = 0, a = 1,. . . ,n. Hence

To summarize, we have proved: Theorem 3.2.2 The basic cohomology ring of W is S(g')G. Equation (3.14) is known as the C a r t a n structure equation and equation (3.15) is known as the Bianchi identity for the Weil algebra. ' In the usual treatments of the Weil algebra, (3.9) is taken as the definition of the Weii algebra, where Who=is defined as in (3.11), that is

Of course, the S(g*) occurring in this version is different from our original S(g'); it is obtained from it by the supersymmetric "change of variables" (3.10). With the generators 19" and pa, the action of 9-1 is defined to be L,B~ = 6: and ~ , = p 0, ~ and the action of go is defined by (3.6) on the Ba and by (3.13) on the pa The linear space spanned by the Ba is just a copy of g* (which generates the subalgebra r\(g*)) and the linear space spanned by the pa is a copy of g* which generates WhorZ S(g*). Both (3.6) and (3.13) describe the standard, coadjoint, action of g on g*. The action of d in the standard treatments is defined to be (3.14) and (3.15) on the generators Ba

where

.

d ~ 8 := a pa,

dKpa = 0.

So we may think of ~ ( g *8) Who,as a copy of the Koszul complex with opera. tor dK. The net effect of the supersymmetric change of variables (3.10) going from our original A(g*) 8 S(g*) to A(g*) 8 Whoris to introduce Lie algebra cohomology into the picture by adding the Chevalley-Eilenberg operator.

3.3

Classifying Maps

In this section we wish to establish the algebraic analogue of Proposition 1.1.1. Recall that this proposition asserts that if G acts freely on a topological space X, and if E is a classifying space for G then there exists a G-equivariant map h : X + E (uniquely constructed up to homotopy) and hence a canonical map f': H(E/G) Hg(X/G).

-

In our algebraic analogue (where arrows are reversed) W will play the role of the classifying space: Let A be a G algebra of type (C). We claim that Theorem 3.3.1 There exists a G* algebra homomorphism p : W two such are chain homotopic.

4

A. Any

Proof. .If we choose R$ E A1 satisfying (2.38) and (2.40), then the map

38

Chapter 3. The Weil Algebra

3.4 W* Modules

extends uniquely to a G* homomorphism, since W ( g )is freely generated as an algebra by the 6)" and dBO. This establishes the existence. If po and pl are two such homomorphiims, define pt, 0 5 t 5 1 by first defining pt :Wl --, A by p t ( w = (1 - t)po(ea)+ t p 1 ( p )

i.e. pt = (1

- t)po + tp1

39

and

~ ~ = e-c:~%. ; Indeed, since La@; = [d, ',lob = ~,deb and

on W1.

algebra homomorphism which we shall also denote by h. Let Q, be the pt chain homotopy defined by

we conclude that dBA differs from - g c ~ , P ~ % by an element of degree two which is horizontal, and which we could define as p i and so get (3.18). Equation (3.19) then follows from (3.18) by applying d and using the Jacobi identity as we did for the case of the Weil algebra. Theorem 3.3.1 can be thought of as saying that the Weil algebra is the simplest G* algebra satisfying condition ( C ) .

on our generators, and extended by (2.37) (with 4 = p,). It clearly satisfies (2.32) on these generators, and so is a chain homotopy relative to pt. Then

3.4

This map satisfies p t ~ c= L C P ~ and is G-equivariant. So it extends to a G'

is the desired chain homotopy between po and pl. Since p is a G* morphism, it maps Wb,into Ah and hence the basic cohomology ring of W into the basic cohomology ring of A. Moreover, since p is unique up to chain homotopy, this map does not depend on p. Hence, by Theorem 3.2.2, we have proved

Theorem 3.3.2 Them. is a canonical map

W* Modules

If A is a G* algebra satisfying condition (C), and if we have chosen connection elements, 01, then the homomorphism p : W + A makes A into an algebra over W; in particular, into a module over W. We want to generalize this notion slightly. To see why, consider the case of a compact Lie group, G acting freely on a non-compact manifold M. We can construct the connection forms e'& E R ( M ) which satisfy (2.38). These forms will not vanish anywhere, in particular do not have compact support. But we may want to consider the algebra, R ( M ) o , of compactly supported forms on M. This algebra does not satisfy condition ( C ) , but we can multiply any element of R(M)o by any of the 0% to get an element of Q(M)o. In other words, Q(M)o is a module over W even though there is no G* homomorphism of W into R(M)o. Armed with this motivation we make the following

We shall call this map the Chern-Weil m a p or characteristic homomorphism. For a slightly different version of it see Section 4.5. Suppose we have chosen the "connection elements7' 5 E A satisfying (2.38) and hence the homomorphism p : W --+ A of the We2 algebra into A with p(Oa) = 85. Define PI := &)(pa).

is a morphism of G modules. A W* algebra is a

Since p is a G*.morphiim, (3.14) and (3.15) imply

module.

Definition 3.4.1 A W' module is a G* module B which is also a module over W in such a way that the map

G algebm which is a W*

Recall that Bhor denotes the set of elements of B which satisfy (2.39). For each mufti-index

These are known as the Cartan equations and the Bianchi identity, or more simply as the Cartan structure equations for A. We could have derived them directly from the defining equations (2.38):

let

.

@I = ,g*~ . .@-

denote the corresponding monomial in the 8'. Since each Ba acts as an operator on B, the monomials 0' act as operators on B.

40

Chapter 3 . The Weil Algebra

Theorem 3.4.1 Every element of a W' module B can be written uniquely as a sum eJh,

Proof. We will prove the following lemma inductively:

Lemma 3.4.1 Every element

of

B can be written uniquely as

a

Chapter 4

sum

eJhj

The Weil Model and the Cartan Model

where J = ( j . . j )

15 j l < . + . < j m I k - 1

and

i,hJ=O,

a=l,

..., k - 1 .

The case k = 1 of the lemma says nothing and hence is automatically true. The case k = n + 1is our theorem. So we assume that the lemma is true for k - 1 and prove it for k. Let

The results of the last chapter suggest that, for any G module B we take B@ W as an algebraic model for the X x E of Chapter 1, and hence Hb, (B @I W) as a definition of the equivariant cohomology of B. In fact, one of the purposes of this chapter will be to justify this definition. However the computation of (38 W)bas is complicated. So we will begin with a theorem of Mathai and Quillen which shows how to find an automorphism of B @ W which simplifies this computation. For technical reasons we will work with W @ B instead of B 8 W and replace W by an arbitrary W* module.

Then Identifying A(@', . ..,en) with ~ ( g *c) W, Theorem 3.4.1 says that the map ~ ( g *8 ) B~~~+ B, el.@h I+ elh (3.20) is bijective.

3.5

4.1

Let A be W* module and let B be a G* module. Let

Bibliographical Notes for Chapter 3

1. Sections 3.1-3.3 are basically just an exposition of Weil's version of Chern-Weil theory. The first account of this theory to appear in print

be connection and curvature generators of the Weil algebra corresponding to a choice of basis, 51,. . .,tn,of g. We define the degree zero endomorphism, 7 E End(A 8 B) by y := 80 8 L,. (4.1)

is contained in Cartan's paper: "Notions d'alghbre diffkrentielle, ...". 2. One important G* module to which this theory applies is the equivariant de Rham complex RK(M), K being a (not necessarily compact) Lie group. If M is a (G x K)-manifold on which G ads freely, there is a Chern-Weil map

whose image- is the ring of equivariant characteristic classes of M / F . (We will discuss (3.21) in more detail in Section 4.6.)

The Mathai-Quillen Isomorphism

.

Notice that its definition is independent of the choice of b&is. It is also invariant under the conjugation action of G. It is nilpotent; indeed -yn+' = 0 since every term in its expansion involves the application of n 0. So q5 E Aut ( A@ B) given by

+ 1 factors of

42

Chapter 4. The Weii Model and the Cartan Model 4.1 The Mathai-Quillen Isomorphism

is a finite sum. The automorphism q5 is known as the Mathai-Quillen isomorphism. It is an automorphism of G-modules. For any p E End(A 8 B) we define

43

as we have checked by applying both sides to z 8 y. This is just our usual rule: moving the ,O past the y costs a sign; this time in the context of the tensor product of two algebras. So we can write the above argument more succinctly as

as usual. Notice that wery term of

vanishes so ad y is nilpotent and we have

as this relation, exp ad = Ad exp, is true in any algebra of endomorphisms when the series on both sides converge. We will now compute six instances of (ad y)k/3: ad-Y(Lb@ 1 ) = -1 €3 Lb a d 7 ( ~ @ ~= ~ 0) V V E A

Proof of (4.5). Suppose that v E A,,

(4-6)

+

ad ~ ( d )= -dBa @ L, P @ L, (ad ~ ) ~ ( d=) - c ~ ~ B "@BLk~ ( a d ~ ) ~= d 0.

(4.7) (4.8) (4-9)

* = (- l)m. Then

Y ( v @ L ~ )= f@V@Lalb

(4-4) (4.5)

(adr)2(y@l) = 0

and let

(V 8 L ~ ) Y =

-uOa @ L ~ L ,

=

T L ~6 ~3 LV~ L ,

i.e. [Y, V 8 ~ b ] . =*gay @ [La,Lb] = 0. 0

Proof of (4.6). Equation (4.6) is an immediate consequence of (4.4) and (4.5).

Proof of (4.4). For x E A k , y E Bm we have Proof of (4.7). The d occurring in the left hand side of (4.7) is d @ 1 ( ~ O ( . h @ l ) ) ( x @ y= ) Y(L~~@Y) = (Oa @ L,)(L~x€3 y ) = ( - i ) k - l ( ~ ~€3bLay) ~

We have b " ' @ ~ ~ , d= @- ~[ ] ~ ~ , 4 = -d@' @L,

while [ g a @ L a 7 l @ d=] B " @ [ L , , ~ I= e a & ~ , .

while

Proof of (4.8). By (4.5) we have ad 7(dOa @ L,) = 0

Subtracting the second result from the iirst gives (4.4). Of course, the role of the x 8 y in the above argument is just a crutch to remind us of the multiplication rule in

namely ( 0 € 3 b ) ( ~ @ 6=) ( - 1 ) W ~ € 3 @ bif deg

= p , deg y = q

so

by (4.7). We have

(add2d=

@ La)

+i

d.

44

Chapter 4. The Weil Model aqd the Cartan Model

4.2 The Cartan Model

Proof of (4.9). This follows from (4.5) and (4.8). S i e y is invariant under conjugation by G we know that

45

and, according to (3.16), d = d w restricted to this subspace is

[.y,La@1+1@La]=O.

It is an instructive exercise for the reader to prove thii result by the above methods. We now obtain the following theorem of Mathai-Quillen [MQ] and Kalkman [Ka].

-

According to (4.11), 4 conjugates d = dw @ 1 + 1 8 dB into BaLa@l+ l @ d s + P @ L a - p a @ ~ a= (P@1)(La@l+18La)+l@dB-Pa@La on W h o =@ B. The space (W @ B)basis just the space of invariant elements of ( W @ B)har. Since 4 is G-equivariant, it carries invariant elements into invariant elements and hence

Theorem 4.1.1 The Mathai-Quillen isomorphism satisfies 4 ( 1 @ ~ ~ + ~ ~ @ 1 )=@L- C l @ ~Vt'EEg

(4.10)

The operator La €3 1

+ 1 @ La vanishes on invariant elements and hence M4-'

and

If A is a W* algebm and B is a G* algebm Men 4 is an algebm automorphism. Proof of (4.10). Applying (4.3), (4.4) and (4.6) to side of (4.10) is

E

= &,,the left-hand

= 1 @ d g pa@^,

on ( S ( g 9 )@I B)G. For any Gf module B the space C G ( B ) := ( S ( g * )@ B ) ~ together with the differential is called 'the C a r t a n model for the equivariant cohomology of B . We can think of an element w E CG ( B )as being an equivariant polynomial map from g to B. With this interpretation, the element ( p a @ L,)W is the map

Proof of (4.11). Apply (4.3) and (4.7),(4.8) and (4.9). The left-hand side of (4.11) becomes

E +, LCW(O. If w is a homogeneous polynomial map then E degree one higher. So if

+,

L < W ( has ~ ) polynomial

w E Sk((g*)@ At then pa L,W E sk+l and the total degree 2k t is increased by one. From the point of view of polynomial maps the differential operator dG is given by

+

The last statement in the theorem follows from the fact that y is a derivation. This is true because in any algebra a derivation followed by multiplication by an element is again a derivation.

4.2

The Cartan Model

Equation (4.10) implies that 4 carries ( A @ B)hor,the horizontal subspace of A 8 B, into Ahor8 B:

Let us apply this to the case A = W. Then we may apply (3.11) which says that Whor = c[/.L1,. . . ,pn] S ( g * )

d c ( w ) ( Q = d s [ w ( O l - LC[W(QI (4.17) and is of degree +1. To summarize, we have proved the following fundamental theorem of Cartan: Theorem 4.2.1 The map 4 carries (W @ B)bas into C G ( B ) and carries Me restriction of d = d w 8 1 1 @ d g into dG. Thus

+

The cohomology on the left is called the Weil model for the equivariant cohomology of B. We will justify its definition a little later on in this chapter by showing that H 8 ( ( E8 B)bas, d ) ) is the same for any choice of acyclic W* algebra, in particular E = W. So the thrust of Cartan's theorem is to say that the Cartan model gives the same cohomology as the Weil model.

Chapter 4. .The Weil Model and the Cartan Model

46

4.3

Equivariant Cohomology of W*Modules

In this section we assume that the group G is compact. As we pointed out in Chapter 1, a key property of equivariant cohomology for topological spaces is the identity

4.3 Equivariant Cohomology of W* Modules

ei,

Since C c if T E C k , k > 0 satisfies d r = 0,we can find a E Ck-' with ( 1 8 d E ) c = T . Averaging o over G, we may assume that it is Ginvaxiant, i.e. lies in Ck-'. Let

HE(M] = H * ( M / G ) if

M is a topological space on which G acts freely.

isj

so that

The main result of this section is an algebraic analogue:

Theorem 4.3.1 Let A be a W* module and E an ayclzc W* algebra. Then

47

cCj+l,

cj

(Attor8~)G=C*=U~j G

gives an increasing filtration of (Ahor8 E ) with Co = Abas. To prove Theorem 4.3.1 it is enough to prove

L e m m a 4.3.2 Ifp

E Cj

satisfies

We will prove this theorem using the Mathai-Quillen isomorphism, Theorem 4.1.1, taking B = E, an acyclic w* algebra. So . then there is a Y E Cj-1 and an a E Ab, with transforming the restriction of d to (A 8 E)bas into Moreover a is unique up to a coboundary, i.e. if

j~

= 0 in (4.23) then

where 61 := 1 8 d E

Proof (by induction on j): Suppose j = 0. Then

and 62:=d~@l+e08L,-pa8h. by Lemma 4.3.1. If

Define

Y

E Co satisfies

C ' : = ( A h o r @ ~ ) ~C ,* : = C O @ c 1 @ so that

6 1 : C ' + ~ ' + 1 and

62:C"-~C'QjC-1,

then 6 1 v = 0 so

Y

= - b @ 1, b E A ~ ,and

and 6; = 0. We can consider dl as a differential on the complex C'. We claim that

Lemma 4.3.1 The cohomology groups of ( C * ,6 1 ) are given by

Proof. Let

C := Ahor @ Et, Since E is acyclic we have 1

Now assume that j > 0 and we know the lemma for j - 1. Let p E C j with 6 p = 0, and write p = p, + . . . where p, E C J and the - . . lies in C,-l. Then G1p, = 0 so p = 61vj-l by Lemma 4.3.1. So

t..:= @Z.i. Since 6 p = 0, we have bw = 0. We now may apply the inductive hypothesis tow. a b The proof given above establishes an isomorphism between Hb,(A) and H G ( A ) in the case that A is a W* module. The isomorphism might appear

Chapter 4. The Weil Model and the Cartan Model

48

4.6 Commuting Actions

to depend on the actual structure of A as a W * module, and not merely on its structure as a G* module. However an analysis of the proof will show that this isomorphism depends only on the G* structure. Thii will become even clearer in the next chapter when we examine the proof of Theorem 4.3.1 from the point of view of the Cartan model. Let

The map commutes with d and hence induces a homomorphism on cohomology. We will show that i, which depends only on the G structure, induces an isomorphism on cohomology by writing down a homotopy inverse for i. See Equations (5.9), (5.10), and (5.11) below.

49

classes of HG(A). In the case that A = Q(M) where M i s a manifold, it has the following alternative description: The unique map

M -+ pt. of M onto the unique, connected, zero-dimensional manifold, pt., induces, by functoriality, a map HE(pt.) -+ HG(M). Hence, if G acts freely on M , a map

Since

4.4

H ((A @ E)bas)does not depend on E

Let E and F be two acyclic W * algebras. Then A @ F is a W' module and

H&(P~.) = Hc(Q(pt.)) = HG(C) = S(g*)G, this identifies our map KG as a map

SO

H{,(A@F@E) =Hbf,(A@F) by Theorem 4.3.1. Interchanging the role of E and F shows that H{,(A@E)

= H{,(A@F).

(4.24)

This is the usual Chern-Weil map. We will discuss the structure of S(g*)G for various important groups G in Chapter 8. This will then yield a description of the msre familiar characteristic classes. To compute KG in the Weil model, observe that the map

Thii provides the justification for using the Weil model

as the definition of equivariant cohomology; as we can replace W in this formula by any acyclic W* algebra.

4.5

The Characteristic Momomorphism

-

Let 4 : A -+ B be a homomorphism of G* algebras. We know that 4 induces a homomorphism 4, : HG(A) -+ HG(B), and that the assignment 4 4, is functorial. We also know that the equivariant cohomology of C , regarded as a trivial G*-module is given by Hc(C) = Hbas(W)= S(g*)G. Suppose that A is a (unital) G* algebra, so has a unit element 1 = la which is G-invariant (and hence basic). The map

given by tensoring by I A maps S(g*)Ginto closed elements in the Weil model, and passing to the cohomology gives KG. Every element of the image of n@ id is fixed by the Mathai-Quillen homomorphism, 4, and so, in the Cartan model, n @ id is again the map given by tensoring the invariant elements of S(g') by l a . Passing to the corresponding cohomology classes then gives nG.

4.6

Commuting Actions

Let M and K be Lie groups. Suppose that G = M x K as a group, wit$ the corresponding decomposition g = m @ k as Lie algebras. Then 7% and k can be regarded as subalgebras of 3 with m-1 @ m o commuting with k-l @ ko. Also, we have the natural decomposition of Weil algebras,

is a homomorphism of G* modules, and hence induces a homomorphism

This map is called the characteristic homomorphism or the Chern-Weil map. The elements of the image of n~ are known as the characteristic

Any G' module A can be thought of as an M* module and as a K* module. The space of elements of A which are basic for the M* action, call it Abas,, , is a submodule for the K* action and vice versa. We have

Chapter 4. The Weil Model and the Cartan Model

50

4.8 Exact Sequences in the obvious notation. We can apply this to A @W ( g )= A @ W(m)@W(k). Suppose that A, and hence A 8 W ( m ) is a a W ( k ) * module. Then, by Theorem 4.3.1,

4.8

51

Exact Sequences

Let G be a compact Lie group and

be an exact sequence of G* modules. Tensoring with S(g*) gives an exact sequence We conclude that

% ( A ) = H ~ ( A b a s)-~ . (4.26) If A is also a W(m)*module (when considered as an M* module), we conclude that (4.27) HK(A~=". ) = H M ( A ~ =).~ . In the case that M is compact we can describe (4.26) in terms of the Cartan map. Indeed, suppose that the Bi are the connection forms that make A into a W ( k ) * module for the K* action. Since M and K commute, we may average these 0's over M using the M action to obtain ones that are M-invariant. Then

and

~ c ( G= ) & ( K ) ( ~ c ( M ) ) = 1 @ ~ c ( M )- 3 ~ ( 7 j ) 7 where (71,.. .qr) is a basis of k and {vl,... ,vr) the corresponding dual basis of k*, and where d C ( ~is)the Cartan d relative to K* of d ~ ( the ~ ) Cartan , d relative t o M* of A. This cohomology is isomorphic to .cohomology relative to d ~ (of~ ) [ s ( m * )€3 AbasK*I M which is just H M ( A h K .). In particular, we have the characteristic homomorphism IEK

:S

( I C * ) ~H ~ ( A b a s ~ . ) . +

The Equivariant Cohomology of Homogeneous Spaces

computing the equivariant cohomology of a homogeneous space.

-+

(A,-I 8 s(g*)lG --+ (A; 8 ~ ( 9 " )-+) (A+1 ~ 8 ~ ( 9 ' )-+ ) .~. . .

We claim that this sequence is also exact. Indeed, suppose that v is in the kernel of ( A , @ ~ ( 9 ' ) ) ~( A + I @ ~ ( g * ) ) ~ :

-

Then there is a P 6 4 - 1 8S(g8) whose image is v. Since v is G invariant, the image of a p is also v for any a E G. Hence, averaging all the a p with respect to Haar measure gives an element of (A,-l 8 ~ ( g * )whose ) ~ image is v. We have thus proved

T h e o r e m 4.8.1 An ezact sequence (4.30) of G* modules gives rise to a n exact sequence of Cartan complexes

In particular, consider a short exact sequence

of G* modules. By Theorem 4.8.1 we get a short exact Sequence

of complexes and hence a long exact sequence in cohomology

Let K be a closed subgroup of the compact group G and apply (4.27) with G x K playing the role of G , where G acts on itself from the left and K from the right, giving commuting free actions of G and K on G. We conclude that

H G ( G / K ) = H K ( G / G )= ~ ( k * ) ~

..

(4.28)

The image of &K is called the ring of M-equivariant characteristic classes.

4.7

and hence a sequence

(4.29)

4.9

Bibliqgraphical Notes for Chapter 4

1. Most of the material in this chapter is due to Cartan and is contained in Sections 5-6 of "La transgression dans un groupe de Lie...". A

52

Chapter 4. The Weil Model and the Cartan Model word of warning: These two sections (which consist of five brief p a r a graphs) don't make for easy reading: they contain the definition of the UTeilmodel (page 62, lines 20-23), the definition of the Cartan model (page 63, lines 30-33), a proof of the equivalence of these two models (page 63, lines 19-37), the definition of what we're really calling a "W' modulen (page 62, line 32), a proof of the isomorphism,

H ((A €3 Elbas) = H(Abas) (page 63, lines 7-17) and most of the results which we'll discuss in the next chapter (page 64, l i e s 1-21).

2. The Mathai-Quillen isomorphism is implicitly in Cartan, is much more acplicitly described in section 5 of [MQ] and is made even more explicit in Kalkman's thesis [Ka]. Our version of Mathai-Quillen is a somewhat simplified form of that in [Ka]. 3. There are some very interesting variants of the Cartan model, due to Berline and Vergne and their co-authors: An element, p, of the Cartan complex (R(M) 8 S(g*))G,can be regarded as an equivariant mapping

which depends polynomially on g, and its equivariant coboundary, dGp, can be defined to be the mapping

This definition, however, doesn't'require p to be a polynomial function of 5. One can .for instance define the equivariant cohomology of M with Cw coefficients: HF(M) to be the cohomology of the complex of smooth mappings, (4.33), with the coboundary operator (4.34) (c.f. [BV], [DV], [BGV]) and one can define an equivariant cohomology of M with distributzonal coefficients

Chapter 5

Cartan's Formula In this chapter we do some more detailed computations in the Cartan model. Recall that . CG(A) = (S(gg) 8 It will be convenient, in order not to have to carry too many tensor product signs, to identify S(g0)8 A with the space of A-valued polynomials. If El,. . .,En is a basis of g, we will let x', ...zn denote the corresponding coordinates, i.e. the corresponding dual basis. (So we are temporarily using xi instead of the p' used in W ( g ) for pedagogical reasons.) The Cartan dserential in this notation is given by

We set

In the current notation, the polynomial

(where I = (il, . . . ,in) is a multi-index) is identified with the element by allowing the mappings (4.33) to be distributional functions of g.

(see 4. The proof of Theorem 4.3.1 can be streamlined a bit by using the spectral sequence techniques that we describe in Chapter 6 . By (4.20) and (4.21) ( A 8 E)bas is a bicomplex with coboundary operators and 62, and by lemma 4.3.1 its spectral sequence collapses at the E2 stage with E;" = Hq(Ab) and Epq = 0 for p # 0.

The fact that A is a W(g)-module means that we have an evaluation map

sending XI 8 ar

+-+

play,

54

Chapter 5. Cartan's Formula

5.1 The Cartan Model for W* Modules

so we will denote the image off ( x ) under this map by f ( p ) . In other words,

55

attempt to eliminate tensor product signs, we write the 1 8 d~ occurring on the right hand side of (4.16) simply as d and the pa8ca as z a c a so that (4.16) becomes dG = d - x"L,.

This can also be written as follows: Let

For any a E CG(A)we have

(6 + Kd)a = -dsrara

So S is an operator on A-valued polynomials. Let

+ Ba,d.cr - era,&

= -&

( - x r b ) ~ = (xrb)oaa,~ = xrara - xrBdbara since +Ba = 6: n(-xrb)a = (-~~d,)(-x+b)a = Brc,a xrOS~rada

be evaluation of a polynomial at 0,so

+

Then

f (PI = P [(expS ) f I

so adding all terms shows that

+

Put "geometrically," the operator expS is just the "translationn x -+ x p in f and p has the effect of setting x = 0. In other words, we are taking the Taylor expansion of f at 0 with p "plugged in". The basic subcomplex C".' c C G ( A )is d e h e d to consist of those maps which satisfy bW = 0, aSw= 0, V T , ~ .

d G K + K d G = E - R. I t follows that &(E

- R) = ( E - R)dG.

(5.5) The operators xrar and PL,commute and map C G ( A )into itself, so we have the simultaneous eigenspace decomposition:

The second equations say that w E C G ( A ) is a constant map, and so is a G-invariant element of A, while the first equations say that w is horizontal. . So C0?O(A)= Aha when G is connected.

5.1

where p is the degree as a polynomial in x and q represents the ''vertical degree" in the sense that an element of C0*Q(A) is a sum of terms of the form

The Cartan Model for W* Modules

0'' .- .OC . w where w

Let us give an alternative proof, using the Cartan mode!, of Theorem 4.3.1, namely that for a W* module, A, we have the formula

S(gW)8 Ahor.

In other words, CP.qis the image of ~ ~ ( g8*SP(g*) ) @J Ahor under the evaluation map W ( g )@ A -+ A. In particular, this notation is consistent with the previous notation in that C".O consists of basic elements of A. Also, introduce the atration corresponding to polynomials of degree at most p:

Suppose that A is a W* module so that there are connection elements Or and their corresponding curvature elements pr acting on A, corresponding to a choice of basis of g. Define the operators K , E, R on C G ( A )by

We have

K lowers filtration degree by 1 , We want to think of E a s the supersymmetric version of the Euler operator, where the {Or) are thought of as odd variables. In our current notational

E

I

dG raises filtration degree by 1,

E preserves filtration degree,

56

Chapter 5. Cartan's Formula

5.2 Cartan's Formula

R lowers filtration degree, (and is nilpotent)

and that this isomorphism depends only on the G structure and not on the W* structure. Proof of (5.9). It follows from (5.5) that

E = ( p + q)id on 0 . 4 . So E - R is invertible on

57

@(p,q)f (0,0) CP.Q.

Let denote projection along @(p,q)f

(0,0)

so multiplying on the right and left by U = J-'

CP3qSO

n = 0 on Pq,(p,q) # (O,O), n = id on

we get

-

@YO.

Notice that on @ Cp*O= ( S ( g * ) the operator a is just evaluation of a polynomial at 0 , in other words, it coincides with the operator p defined in the introduction to this chapter. In particular, formula (5.2) holds on ( S ( g 8 )8 Ah,,)G with p = X . The operator J:=E+x-R

Now dc maps CO*O into itself and so dG o n : C G ( A ) CO.Oand so does r and U = id on CO.O.So we can and hence 1~ o dG. So IdG,a] : CG + simplify the last equation to

we have Since K = 0 on COsO

is invertible, and we can write (5.4) as

Then

Let

F := ( E + a)-'

and define

The series on the right is locally finite since RF lowers filtration degree. Let Q := KU.

proving (5.9).

(5.8)

5.2

We will prove that

+

dGQ QdG = I - n u . (5.9) If we let i : CO*O-+ C G ( A ) denote the inclusion, then i o n = a so we can write (5.9) as dGQ QdG = I - i o ( x u ) . (5.10)

Cartan's Formula

We now do a more careful analysis which will lead to a rather explicit formula for the operator n u . We have

+

On the other hand U = id on

p,O so

from (5.7) since F preserves the bigradation and equals the identity on CovO. We may write .

R=S-T on COvO= Abas Thus the maps i and aU are homotopy inverses of one another, and hence induce isomorphisms on cohomology. In other words, the formula (5.9) implies that

where and

s=

. .

:

C.J

1 : T = _,-,O,,g'paa 2

+~

c t .. 3.

- l . j .

, ~i-l.3+2

Chapter 5. Cartan's Formula

58

5.3 Bibliographical Notes Eor Chapter 5

Now T increases the q degree by two and S does not decrease it so we can write ~ u = ~ + ~ s F + ~ ( s F ) .~ + - . .

59

R o m a we get a bijection

+

The operator S decreases total degree, f! = p q by one and F takes the value l/f!on elements of total degree !f # 0. Also x = 0 on elements of total degree not equal to 0. Hence

Combining the first map, restricted to the invariants, with the inverse of the second map, we obtain a map

This is the manifold version of xU in the theorem. The restriction of this map to S(g*)Gis the Chern-Weil map at the level of forms. It induces a map on cohomology ~ ( g *+) H~* ( X )

and so xU = nexpS. S , and hence exp S does not change the q degree. So

which is the Chern-Weil map that we discussed in section 4.5. For

-

n u = x e x p s o Hor where Hor denotes the projection CG(A) @, O ' v O . NOWevery element of 0'*O i s a sum of terms of the form x'w where x' is a monomial of total degree 111 = p and w a horizontal element of A. On such a term we have nSc = 0 unless f! = p and then &xse(xiw)= piw. So

p = p(xl, . . . ,2") E ~ ( g * ) ~ its image under

KG

is given by the cohomology class of

p(/J1,...,/J")This map is well defined and independent of the choice of connection.

as we observed in the introduction, equation (5.2). In any event, we have proved the following theorem due to Cartan:

Theorem 5.2.1 The Cartan opemtor nU is the wmpositzon of the projection operator Hor : CG(A)+ @ c".' = (S(g*) 8 ~

h , ) ~

I

and the map

.

(S(9') @ A ~ O I ) ~ Abas wming from the W* module structure, i.e. the "evaluation map" +

Let us see what the Cartan map looks like in the case of manifolds. Let M be a G-manifold on which G acts freely, let X := M/G, and let n : M + X be the map which assigns to every m E M its G-orbit. We can t h i i of this situation as a principal G-bundle. Equipping this bundle with a connection, we get curvature forms P' E R(M)hor and a G-equivariant map

5.3 Bibliographical Notes for Chapter 5 1. A close textual reading of "La transgression dans une groupe de Lie...", page 64, lines 1-21 seveals that the proof of the Cartan isomorphism: HG(A)= H(Ab,), which we've given in this chapter is that envisaged by Cartan. We are grateful to Hans Duistermaat for explaining this proof to us. The construction of the homotopy operator in Section 5.1 is based on some unpublished notes of his as is the beautiful identity, nU = n exp S in section 5.2. 2. Not only can the Cartan formula be used to reprove the isomorphism: HG(A)= H(Ab,) but it has many other applications besides. We will discuss a csuple of these in Chapter 8 and Chapter 10. In Chapter 8 we will use the Cartan formula to give simple proofs of two well-known theorems in symplectic geometry: the Duistermaat-Heckmann Theorem and the minimal coupling theorem. In Chapter 10 we will use it to obtain a formula of Mathai-Quillen for the Thom form of an equivariant vector bundle in terms of the curvature forms of the bundle.

Chapter 6

Spectral Sequences We begin this chapter with a review of the theory of spectral sequences in the special context of double complexes. We then apply these results to equivariant cohomology: We will show that if a G* morphism between two G* modules induces an isomorphism on cohomology it induces an isomorphism on equivariant cohomology. Given a G* module A, we will discuss the structure of HG(A) as an'S(g*)G-module, and show that if the spectral sequence associated with A collapses at its El stage then HG(A) is free as an module. Finally, we will prove an abelianization theorem which says that HG(A) = H T ( A ) ~ where T is a Cartan subgroup (maximal torus) of G and W its Weyl group.

6.11

Spectral Sequences of Double Complexes

A d o u b l e complex is a bigrrided vector space

with coboundary operators

satisfying d2 = 0,

db + 6d = 0,

The associated total complex is defined by

with coboundary d + b : Cn + C+'.

J2 = 0.

62

Chapter 6. Spectral Sequences

6.1 Spectral Sequences of Double Complexes

63

We will construct a sequence of complexes (E,, 6,) such that each E,+I is the cohomology of the preceding one, Er+l = H(Er, 6,), and (under suitable hypotheses) the "limitn of these complexes are the quotients HP.4. To get started on all this, we will need a better description of these quotients: Any element of C has all .its elements on the (anti-)diagonal line @i+j=, Cf and will have a "leading term" at position (p, q) where p denotes the smallest i where the component does not vanish. Let ZP.4 denote the set of such components of cocycles at position (p, q). In other words, P . 4 denotes the set of all a E CP74 with the property that the system of equations Let p q , P+q=n, p2k so Ct consists of all elements of C" which live to the right of a vertical line. :=

admits a solution (al, az, . . :)

where

%E

CP+"q-'.

In other words, a E P . 9 C can be ratcheted by a sequence of zigzags to any position' below a on the (anti-)diagonal line e through a, where L = {(i,j)li j = p q ) : 0 0 ' 8 9

,

+

+

Let Z,":={zEq,

(d+b)z=O),

Bn:=(d+6)C-I

The map

z;

+G/(Bn

nq )

gives an decreasing filtration

of the cohomology group Hn(C, d and let by ~

~

l

~

-

+ 6). We denote the successive quotients

~

The spectral sequence we are about to describe is a scheme for computing these quotients starting from the cohomology groups of the ''vertical complexes" (Cry*,d ) .

In the examples we will be encountering,

for some mc and hence the system (6.2) will be solvable for all i provided that it is solvable for a bounded range of i. To repeat: the equations (6.2) say that the element z:=a@al@az@... ' lies in 2; and has "leading coefficient" a.

64.

Chapter 6. Spectral Sequences 6.1 Spectral Sequences of Double Complexes

Let

BP.4

c

0 . q

consist of all b with the property that the system of

65

and c-; = 0 for i 2 r. So let us define

equations

be the set of all b E CP94for which there is a solution of (6.4) with c-, = 0 for i 2 r. Then we have proved Theorem 6.1.1 Let a E ZPQ. Then admits a solution (Q, C-1, c-2,

...)

for any solutions (al,. ..,%-I) of the first r - 1 equations of (6.2).

with c-, E CP-'.'J+i-l.

Notice that since 6a,-i satisfies the system of equations

Once again, if the boundedness condition (6.3) holds, it sutEces to solve these equations for a bounded range of i. It is easy to check that the quotients HP.9 defined in (6.1) can also be described as H p . 9 = ZP.Q/BP,Q. (6.5) Let us try to compute these quotients by solving the system (6.2) inductively. consist of those a E CPlg for which the first r - 1 of the Let Zr>gc equations (6.2) can be solved. In other words, a E Zpg if and only if it can E Chlgr where be joined by a sequence of zigzags to an element

- 1 units down the diagonal e from (p,q). When can such an a be joined by a sequence of r zigzags to an element of C p y f',qy-'? In order to do so, we may have to retrace our steps and replace the partial solution ( a l , . . . ,a,-l) by a different partial solution (a;, . . . ,a:-,) so that the differences, a: = a: - ai satisfy is the point r

'.Q'.Indeed, every element of

itself is in B,"$ we see that be written as a sum of the form

.

can

with c E CP,+ 1 . q ~ - 1 and ( a ,a l , . . . ,a,-1) a solution of the first r - 1 equations of (6.2). From this we can draw a number of conclusions: 1. Let E74

:= Z,P.q/B?Q.

(6.10)

we see that 6are1 projects onto an element bar-,+da:-,

= 0

and to zigzag one step further down we need an a: E CPr+l+qr-l SU& that = -6aLl

- da;.

Theorem 6.1.1 can be rephrased as saying that an a E Zp4 lies i n Z,";-"l if and only i f 6,a = 0. 2. It is clear that 6,a only depends on the class of a modulo BP.4. Since BPlQ> BPQ we can consider 6# to be a map

Let us set b : = 6 a T - I , %:=-a:,

c-,:=-a:-,,

i = l ,...,r - 1

66

Chapter 6 . Spectral Sequences

6.3 The Long Exact Sequence

3. By (6.9), the image of this map is the projection of B%:~-'+' into Ec+~+~-'+' and, by Theorem 6.1.1, the kernel of this map is the projection of Z,"cl into @ ? q . Thus the sequence

has the property that ker 6,

> im 6,

6.3

67

The Long Exact Sequence

Let ( C ,d, 6) be a double complex with only three non-vanishing columns corresponding to p = 0 , 1 , or 2, In other words, we assume that

and

(ker 6,) / (im 6,) = gT1

(6.13)

The El term of this spectral sequence is

in position (p, q). with i = 61, j = 6 i . Thus

In other words, the sequence of complexes (Er,6,),

~ = 1 , 2 , ,... 3

,$94

has the property that H ( E r , 6,) = -&+I.

= ker i : Hq ( e l * , d )

+

Hq ( C 1 , * , d )

E:,Q = ker j : Hq (C1'*,d ) + Hq (C2**,d ) im i : Hq (CDl*, d ) + Hq ( C 1 + *d ,)

(6.14)

By construction, these complexes are bigraded and 6, is of bidegree ( T , -(T

- 1)).

Moreover, if condition (6.3) is satisfied for all diagonals, sequence" eventually stabilizes with

e, the

"spectral

EP.9 = EPcl = .. . for r large enough (depending on p and q). Moreover, this "limiting" complex, according to (6.5), is given by EP&'

6.2

= limE,P'P = HPA'.

~ 2 ~ .

~21'~

(6.15) is a short exact sequence of complexes. If we interchange rows and coiumns, we get a double complex whose columns are exact, hence a double complex having its El = 0. Thus we must have E3 = 0 in our original double complex. Hence we must have E?=O

The First Term

The case r = 1 of (6.14) is easy to describe: By definition, EFq = H q ( P * ,d )

Gvq ~ 2 2 ' ~ - ~

The coboundary operator, 62 maps + and vanishes on and on For T > 2 we, have 6, = 0,so this spectral sequence "collapses at its E3 stage". Suppose now that the rows of the double complex are exact. In other words, suppose that

(6.16)

is the vertical cohomology of each column. ~ o r & v e r since , d and 6 commute, one gets from 6 an induced map on cohomology

and is an isomorphism. So if we define the "connecting homomorphismn A a s

Hq(CP**, d ) 4 Hq(Cp+',*,d ) and this is the induced map 61. So we have described ( E l , & ) and hence E2. The 6, for r 2 2 are more complicated but can be thought of as being generalizations of the connecting homomorphisms in the long exact .sequence in cohomology associated with a short exact sequence of cochain complexes as is demonstrated by the following example:

we get the long exact sequence

in cohomology.

68

6.4 6.4.1

Chapter 6. Spectral Sequences

6.5 The Cartan Model as a Double Complex

Useful Facts for Doing Computations

6.4.3

Fhnctorial Behavior

Let ( C ,d , 6 ) and (C',d', 6') be double complexes, and p : C -+ C' a morphism of double complexes of bidegree (m,n) which intertwines d with d', and 6 with 6'. This give rise to a cochain map

p : ( C ,d of degree m

+6 )

-+

(C', d'

+ 6')

+ n. It induces a map pn on the total cohomology:

of degree m + n and consistent with the filtrations on both sides. Similarly p maps the cochain complex (CP?*, d ) into the cochain complex ((C1)P+*.*, dl) and hence induces a map on cohomology

.

69

Switching Rows and Columns

We have already used this technique in our discussion of the long exact sequence. The point is that switching p and q, and hence d and 6 does not change the total complex, but the spectral sequence of the switched double complex can be quite different from that of the original. We will use this technique below in studying the spectral sequence that computes equivariant cohomology. Another illustration is Weil's famous proof of the de Rham theorem, [We]:

Theorem 6.4.4 Let ( C , d , 6 ) be a double complex all of whose columns are exact except the p = 0 column, and all of whose nnus are exact except the q = 0 row. Then H (COX*, d ) = H ( c * , ~ 6). , (6.19) Proof. The E l term of the spectral sequence associated with ( C ,d , 6 ) has only one non-zero column, the column p = 0, and in that column the entries d ) . Hence 6, = 0 for r > 2 and are the cohomology groups of

(e.",

H(C,d + 6 ) = H(@*,d). of bidegree ( m , n) which intertwines 61 with 6;. Inductively we get maps Switching rows and columns fields Here p,+, is the map on cohomology induced from p, where, we recall, E,+l = H(E,, 6,). It also is clear that

Putting these two facts together produces the isomorphism stated in the theorem.

Theorem 6.4.1 If the two spectral sequences conuerge, then (6.18)

lim p, = gr pb In particular,

Theorem 6.4.2 If p, is an isomorphism for some r = ro then i t is an isomorphism for all r > ro and so, ij both spectml sequences converge, then is a n isomorphism.

6.4.2

Gaps

The Cartan Model as a Double Complei

6.5

Let G be a compact Lie group and let A = @ Ak be a GI module. Its Cartan complex C c ( A ) = ( S ( g 8 )8 can be thought of as a double complex with bigrading

and with vertical and horizontal operators given by

Sometimes a pattern of zeros among the Ep4 allows for easy conclusions. Here iS a typical example:

Theorem 6.4.3 Suppose that E,P.q = 0 when p + q is odd. Then the spectml sequence "collapses at the E, stage", i.e. E , = Er+l = . . .. Proof. 6, : Epq -+ E:+r,9-r+1 SO changes the parity of p its domain or its range is 0. So 6, = 0.

+ q.

Thus either

and

6 : = pa@^a. Notice that in the bigrading (6.20) the subspace (SP(g*)8 A , " ) ~has bidegree (Ip, m p ) and hence total degree 2p + rn which is the grading that we have been using on ( S ( g * )8 A ) as~ a commutative superalgebra.

+

70

Chapter 6. Spectral Sequences

If Ak = 0 for k < 0, which will be the case in all our examples, then the double complex satisfies our diagonal boundedness condition (6.3), and so, under this assumption, the spectral sequence associated to the double complex (6.20),(6.21), and (6.22) will converge. We begin by evaluating the El term:

We assume from now on that G is compact and connected so that (6.25) holds. If f E ~ ( g * )the ~ multiplication , operator

Theorem 6.5.1 The El term an the speetrd sequence of (6.20) is

More eqlicitly

Erg= (SP(g*) 8 H P - P ( A ) ) ~ .

(6.23)

Proof. The complex C = (S(g*) 8 A ) with ~ boundary operator 18dA sits inside the complex (6.24) (S(g*) 8 A, 1 8 d ~ ) 18 dA) are and (by averaging over the group) the cohomology groups of (C, just the G-invariant components of the cohomology of (6.24) which are the appropriately graded components of S(g*) 8 H(A). 0 To compute the right-hand side of (6.23) we use (cf. Remark 3 in Section 2.3.1) Proposition 6.5.1 The connected component of the identity in G a d s t r i v idly on H(A). Proof. It suffices to show that all the operators La act trivially on on H(A). But La = ~~d d~~ says that La is chain homotopic to 0 in A. So we get

+

.

Theorem 6.5.2 If G is connected then

EFq= S P ( ~ *@I) Hq-P(A). ~

(6.25)

We now may apply the gap method to conclude Theorem 6.5.3 If G is connected and HP(A) = 0 forp odd, then the spectral sequence of the Cartan complez of A collapses at the El stage. Proof. By (6.25), Erg = 0 when p + q is odd. Let M. be a G-manifold on which G acts freely and let A be the de Rham complex, A = R(M) so that HG(A) = H*(X) where X = MIG. Theorem 6.5.1 gives a spectral sequence whose El term is ~ ( 9 ' €9 ) ~H*( M ) and whose E, terms is a graded version of H*(X). The topological version of this spectral sequence is the Leray-Serre spectral sequence associated with the fibration (1.4). See [BT],gage 169. The notation there is a bit different. They use a slightly different bicomplex so that their E2term corresponds to our El term.

is a morphism of the double complex (C,d, 6 ) given by (6.20), (6.21), and (6.22) of bidegree (m,m) and so it induces a map of HG(A) into itself. In other words, we have given HG(A) the structure of an S(g*)G-module. Also, all the E,'s in the spectral sequence become S(g*)G-modules. Under the identification (6.25) of El this module structure is just multiplication on the left factor of the right hand side of (6.25), which shows that El is a free S(g*)G-module. Now S ( S * ) is ~ Noetherian; see [Chev]. So if H(A) is finite dimensional, all of its subquotients, in particular all the E,'s are finitely generated as ~(9,)~-modules.Since the spectral sequence converges to a graded version of HG(A), we conclude Theorem 6.6.1 If dim H(A) is finite, then Hc(A) is finitely generated as an S(g*)G-module. Another useful fad that we can extract from this argument is: Theorem 6.6.2 If the spectral sequence of the Carton double complex collapses at the El stage, then HG(A) is a free S(g*)G-module. Proof. Equation (6.25) shows that El is free as an S(g*)G-module, and if the spectral sequence collapses at the El stage, then El 2 gr HG(A) and this isomorphism is an isomorphism of S(g*)G-modulesby Theorem 6.4.2. So gr HG(A) is a free S(g*)G-moduleand hence is so is HG(A). 0

6.7

Morphisms of G* Modules

Let p:A+B be a morphism of degree zero between two Gt modules. We get an induced morphism between the corresponding Cartan .double complexes and hence induced maps p, : H(A, d) -+ H(B, d) on the ordinary cohomobgy and

on the equivariant cohomology. From Theorems 6.4.2 and 6.5.2 we conclude: Theorem 6.7.1 If the induced map p. on ordinary cohomology is bijective, then so is the induced map pt on equivariant cohomology.

72

6.8

Chapter 6. Spectral Sequences

6.8 Restricting the Group

Restricting the Group

Suppose that G is a compact connected Lie group and that K is a closed subgroup of G (not necessarily connected). We then get an injection of Lie algebras k-+g and of superalgebras

I+g

-

so every G* module becomes a K* module by restriction. Also, the injection k g induces a projection g* -+ k*

73

Unfortunately, there is only one non-trivial example we know of for which the hypothesis of the theorem is fulfilled, but this is a very important example. Let T be a Cartan subgroup of G and let K = N(T) be its normalizer. The quotient group W = KIT is the Weyl group of G. It is a finite group so the Lie algebra of K is the same as the Lie algebra of T. Since T is abelian, its action on t*, hence on S(t*), is trivial. So

~ ( k *=) ~~ ( t *=) S(t*)W. ~ According to a theorem of Chevalley, see for example [Helg] (Chapter X Theorem 6.1), the restriction

which extends to a map S(g*) -+ S(k*) and then to a map

which is easily checked to be a morphiim of complexes, in fact of double complexes CG(A) -, CK(A). We thus get an restriction mapping

which induces a morphism

-

-+

and also a morphism at each stage of the spectral sequences. At the El level this is just the identity morphism

and also a restriction morphism at each stage of the corresponding spectral sequences. Now since G acts trivially on H(A), being connected, and K is a subgroup of G, we conclude that K also acts trivially on H(A) even though it need not be connected. In particular the conclusion of Theorem 6.5.2 applies to K as well, and hence the restriction morphiim on the El level is just the restriction applied to the left hand factors in s ( ~ *8 ) H(A) ~

is bijective so Theorem 6.8.1 applies. We can do a bit more: n o m the inclusion T K we get a morphism of double complexes CK(A) ~ d - 4 ) ~

-

~ ( k *8)H(A). ~

Therefore, by Theorem 6.4.2 we conclude:

and hence another application of Theorem 6.4.2 yields HK(A) = H~(A)W. Putting this together with the isomorphism coming from Theorem 6.8.1 we obtain the important result:

Theorem 6.8.2 Let G be a connected compact Lie group, T a maximal torus and W its Weyl p u p . Then for any G*-module A we have

Theorem 6.8.1 Suppose that the restriction map

-

s ( ~ * ) ~S(k*)K

This result can actually be strengthened a bit: The tensor product

is btjectzve. Then the restriction map

HG(A) --,HK(A) in equivariant whomology is bijective.

is also a bicomplex since the coboundary operators on CK(A) are S(t*)Wmodule morphisms. Moreover there is a c a n o n i d morphism

74

Chapter 6. Spectral Sequences

6.9 Bibliographical Notes for Chapter 6

The spectral sequence associated with the bicomplex CK(A)@ S ( ~ - ) WS(t*) converges to HK( A )@.s(t-)w S(t.1. Fkom (6.27) one gets a morphism of spectral sequences which is an isomorphism at the El level. Hence, at the Em level we have HG(A) B ~ ~~ (~t *2 . )H=(A). ) ~

(6.28)

Theorem 6.8.2 has an interesting application in topology: the splitting principle. Let M be a G-manifold on which G acts freely. Let

Theorem 6.8.4 H*(Y) is the quotient of the ring

by the ideal generated by the expressions on the left-hand side of (6.29).

For more details on the splitting principle in topology, see Section 8.6 below.

6.9

Y := M / T

X := M / G , and let

7r:Y-rX be the map which assigns to every T-orbit the corresponding G-orbit. This is a differentiable fibration with typical fiber G/T.One gets from it a map a* : H*(X) + H*(Y).

Moreover there is a natural action of the Weyl group W on Y which leaves fixed the fibers of A. Hence a* maps H *(X) into H*( Y ) W .

75

Bibliographical Notes for Chapter 6

1. There are several other versions of the theory of spectral sequences besides the "bicomplex version" that we've presented here. For the Massey version (the spectral sequence associated with an exact couple) see [Ma] or [BT], and for the Koszul version (the spectral sequence associated with a filtered cochain complex) see [Go] or [Sp]. The oldest and most venerable topological example of a spedtral sequence is the Serre-Leray spectral sequence associated with a fibration

Theorem 6.8.3 The map T* :

If B is simply connected, the E2 term of this spectral sequence is the tensor product of H(B) and H ( F ) and the Em term is a graded version of H(X). (For a description of this sequence as the spectral sequence ' of a bicomplex, see [BT] page 169.)

H*(x) -,H * ( Y ) ~

is a bijection. Proof. This follows from (6.26) and the identifications

HG(M) H*(x) H = ( M ) r H*(Y) given by the Cartan isomorphism. We can sharpen this result. The theorem of Chevalley cited above also asserts that S(t*)W is finitely generated, and is, in fact, a polynomial ring in finitely many generators. Let xl,. . . ,xr be a basis of t* and let i = l , ...k

p i ( ~,..., l xr),

be generators of S(t*)W. The Chern-Weil map nT : S(t*) -+

H*(Y)

maps xl,.. . ,xr into cohomology classes ol,. . . ,or and the Chern-Weil map KG

: S(gW) =~

( t *+) H*(X) ~

From (6.28) one easily deduces

and its Em term is HG(M). Since G acts free1y.o~E, H ( E / G ) = HG(E); and since E is contractible. HG(E) = HG(pt,) = s ( ~ * ) so ~; (6.30) is equal to: s(g*IG@ H(M) i.e., is equal to the El term of the spectral sequence associated with the Cartan model (S(!3*) @ fl(WIG . (The E, term is, of course, the same:

HG(M).)

3. If this spectral sequence collapses at its El stage M is said to be equivanantly formal. Goresky, Kottwitz, and MacPherson examine this

maps pl, . . . ,pk into cohomology classes cl,. . .ck satisfying a*&- pi(ul,.. . ,ur)= 0.

2. For example, the E2 termrof the spectral sequence associated with the fibration (1.6) is H(EIG) @ H ( M ) , (6.30)

(6.29)

property in detail (in a much broader context than ours) in [GKM] and derive a number inequivalent sufficient conditions for it to hold. In particular they prove

76

Chapter 6. Spectral Sequences

Theorem 6.9.1 Suppose the ordinary homology of M, Hk(M,R), is genemted by classes which are mpresentable by cycles, t , each of which is invariant under the action of K. Then M is equivariantly formal.

We will discuss some of their other necessary and sufficient conditions in the Bibliographical Notes to chapter 11. 4. An important example of equivariant formality was discovered by Kirwan [Ki] and, independently, by Ginzburg [Gi]: M is equivariantly formal if it is compact and admits an equivariant symplectic form.

5. If M is equivariantly formal, then as an S(g*)Gmodule

by the remarks in section 6.6. Tensoring this identity with the trivial S(g*)Gmodule C gives

expressing the ordinary de Rham cohomology of M in terms of its equivariant cohomology.

6. Let G be a compact Lie group, T a Cartan subgroup and K a closed subgroup of G with T c K c G. Combining (6.28) with (4.29) we get

where W Kand WG are the Weyl groups of K and G. Using note 5, we get the following expression for the ordinary de Rham cohomology of GIK: H ( G / K )= ~ ( t * ) ~ ~ / r n ~ ( t * ) ~ ~ where m = ( S ( t * ) W ~ )the + , maximal ideal of S ( t * ) W at ~ zero, cf. [ G w Chapter X Theorem XI (p. 442). 7. The relation of the splitting principle described in Section 6.8 to the usual splitting principle for vector bundles will be explained in Section 8.6.

8. In [AB] section 2, Atiyah and Bott give a purely topological proof of Theorem 6.8.3 and then, by reversing the sense of our argument, deduce Theorem 6.8.1.

Chapter 7

Fermionic Integration Fermionic integration was introduced by Berezin [Be] and is part of the standard repertoire of elementary particle physicists. It is not all that familiar to mathematicians. However it was used by Mathai and Quillen [MQ] in their path breaking paper constructing a "universal Thom form". In this chap ter we will develop enough of Berezin's formalism to reproduce the MathaiQuillen result. We will also discuss the Fermionic Fourier transform and combine Bosonic and Fermionic Fourier transforms into a single "super" Fourier transform. We will see that there is an equivariant analogue of compactly supported cohomology which can be obtained from the Koszul complex by using this super Fourier transform, and use this to explain the Mathai-Quillen formula. In Chapter 10 we will apply these results to obtain localization theorems in'equivariant cohomology.

7.1

Definition a d El~mentaryProperties

Let V be an d-dimensional real vector space equipped with an oriented volume element, that is, a chosen basis element, vol, of A ~ V .A preferred basis of V is a basis such that {+I,. ..

Elements f E AV are thought of as "functions in the odd variables $J" in that every such element can be written as

. When I = ( l , . .,n), $1 = v01,

78

Chapter 7. Fermionic Integration 7.1 Definition and Elementary Properties

and the coefficient fWl is called the Be-in integral of f :

]f (

. .,d

:= f

and, in particular when f = fI@.

If

T

vol H (det A) vol

(7-2)

Various familiar formulas in the integral calculus have their analogues for .this notion of integration, but with certain characteristic changes: .-

7.1.1

Integration by Parts

E V* then interior product by T is a derivation of degree -1 of AV:

where A = (a;) .

More abstractly, we are considering a linear map of V -+ V and extending to an algebra homomorphism of AV -+ AV which exists and is uniquely determined by the universal property of the exterior algebra In any event, we can write the preceding equality as

1 (C

a:@,

.. .,

a,d~)d+ = det A

: Akv+ Ak-'V.

In particular,

L,

79

integral or the Fermionic

I

f ($I, .. .,~ ) ~ ) d + .

(7.5)

Notice an important differences between this Fermionic "change of variables" law and the standard ("Bosonic") rule. In ordinary (Bosonic) integration the rule for a linear change of variables would have a (det A)-' on the right-hand side. We will have occasion to use both Bosonic and Fermionic linear changes of variables in what follows.

f has no component in degree n, and hence

If TI,...,r d is the basis dual to $I,. .. ,qhd, let us denote the operation bsby

a/av:

7.1.3

d .-

a@

'-

Gaussian Integrals

Let d = 2m and let q E A ~ SO V

LC-

So we have /&,f(*l,...,*d)d+=o.

q = ,J(*l,.

(7-3)

. , ,* d )

We can apply this to a product: Recall that

in terms of a preferred basis. Then, writing

is the Z/2Z-gradation of AV. Then using (7.3) and the fact that d / a v is a derivation, we get

we have

/ilp

1

-f

where deg u = 0 if u E (AV)o and deg u = 1 if u E ( A V ) ~ .

7.1.2

Change of Variables

qij = -pi,.

= qij?,v*j,

q($l,. . . ,rld)d$ = (det Q)+.

(7.6)

Here exp is the exponential in the exterior algebra, given by its usual power series formula which becomes a polynomial since q is nilpotent. Proof. We may apply a linear transformation with determinant one to bring Q to the normal form

Consider a "linear change of variables" of the form

This then induces, by multiplication, a map 4

AV -+ AV

1

so that

-2

2

= ~~*l"jl*+ ~

+

~ * ~ * . .4.

+

80

Chapter 7. Fwmionic Integration

Then the component of exp -5 lying in Ad(V) = A ~ ~ ( isV (by ) the multinomial formula)

7.1 Definition and Elementary Properties

81

and let us choose the volume element on U 83 V so that

in the obvious notation.

7.1.5 On the other hand, the determinant of Q as given in the above normal form is clearly

x;...x:

.

Notice the contrast with the "Bosonic" Gaussian integral,

where qij = qji is symmetric and must be positive-definite for the integral to converge. he result is ( 2 ~ ) - det ~ l QI-4 The factors of 27~are conventional - due to our choice of normalization of Lebesgue measure so that the unit cube have volume one. The key difference between the Bosonic and Fermionic Gaussian integrals is that I det Q I occurs ~ in the denominator in the Bosonic case and in the numerator in the Fermionic case. If V carries a scalar product we can think of q E. A2V as an element of g = o(n),and the formula 7.6 becomes the supersymmetric definition of the Pfafian of q. ( For an alternative non-supersymmetric definition see $8.2.3 below.) Thus we can write (7.6) as

7.1.4 Iterated Integrals Let A be an arbitrary commutative superalgebra and consider elements of A @ AV as the exterior algebra analogue of functions with values in A. So we may consider the expression (7.1) as an element of A @ AV where now the "coefficients" j~are elements of A. We then can use exactly the saine definition, (7.2), for the integral; the end result of the integration yielding an element of A. The operator 1, is interpreted as 1 @J L, and then the integration-by-parts formula (7.4) continues to hold, where deg u now means the Z/2Z-degree of u as an element of A @ AV. The operation of integration is even or odd depending on the dimension of the vector space V. In particular, we can take A = AU where U is a second vector space with preferr4 volume element. We have

The Fourier Transform

In this subsection we wish to develop the Fermionic analogue of the Fourier transform, using Fermionic integration. We begin by recalling some basic facts about the classical (Bosonic) Fourier transform: Let V be a & dimensional real vector space space with volume element du and suppose that we have chosen linear coordinates u' ,.. .ud SO that

We let S(V) denote the Schwartz space of rapidly decreasing smooth functions on V. 1. For f E S(V), its Fourier transform

f

E S(V*) is defined by

where ( , ) denotes the pairing between V and V*. 2. If yl, . . . ,yd denote the coordinates on V' dual to u', tegration by parts gives

and 3. Differentiation under the integral sign gives

4. The Fourier inversion formula asserts that

and hence, if the map

f-f is denoted by F, that

. . . ,ud,then in-

82

Chapter 7. Fermionic Integration

7.1 Definition and Elementary Properties

83

Proof. As in the Bosonic case, (7.12) is proved by integration by parts: By 'nearity, it is enough to check this formula for f E APV. We have There are various choices of convention that have been made here - the use of -2 in 1) and hence of i in 4) and the placement of the factors of 2 ~ . We now wish to develop an analogue in the Fermionic case. We do this by taking the A of the preceding subsection t o be A(V*),the exterior algebra of the dual space of V. Here A is generated by the basis T I , . .. ,i-d of V*, dual to the basis $ I , . .. ,tjd of V . Define the (Fermionic) Fourier transform

I a exp w = - ( - l ) ~ [ fr d +

Similarly, (7.13) is verified by differentiating under the integral sign:

I

The map F is clearly linear. Define w E A(V*)@ A ( V )by

The definition of w is independent of the the choice of basis and we have defined the Fourier transform as

Notice that w is an even element of A(V*)@ A ( V ) and that

.

I Proposition 7.1.2

Therefore

F' = id if dim V is even and F2 = i id

In analogy to 2) and 3) above we have Proposition 7.1.1 For f

E

AV we have

I

if dim V is odd.

(7.14)

Proof. Let us first verify this formula when applied to the element 1. We have

I

and

where s(d) := ( - l ) f d ( d - l )ik the sign involved in the equation

84

Chapter 7. Fermionic Integration

7.2 The Mathai-Quillen Construction

85

We have

So ids(d) = 1 or i according to whether d is even or odd. Applying the Fourier transform again gives

7.2

The Mathai-Quillen Construction

Let V be a d-dimensional vector space over R equipped with a positive def) an oriented orinite inner product and an orientation. Let {$I,. . .$ J ~be thonormal basis of V and g := o ( V ) , the Lie algebra of endomorphisms of V which are skew symmetric with respect to the inner product. We want to consider Fermionic integrals of expressions in A 8 AV, where A = R G ( V ) . We let &, ... ,tnbe a basis of g, n = i d ( d - 1 ) . Each E g is represented on V by a linear transformation ME whose matrix is skew-symmetric in terms of the basis $ l , . . . ,lid. In other words,


0, Vi.

131

Proof. In terms of our coordinates, the representation given by (9.47) is a representation of T~ on V and we can regard G as a subgroup of Td. The representation given by (9.47) restricts to the given representation of G, so there is no harm in denoting this extended representation also by p.

132

Chapter 9. Equivariant Syrnplectic Forms

9.9 Equivariant Duistermaat-Heckman

This action of Td commutes with the action of G and induces a Hamiltonian action of Td/G on Xx. Now Td acts freely near any point z E V if all its coordinates .q # 0, and hence Td acts freely on an open dense subset of Zx Consequently T ~ / Gact freely on an open dense subset of X x . Since 2n = dimXx = 2dhii?/G where n = d - r the action of PIG is a Delzant action, cf. [Gull. The computation of its moment polytope, which we will omit, involves staring carefully at the Delzant construction which we outlined in the preceding section. 0 Finally, let ( M , w ) be an arbitrary Harniltonian G-space with moment map Q : M -+g*. Let p be an extremal fixed point of G. Near the point p the action of G is isomorphic to the linear isotropy action of G on the tangent space T, by the equivariant Darboux theorem. Therefore, taking V = T,, the conclusions of the preceding theorem are valid for regular values, A, of I$ providing that X is sufficiently close to 4(p). Thii proves Theorem 9.8.3. In particular, for such values X we can compute the cohomology ring of Xx by Theorem 9.8.6.

9.9

Equivariant Duistermaat-Heckman

Let G and K be tori, and let (M, p ) be a Hamiltonian (G x K)-space with moment map ($,$I : M - + g *@k*,

where

r : QZXK(M)4 &(M)

-

zs

the "forgetfiLLvmap, i.e. the map corresponding to the inclusion K +

ExK. Suppose that Q is proper. Let &, . . . ,{,, be a basis of g, and E l , . .. ,E,, the equivariant Chern classes associated with the fibration

The equivariant version of the Duistermaat-Heckman theorem asserts the following: T h e o r e m 9.9.2 Equivariant Duistermaat-Heckman. There exists a neighborhood U of the origin in g* such that for all E E U, .

as K -manifolds, and

[Cia+,] = [Pa] +

P

52&xK(M) be the corresponding G x K equivariant symplectic form Let a be a regular value of and define Xa := Za/G.

where E = C E'&.

n

i:Za-+M K

:

za

--r

(9.58)

which is an analytic function on k. From the definition of ma, we may write this function as

Let be the inclusion and

C E'&,

Here is an important application of this formula. Let ma be the DuistermaatHeckman measure on k* associated with the action of K on Xa. As this measure is compactly supported, it has a well defined Fourier transform

and let

Z := ( a ) ,

133

+ +

exp(i~)(s).

More generally, we may allow a to vary in a small neighborhood and use (9.58) to write

Xa

the projection. The group K acts on these spaces. The equivariant version of the Marsden-Weinstein reduction theorem asserts that Theorem 9.9.1 Equiimriant Marsden- Weinstein. equivariant symplectic form

There exists a K

As in section 9.8 we can use this to evaluate the equivariant characteristic numbers

~(21,. . .,L)(=pFa)l(n),

where p is a polynomial in n-variables: just apply the differential operator such that

'I

**Fa = ifr(fi)

134

i

Chapter 9. Equivariant Symplectic Forms

1 f

to (9.60) and set e = 0 to obtain

i '

As a special case of this formula we get an interesting equivariant analogue of (9.51). Let A = Ax be the convex polytope (9.44) and let 6(X,q ) be the integral of e'vs over Ax with respect to the standard Lebesgue measure, ds. Then by theorem (9.8.4), completed with the identity above,

I I

i i J

9.10 Group Valued Moment Maps

135

So long as takes values in a neighborhood of the origin where exp is a diffeomorphism, we can translate properties of the moment map @ into properties of v and vice versa. (For example, adjoint orbits go into conjugacy classes.) These translations of properties of Qi turn out to involve the equivariant form XG mentioned above. But these properties make sense in their own right, and are the subject of study of the recent paper by Alekseev, Malkin and Meinrenken [AMMIwhere many important applications of thesewegoupvalued moment maps" are given. This section consists of an introduction to their paper. In most of what follows the group G need not be compact and the form ( , ) need not be positive definite, only non-singular.

I

This formula was used in [ G u ~to ] compute the equivariant Riemann-Roch number of Xa and thereby obtain a generalized "Euler-Maclaurin" formula for the sum xeqK, K E L'nA.

!

I

9.10.1

The Canonical Equivariant Closed Three-Form on G

Suppose that the Lie algebra g possesses an invariant, non-degenerate symmetric bilinear form ( , ), so

(See also [CS]:)

9.10

Group Valued Moment Maps

Let G be a compact Lie group and suppose that we put a G invariant scalar product ( , ) on its Lie algebra g. (In case G is simple, this scalar product is unique up to positive multiple.) Let 0 E R1(G,g) denote the left invariant Maurer-Cartan form. Then it is well known that the three-form

This means that the trilinear map

is antisymmetric and invariant, i.e.

We have, for v,C,E,s E g, using the invariance of ( identity is closed and bi-invariant. (We will review the proof of this fact below.) It has recently been observed cf. [AMM]that there is a equivariant version of this three-form, i.e. an equivariant three-form XG E RG(G) relative to tKe conjugation action of G on itself which is dG closed. We shall describe this below. Suppose that ( M , w ,4) is a Hamiltonian G-space. The scalar product gives an isomorphism of g* --, g, and composing this isomorphism with the moment map 4 : M --, g* we obtain a map : M -+g which we may also call the moment map. We have the exponential map

, ) and Jacobi's

([u,CI, [E,71) = (v, [C,[E,711) = (v, [[C>E1?7II) + (v, It>I/ 1 (pour m= 0, He(W(G)) s'identifie kvidemment au corps des scalaires). De mdme, l'algibre de cohomologie dc la sous-alglbrc Iw'(G) est triviale. Ce th6orhme vaut sans aucune hypothkse restrictive sur lialg&-brede Lie a(G) . I1 se dkmontre comme suit : soit k l'antiderivation de W (G) , de degrB -1, nulle sur A (G) , et d6finie

-

sur S'(G) par k(i)=& (autrement dit : l'endomorphisme cornpod kh est lYidentit6sur A1(G)). L'op6rateur k commute avec les transformations infinithimales 0 (2) , par suite k ophre dans la sous-algbbre Iw(G) des ClBments invariants de W (G) . 6k+ k6 est une ddrivation; elle est entikrement d6finie quand on la connaft sur A1(G) et sur S'(G): or elle trans-

-

-

forme tout z' E A' (G) en z' lui-m&me, et tout d E S' (G) en z' - d,d. Appelons poids d'un 61Cment de W (G) le plus grand des entiers q tels que sa compomnte &ns A (G) 8 S9(G) ne soit pas nulle (le poids ktant, par dkfinition, - 1 si 1'816ment consid6r6 est nul). Soit alors u un element homogbne de d e g d it1 (rn 1) de W (G) ; 6ku k6u est homogbne de degrk m. Soit q 0 le poids de u (u Btant suppod 0) ; le poids de

> >

+

+

est \< q- 1. Le processus qui fait passer de u 1'61Cment zj de poids strictement plus petit peut Qre-itBrC, et conduira finalement' ? uni 6lCment'nul. Supposoils que u soit uit cocyclc : 6u=0; alors v est un cocycle homologue B u, et de proche en proche on voit que u est le cobord d'un ClCment de W (G). Ceci montre bien - que Hm(W (G) ) est nul. Si en outre u est un cocycle invariant, le processus montre que u est le cobord d'un Clement invariant de W (G); donc Hm(I, (G) ) est nul.

Soit u E IsP(G) ( p >/ 1). Puisque c'est un cocycle de deer6 2 p de l'algkbre I,(G), il existe, d'aprks le thkorkme 1, un w f I,(G), de degrC 2 p- 1, tel que 6w = u. L3. projectio~~ canonique de W (G) sur A (G) tnnsforme w en un BlBment zc9, de IA(G), de degrC 2 p - 1. Cet ClCment ne depend pas drl choix de w; car si 6w1= 6w, il existe un v E I,(G) tel que w'- w =8v. hlors w,'- w, =d , ~ , , et comme t', E I,(G), d,v, est nul.

207

En associant ainsi i chaque u E r? (G) 1'616ment w, E IAZD-'(G),on d6finit une application lindaire canonique de IsP(G) duns IAZF'(G), pour toute valeur de l'entier p>/ 1; cette application sera not6e p. Les ClCments de I'image de cet homomorphisme jouissent de la propriBt6 d'&tre transgressifs dans l'alghbre I,(G). Voici ce qu'on entend par 1% : un BlBment a E I19(G) est dit transgressif s'il est l'image, par la projection canonique de W(G) sur A (G) , d'un Blkment w E I (G) dont le cobord 6w soit dans S(G), et par suite dans Is(G). Alors w s'appelle une cochaine de transgression pour a ; u n BlCment transgressif a peut avoir plusieurs cochaines de transgression. Tout ildrnent transgressif non nu1 est de degrd impair : car si a transgressif est de degrB pair, 6w est de degrC impair, et comme 6w est dans Is(G) dont tous les degCs sont pairs, 6w est nul. D'aprb le thCor&me 1, il existe alors un v E Iw(G) tel que 6v = w ; d ' o ~a = w, =dAvA,et comme v, E I, (G) , cela implique d,v, =0. Les ClCments transgressifs de I, (G) forment un sous-espace vectoriel T, (G) , engendrk par des BlCments de degr6s impairs; T,(G) est le sous-espace de 1,(G), image de l'application p. Prenons une base homogkne de T,(G), et, chaque 6ICment a de cette base, associons le cobord 6w E I,(G) d'une cochaine de transgression w. On obtient une application linhire de TA(G) dans Is(G), qu'on appellera une transgression. On peut encore dBfinir une transgression comme suit : c'est une application linthire 7 de T, (G) dans I, (G) , qui, suivie de l'application canonique p, donne l'application identique de T,(G).

Rappelons d'abord le thborkme. de Hopf ( I ) : si a (G) est rdductive, l'alghbre IA(G)s'identifie B l'algibre eztdrieure d'un sous-espace bien dBterminC PA(G) de I,(G) ; l'espace P,(G) est engendrC par des 6Mments homogBnes de degr4s impairs, appe16s ClBments primitifs de IA(G); la dimension de P,(G) est le rang r(G) du groupe G. Voici comment on dkfinit un Bl6ment homogene primitif : considkrons l'algkbre exterieure A,(G) de l'alg8bre de Lie a(G) (algkbre des chaines du groupe G) , et la sous-algkbre 1, (G) des BlBments invariants de A. !G j . L'hypothbe de rkductivitk entraine que la dualit6 canonique entre A,(G) et A(G) induit une dualit6 entre I, (G) et IA(G); ((

(') Voir thbse de Koszm., chap.

IT,S 10.

),

208

Henri Cartan

La transgression dans un groupe de Lie

=la Ptant, un element homoene de IAv(G) (p>, 1) est appel6 primitif s'il est orthogonal aux 6lCments (de degr6 p) de'composables de I,(G) (dans une algbbre graduBe quelconque, u n PlCment homogkne de degr6 p est decomposable s'il est somme de produits d'616ments homoghnes de degr6s strictement plus pelits que p). Ceci etant rappele, revenons 5 l'application p et A la transgression :

Soit alors a un element primitif de IA(G) (cocycle invariant de la fibre de l'espace 6 j ; choisissons une cochaine de transgression w (comme il a CtB dit au 5 2 ) ; alors f(w) est un 616ment de E qui induit ,, le cocycle a sur chaque fibre. Sa diff6rentielle df(w)= f (8w) est 1'616ment de B (alghbre des formes diffhrentielles de I'espace de base 03) que la connexion associe l'element 6w de I,(G). Ainsi la forme differentielle j ( w ) ( c c forme de transgression n) a pour diffPrentielle un cocycle de l'espace de base. I1 est ainsi prouvC que les cocycles invariants primitifs de la fibre sont transgressifs dans l'espace fibre principal 6 ; fait qui a d'abord kt15 mis en evidence par Koszul dans le cas particulier o ? ~ G est l'espace d'un groupe de Lie dont G est un sous-groupe (I), puis a kt15 g6neralis6 par A. Weil en se servant de la transgression dans W ( G ) , comme il vient d ' b e expliqu6. Faisons choix une fois pour toutes d'une transgression s dans W (6);alors le choix d'une connexion f dans E definit une forme de transgression y ( w ) pour chaque cocycle invariant primitif a E PA(G) ; l'application lineaire a 4 dj(w) de PA(G) dans B, appel6e cc transgression dans l'espace fibre, applique PAzv-'(G) dans BZv;elle est composee de la transgression T : PAzv-'(G) +IsD(G) , et de l'application Isv(G)+ Bz' definie par la connexion (cf. premiere confPrence, S 7). Soit y~ l'application linCaire PA(G)--+ B ainsi obtenue. Sur l'algbbre graduee IA(G)@B, il existe une antiderivation et une seule qui, sur le sous-espace PA(G) de I,(G), soit @le B a, et, sur B, soit Cgale B la diffkrentielle d de B. Cette antiddrivation A est de d q r 6 1, et son carre est nu1 : c'est une differentielle. Un thCor6me de Chevalley (') permet d'affirmer, lorsque G est un groupe compact (connexe), que l'alghbre de cohomologie de I,(G) €3B, pour la diffkrentielle A, s'identifie canoniquement B l'alghbre de cohomologie de la sous-algkbre I, des 616ments invariants de E. D'ailleurs H(1,) s'identifie canoniquement B l'alghbre H(E), algbbre de cohomologie de l'espace fibrC 6. En resum6 : la connaissance de Z'homomo~phisme I, (G)+B de'fini par une connezion de l'espace fibre' permet de difinir, sur l'alglbre I, (G) €3B, une difftrentielle pour laquelle 1'alg)bre de cohomologie s'identifie B l'alglbre de cohomologie fre'elle) de l'espace fibre'. En particulier : quand on connait I'espace de base U3, et l'homomorphisme I,(G)+ B dBfini par une connexion, on connait l'alghbre de cohomologie (rkelle) de l'espace fibre. ((

TIIBOX&ME 2. - Si l'alghbre de Lie est re'ductive, l'image de l'application canonique p est l'espace P,(G) des tle'ments primitifs de IA(G) (autrement dit, PA(G) est identique d l'espace T,(G) des tle'ments transgressifs). Le noyau de l'application p est forme' des tle'ments dicomposables de I,(G). Ce thhrbme a d'abord Ctt5 conjecture5 par A. Weil en mai 1949; le fait que tout element primitif est transgressif a nussitbt kt6 prouv6 par Chevalley, s'inspirant d'une demonstration donnee par Koszul du theorbme de transgression de sa these (th. 18.3). Puis H. Cartan a d8montr6 qu'il n'y a, dans l'image de p, que des BlCments primitifs, et que le noyau est forme exactement des BlBments d6composables de I,(G). Le thborbme 2 (qu'il n'est pas question de demontrer ici) entraine ceci : pour toute transgression T : P,(G) +I,(G), l'image de .; engendre (nu sens multiplicatif) l'alghbre (commutative) I, (G) . Une Ptude plus approfondie (Chevalley, Koszul; cf. la conference de Koszul A ce Colloque) montre que les transform6s, par une transgression r , des dlements d'une base homogbne de PA(G) , sont alge'briquement jnde'pendants dans Is(G) ; par suite I,(G) a la structure d'une algZbre de polyndmes A r(G) variables (r(G) : rang du groupe G). D'une facon plus precise, le nombre des gBnCrateurs de poids p de l'algkbre I,(G] est Cgal B la dimension de l'espace des 6lBments primitifs de I, (G) , de degrk 2 p - 1. Ce resultat relatif B la structure de l'algbbre I,(G) est le pendant du theorbme de Hopf sur la structure de l'algbbre I* (GI-

L'algbbre de Lie est desormais supposee re'ductive. Soit, avec les notations de la premihre conference, E l'algbbre des formes diff6rentielles d'un espace fibre principal 6, de groupe G (groupe de Lie connexe, tel que son alghbre de Lie a ( G ) soit reductive). Choisissons une connexion infinit& simale dans 6;. elle dCfinit un homomorphisme f de W(G) dans E, compatible avec les graduations et tous les op6rateurs.

209

((

))

)>

+

,

(') T W e . theoreme 18.3. (') V ~ i rla conference de Koszul

A ce Colloque.

210

Henri Cartan

La transgression dans un groupe de Lie

Nous nous inthressons dksormais au problbme inverse du preddent : il s'agit de trouver un processus qui permette, de la cohomologie H(E) de l'espace fibrh, de passer B la cohomologie H(B) de l'espace de base. Pour cela, nous nous placerons dans le cadre algkbrique gknhral : E est une algbbre diffhntielle gradu6e dans laquelle opbre u n groupe de Lie G (dans le sens du S 4 de la premi6re confhrence); B est alors la sous-algebre des Cl6mcnis basiques de E. Considhrons le produit tensoriel E 8 W (G) (produit tensoriel d'algbbres gndukes) . C'est une alg6bre m d u b e , sur lnquelle nous considerons la diffkrentielle-8 qui-prolonge la differentielle d de E et la diffkrentielle 6 de W (G). nlus. , - , De - - r---, les antidkrivations i(z) (d6jB dkfinies sur E et sur W (G)) se prolongent en antidCrivations de E@W(G), que l'on notera encore i ( r j ; on definit de meme les derivations 9(z) sur E 8 \V (G). I1 est clair que les relations (I), (11) et (In) de la premiere conf6rence (ofi d serait remplack par g) sont satisfaites sur E @ W (G) , puisqu'elles le sont sur E et sur W (G) . Soit B la sous-alpebre des elements basiques de EO W (G) : elements annulks par les i(z) et les b ( t ) . Elle est stable pour%, et l'on peut considerer l'algkbre cle col~omologieH(B). Si I'on songe l'interpr6tation dc W(G] rollll~~e alghbre de cochdnes d'un espace f i r 6 universe1 (cf. S 8 clc la premiere confkrence), les OHrations prickientcs admettent I'interpretation gbm6trique suivante : soit &I un espace fibri classifiaut; considerons I'espace produit 6 X &I, ct faisons-y op6rer Ie grnupc (; par la loi : (P, PO + (P - s, PI . s). L'espace quotient 3; cst un esparc fibr6 dc m&mebase (A et de fibre &. L'alglrbre E OW(G) Jooc nlors lc r61c tlc I'alghbre des cochaines de I'espace & X & I , et B joue Ic rdle de l'alglrbre des cochaines de l'espace fibre 5 . Or, dans un sens :I prkiser, la fibre 6 1 est cc triviale a; cela laisse supposer que la coholnologie de 3 s'identific B la whomologie de l'espace de base Cij En fait,-nous allons voir que, sous certaines hypoth6~t.s.0 1 1 rwul identifier H(B) ct H(B).

Les algbbres differentielles. B et I,(G) (lYopCrateurdiffkrentiel de la seconde est d'nilleurs nnl) s'identifient canoniquement ti des sous-algbbres de l'alg2tbre diffhrentielle E. On en dbduit des homomorphismes crrnoi~iques TH~ORBME 3. - S'il eziste uite I, coi~i~ezion(au sens algkbrique du mot) dons E, Z'l,o~nornorpl~isme H (B) 4 H (E) est zzn isomorpltisme de H (B) sur H(B). ,)

.

211

Ceci ktant ndmis, le second homomorphisme (1) donne u n homomorphisme I,(G)--t H (B) , et on voit kcilement que c'est pdcishment celui que dCfinit la connexion (premibre confkrence, 5 7). Par condquent, l'homomorphisme dkfini par une connexion est indipendant dc la connezion, comme il avait kt6 annonc6. Pour demontrer le thkodme 3, on utilise sur E@W(G)=E@A(G)@S(G), I'antidhrivation k, de d-6 -1, qui est nulle sur E et A(G), et est dkfinie, sur S1(G), par k(lBl@d)=i@d@l-f (~')818i,

-

f designant l'application A' (G)+ E' d6finie par la connexion. On raisonne alors comme dans la dkmonstration du thCorbme 1, en considkrant la derivation xk kE; la dcurrence est un peu plus subtile, elle permet de montrer que tout Blkment de B dont la diffhrentielle est dans B est la sopme d'un BlBment de B et de la diffkrentielle d'un klkment de B.

+

-B est

contenue dans la sous-algbbre de E 8 A(G) 8 S (G) formke des BlCments annul& par les produits intkrieurs i(z), c'est-B-dire dans le produit tensoriel F@S(G), F designant la sous-algbbre de E B A (G) formke des BlCments nnnulhs par les i(z). D'une facon prkcise, @ s'identifie B la sous-algbbre des 6Mments invariants de F 8S (G) . Pour interpniter F, considkrons la projection anonique de EOA(G) sur E; elle commute avec les Q(z)et applique buznivoquement la sous-algbbre F sur E, comme on s'en assure aiskment. D'oh un isomorphisme canonique de F sur E, qui qrmet d'identifier B h la soos-nlgbbre C des Clkments inaunants de E@S(G). Reste ii erpliciter la diffkrentielle A que l'on obtient en transportant B C la diffkrentielle de % : on trouve .que A est induite, sur C, par la diff6rence d- h des antidhrivations d et 12 de E@ S (G) que voici : d se rhduit, sur E, & la diffkrentielle de E, et est nulle sur S(G); h est nulle sur S (G) , et est donnke par la formule

-

IL

= .

(oh (z,) et

(2,')

i (r.) e (o:)

(2)

k

sonF deux bases duales de a(G) et A1(G),

212

-

Henri Cartan

z,'

dksignant la multiplication par dans I'alghbre e(z,') EBS(G)). On notera que le carrk de d - h n'est pas nu1 en gknkral; mais sa restriction A ?t la sous-algebre C a un c a r d nul. Reste B voir ce que deviennent les homomorphismes 1, (G)+ H (B) et . H(B)--t H (1,) quand on identifie H (B) B H (C) . On voit aussit6t que le premier est dCfini en consid6rant I,(G) comme une sous-alggbre de C, tandis que le second s'obtient B partir de l'application de C sur I, dkfinie par la projection canonique de EBS(G) sur E. RCsumons :

'

THEOREME4. - S'il eziste u n e connezion dans E, l'algLbre de cohornologie H(B) de la sous-algibre B des tltments basiques de E s'identifie canoniquement d l'alglbre de wlzomologie H(C) de la sous-algibre C des kltments invariants de E @S(G), rnunie de la diffkrentielle A ezplicitb ci-dessus. Par devient cette identification, l'homomorphisme I,(G)+B(B) l'hornomorphisrne I, (G)+ H (C) obtenu en considtrant I, (GI wrnrne sous-alg6bre de C, et l'hornomorphisrne H(B)--t H(I,'J devient l'homomorphisrne H (C)+ H (I,) obtenu en considCrant I, comme algibre quotient de C. Observons que si G est un proupe compact (connexe), ou si, E etant de dimension finie, a ( G ) est rkductive, H(1,) s'identifie canoniquement ? Hi (E) . Remargue. - Examinons le cas o i ~E est l'nlgsbre A(GI elle-mCme, la connexion Ctant dkfinie par l'application identique de A1(G) dans A1(G) (cf. premitre conference, fin du $ 6). Alors C est la sous-alglrbre des Blements invariants de A (GI@ S (G) , c'est-A-dire la sous-alghbre I,(G) ; I'opCrateur Ir est le mCme que celui dCfini par la formule ( I f ? ) de la premigre confkrence; et on constate que A =d - h , sur I, (G) , est Cgale 3 - 8, 8 Ctant la diffkrentielle de l'nlggbre de \Veil W (G) .

La transgression dans un groupe de Lie

213

due H(E) B I,(G) sur un sous-espace vectoriel de I'espace vectoriel graduk C, application qui conserve les degr6s et possede les proprikt6s suivantes : 1. Sur I,(G) (sous-alggbre de H (E) BI,(G) ) , 5 se reduit ?t l'application identique (I,(G) Ctant aussi identifiie B une sous-algtbre de C) ; 2. q~ applique chaque ClCment a €3 1 de H (E) €3 1 sur un ilkment de C dont la projection canonique (ClCment de I,) est u n cocycle de la classe de a; 3. L'image de H (E) €3 I, (G) par (s est stable pour 8. G r b e B 9,on peut alors identifier H (E)@ I, (G) B un sousespace vectoriel de C, ce qui dCfinit sur H (E)€3 I, (G) un operateur cobord (de carrk nul, obtenu en transportant A); et l'on montre que l'application de l'espace de cohomologie de H(E) @ Is(G) (relatif B cet operateur cobord) dans H(C) est biunivoque sur. (Par contre, comme (G n'est pas, en gknCra1. u n homomorphisme multiplimtif, il n'y a pas de structure multiplicative sur H (H (E) @ I, (G) .) Finalement, on voit qu'il existe sur H (E) €31s(G) un opkrateur cobord, de degre 1, nu1 sur I, (G) , et c~uiapplique H(E) dans l'id6al engendrC par I,"(G) ; de plus, il existe un isomorphisme de l'espace de cohomologie H (H (E) €3 Is(G) ) sur H(B), compatible avec les homomorphismes du diagramme H (H(E) €3 Is (GI)

+

7

I

\

H(R) Application. - Sypposons que b soit un espace fibrC principal de groupe compact connexe G, classijiant pour la dimension N (cf. S 8 de la premihre confkrence). Alors les espaces de cohomologie Hm(E) sont nuls pour 1 m \( N , et Ho(E) Se rkduit au corps des scalaires. Dans ces conditions, l'espace de cohomologie de H(E) @ I,(G) s'identifie 21 Is(G) pour tous les d e e d s m< N. Ceci prouve que l'homomorphisme I s ( G ) - + H (B) est un isomorphisrne de Isp(G) sur H" (B) pour 2 p K, et que Hm(B) est nu1 pour les valeurs irnpaires de In N. C'est le rksultat annonce B la fin de la premiere confkrence.




1

I., (G) = H (G) Le noyau de l'hotno~i-rorphisti~c I (g) -+ H (G/g ) csl I'idPnl J ettgendri, duns l'alg$bre I (g-), pcrr l'i,t~~agr dc. I+(G). Doric la sous-algPbre curactiristii~ucde H (Gig) est canonicluement

'

*.=Z!q 9 4

d

=4

218

Henri Cartan

La transgression dans un groupe de Lie

isomorphe B I'nlghbre quotient I(g) l J . Rnppelons yue ses 614ments sont de degrts pairs. L'itnage de H(G/g) dans H(G) est une sous-alghbre (que nous noterons H, (G) ) engenddr par un sous-espace P,(G) de I'espice P(G) des elements primitifs de H(G) ('). On obtient P,(G) de la facon suivante : la differentielle E applique P(G! sur un sous-espilce \- de I ( g ) ; soit J1 l'id6al de I ( g ) , form6 des combinnisons linkires d'elfments de 1- B coefficients dans I+(g); J1 est contenu dans J et independant du choix de E. Alors P, (G) est 2e sous-espacc de P (G) fortnd dcs Pltr~zentsque E applique duns J'. La dimension de I'espace vectoriel P,(G) est au plus tgale 6 la diffkrei~cer(G)- r ( g ) des rungs de G et de g. D'autre part, l'image de I+(g) dans H(G/g) est toujours contenue dans le noyau de l'homomorphisme H(G/g)--tH(G); pour que ce noyatr soif ezactcment l'ide'al engeizdri par les e'liments de degrP' 0 de In sous-algibre curocte'ristique. il faut et il suffit que dim P, (G)= r (G)- r (g) . (3)

>

La condition (3) est trivialement v6rifiCe si r ( g ) = r ( G ) ; dans ce cas, H (GIg) est canoniquement isomorphe i I(g) /J. Elle est aussi verifiCe quand l'espace homoghne G/g est synzPtrigue (au sens de E. Cartan). Chaque fois qu'elle est vtrifide, H(G/g) s'identifie au produit tensoriel d'algbbres ( I ( g ) / J ) @H"(G) , et on a, d'aprbs Koszul, une (I formule de Hirsch qui donne le polyn8me de PoincarC de I(g) / J (cf. confPrence de IEioszul) connaissant les polyn6mes de PoincnrC de H(G) et de H ( g i , ainsi que les degrCs des Clements primitifs de H,(G), on trouve immtdintement le polynbme de Poincar6 de H(G/g). Si dim Po(G) r (G)- r (g) , I'algbbre H (G/g) est encore isomorphe i un prodrlit tensoriel I\; @ H, ( G ), mais la structure de l'algghre I; est plus compliqu6e que lorsque (3) a lieu : K contient alors, outre une sous-alghbre isomorphe i I ( g ) / J . des genernteurs de degre' itnpair. Signalons qu'il esisle des a s simples (J. Leray, -4. BorelY oil r ( g ) r(G) , et oil nkanmoins H,(G) est rPduit B 0. En application des rCsultnts relntifs au cas oil (3) a lieu. on peut dgterminer explicitement les polyn8mes de Poincnd des gm~smanniennesr6elles G,,, (il s'agit des prassmnnniennes orientCes u : G,,. dCsigne I'espace des sous-espnces vertoriel~ ))