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 <,, i = 1,. . . ,n be a basis of g, xi, i = 1,. . . ,n the dual basis of g* and 4, the 6-th component of 4. The identities (0.27) can be interpreted as saying that the equivariant two-form
.
is closed. This trivial fact has a number of surprisingly deep applications and we will discuss three of them in Chapter 9: the Kostant-Kirillov theorem, the Duistermaat-Heckmann theorem and its consequences, and the "minimal coupling" theorem of Sternberg. We also give a short introduction to the notion of groupvalued moment map recently introduced by Alekseev, Mallcin and Meinrenken [AMM].
xx
Introduction
Introduction
0.9 The last two chapters of this monograph will deal with localiation theorems. In Chapter 10 we will discuss the well-known Abelian localiation theorem of Berline-Vergne and Atiyah-Bott and in Chapter 11 a related "abstractn localization theorem of Bore1 and Hsiang. &om now on we will assume that G is abe1ian.l Let M be a compact oriented d-dimensional G-manifold. The integration map
is a morphism of G* modules, so it induces a map on cohomology
and the localization theorem is an explicit formula for (0.29) in terms of fixed point data. If MG is finite it asserts that
p being a closed equivariant form, i;p its restriction t o p and a,,,, i = 1,. . .d, the weights of the isotropy representation of G on the tangent space to p. (More generally, if M G is infinite, it asserts that
the Fklsbeing the connected components of M~ and ek being the equivariant Euler class of the normal bundle of Fk.)TO prove this formula we will fist +of all describe how to define "push-forward" operations (or ILGysinmaps") in equivariant de Rharn theory; i.e., we will show that if MI and Mz are G-manifolds and f : MI 4 Mz a G-map which is proper there is a natural "push-forward" f* : H ~ ( M ~ ) H&+'(M~), (0.32) t? being the difference between the dimension of M2 and the dimension of M1. To construct this map we will need to define the equivariant Thom form for a pair, (M, E), consisting of a G-manifold, M, and a vector bundle E over M on which G acts by vector bundle automorphisms; and, following Mathai-Quillen, we will show how this can be defined in terms of the unzversal 'We will prove in Chapter 10 that for a localization theorem of the form (0.31) to be true, the Euler class of the normal bundle of M G has to be invertible, and that this more or less forces G to be Abelian. For G non-Abelian there is a more complicated localization theorem due to Witten [Wi] and Jeffrey-Kirwan [JK] in which the integration operation (0.29) gets replaced by a more subtle integration operation called "Kirwan integration".
'
xxi
equivariant Thom form described above. We will then show, following AtiyahBott, that the localization theorem is equivalent to the identity
.
;
i being the inclusion map of MG into M and e being the equivariant Euler class of the normal bundle of MG.
The following theorem of Borel and Hsiang, which we will discuss in Chapter 11, is a kind of "raison d'6tren for formulas of the type (0.30) and (0.31). T h e o r e m 0.0.2 Abstract localization theorem The kernel of the restriction map i* : HG(M) -+ H G ( M ~ ) (0.34) is the set of torsion elements in HG(M), i.e., b is in this kernel if and only if there exists a p E S(g*) with p # 0 and pb = 0 .
From this the identities (0.30) and (0.31) can be deduced as follows. It is clear that the map (0.29) iszero on torsion elements; so it factors through the map (0.34). In other words there is a formal integration operation
whose composition with i* is the map (0.29); and, given the fact that such an operation exists, it is not hard to deduce the formula (0.31) by checking what it does on Thom classes. Another application of the abstract localization theorem is the following: We recall that there is a spectral sequence whose El term is the tensor product (0.20) and whose E-term is HG(M). Following Goresky-KottwitzMacPherson, we will say that M is equivariantly formal if this spectral sequence collapses. (See [GKM], Theorem 14.1 for a number of alternative characterizations of this property. We will discuss several of these alternative formulations in the Bibliographical Notes to Chapter 11.) If M is equivariantly formal, then by (0.20) the cohomology groups of M are
and in fact we will prove in Chapter 5 that if M is equivariantly formal,
as an S(g')-module. We will now show that the Borel-Hsiang theorem gives one some information about the ring structure of HG(M). If M is equivariantly formal, then, by (0.37), HG(M) is free as an S(ga) module; so the
xxii
Introduction
Introduction
submodule of torsion elements is (0). Hence, by Borel-Hsiang, the map
and hence
i* : H G ( M ) --+ H ~ ( M ~ )
H G ( M ~=) Maps (Vr, S ( g * ) )
a
k
is injective. However, the structure of the ring H c ( M G ) is much simpler than that of H G ( M ) ;namely
,
where Vr is the set of vertices of
a
is in the image of the embedding
n
if and only if, for every edge, C, of the intersection graph, I?, i t satisfies the compatibility condition r h ~ ( v 1= ) rh~(v2) (0.44)
H
For every codimension-one subtorus, H
, of G ,dim M~ 5 2 .
I
vl and v2 being the vertices of C, and h being the Lie algebra of the p u p (0.42) and (0.45) T,, : S(g8)4 S ( h f ) being the restriction map. !
(0.40)
Given this assumption, one can show that there are a finite number of codimension-one subtori H * , i = l , ... , N with the property dim M ~ =. 2 , and if H is not one of these exce~tionalsubtori M H = MG. Moreover, if H is one of these exceptional subtori, the connected components, Ci,j,of MH' are 2-spheres, and each of these 2-spheres intersects M G in exactly two points (a "north pole" and a "south pole"). For i fixed, the Xij's can't intersect each other; however, for different i's, they can intersect at points of M G ; and their intersection properties can be described by an "intersection graph", I?, whose edges are the C,,,'s and whose vertices are the points of MG. (Two vertices, p and q, of are joined by an edge, C, if C n MG = {p,q).) Moreover, for each C there is a unique, Hi, on the list (0.41.) for which
so the edges of I? are labeled by the Hi's on this list. Since M G is finite,
r.
Theorem 0.0.3 ([GKM])A n element, p, of the ring
f
so one will have more or less unraveled the ring structure of H G ( M ) if one can describe how the image of in sits inside this ring. Fortunately there is a very nice description of the image of i*, due to Chang and Skjelbred, which says that i * H G ( M )= i ; H G ( M H ) (0.39) the intersection being over all codirnension-one subtori, H, of G and i H being the inclusion map of MG into M H . ( A proof of this using de Rham-theoretic techniques, by Michel Brion and Michele Vergne, will be given in Chapter 11.) If one is willing to strengthen a bit the assumption of "equivalently formal" one can give a much more precise description of the right hand side of (0.39). Let us assume that MG is finite and in addition let us make the assumption
xxiii
.
As we mentioned at the beginning of this introduction, the results that we've described above involve contributions by many people. The issue of provenance--who contributed what-is not easy to sort out in an area as active as this; however, we've added a bibliographical appendix to each chapter in which we attempt to set straight the historical record in so far as we can. (There is also a more personal historical record consisting of the contributions of our friends and colleagues to this project. This record is harder to set straighi; however, there is one person above all to whom we would l i i to express our gratitude: It is to Rmul Bott that we owe our initiation into the mysteries of this subject many years ago, in the Spring of 1982 at Bures-sur Yvette just after he and Atiyah had discovered their version of the localization theorem. The ur-draft of this manuscript was twenty pages of handwritten notes based on his lectures to us at that time. We would also like to thank Matthew Leingang and C Z t S i Zara for helping us to revise the first draft of this monograph and for suggesting a large number of improvements in style and content .)
Chapter 1
Equivariant Cohomology in Topology Let G be a compact Lie group acting on a topological space X . We say that this action is free if, for every p E X , the stabilizer group of p consists solely of the identity. In other words, the action is free if, for wery a E G, a # e, the action of a on X has no fixed points. If G acts freely on X then the quotient space X / G .is usually as nice a topological space as X itself. For instance, if X is a manifold then so is X/G. The definition of the equivariant cohomology group, H z ( X ) is motivated by the principle that if G acts freely on X , then the equivariant cohomology groups of X should be just the cohomology groups of X/G:
H&(X)= F ( X / G ) when the action is free.
(1.1)
For example, if we let G act on itself by left multiplication this implies that
If the action is not free, the space X / G might be somewhat pathological from the point of view of cohomology theory. Then the idea is that H z ( X ) is the "correctn substitute for H*(X/G).
1.B
Equivariant Cohomology via Classifying Bundles
Cohomology is unchanged by homotopy equivalence. So our motivating principle suggests that the equivariant cohomology of X should be the ordinary cohomology of X * / G where X* is a topological space homotopy equivalent to X and on which G does act freely. The standard way of constructing
2
Chapter 1. Equivariant Cohomology in Topology
1.1 Equivariant Cohomology via Classifying Bundles
such a space is to take it to be the product X * = X x E where E is a contractible space on which G acts freely. Thus the standard way of defining the equivariant cohomology groups of X is by the recipe H E ( X ) := H* ( ( X x E ) / G ) .
onto the first factor gives rise to a map
which is a fibration with typical fiber E . Since E is contractible we conclude that H E ( X ) = H* ( ( X x E ) / G ) = H * ( X / G ) , in compliance with (1.1). Notice also that since (1.4) is a fiber bundle over X / G with contractible fiber, it admits a global crosssection
X-E
h
-
commute. Conversely, every G-equiuariant map h : X 4 E determanes a section s : X / G ( X x E ) / G and a map f which makes (1.8) commute. Any two such sections are homotopic and hence the homotopy class of ( f ,h ) is unique, independent of the choice o f s .
Proof. Let y E X / G and consider the preirnage of y in ( X x E ) / G . This preimage consists of all pairs
modulo the equivalence relation ( x ,e ) .- (ax,ae), a E G.
Each such equivalence class can be thought of as the graph of a G-equivariant map T - ' ( ~ )-+ E.
The projection XxE-.E
gives rise to a map
which makes the diagram
(1.3)
We wiIl discuss the legitimacy of this definition below. We must show that it does not depend on the choice of E . Before doing so we note that if G acts freely on X then the projection
onto the second f&r
3
.
So s(y) determines such a map for every y. In other words we have defined h : X -+ E by the formula
Composing (1.6) with the section s gives rise to a map
Let
-
where [ ] denotes equivalence class modulo G. The deiinition (1.7) of f then says that the square (1.8) commutes. Since the fibers of ( X x E ) / G X / G are contractible, any two cross-sections are homotopic, proving the last assertion in the proposition. Proposition 1.1.1 is usually stated as a theorem about principal bundles: Since G acts freely on X we can consider X as a principal bundle over
be the projections of X and E onto their quotient spaces under the respective G-actions. Proposition 1.1.1 Suppose that G acts freely on X and that E is a contmdible space on which G acts freely. Any cross-section s : X / G 4 ( X x E ) / G determines a unique G-equivariant map
S i a r l y we can regard E as a principal bundle over
B := E / G . Proposition 1.1.1 is then equivalent to the following "classiiication theorem" for principal bundles:
4
Chapter 1. Equivariant Cohomology in Topology
Theorem 1.1.1 Let Y be a topological space and G-bundle. Then there ezists a map
?r
:
1.2 Existence of Classifying Spaces
X
-+
Y a prineiM
1.2
E
j and an isomorphism of principal bundles
i"
-+
B to X. Moreover f and @
I
!
F
Remarks.
the space of square integrable functions on the positive real numbers relative to Lebesgue measure. But of course all separable Hilbert spaces are isomorphic. Let E consist of the set of all n-tuples
1. f * E = {(y, e)Jf (y) = p(e)) so the projection (y, e) H y makes f ' E into a principal G-bundle over X. This is the construction of the pull-back bundle. 2. We can reformulate Theorem 1.1.1 as saying that there is a one-to-one correspondence between equivalence classes of principal G-bundles and homotopy classes of mappings f : Y -+ B. In other words, Theorem 1.1.1 reduces the classification problem for principal G bundles over Y to the homotopy problem of classifying maps of Y into B up to homotopy. For this reason the space B is called the classifying space for G and the bundle E -+ B is called the classifyzng bundle .
The g o u p U ( n )acts on E by
Av = w = (wl,. . . , w,,),
q3uJ.
(1.9)
This action is clearly free. So we will have proved the existence of classifying spaces for any compact Lie group once we prove:
Theorem 1.1.2 If El and E2 are contractible spaces on which G acts freely, they are equivalent as G-spaces. In other words there exist G-equivariant maps + E2,
w, = 3
One important consequence of Thbrem 1.1.1 is:
4 : El
Existence of Classifying Spaces
Theorem 1.1.3 says that our definition of equivariant cohomology does not depend on which E we choose. But does such an E exist? In other words, given a compact Lie group G can we find a contractible space E on which G acts freely? If G is a subgroup of the compact Lie group K and we have found an E that ' h r k s " for K, then restricting the K-action to the subgroup G produces a free G-action. Every compact Lie group has a faithful linear representation, which means that it can be embedded as a subgroup of U ( n ) for large enough n. So it is enough for us to construct a space E which is contractible and on which U ( n ) acts freely. Let V be an infinite dimensional separable Hilbert space. To be precise, take v = L~[o,oo),
I
where f*E is the 'bull-back" of the bundle E are unique up to homotopy.
5
Proposition 1.2.1 The space E is contractible. We reduce the proof to two steps. To emphasize that we are working within the model where V = L2[0,oo) we will denote elements of V by f or g. Let E' C E consist of n-tuples of functions which all vanish on the interval [O, 11.
4 : E2 -+ El
with G-equivariant homotopies
Lemma 1.2.1 There is a defownatzon retract of E onto E' Proof. For any f E V define Ttf by The existence of q5 follows from Theorem 1.1.1 with X = El Proof. and E = EZ.Similarly the existence of $ follows from Theorem 1.1.1 with X = E2 and E = El. Both idE, and $04are maps of El + El satisfying the conditions of Theorem 1.1.1 and so are homotopic to one another. Similarly for the homotopy 4 o $ id&. A consequence of Theorem 1.1.2 is: Theorem 1.1.3 The definition (1.3) is independent of the choice of E .
T t f ( z )= Define
,
{
Of ( x - t )
for
0 < x < t;
for
t<x
Ttf = ( T t f i , .. . , Ttf,), for f = ( f i , . . . ,f,). Since Tt preserves scalar products we see that Tt is a deformation retract of E onto E'.
6
Chapter 1. Equivariant Cohomology in Topology
1.3 Bibliographical Notes for Chapter 1
Notice that wery component of f is orthogonal in V to any function g E V which is supported in [0, 11. Therefore if f E E' and g E E has all its components supported in [O,1] the "rotated frame7' given by f
=
(
A
-2t ) h
A
+ (sin -t)gl.. 2
A
. . ,(COS?t)fn
A
+ (sin -t)gn) 2
7
of ,(Xx E)/G and hence give a de %am-theoretic definition of the cohomology groups (1.3). The details will be described in Chapter 2.
5. For G = S1, Ek is just the (2k + 1)-sphere
belongs to E for all t.
Lemma 1.2.2 E' is contractible to a point within E. Proof. Pick a point g all of whose components are supported in [O, I]. Then for any f E E' the curve rtf as defined above starts at f when t = 0 and ends atgwhent=l.
1.3 Bibliographical Notes for Chapter 1
Consider the map of S2k+'onto the standard k-simplex
-
One can reconstruct S2k+1 from this map by considering the relation: z z1 iff y(z) = y(zl). This gives one a description of SZk+'as the product
1. The definition (1.3) and most of the results outlined in this chapter are
due to Borel (See [Bo]). The proof we've given of the contractibility of the space of orthonormal n-frames in L2[0,m) is related to Kuiper's proof ([Ku]) of the contractibility of the unitary group of Hiibert space. 2. The space E that we have constructed is not finite-dimensional, in particular not a finite-dimensional manifold. In order to obtain an object which can play the role of E in de Rham theory, we will be forced to reformulate some of the properties of G-actians on manifolds, like "free-ness" and "contractibility" in a more algebraic language. Having done this (in chapter 2), we will come back to the question of how to give a de Rham theoretic definition of the cohomology groups H E ( M ) . 3. Let C be a category of topological spaces (e.g. differentiable manifolds, finite CW complexes, ...). A topological space E is said to be contractible with respect t o C if, for X € C, every continuous map of X into E is contractible to a point. In our definition (1.3) one can weaken the assumption that E be contractible. If X E C it suffices to assume that E is contractible with respect to C. (It's easy to see that the proof of the theorems of this chapter are unaffected by this assumption.) 4. For the category C of finite dimensional manifolds a standard choice of
E is the direct limit lim Ek,
k-w
Ek being the space of orthonormal n-frames in Ck+', k 2 n. This space has a slightly nicer topology than does the "E" described in section 1.2. Moreover, even though this space is not a finite-dimensional manifold, it does have a nice de Rham complex. In fact, for any finitedimensional manifold, X, we will be able to define the de Rham complex
modulo the identifications (z, t)
-
(z', t') iff ti = t: and q = zi where t, # 0 .
Milnor observed that if one replaces S1 by G in this construction, one gets a topological space E6 on which G acts freely (by its diagonal action on Gk+'): Moreover, he proves that if X is a finite CW-complex, then, for k sufficiently large, every continuous map of X into E i is contractible to a point. (For more about this beautifuI construction see [Mil-1 6. Except for the material that we have already codered in this chapter,
the rest of the book will be devoted to the study of the equivariant cohomology groups of manifolds as defined by Cartan and Weil using equivariant de %am theory. In particular, we will be essentially ignoring the purely topological side of the subject, in which the objects studied are arbitrary topological spaces X with group actions, and Hc(X) is defined by the method of Borel as described in this chapter. For an introduction to the topological side of the subject, the two basic classical references are [Bo] and [Hs]. A very good modern treatment of the subject is to be found in [AP].
Chapter 2
G* Modules Throughout the rest of this monograph we will use a restricted version of the Einstein summation convention : A summation is implied whenever a repeated Latin letter occurs as a superscript and a subscript, but not if the repeated index is a Greek letter. So, for example, if g is a Lie algebra, and we have fixed a basis, El,. .. ,Enof g, we have
where the basis.
2.1
ej are called the structure constants of g relative to our chosen
Differential-Geo'metric Identities
Let G be a Lie group with Lie algebra g, and suppose that we are given a smooth action of G on a differentiable manifold M. So to each a E G we have a smooth transformation
such that 4ob = 4a
O
4b-
Let R(M) denote the de Rham complex of M , i.e., the ring of differential for& together with the operator d. We get a representation p = pM of G on R(M) where paw = (4i1)*w, a E G, w E Q ( M ) . We will usually drop the symbol p and simply write
10
Chapter 2.
GtModules
2.2 The Language of Superalgebra
-
We get a corresponding representation of the Lie algebra g of G which we denote by LC,where
<
11
which is an odd derivation (of degree +l) in that
These operators satisfy the following fundamental differential-geometric identities (the Weil equations): The operator LC : R(M) -, R(M) is an even derivation (more precisely, a derivation of degree zero) in that
and where we have dropped the usual wedge product sign in the multiplication in R(M). Let us be explicit about the convention we are using in (2.1) and will follow hereafter: The element E g defines a one parameter subgroup t I-+ exp t< of G, and hence the action of G on M restricts to an action of this one parameter on M. This one parameter group of transformations has an " W t e s i m a l generatorn, that is, a vector field which generates it. We may denote this vector field by <& so that the value of <& at x E M is given by
<
Furthermore, pa
0
L~ o p a l = Lado€
and pa OP;' = (.Ad,€ (2.9) where Ad denotes the adjoint representation of G on g. In terms of our basis, we will always use the shortcut notation
L3 := LC,
and
L3
:= LC,
and so can write equations (2.2)-(2.7) as However the representation pa is given by paw = q5z1*w and hence, to get an action of g on R(M) we must consider t h e Lie derivative with respect to the infinitesimal generator of the one parameter group t H exp(-Y), which is the vector field '
+ <w= -<M.
<
and is an odd derivation (more precisely, a derivation of degree -1) in the sense that
Finally, we have the exterior difFerential I
d :R ~ ( M ) a k + l ( ~ ) -+
+LjL,
L,Lj - LjL, d~i+r,d dLi - Lid d2
<
We wi!l call this vector field the "vector field corresponding to on M," and, as above, write LC for the Lie derivative with respect to this vector field, instead of the more awkward LC;. We also have the operation of interior product by the vector field correWe denote it by L C . SO,for each E g, sponding to
<.
LiLj
= 0,
LiLj - LjLi = cfj 6 k r = Ck,~k, =
L;,
= 0 i 0.
One of the key ideas of Cartan's papers was to regard these identities as being more or less the definition of a G-action on M . Nowadays, we would use the language of "super" mathematics and express equations (2.2)-(2.7) or (2.10)-(2.15) as defining a Lie superalgebra. We pause to review this language. \
2.2
The Language of Superalgebra
In the world of "super" mathematics all vector spaces and algebras are graded over 2/22. So a supervedor space, or simply a vector s p x e is a vector space V with a 2 / 2 2 gradation:
Chapter 2. ' G Modules
12
2.2 The Language of Superalgebra
where 2 / 2 2 = {0,1) in the obvious notation. An element of Vo is called even, and an element of VI is called odd. Most of the time, our vector spaces will come equipped with a Zgradation
in which case it is understood that an element of Vzj is even:
[L,M] := L M - ( - I ) ~ ~ M Lif L E Endi V, M E Endj V.
13 (2.16)
We now recognize all the expressions on the left hand sides of equations (2.2)-(2.7) as commutators. More generally, we define the commutator of any two elements in any associative superalgebra A in exactly the same way:
[L,MI := L M - ( - I ) ~ ~ M L if L E Ai, M E Aj. An associative algebra is called (super)commutative if the commu-
and an element of V23+1is odd:
tator of any two elements vanishes. So, for example, the algebra R(M) of all differential forms on a manifold is a commutative superalgebra.
An element of is said to have degree i. A superalgebra (or just algebra) is a supervector space A with a multi-
A (Z-graded) Lie superalgebra is a Zgraded vector space
plication satisfying equipped with a bracket operation if A is Zgraded. For example, if V is a supervector space then EndV is a superalgebra where (End V)i := {A
E
End VIA : Vj
4
or (EndV), := {A E End VIA I V, -+ T + ~ ) in the Zgraded case, if only finitely many of the V , # (0) (which will frequently be the case in our applications). We will also write Endi(V) instead of (End V), as a more pleasant notation. (In the case that infinitely many of the V, # 0, End V is not the direct sum of the End, V: an element of EndV might, for example, have infinitely many different degrees even if it were homogeneous on each V,. In this case, we define Endz V := @ Endi V.) The basic rule in supermathematics (Quillen's law) is that all definitions which involve moving one symbol past another (in ordinary mathematics) cost a sign when both symbols are odd in supermathematics. We now turn to a list of examples of Quillen's law, all of which we will use later on: Examples. o
which is (super) anticommutative in the sense that
The supercommutator (or just the commutator) of two endomorphisms of a (super)vector space is defined a s
and satisfies the s u p e r version of t h e Jacobi identity
For example, if g is an ordinary Lie algebra in the old-fashioned sense, and we have +osen a basis, G , ...,En of g, define 3 to be the Lie superalgebra 3:=g-1@90@91 where g-1 is an n-dimensional vector space with basis L I , . ..,L,,, where go is an n-dimensional vector space with basis L1,. . .,Ln and where gl is a one-dimensional vector space with basis d. The bracket is defined in terms of this basis by
14
Chapter 2. G* Modules 2.2 The Language of Superalgebra
Notice that this is just a transcription of (2.10)-(2.15) with commut& tors replaced by brackets,
0
The Lie superalgebra, 8, will be the fundamental object in the rest of this monograph. We repeat its definition in basis-free language: The assertion B = 9-1 @go @gi
Four important facts about derivations are used repeatedly: 1) if two derivations agree on a system of generators of an algebra, they agree throughout; and 2 ) The field of scalars lies in A. and D a = 0 if D is a derivation and a is a scalar since D l = D l Z = 2 0 1 and our field is not of characteristic two. 3) Der A is.a Lie subalgebra of End A under commutator brackets, i.e. the commutator of two derivations is again a derivation. We illustrate by proving this last assertion for the case of two odd derivations, dl and d2: Let u be an element of degree m. We have
as a Zgraded algebra implies that
[g-1,g-11 = 0 and
15
[gl,gl]= 0.
The subalgebra go is isomorphic to g; we if we denote the typical element of go by LC, S E g, then
Lvl = L ~ . s l gives the bracket [ , ] : go x go --,go. The space g-1 is isomorphic to g as a vector space, and [ ] : go x g-l + g-1 is the adjoint representation: if we denote an element of g-1 by L,,, q E g then
The bracket [ , ] : go x gl is given by
-+
Interchanging dl and d2 and adding gives
gl is 0,and the bracket [ , ] :g - ~x gl + go
In particular, the square of an odd derivation is an even derivation. So, by combining 1) and 3):
[~F,dl= LF.
4) If D is an odd derivation, to verify that D2 = 0,it is enough to check this on generators.
If A is a superalgebra (not necessarily associative) then Der A is the subspace of End A where e
consists of those endomorphisms D which satisfy D(uv-) = (Du)v
An ordinary algebra which is graded over Z can be made into a superalgebra with only even non-zero elements by doubling the original degrees of every element. If the original algebra was commutative in the ordinary sense, this superalgebra (with only even non-zero elements) is supercommutative. An example that we will use frequently is the symmetric algebra, S ( V ) of an ordinary vector space, V. We may think of an element of Sk(V)as a homogeneous polynomial of ordinary degree k on V*. But we assign degree 2k to such an element in our supermathematical setting. Then S ( V ) becomes a commutative superalgebra. Similarly, an ordinary Lie algebra which is graded over Z becomes a Lie superalgebra by doubling the degree of every element.
+ ( - l ) k m u ( ~ v ) , when u E A,.
Similarly for the Z-graded case. An element of DerkA is called a derivation of degree k, even or odd as the case may be. For example, in the geometric situation studied in the preceding section, the elements of g, act as derivations of degree i on R(M). So we can formulate equations (2.10)-(2.15) as saying that the Lie superalgebra, ij acts as derivations on the commutative algebra A = R(M) whenever we are given an action of G on M.
A second important example of a derivation is bracket by iin element in a Lie superalgebra. Indeed, the super version of the Jacobi identity given above can be formulated as saying that for any fixed u E hi, the map P [u,~1 of the Lie superalgebra h into itself is a derivation of degree i.
-
An ungraded algebra can be considered as a superalgebra by declaring that all its (non-zero) elements are even and there are no non-zero elements of odd degree.
r
If A and B are (super) algebras, the product law on A €9 B is defined
where deg a2 = a and deg bl = j. With this definition, the tensor product of two commutative algebras is again commutative. Our definition
16 . Chapter 2. G* Modules
2.3 From Geometry to Algebra
of multiplication is the unique definition such that the maps A+A@B B+A@B
2.3
a ~ a @ l b ~ l 8 b
17
From Geometry to Algebra
Motivated by the geometric example, where G is a Lie group acting on a manifold, and A = a ( M ) with the Lie derivatives and interior products as described above, we make the following general definition: Let G be any Lie group, let g be its Lie algebra, and ij the corresponding Lie superalgebra as constructed above.
are algebra monomorphisms and such that
For example, let V and W be (ordinary) vector spaces. We can choose abasisel ..., e,,fi ... j , o f V @ W w i t h t h e e i ~ V a n d t h ef j E W . Thus monomials of the form
Definition 2.3.1 A G* algebra is a commutative supemlgebm A, together with a representation p of G as automorphisms of A and an action of ij as (super)derivations of A which are consistent in the sense that
e,, A . . . A e,, A fj, A ...f,, constitute a basis of A(V @ W). This shows that in our category of superalgebras we have A(V @ W) = A(V) 8 ~ ( w ) If . M and N are smooth manifolds, then R(M) 8 R(N) is a subalgebra of R(M x N ) which is dense in the Cm topology. Our definition of the tensor product of two superalgebras and the attendant multiplication has the following universal property: Let u:A-tC,
for all a E G, E E g. A G* module is a supervector space A together with a linear representation of G on A and a homomorphism ij -+ End(A) such that (2.23)-(2.26) hold. So a G* algebra is a commutative superalgebm which is a G* module with the additional condition that G acts as algebra automorphisms and ij acts as ~u~erderivations.
v:B+C
be morphisms of superalgebras such that
[u(a),v(b)] = 0,
Va E A, b E B.
Then there exists a unique superalgebra morphism '
w:A@B+C such that w(a 8 1)= u(a),
w(1 8 b) = v(b).
If V and W are supervector spaces, we can regard End(V) 8 End(W) as a subspace of End(V 8 W) according to the rule (a 8 b)(x €3y) = (-l)qPax 8 by, deg b = q, deg s = p. Our law for the tensor product of two algebras ensures that End V 8 End W is, in fact, a subalgebra of End(V 8 W). Indeed, (a1 8 bl) ((a2 8 b2)(x 8 Y)) = ( - l ) W ( a ~8 61) (a2x 8 b2y) = (-l)~~(-l)j(p+~)alazx 8blby where deg x = p, deg bz = q, deg bl = j and deg a2 = i, whiie ((a1 8 bl)(az 8 b2)) (x 8 Y) = (-l)'j(alaz 8blbz)(z 8 Y) = (-l)'j(-l)(j+q)palalz @ blby
so the multiplication on End(V8W) restricts to that of End V@EndW.
Remarks. 1. In order for (2.23) to make sense the derivative occurring on the left side of (2.23) has to be defined. This we can do either by assuming that A possesses some kind of topology or by assuming that every element of A is G-hite, i.e. is contained in a h i t e G-invariant subspace of A. An example of an algebra of the first type is the de Rham complex R(M), and of the second type is the symmetric algebra S(g8) = $S(g*). (The tensor product R(M) @ S(g*), which will figure prominently in our discussion of the Cartan model in chapter 4, is an amalgam of an algebra of the first type and the second type.) 2. This question of A having a topology (or being generated by its G-finite elements) will also come up in the next section when we consider the averaging operator a EA
IG
P(s)" dg
dg being the Haar measure. 3. If A doesn't have a topology one should, strictly speaking, q u a l i i every assertion involving the differentiation operation (2.23) or the integration operation by adding the phrase "for G-finite elements of A"; however, we will deliberately be a bit sloppy about this.
18
Chapter 2. G* Modules 2.3 From Geometry to Algebra
19
4. Notice that if G is connected, the last three conditions, (2.24)-(2.26),
are consequences of the first condition, (2.23). For example, to verify (2.25) in the connected case, it is enough to verify it for a of the form a = exp tC, C E g. It follows from (2.23) that
for all t , and hence
by the fact that we have an action of j . Taking a = expt< and Ad,-I 7 proves (2.25). A similar argument proves (2.26).
<=
5. Clearly a G* algebra is a G module if we forget about the multiplicative structure. We want to make the set of G modules and the set of G* algebras into a category, so we must define what we mean by a morphism. So let A and B be G* modules and f :A+B
Or, more informally, we could say that f preserves the G* action. It is clear that the composite of two G* module morphisms is again a G* module morphism, and hence that we have made the set of G* module morphisms into a category. We define a morphism between G* algebras to be a map f : A + B which is an algebra homomorphism and satisfies (2.27)-(2.30). This makes the set of G* algebras into a category. We can make the analogous definitions for Z-graded G* modules, algebras and morphisms. If we have a G-action on a manifold, M, then R(M) is a G* algebra in a canonical way. If M and N are G-manifolds and F : M -+ N is a Gequivariant smooth map, then the pullback map F* : R ( N ) -, R(N) is a morphism of G* algebras. So the category of G* algebras can be considered as an algebraic generalization of the category of G-manifolds. Our immediate task will be to translate various concepts from geometry to algebra:
2.3.1
a (continuous) linear map. Definition 2.3.2 W e say that f is a morphism of G* modules if for all x E A , a ~ G , t € gwe have
Cohomology
By definition, the element d acts a s a derivation of degree +1 with d2 = 0 on A. So A is a cochain complex. We define H(A) = H(A,d) to be the cohomology of A relative to the differential d. In case A = R(M) de Rharn's theorem says that this is equal to HB(M).
Remarks.
Notice that (2.28) is a consequence of (2.27) because of (2.23). If G is connected, (2.27) is a consequence of (2.28) for the same reason. If, for all i, .
f : A, -.B,+k
we say that f has degree k, with similar notation in the (Z/2Z)-graded case. We say that a morphism of degree k is even if k = 0 and o d d if k = 1. If the morphism is even (especially if it is of degree zero which will frequently be the case) we could write conditions (2.27)-(2.30) as saying that Va€G,t€g,
1. H*(A) is a supervector space, and a superalgebra if A is. It is Z-graded if A is.
2. A morphism f : A + B induces a map f. : H8(A) -+ H8(B) which is an algebra homomorphism in the algebra case. It is Zgraded in case we are in the category of Z-graded modules or algebras: 3. Condition (2.26) implies-that H*(A) inherits the structure of a Gmodule. But notice that the connected component of the identity of G acts trivially. Indeed, if w E A satisfies dw = 0, then, for any E g we have, by (2.51, Lcw = ~ L < W
<
so the cohomology class represented by LCw vanishes.
Chapter 2. G* Modules
20
2.3 !3om Geometry to Algebra
4. If f : A -+ B is a morphism, then the induced morphism
Let us redo the above argument in superlanguage: Since Q is odd, condition (2.32) says [Le,QI = o QtE g and the definition (2.34) can be written as
f. : H * ( A ) + H * ( B ) is a morphism of G modules.
2.3.2
Acyclicity
:= [d,Q]
T
If M is contractible, the de Rharn complex ( R ( M ) , d ) is acyclic, i.e., A = R ( M ) satisfies k = 0, ~ ' ( A y d= ) k+0. (2.31)
{r
and (2.33) as
FEE-
[Le,Ql = O By construction
T
is an even G-morphism so [LC, T ] = 0 for all t E g. Also
where F is the ground field, which is C in our case. We take this as the definition of acyclicity for a general A.
2.3.3
21
[LC,
TI
[LC, [d,
= =
Q11
[[he,dl, Ql - [d, [LC, Q] = [L<,Ql-0 = 0,
Chain Homotopies
Let A and B be two G modules. A linear map while
Q:A+B
[d, T ] = [d, [d,Q]] = [Id,dl, Q] - [d, [d, Q]] = -[d,?
is called a chain homotopy if it is odd, G-equivariant, and satisfies
We say that two morphis&s and write If A and B are Zgraded (as we shall usually assume) we require that Q be of degree -1 in the Zgradation. The G-equivariance implies that LcQ
Proposition 2.3.1 If Q : A
-t
- QLc
= 0 V[ E g.
and
if there is a chain homotopy Q :A 7 1 -TO
TO
(2.34)
--+
B are chain homotopic
--,B
such that
= Qd+dQ.
(2235)
z TI
+-TO.
= TI,.
(2.36)
We pause to remind the reader how chain homotopies arise in de-Rham theory: Suppose that
B
A = R(Z),
Proof. We have dr=dQd=rd
+
L E ~ QY Q ~
= ~ e d Q- Q ~ c d = - ~ L < Q LeQ
+
= (dQ +Qd)L< - TLc.
=R(W)
where Z and W are smooth manifolds. Suppose that
and Q is assumed to be G-equivariant hence gives a go-morphism. We must check.that L E T = T L E 'dt E g. We have =
:A
Notice that this implies that the induced maps on cohomology are equal:
is a morphiim of G* modules.
L ~ T
TI
To 21 T l
(2.33)
B is a chain homotopy then r:=dQ+Qd
TO
= 0-
Q50 : W
-+
2,
and
41 : W + Z
are smooth maps, and let T,
+ Q d ~ c- Q L E 'We say that 4o and
:= 4: : A
+
B, i = 0,l.
bl are smoothly homotopic if there is a smooth map 4 : W x I + Z
22
Chapter 2. G* Modules
2.3 From Geometry to Algebra
where I is the unit interval, and
40 =
d . 9
We claim that this implies that
O),
70
d l = d(., 1).
since Q and
and define &:W+TZ by letting & ( w ) be the tangent vector to the curire s H d(w, S ) at s = t . For u E Q k + ' ( Z ) define
4 f ( i ( E t ) ~E) Q k ( W
by
The "basic formula of differential calculus" asserts that
(For a proof of a slightly more general formula, see [GS] page 158.) Define Q:A+Bby &a:=
chain homotopy. Indeed, we must show that if (2.32) holds when evaluated at x and at y then it holds when evaluated on xy. We have, using (2.37),
and 7 1 are chain homotopic.
Proof. For general t E I, define
-
23
1'
gjf(~(E~)u)dt.
Integrating the preceding equation from 0 to 1 shows that ( 2 . 3 5 ) holds. All the above is completely standard. Now suppose that Z and W are Gmanifolds and that all the maps in question; do, dl,@,are G-equivariant. Then A and B are G modules, ro and rl are G* morphiim, and it follows from the above definition of Q that (2.32) holds, i.e. that Q is a chain homotopy. Suppose that A and B are G* algebras, and we are given an algebra homomorphism 4 : A + B which is a G* morphism. We say that Q is a chain homotopy relative t o 4 or a qLhomotopy if, in addition to ( 2 . 3 2 ) , Q satisfies the derivation identity
This condition implies that Q is.determined by its values on the generators of A. Conversely, suppose that we are given 4 and a linear map Q : A + B satisfying (2.37) and which satisfies ( 2 . 3 2 ) on the generators. Then Q is a
LC
are both odd. On the other hand
and upon adding, the middle terms cancel.
2.3.4
Free Actions and the Condition (C)
There is no easy way in de Rharn theory of detecting whether or not an action is free. But it is useful to weaken this condition to one that can be detected at the infinitesimal level: Definition 2.3.3 An action of G on M is said to be locally free if, the corresponding infinitesimal action of g is free, i.e., i i for every 5 # 0 G g, the vector field generating the one parameter p u p t ++ exp -t< of tmnsfomatiolls on M ES nowhere vanishing. If the action is locally free, we can find linear differential forms, ol, - ..,On on M which are everywhere dual to our basis &, .. . ,<, in the sense that
Conversely, if we have a G-action on a manifold on which there exist forms Ba satisfying (2.38) then it is clear that the action is locally free. A linear differential form w is called horizontal if it satisfies
The local-freeness assumption says that the horizontal linear differential forms span a sub-bundle of the cotangent bundle, whose fiber at each point consist; of covectors which vanish on the values of the vector fields coming from g. In other words, it says that the values of the vector fields coming from g form a vector sub-bundle of the tangent bundle, T M . The sub-bundle of T'M spanned by the horizontal differential forms is called the horizontal bundle. If the sub-bundle sp&ned by the forms satisfymg (2.38) is G-invariant, then the forms 8' are usually called connection forms; at least this is the standard terminology when the G-action makes M into a principal bundle over some base B (so that the action is free and not just locally free). In the standard terminology, one usually considers a "connection form" to be a gvalued one form O E R1(M) 63g. Relative to our chosen basis of g, o = o'@& where the Ba are the connection forms defined above.
24
Chapter 2. G* Modules
Suppose we have a locally free action of G on M, and we put a Riemann metric on M. This splits the cotangent bundle into a subbundle C complementary to the horizontal bundle whose fiber at each point is isomorphic to g*. Hence our basis of g picks out a dual basis of the fiber of C at each point, i.e. a set of linear differential forms satisfying (2.38). In general, the sub-bundle C will not be G-invariant. But if the group G is compact, we can choose our Riemann metric to be G-invariant by averaging over the group, in which case the sub-bundle C will also be G-invariant. Since L ~ O Jis constant, we have
where wd is horizontal, i.e. satisfies (2.39). If the sub-bundle C is G-invariant, then all the 3, = 0 and we get
2.3 F'rorn Geometry to Algebra
25
If we are given 8' E A1 satisfying (2.38) and (2.40) then, as we have seen, (2.41) and (2.42) are consequences. If we apply d to (2.41) we h d (using Jacobi's identity) that dpa = -c:j8'p3 (2.43) and from this equation and from (2.5) and (2.42) that
If A is any G* algebra and B is a G* algebra of type (C), with connection elements 8; then A @I B is again an algebra of type (C) with connection elements 1 @I 8;. Let us return to conditions (2.38) and (2.40). Consider the map C : g* + Al, given by c(x') = 8' where x l , . . . ,xn is the basis of g* dual to the basis El,. . .,En that we have chosen of g. Thus the subspace, C, spanned by the 8' is just the image of C,
Condition (2.38) is then equivalent to Abstracting from these properties, we make the following definition: Definition 2.3.4 A G* algebra A is said to be of type (C) if there are elements 8' E A1 (called connection elements) which satisfy (2.38), and such that the subspace C C A1 that they span is invariant under G.
Notice that if C satisfies this equation, so does
If G is connected, condition (2.40) implies that the space spanned by the 8' is G-invariant. So if G is connected then being of type (C) amounts to the existence of 8' satisfying (2.38) and (2.40). Usually the properties of A that we will study wiil be independent of the specific choice of the connection elements, 8'. This is in analogy to the geometrical case where the topological properties of a principal bundle are independent of the choice of connection. It follows from (2.38) and (2.40) that
where dl denotes the co-adjoint representation, the representation of G on g* contragredient to the adjoint representation:
where the pa are two-forms satisfying
(In passing from the first to the second line we are making the mild assump tion that G a d s trivially on the scalars, considered as a onedimensional subspace of Ao. This is usually what is meant when we talk an automorphism of an algebra with unit - that the automorphism preserve the unit.) The condition that C be invariant is the same as the condition that C be equivariant, i.e. that
In the case of principal bundles and connection forms, the forms pa are called the curvature forms associated to the given connection. For general algebras of type (C) we wiil call the elements pa occurring in (2.41) 'the curvature elements corresponding to the connection elements {Ba).
aoCoAd!-,
Indeed,
26
Chapter 2. G Modules
2.4 Quivariant Cohomology of G* Algebras
If G is compact, and we are given a C satisfying (2.45), then averaging C, := a o C o ~ d t - ,over the group, i.e. considering the integral
27
Let 9 : A + B be a morphism of G* modules. It follows immediately from the definitions that $(Abas) C Bbar and hence that ~5 induces a linear map
with respect to Haar measure gives a new C which is equivariant. So in the case that G is compact, a G* algebra is of type (C) if and only if there exist elements satisfying (2.38).
2.3.5
In case q.5 is a homomorphism (and morphism) of G algebras, the induced map q5b is an algebra homomorphism.
The Basic Subcomplex
If the action of G on M is free and G is compact, the quotient space X = M/G is a manifold and the projection
is a principal G-fibration. The subcomplex
2.4
Equivariant Cohomology of G* Algebras
Let E be a G* algebra which is acyclic and satisfies condition (C). Given any G* algebra A we will define its equivariant cohomology ring HG(A) to be the cohomology ring of the basic subcomplex of A 8 E: HG(A) := &-(A
is called the complex of basic forms since they are images of forms coming from the base X under the injective map T*.Since r* is injective, the complex of basic forms is isomorphic to R(X). It is easy to detect when a form is basic: w is basic if and only if it is G-invariant and horizontal, i.e. satisfies (2.39). Moreover, if G is connected, being G-invariant is equivalent to satisfying
For an arbitrary G* module A we define Ab, to be the set of all elements which are G-invariant and satisfy (2.39). If G is connected we can replace Ginvariance by (2.46). 'The set of elements of Ah are called basic. It follows from (2.5) ' dAb, C Abas, in other words Ab, is a subcomplex of A. We will call its cohomology the basic cohomology of A. We will denote this basic cohomology by
or, more simply, by Hb,(A). By definition, p(a), a E G, LC and ~ e J, E g all act trivially on Abm. SO Ab, is a G* submodule of A, but the only non-trivial action is that of d. In the case that A is a G* algebra, it follows from the fact that G acts as automorphiims and g-1 as derivations that Ab, is a @ subalgebra. In this case, Hb, inherits an algebra structure.
8 E ) = H ( ( A8 E)b,, d) .
(2.47)
We make the same definition (without the algebra structure) in the case of G* modules. Notice that this definition mimics the definition (1.3) in the framework of G* algebras: we have replaced the space M x E where E is a classifying space (a free,acyclic G-space) by R(M) 8 E where E is an acyclic G* algebra of type (C), and then Q(M) by a general G* module A. We have replaced the cohomology of the quotient by the basic cohomology. To show that the definition (2.47) is legitimate we wilI have to address the same issues we faced in Chapter 1: Does such an E exist and is the definition independent of the choice of E? We will postpone the independence question until Section 4.4. In the next section we will construct a rather complicated acyclic G* algebra satisfying condition (C), but one which is closely related to the geometric construction in Chapter 1. We will use it to prove that the equivariant cohomology of a manifold M (as a topological space) is the same as the equivariant cohomology of R(M) (as a G* module). In the next chapter we will introduce the Weil algebra which is the most economical choice of acyclic G* algebra satisfying condition (C); most economical in a sense that we will make precise. Let us continue to assume that the definition (2.47) is legitimate. If q.5 : A --+ B is a morphism of G* modules, we may choose the same E to compute the equivariant cohomology of both A and B. Then
is a morphism of G* modules and we may try to define
Chapter 2. G' Modules
28
2.5 The Equivariant de Rham Theorem
The proof that we will give of the legitimacy of (2.47) will also show that this definition is independent of the choice of E. It then follows that
Proposition 2.5.1 Let X be a wmpact m-dimensional manifold. Every continuous map f : X -+ E is contractible to a point. Proof. We know that f (X) C Ek for some k. To prove the proposition it suffices to prove
is a functor. We leave the proof of the following as an easy exercise for the reader:
Proposition 2.5.2 A continuous map f : X a point if k 2 m + n .
Proposition 2.4.1 If two G' morphisms, di : A homotopic, then (40)~ = (41)~-
Proof (by induction on n): Consider the fibration
2.5
-+
B, i = 0 , l are chain
The Equivariant de Rham Theorem
Theorem 2.5.1 Let G be a compact Lie group acting on a smooth wmpact manifold M. then = HG (R(M)). (2.48)
29
-+
Ek =
EP)is contractible to
By induction, y o f is contractible to a point po and hence, by the covering homotopy property (see [BT]page 199) f is homotopic to a map h : X -+ whose image sits in the fiber over po. But this fiber is asphere S2(k-(n-1))-1; so if k 2 n + m this map is contractible to a point. Let R(E) be the inverse limit of the sequence of projections
&p)
Without loss of generality we can assume that G is a closed subgroup of
U(n). Let Cmdenote the space of all sequences (q,z2 .. . ,z,,,. ..) with zi = 0 for i sufficiently large. So Cm = UCk where C k consists of all sequences with q = 0, i > 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,
<
This proves the formula for f = 1. We can now proceed inductively on the degree of f . We have since, writing ( M j ) for the matrix of ME relative to our orthonormal basis,
. we have
(We are indebted for this organization of the proof to Matt Leingang.) Finally, let us compute the Fourier transform of a "Gaussiann: Suppose d=2mand
as above. We assume that Q is non-degenerate, hence induces an isomorphism of V onto V', and therefore a non-degenerate element, call it q* E A ~ v * . Proposition 7.1.3 The Fourier trakfonn of exp -:($I,
Proof.We may assume that we have brought an so by iterated integration it suffices to = I,this case
-:
-3
.. . ,qhd) is
to the normal form (7.7) (7.15) when d = 2 and
As usual, we will write Ma for ME- so we write
Let {xl,. . . ,sn)be the basis of g* dual to {&,. .. ,Cn} and let { u ~.,. .ud) be the (Bosonic) coordinates on V associated with the basis {$I,. .. ,$d) of V . Let A = R G ( V ) . We will consider Fermionic integration in A 8 AV followed by the usual (Bosonic) integration over V. For applications to geometry, we will want to construct an equivariantly closed differential form on V which vanishes rapidly at infinity and whose
86
Chapter 7. Fermionic Integration
(Bosonic) integral over V does not vanish. We will call such a form a uni-
versal Thom form for reasons which will become apparent in the geometric
7.2 The Mathai-Quillen Construction
87
Thus
applications. Consider the expression
Up to sign and factors of powers of i and (2s)'l2, we will h d that the Fermionic integral
Both sides of (7.18) are derivations (of odd degree) in the algebra O G ( V ) @ ~ V applied to the element a . Since a is an even element of this commutative superalgebra, any derivation, D (even or odd) satisfies
r
is our desired universal Thom form. We must show that it is &-closed and that its (Bosonic) integral over V does not vanish. We consider the operator
and hence D (exp o) = exp aDa. So (7.18) implies that dG(expa) =
as acting on S(gg)@ R(V) 8 AV by letting it act trivially on the last factor, i.e. by acting as dG @ 1. We first compute dGa. Let us consider the two parts of dG separately. The operator d applies to the u variables in (7.16) yielding
expo.
Since the derivation dG on the left of (7.18) does not involve the 1C, variables, we can pass it inside the Fermionic integration with respect to 1C, to obtain
= 0
by (7.3). This proves that the form (7.17) is dG-closed. We now must evaluate its integral over V, and, in particular, show that the value of this integral is not zero. When we compute this integral, we must extract the coefficient of dul . . -dud in (7.17). We can write (7.17) as
where the remaining terms involve fewer than d factors of the dukrand can be ignored. The Fermionic integral in this last expression yields
On the other hand
and
a
k
Similarly the operators L, only see the second term in (7.16). Since L , , ~ z L ~ = Lauk and the are derivations of odd degree,
the last equation stemming from the fact that the uk are linear coordinates dual to the @. Since the u, are even in our superalgebra, we obtain
auk-
and so the integral of (7.17) over V is
a
-(illkduk)
= iduj.
/
(-i)ds(d) v e - i x . ':dul
- - dud = ( - ~ ) ~ s ( d ) ( 2 n ) ~#/0. '
88
Chapter 7. Fermionic Integration
7.3 The Fourier Transform of the Koszul Complex
If we want the integral to come out to be one we must divide by this non-zero constant. Thus
is the u n i v d Thom form as constructed by Mathai and Quillen. Now suppose that d is even. Let j : {0}
4
7d
where we have identified AV* with differential forms on V which are linear combinations of the dur with constant coefficients. The elements of Qc(V), are ail integrable and hence we get an integration map
The main goal of this section is to show that this map is a bijection, and, in particular, that
V
be the inclusion, and define the normalizing constant
89
by
If we apply j*to u,all expressions involving ui and dui go to zero, e d hence, applying (7.8) to q = Ckq!~~z"M,,@~we obtain
In other words, up to the factor 7 d , j'v is the element Pfaff. The whole discussion above applies to a subgroup K of O(V): If i denotes the inclusion i : k -+ o(V) and i* the dual map fiom the ring of invariant polynomials on o(V) to S(k*)K we obtain
Let us first examine thii assertion in the non-equivariant case, i.e. where we take G = {e) to be trivial and so are considering
consisting of all differential forms which are of the form pre-u2/2dur where the p' are polynomials on V. We want to t h i i of R(V), a s a kind of substitute for R(V),, the spice of differential forms of compact support. The analogous theorem for the compactly supported cohomology asserts that H'(V), vanishes for .all i # d, and that H ~ ( v ) , = C. The standard way of proving this is to identify V with R d using the coordinates (ul, . . . ,ud) and proving by induction that the fibrations
where i* : S(O(V)*)~'(") -, ~ ( k * ) ~ .
induce isomorphiisms on cohomology
7.3
The Fourier Transform of the ~ o s z u Complex l
Let V be a d-dimensional vector space on which G acts in a linear fashion, preserving a positive definite quadratic form u2 = u : + . . - + u ; in terms of coordinates associated with an orthonormal basis @', . . . ,qbd. Let ilc(V)e
c Rc(V)
consist of all equivariant differential forms whose co&cients, in terms of the differential forms duI, are of the form p(ul, . . . ,u,j)e-u2/2. In other words, $
See, for example, [BT], page 39. There doesn't appear to be any analogue of this argument in equivariant cohomology which is why we have replaced R(V), by O(V),. With this replacement we can give an alternative argument using the Fourier transform. . So our first order of business will be to prove (7.21) for trivial G. Let y l . . . ,yd be the coordinates in V* dual to ~ 1 ,. ..,ud and let y2 = (yl)' . .+(yd)' ,the dual quadratic form. The ordinary (Bosonic) Fourier transform maps functions of the form
+
into functions of the form
90
Chapter 7 . Fermionic Integration 7.3 The Fourier Transform of the Koszul Complex
91
where 13 E S 8 ( V ) is a polynomial in y. We thus get a map
Fb: S ( V f ) -* S ( V ) ,
p
I+
In other words, we have shown that (7.21) holds for G = {e), the trivial group, and that this isomorphism is realized by integration. We now show how to modify this argument so as to make it work in the equivariant setting. Let El,. . .,En be a basis of g and z ' , ... , z n the dual basis of g* = S1(g'). Let
j.
We may also call this the Bosonic Fourier transform by abuse of language. It is the usual Fourier transform where we have suppressed the exponential factor on both sides. With this suppression of the exponential factor we may identify R ( V ) , with S ( V * ) 8 A(V*) and R(V*), with S ( V ) 8 AV. We may now combine the Bosonic and Fermionic Fourier transforms so as to get a super Fourier transform
Under our identifications, the basis elements + I , . ..,qjd of A'V are identified with the differential forms d y l , . . . ,dyd on V * . With this in mind, we see that i in (7.9) cancels the -i in (7.13) so that the differential operator
-
a
C=&uk&
be the vector fieid on V corresponding to Ea.. The super Fourier transform
l
extends to a super Fourier transform
and converts the operator dG = d
acting on R ( V ) , is carried by (conjugation by) F = Fb 8Ff into the operator
I
-Z ~ L ,
into
To simplify this expression note that In fact, 6 is just the Koszul operator, d K , of Section 3.1. Let
I We can write the term-inside the parentheses in (7.25) as
Notice that Q is exactly the operator we introduced in Section 3.1 modulo some changes in notation for the variables. We thus see that and hence the conjugation by e - ~ ' /in~ (7.25) selds
I
is the derivation given by
Let a s in Section 3.1. In particular, ( R ( V g ) , 6, ) is acyclic, all its cohomology being concentrated in bidegree (0,O). Now our super Fourier transform F carries S'(V8) @ nd(V*) to 5" ( V )@ A O ( V ) sending p$l
...$d
1
p:=-x xa~a+k$k k
so that
I+@
and p(u)e-u2f2dul. . -dud = $(0).
L
We define
a +Pa- a a+" aya'
8: := e-P6GeP = ya-
92
Chapter 7. Fermionic Integration 7.4 Bibliographical Notes for Chapter 7
'
93
where
We can regard the derivations Q defined by (7.23) and E defined by (7.24) a s being derivations of RG(V*),. Since Q and Pa don't depend on the Q supercornmutes with P& and hence [Q,@] = [Q, a] = E
(7.30)
proving the acyclicity of the complex ( ~ G ( v * ) e6g) > as a module over S(g*)G. This result explains the rather mysterious formula (7.19): The generator of the zero dimensional cohomology group of ( R G ( V ' ) , , ~ ~is) clearly the constant function, 1. Hence the generator of the d-th cohomology group of (RG(V),, dG) is the inverse super Fourier transform of exp(-$ + 8):
since, by the change of variables q!~k
H
-qbk we have
section 7.2 is taken from [MQ] section 6. The alternative construction in Section 7.3, using the super Fourier transform, is closely related to the quantum field theoretic construction of this form by Kallunan in [Ka] section 3.3.
3. Let B be the open unit ball, llvll < 1, in V and let 7 : B
+
V be the
map 7(v) =
1 - llvl12-
The pull-back by this map of the form (7.19) is an equivariant form on B which vanishes to infinite order at boundary points and hence can be extended to a compactly supported form on all of V by setting it equal to zero on the complement of B. This form is the compactly supported version of the universal Mathai-Quillen form (see Section 10.3 for more details). 4. Let M be a compact G-manifold on which G acts freely, and let G act on M x V by its diagonal action. Let X = M / G and E = (M x V)/G. E is a vector bundle over X with a "typical fibern V, and X can be embedded in E by identifying it with the zero section. Mathai and Quillen show that one can construct an explicit Thom form representing the cohomology class in HE(E) dual to[X] a s follows: Pull the universal Thom form (7.19) back to M x V by the projection, M x V -+ V. This gives one an element, v', of Rc(M x V). Now apply to v' the Cartan map QG(M x V) -+ R(E). The image of v' under this map represents the dual class to [XI in H,'(E). (For details see Section 10.4 below.)
and (-l)d is the determinant of this change of variables. This is the expression given by (7.19) up to non-zero constants, i.e.
7.4
Bibliographical Notes for Chapter 7
1. The role of the Berezin integral within the context of integration on supermanifolds is discussed in Berezin's book [Be]. A nice discussion of its application to physics can be found in the book, "Supermanifolds" by Bryce de Witt [dW]. 2. Our treatment of the Fermionic Fourier transform and the super Fourier transform is taken from Kalkmh's thesis, [Ka], Sectionl.3. The material on the Mathai-Quillen construction of the universal Thom form in
5. Using the Mathai-Quillen construction, Atiyah and Jeffrey show that Witten's formula for the Donaldson invariants of a four-manifold has a beautiful interpretation in terms of Euler numbers of (infinite dimensional) vector bundles. (See [AJ].) Their basic observation, is that the construction described in Section 4, when appropriately interpreted, even makes sense when V is infinitedimensional!
Chapter 8
Characteristic Classes Recall from section 4.5 that if A is a G module, then we have a characteristic homomorphism G : ~ ( g * ) HG(A), ~ +
and that the elements of the image of n, are known as characteristic classes. But we have not really written down what the ring s ( ~ *is ) =for any group G. The main function of this chapter is to remedy this by summarizing standard computations of s ( ~ for * various ) ~ important groups. Suppose that q5 : K -+ G is a Lie group homomorphism, and let k denote the Lie algebra . of K. The induced Lie algebra map k -+ g dualizes to a map g* --,k* which extends to an algebra homomorphism S(g')G -+ S(k')K. We will examine this homomorphiim for various examples of inclusions of classical groups. ' In important geometric applications, we want to apply the notion of characteristic classes to the the study of vector bundles. So we begin this chapter with a review of standard geometrical constructions which motivate the choice of groups and inclusions we study.
8.1
Vector Bundles
Let E -+ X be a complex vector bundle. Choose a Hermitian structure on E and let M = 3(E)denote its bundle of unitary frames. So a point of M consists of a pair (x,e) where x E X and e = (el,...,e,,) is an orthonormal basis of E,. The group G = U ( n ) acts on the right, where
This makes M into a principal G-bundle over X and hence we get a map 6 :
s ( ~+ * H*(X) ) ~ =TG(M).
.
The elements of the image of this map gives a subring of the cohornology ring of X called the ring of characteristic classes of the vector bundle E. Its
96
Chapter 8. Characteristic Classes
8.2 The Invariants
definition depended on the choice of a Hermitian metric. Let us show that the functorial properties of equivariant cohomology imply that n does not depend on this choice so the terminology "characteristic" is justified. So let ho and hl be two choices of Hermitian structures. Then for every t E [O,1] = I ht := (1 - t)ho +thl
Define the polynomial q,of degree i in A to be the coefficient of (-l)'Xn-' in the characteristic polynomial of A: det(X - A) = An
- s ( ~ ) ~ n -+l -.-+ (-l)"c,,(A).
For instance, cl (A) = t r A and %(A) = det A. The polynomials q are clearly invariant under the adjoint representation, as the characteristic polynomial is. It is a theorem that they generate the ring of invariants. The characteristic classes corresponding to the q for a complex vector bundle are called its Chern classes.
is a Hermitian structure. Let M = F(E)denote the bundle over X x I whose fiber over (x, t) consists of all frames,e of E, which are orthonormal with
respect to ht. It is a principal G bundle, and we have the injections of Mo and MI into M setting t = 0 or t = 1 where Mo is the frame bundle associated to ho and Ml the frame bundle associated to hl. Let jo and jl denote the corresponding injections of X + X x I, so jo(x) = (x,O), jl(xj = (x, 1). Then functoriality implies that
We may identify 9 and hence g* with the space of skew adjoint matrices. For such a matrix we have det(X - A) = det(X - At) = det(X
in the obvious notation. But since jo and jl are homotopic they induce the same homomorphism on cohomology. In the above discussion we could have taken E to be a real vedor bundle, h a real scalar product, and M to be the bundle of orthonormal frames. The group G is then the orthogonal group O(n). Similarly, we could have taken E to be an oriented red vector bundle, h a real scalar product, M to be the bundle of oriented orthonormal frames and G = SO(n) the special orthogonal group. In all three cases, U(n), O(n), SO(n) we must examine the ring S(g*)Gwhose structure we will recall below. We state these results and refer to standard references such as [Chev] for the proofs. In each case there are standard generators, whose images under IE are called Chern classes for the case of U(n), Pontryagin classes for the case of O(n) and one class, called the Pfaffian, in addition to the Pontryagin classes in the case of SO(2n). We will also examine the case of a symplectic vector bundle. We will find there are standard ways of putting a complex structure on such a vector bundle, and that two such ways differ by a homotopy. Therefore the characteristic classes associated to these choices of complex structures agree.
The Invariants
We may identify the Lie algebra of U(n) with space of all matrices of the form iA where A is self adjoint, and hence (using the trace pairing and forgetting about the inessential a) may identify g* with the space of self adjoint matrices with the cpadjoint action being conjugation:
+ A)
so all the coeffitiients of An-' in the characteristic polynomial of A vanish when i is odd. We may write
det(X - A) = An
,
+ pl(A)Xn-2 + p z ( ~ ) X " - +~ . . . .
The polynomials p, of degree 2i generate the ring s ( ~ * ) ~ The . corresponding characteristic cl&ses for a real vector bundle are called its Pontryagin classes.
8.2.3
G = SO(2n)
The Lie algebra g is the same as for O(2n) so all the pi are invariant polynomials. There is one additional invariant which is not a polynomial in the p, called the Pfaffian. It is defined as follows: To each A E g and v,W ~ EV = R2"set wA(u,w) := (Av, w) where ( , ) denotes the scalar product. We have wA(w,U)= (u, Aw) = -(Au,
8.2
97
W)
= -WA(V,
W)
so W A E A ~ ( V *is ) an alternating b i i e a r form and the map A linear isomorphism. The element
I-+
WA
is a
is an element of A~"(V*)and the map A c AW: depends only on the scalar product and so is O(2n) invariant. However the group SO(2n) preserves a , basis element, vol, of A~"(V*)where vol := e; A e;
A
.-.A. , ;e
98
8.3 Relations Between the Invariants
Chapter 8. Characteristic Classes
Here el, ez, ... ,ezn is any oriented orthonormal basis and e;, . ..,e& the dual basis. We may then define PfaE(A) by
It is a polynomial function of A of degree n which is SO(2n) invariant. For any A we can find an oriented orthonormal basis relative to which the matrix A takes the form
If M = M is a real matrix then this becomes
which is the condition for a real matrix to be orthogonal. The inclusion o(n) -, u ( n ) induces a homomorphism S ( U ( ~ ) * )-~ , S(o(n)')O("). (~) We will check below what the images of the ci are under this restriction map. Any complex vector space of dimension n can be regarded as a real vector space dimension 2n. The complex structure induces an orientation on this real vector space. If the complex vector space has a Hermitian structure, the real part of this structure gives a real scalar product. These two h t s combine to give an embedding U ( n )-, SO(2n). We will also examine the restriction O, ~ ( ~( u~ () n ) * ) ~corresponding (") to this embedding. map ~ ( 0 ( 2 n ) * ) ~-
8.3.1 Relative to this orthonormal basis we have
99
Restriction from U ( n ) to O ( n )
We have identiied the Lie algebra u(n) with the self-adjoint matrices, where A generates the one parameter group expitA. For these group elements to be real matrices, we must have
and hence Pfaff ( A ) = & . - . A n . On the other hand det(A) = A:
. . . A:.
So we see that we have the general formula ~ f a E = det . Thus the square of ( 2 ~ )Pfaff - ~ is equal top,. Notice that for odd dimensions the determinant of any antisymmetric matrix vanishes as we have seen above, so this phenomenon does not occur. The characteristic class corresponding to the PfaEan for a real, 2n dimensional, oriented vector bundle E is called the Euler .class of the vector bundle and denoted by e ( E )E H ~ , ( X ) .
8.3
Relations Between the Invariants
We can regard O ( n ) as the subgroup of U ( n ) consisting of real matrices: U ( n )consists of all complex matrices satisfying
MM* = I where M* = M t .
where B is real and
Bt = -B. We get
so under the restriction map we get
This has the following consequence for characteristic classes: Let E -4 X be a real vector bundle and E @ C its complexification. The Chern classes of this complexiiied bundle are related to the Pontryagin classes of the original bundle by ( EC ) = ( - l ) i p i ( E ) . (8.4) c2i+l(E @ C ) = 0, C Z ~ @
100
8.3.2
Chapter 8. Characteristic Classes
8.4 Symplectic Vector Bundles
Restriction from SO(2n) t o U ( n )
We recall two facts: a) Let V be a complex vector space and C : V + V a complex linear transformation. We can regard V as a real vector space (of twice the dimension) and C as a real linear transformation. The relation between the determinant of C regarded as a real linear transformation and its determinant regarded as a complex linear transformation is given by
So if we use the notation c,,, to denote the Chern polynomials associated to U@),the restriction map from ~ ( u ( n ) * ) ~--t( S~ () U ( ~ ) * ) 8 ~ (S~( U ) (C)*)~(~) is given by Cj,n G-,~CS,L (8-8)
,
++
C
r+r=j
Applied to complex vector bundles we get: Let E = El @ l& be a decomposition of a vector bundle of rank n into a direct sum of vector bundles of rank k and C. Then
det R(C) = I det c(C)12 = det c C . det cC.
b) If A is a self-adjoint transformation (relative to some Hermitian form) on a complex vector space then its determinant (over C) is real. Indeed, we may diagonalize any such operator and its eigenvalues are real. By the same argument, the coefficients of the characteristic polynomial det(A - A) = An - cl (A)An-'
101
We can write this more succinctly as follows. For any complex vector bundle, E, define its total Chern class as
+ c ~ ( A ) x -~ - ~
are all real. The subalgebra u(n) C o(2n) consists of complex linear transformations B where A = iB is self-adjoint. Hence detR(A - B ) =
Then c(E) = c(Ei)c(Ez) when .E=E1@Ez
so the restriction map in question is given by '
where
= 1 and, of course, c, = 0, m
> n. '1n particular
and hence (we may choose the orientation so that)
Applied to a complex vector bundle regarded as an oriented real vector bundle we get the the corresponding equations relating the Pontryagin classes to the Chern classes. In particular, the Euler class is given by
Using the relation between Chern classes of the complexification of a real vector bundle and the Pantryagin classes given above, (8.4), we get an analogous formula for the Pontryagin classes. Equation (8.9) generalizes in the obvious way when E is decomposed into a direct sum of several vector bundles. At the extreme, suppose that E splits as a direct sum of l i e bundles,
Then c(E) = (1
where uk denotes the k-th elementary symmetric function.
8.4 8.3.3
Restriction from U ( n ) to U ( k ) x U ( e )
Consider a self-adjoint operator of the form A = A1 @ A2 relative to the direct sum decomposition Cn= Ck@ Ce where k + E = n. Then
+ cl(Li))(l+ C I ( L ~ ).).. (1 + cl(Ln))
Symplectic Vector Bundles
8.4.1 Consistent Complex Structures Let V be a real finite dimensional vector space and let ( , )o be a positive definite scalar product on V. We let A denote the space of linear transformations which are~self-adjointrelative to this scalar product, so satisfy
102
Chapter 8. Characteristic Classes
8.4 Symplectic Vector Bundles
We let P denote the open subset of A consisting of all positive definite selfadjoint linear transformations, so A E P if and only if
L e m m a 8.4.1 The map Sq : A
I-+
Since B and C commute we have J~ = -1 C J = JC B = JC.
A2 is a diffeomorphism of P onto itself.
Proof. Let el,. ..,e, be an orthonormal basis of eigewalues of A with eigenvalues XI,. ..,A, so all the X i > 0. Then any A E P has a unique positive definite square root, namely the operator with the same eigenvectors and .. where we take the positive square roots. This with eigenvalues shows that the map Sq is bijective. We must show that it is a diEeomorphism. The tangent space to P is A and we must show that for any A E P the map
A,. ,a
103
We have W(U,v) = (u, BV)O= (u,C J V ) ~ so if we define a new scaIar product ( , ) by (u, v ) := (u, Cv)o we have W(U,
is injective. We have
v ) = (u,Jv)
so
d(Sq)*(W)= AW
+ WA,
(Ju,
W E A.
V)
= -(u, Jv)
and Suppose the right hand side of this equation were 0. We would then get
(u,v ) = w(Ju, v ) . We may think of J as defining a complex structure on V and then may define
If W e i # 0 it would be an eigenvector of A with a negative eigenvalue which is impossible. Hence W = 0. As a corollary we conclude that the inverse map
h(u,v ) = (u,V)+ W ( U , v).
(8.18)
It is easy to check that h is a Hermitian form relative to J, that is that
is smooth.
Suppose that w is a symplectic form on V (so V is even dimensional and w is a non-degenerate antisymmetric bilinear form). Then there is a unique anti-symmetric linear operator, B : V
+
V such that
w(u, v ) = (u, B U ) ~'d u, v E V. Notice for future use that B depends linearly on w and smoothly on the choice of scalar product, ( , )o. The operator BtB = -B2 is positive definite and and has a unique square root, C which depends smoothly on B and hence on w. So
Let
Notice that J and h depend smoothly on ( , )o and w and if w and ( , )o are invariant under the action of some group G then so are 3 and h.
8.4.2
Characteristic Classes of Symplectic Vector Bundles
Let E + X be a symplectic vector bundle. This means that each fiber E, has a symplectic form which varies smoothly in the usual sense. We may put a scalar product on this vector bundle which then determines a complex structure and an Hermitian structure (depending on our choice of scalar product). A homotopy between two diffeient choices of scalarproduct induces a homotopy between the corresponding complex and Hermitian structures. So the characteristic classes associated to the corresponding unitary frame bundles are the same. In this way the Chern classes (for any choice of real scalar product) are invariants of the symplectic structure.
104
Chapter 8. Characteristic Classes
8.5
Equivariant Characteristic Classes
8.5 Equivariant Characteristic Classes
105
This gives a homomorphism of the rings of invariants:
Let K and G be compact Lie groups and set H := K x G. If M is an Hmanifold, we can regard M as a K-manifold on which G acts, the G action commuting with the K action. If the K action is free, one gets an induced
and the equivariant Chern classes are just the images of the q E S(k*)K.
action of G on the quotient manifold
8.5.3 and, by (4.28), a Chern-Weil map
Suppose that G has a positive dimensional center. In other words, suppose l Let X be a Gthat the circle group S1 sits inside G as a ~ e n t r asubgroup. manifold, Y a connected component of XS , and E 4 Y the normal bundle of Y in X . Since the action of S1 commutes with the action of G, we get an action of G on E as vector bundle automorphisms and an action of S1which commutes with this G-action. We claim that this S1-action endows E with a complex structure which is preserved by G. For this we'll need the following lemma:
The elements of the image of this map are called the eqvivariant characteristic classes. Here are some important examples:
8.5.1
Equivariant Chern classes
Let X be a G-manifold and E + X a complex rank n vector bundle on which G acts as vector bundle automorphisms. Thus, if x E X and a E G, the action of a on E maps the fiber of E over x linearly onto the fiber over ax. If we equip E with a G-invariant Hermitian inner product, we get a G-adion on the associated unitary frame bundle F ( E ) which commutes with the action of U ( n ) as described in Section 8.1. Thus, if we take K = U ( n ) we get from (8.19) a map from S(k*)K to H G ( X ) . The images under this map of the elements cj described in subsection 8.2.1 will be called the equivariant Chern classes. Just as in Section 8.1 one can prove that they are independent of the choice of Hermitian inner product. If the vktor bundle E + X is real, we can define the equlvariant versions of the Pontryagin classes by the same method, and if E is an oriented real vector bundle of even rank we get an equivariant Euler class - either by mimicking the construction above with K = S O ( n ) or, if E has an underlying complex structure, by defining the' equivariant Euler class in terms of the equivariant Chern classes as in 8.7. These classes satisfy the same identities as those described in Section 8.3, e.g. the identities (8.4), (8.7), and (8.9).
8.5.2
Lemma 8.5.1 Let V be a vector space over R and p : S1 -+ GL(V) a representation of S' on V which leaves no vector &ed except 0. Let
d A = -p(e dt
it
)It=o.
Then there exiscists.a unique decomposition:
and posztzve integers, 0 < m l < J,2 = -I.
. . . < mk
such that A = m,J, on V , with
Proof. Equip V with an S1-invariant inner product, i.e., an inner product satisfying:
(p(eat)vlp(eft)w) =
( v ,w ).
Differentiating and setting t = 0,
( A v ,w )+ ( v ,Aw) = 0
Equivariant Characteristic Classes of a Vector Bundle Over a Point
If the vector bundle E 4 X is topologically trivial, its characteristic classes vanish. But this need not be true of its equivariant characteristic classes. For example, consider a vector bundle over a point E -+ pt. This is just an ordinary n-dimensional vector space on which G acts as linear automorphisms. Equipping E with a G-invariant metric, we can regard the representation of G on E as a homomotphism
Equivariant Characteristic Classes as Fixed Point Data
i.e., At = -A. Since vS1= {0), A is invertible and hence A t A is positive definite. Let 0 < X I < - . < X k be the distinct eigenvalues of A t A and V=V,@...@V,
q
the decomposition of V into the eigenspaces corresponding t o these eigenand J: = -I. values. On V1,, A2 = -Air, SO Ai = miJi, with mi = The eigenvalues of J, are *fl so the eigenvalues of exp(2aA) on V , are exp(*2nm,&f). However, exp 2aA = I ; so the mi's are integers. The uniqueness of the decomposition (8.20) follows from the fact that V , is the kernel of A2 + m: I .
106
Chapter 8. Characteristic Classes 8.6 The Splitting Principle in Topology
Corollary 8.5.1 V admits a canonical complex structure. I n particular V is even dimensional and has a canonical orientation. Proof. A canonical complex structure on V is defined by J = Jl @ . -.@ Jk. D Now, since every y E Y is fixed by S1, we get a representation of S1 on E, with no trivial component. So we can decompose E, into S1-invariant subspaces
E, = @E*,k where the action of S1on Ey,k is given by
107
A manifold Y with these properties is called a splitting manifold for E -+ X . We will deduce the existence of such a manifold from the abstract splitting principle which we established in section 6.8. As in section 8.1, let 3 ( E ) be the unitary frame bundle of E relative to a choice of Hermitian metric. Let
M := 3 ( E ) , T := a Cartan subgroup of U ( n ) , Y := M / T . Since we may identify X with M/U(n), we get a fibration
t E Rl2xZ
t H exp(aktJ,,k),
where the ak are positive integers, aj # ak for j # k and
and Theorem 6.8.3 establishes property 1) in our theorem. Furthermore, not only is A* : H*(X) + H * ( Y ) injective, we have the identification
J ; , ~= -I. The E,,k and position
Jy.k
depend smoothly on y and so define a canonical decom-
E=@E; of E into complex vector bundles. The equivariant Chern classes of these vector bundles are important topological invariants of X with its G-action. In particular; the equivariant Euler class defined by (8.7) will play a fundamental role in the "localization thecrem" which we will discuss in Chapter 10. .
8.6
The Splitting Principle in Topology
Let X be a mmifold and let E -+ X a complex rank n vector bundle over X. Given a manifold Y and a smooth map y : Y --+ X, the pull-back y*E of E to Y is defined to be the set y * E := {(Y, el, Y E
Y,e E E7(Y))
with the obvious projection (onto the first factor). The splitting principle (cf.[BT] Section 21) asserts:
Theorem 8.6.1 For every vector bundle E -r X there ensts a manzfold Y and a fibratton
A
:Y +X
such that
where W is the Weyl group. We now show that Y is a splitting manifold. For U(n), We may choose T to be the group of diagonal unitary matrices, so two orthonormal frames e, e' € 3 ( E ) over p E X
lie in the same T-orbit if and only if there exist {Bk) such that
i.e. if and only if ek and e i span the same one dimensional subspace L,(k) E,. Thus a T-orbit in 3 ( E ) p defines a decomposition,
C
into mutually orthogonal one-dimensional subspaces. Conversely, given such a decomposition, choosing a unit vector in each summand gives an orthonorma1 frame, e, and two different choices differ by a transformation of the form (8.22). So the decomposition (8.23) defines a T-orbit in F(E). We have thus proved Proposition 8.6.1 There is a one-to-one correspondence between orthogonal decompositions (8.23) and points on the fiber of Y over p. From this description of Y we see that
1. x* : H*(X) + H*(Y) w znjectzve, and
2. n*E splits znto a dzrect sum of lzne bundles.
where the Ll are the L%autological"line bundles associated with the splitting (8.23). This establishes property 2) and proves the theorem.
108
Chapter 8. Characteristic Classes
8.7 Bibliographical Notes for Chapter 8
The Weyl group W is just the group of permutations of (1, . . . ,n); it acts on Y by permuting the summands in (8.23). In other words, T E W = S, sends the point of Y represented by the decomposition (8.23) into the point represented by the decomposition
109
2. The characteristic classes of a symplectic vector bundle have generated a lot of interest lately because of their role in the proof of the "quantization commutes with reduction" theorem and its many variants. See Meinrenken, [Mel] and [Me2], Vergne [Ve] and Duistermaat-GuilleminMeinrenken-Wu [DGMW]. 3. To prove that for G = U(n) the Q'S generate S(g*)G,we note that by Chevalley's theorem (see section 6.8) the map
In particular. r*Lk = Lr(k).
Let c(Lk) denote the Chern class of Lk. It follows that
By Theorem 6.8.4, the c(Lk) generate H*(Y) as a ring over H*(X) and so (8.26) specifies the action of W on H*(Y). A s an independent confirmation of (8.21), we note that by (8.10),
where uk is the k-th symmetric function. By the basic theorem of symmetric functions, ([VdW] page 78), evey symmetric polynomial is a polynomial function of 01,. . . ,on. Hence, by (8.26) and (8.27) every W-invariant of H*(Y) is in a*H*(X). If X is a G-manifold and E + X is a vector bundle on which G acts by vector bundle automorphisms, there is an equivariant version of the splitting principle: As we pointed out in Section 8.5.1, the action of G on E lifts t o an action of G on F ( E ) which commutes with the action of U(n). Hence one gets an action of G on the quotient Y = .F(E)/T. The fibration T : Y -+ X becomes an equivariant fibration and hence induces a map
is bijective (t being the Lie algebra of the Cartan subgroup, T, of G and W being the Weyl group). For G = U(n), T is the group of diagonal unitary matrices, i.e.,
T = S1 x . . . x S1
(n copies)
Moreover, W is the group, C,, of permutations and t = -Rn. of the set {1,2, . . . ,n) and acts on t by permuting the coordinates, ( X ~ , . . . , X , )o f x E -Rn. Thus
and under this identification the q's go into the elementary symmetric polynomials in 1 , . . . ,x,. Hence to prove that the q generate S(g*)G it suffices to show that the elementary symmetric polynomials generate the ring C[xl,. .- ,xnICn. For a proof of this see, for instance, [VdW] section 26. 4. Oui assertion that the pi's generate S(g*)G when G = O(n) can be
proved by a similar argument. Let W be the semi-direct product
By Theorem 6.8.2 this map is injective . It is also clear that splitting (8.24) is an equivariant splitting. So we have proved
and let W act on C[x,, . . . ,x,] by letting
Theorem 8.6.2 There exists an equivariant splitting manifold of E . That is, there exists a G-manifold Y and a G-fibmtion T : Y X such that
a being a permutation of {1,2, . . . ,n) and E = ( f 1,. - . , f 1 ) an element
-+
1. a* : HG(X) -+ HG(Y) is injective, and
2. a*E splits equivalzantly into a direct sum of G-line bundles.
8-7
Bibliographical Notes for Chapter 8
1. Most of the material in this section is fairly standard. For a more detailed treatment see [MS] sections 1415, or [BT] chapter IV.
of Zy. To prove that the pj's generate S(g*)G it suffices to show that C [XI,.. . ,x,lW is generated by the elementary symmetric functions in x:, . . . ,x; and this follows easily from the results in [VdW], Section 26, that we cited above. 5. An alternative description of the PfafFian is the description in section 7.1.3 as the Gaussian integral (7.6). 6. For a different approach to the theory of equivariant characteristic, classes see [BGV] Section 7.1. (However, their approach also involves "super" ,ideas: in particular, supercomections on vector bundles).
110
Chapter 8. Characteristic Classes
7. Bott and Tu [BT] give the following pragmatic formulation of the splitting principle: "To prove a polynomial identity in the Chern classes of complex vector bundles it suffices to prove it under the as sumption that the vector bundles are sums of line bundles." In addition, if the vector bundles are G-vector bundles the same is true of the equzvariant Chern classes.
Chapter 9
Equivariant Symplectic Forms 9.1
Equivariantly Closed Two-Forms
Suppose we are given an action G x M element
-+
M . In the Cartan model, an
can be written a s
D=w-4 where w E R 2 ( M ) is a two-form invariant under G and q5 E ( R O ( M8 ) g*) can be considered as a G equivariant map,
G
from the Lie algebra, g to the space of smooth functions on M. For each E g, q5(E) is a smooth function on M , and this function depends linearly on <. Therefore, for each m E M, the value Q ( < ) ( m )depends linearly on <, so we can think of q5 a s defining a map from M to the dual space g* of the Lie algebra of g: d :M g*, ( d ( m ) , E ):= d(E)(m). We will also use the notation
<
+
dc for
d(E).
The condition that D be equivariantly closed now translates into two conditions, dw = 0 and - L ~ W - d$(<) = 0. In other words, t
112.
Chapter 9. Equivariant Symplectic Forms
9.2 The Case M = G
In the language of symplectic geometry this says that q5 is a moment map for the action of g and the closed form w. If, in addition to being closed, the form w is non-degenerate (and so symplectic) we say that the dG closed form 3 is an equivariant symplectic form. In other words, an equivariant syrnplectic form is a G-invariant symplectic form together with a moment map. Even if the form w is not symplectic, we can call the 4 occurring in 3 = w - 4 the moment map.
9.2
The Case M = G
So we have y!Je(a) =
.
die.
113
(9.4)
Let Ge denote the stabilizer group o f t in the coadjoint representation. Then (9.4) implies that y!Jc is constant on left cosets aGc. In other words, is in effect a map +c : GIGt -,g*. (9.5) We now prove a similar result for dBc: Theorem 9.2.1 With the above notations
Suppose that 3 is an equivariantly closed two-form on G (with G acting on itself by left multiplication). We know that PG(G) = H2(G/G) = H 2 ( ~ t= ) 0 , so every equivariantly closed two-form is equivariantly exact, i.e.
I . dBc is the pullback via the canonical pwjection of a closed two-form we E R2(G/Gt). 2. we is symplectic.
for some equivariant oneform 8. But an equivariant oneform is just an invariant oneform. In other words the above equation holds with 8 some left-invariant oneform. Now
where tR is the vector field corresponding to the left multiplication action of G on itself (and so is right invariant). In particular, the B occurring in (9.1) is unique, since dG8 = 0 implies that y , B = 0, and the span the tangent space at each point of G. A left invariant one-form 8 is determined by its value, P = B(e) E TG: = g* at the identity, and every e E g* gives rise to a left invariant one form which we shall denote by Be. Thus the most general equivariantly closed two-form on G is given by
{cR)
dc8r = dQe- 1Clc where (+eye) = ~ ~ ~ ( e t ) .
To prove part 1) it suffices to prove that dBc E R2(G)b, i.e to prove that doe is Gpinvariant and satisfies
<
<
for all E ge, where, for any E G, we let
Lemma 9.2.1 For any
< E g, (9.6) holds if and only if < E gr.
To prove the lemma we use the identity (9.2)
d e t ( t ~ , v= ~ )L~,.lBe(q~)l - L,, [Be(EL)I- QI([
Notice that The G-equivariance of $e says that for any b f G and any
Proof. Let Gr act on G on the right, and consider G as a principal Gc bundle: n : G 4 GIG[.
5 E g,
Since Be, EL and q~ are all left invariant, the first two terms on the right vanish and the third term is constant. Evaluating it at e give
Taking b = a-' and evaluating at e gives
thus (9.6) holds if and only if
The coadjoint representation is defined a s the contragredient of the adjoint repr,esentation and is given by
which says that E ge. This proves the lemma, and with it statement 1) of the theorem. 1 So there is an we E Q(G/Ge) with T * W ~ = dBc. We must prove that it is symplectic. For this it is enough to show that if a*we = d& is annihilated
114
Chapter 9. Equivariant Symplectic Forms
by a left invariant vector field, this vector field is tangent to the fibers of 0. G -, G / G e . But this is precisely the assertion of our lemma. Combining we with the map (9.5) gives an equivariant symplectic form
9.4 The Compact Case
115
and the commutative diagram
on G/Gf such that n8Gf = dGBt. We can state Theorem 9.2.1 in terms of coadjoint orbits, which was the original formulation by Kirillov, Kostant, and Souriau: Let 0 denote the coadjoint orbit containing l . By (9.4) and (9.5), $Jc is a G-equivariant d8eomorphism of G / G f onto 0.Let
denote the inverse diffeomorphism, so that
is the inclusion map of 0 into g*. Let
If we identify G I G t with 0 e we may think of q5( as the identity map, and hence the preceding equation implies that we may identify p with 4. We have proved: Let 3 = w - 4 be a closed equivariant two-form on G / K . Then C$ is a G-equivariant map of G I K onto some coadjoint orbit O = G . e = Oe and
9.4
Theorem 9.2.2 Kirillov-Kostant-Souriau There is a unique equivariant symplectic form 60 E R g ( 0 ) with moment map given'by the inclusion, io : 0 --+ g * .
Remark. Notice that in this section we did not need to assume that G was compact. There are, however, some special features of the orbit picture which are particularly nice when G is compact; see Section 9.4 below.
Equivariantly Closed Two-Forms on Homogeneous Spaces
Let K be a closed subgroup of G and suppose that 3 is an equivariantly closed two-form on G I K . If n : G -+ G / K is the projection onto cosets, then x'3 is a closed equivariant two-form on G and hence of the form dG&. If we write G=w'-
.
5 = pBGc = pawc - p'&.
I n particular, 3 is syrnplectic i f and only if the map
Then Theorem 9.2.1 becomes
9.3
It follows that
4
then x*q5 = qe where $,(a) = ~ d f j e .In particular, at the identity coset, K = eK we have + ( K ) = e and, by the equivariance of q5 : G / K -+ g* we see that K , the stabilizer group of the identity coset, must be contained in Ge, the stabilizer group of e. So we have the projection map
4 is a covering map.
The Compact Case
In case G (and hence K ) are compact, and G is connected, the preceding results can be strengthened. We will prove that every coadjoint orbit is simply connected which will imply that any cover (as in the last section) must be a bijection and that every G e is connected. We begin with a result which is of interest in its own right. Let T be a maximal t o m of the connected compact Lie p u p , G and let 0 be a coadjoint orbit. Then the action of T on 0 has a finite number of fized points which are all non-degenerate, in the sense that the linear action of T on the tangent space at each fized point has no non-zem &ed vectors.
Proof. The non-degeneracy assertion follows from the finiteness of the fixed points, since we may choose a T-invariant Riemann metric on 0 and then any line of fixed vectors in the tangent space is carried by the exponential map of this metric into a whole curve of fixed points. By a choice of an invariant scalar product on g we may identify the adjoint and coadjoint representation, and by the conjugacy theorem for Cartan subalgebras we may assume that 0 = with T C G f . So T is a maximal torus of Ge. A coset bGe is fixed by T if and only if b-'Tb c G c and hence, by the conjugacy theorem for maximal compact subgroups of G e , there exists an a E Gc such that a-lb-'Tba = T. Thus bGe = baGc and ba normalizes T. But N ( T ) / T = W(T), the Weyl group, is finite. Hence there are finitely many fixed cosets, bGc.
116
Chapter 9. Equivariant Sympledic Forms
Let V be the tangent space at a fixed point. Then V decomposes under T into a direct sum of two dimensional subspaces. The action on each two dimensional subspace is rotation through angle a - B where a = (al,. . . ,a,) is a row vector with integer coordinates not all zero and B = ( B 1 , . . . ,Bn)t is typical element of T = (S1)n. We may choose a = (el,. .. ,en)+ E t , the Lie algebra of T, such that the one parameter group generated by E is dense in T and such that a(<) # 0 for any a at any fixed point. The only zeros of the vector field on U corresponding to are the fixed points of T, and these are all non-degenerate zeros of E. Then 4c is a non-degenerate Morse function with critical points at the fixed points of T, and the index at each critical point is even. It follows that the Morse-Whitney stratification of 0 associated with 46 consists of a single open cell W whose complement has codiiension two. Hence every closed curve in 0 can be deformed to a curve in W and then contracted to a point. Hence O is simply connected. It follows from the homotopy long exact sequence for the fibration G -+ GIGe that Gl is connected. We recall the argument in our special case. We consider Ge as the fiber over the identity coset, eGr. Given any two points P and Q in Gc we can connect them by a curve in G since G is assumed to be connected. This curve projects onto a closed curve y in G/Gt based at the identity coset. By the simple connectedness of GIGe we can find a homotopy of this curve to the trivial curve. In other words there is a map of the unit square, 0,into GIG( whose restriction to the bottom edge is y and whose restriction to the top edge is the constant curve, taking the constant value Ge. The lifting property for fibrations implies that we.can lift this to a map of into G whose restriction to the left hand side and top is identically P and whose restriction to the bottom is the curve from P to Q that we started with. The restriction to the right-hand side then gives a curve joining P to Q in Ge.
e
<
9.5
9.6 Symplectic Reduction
+
W is a "twisted product" of X and F. If X and F are symplectic manifolds, then there is a product symplectic structure on X x F. In this section we shall prove an analogous result for the twisted product, under the assumption that we have a symplectic form p on X and an equivariant symplectic form 3 on F. Let pr, and pr, denote the projections of M x F onto the first and second factors. Then for each e E R we get the equivariant two-form
on M x F. Let us equip M with a connection, and let B', . .. ,On be the connection forms (relative to a choice of basis of g). The connection on M induces a connection on the bundle
MxF+W whose connection forms are (prl)*B1,.. . , (prl)*Bn. As we proved in Chapter 5, this connection allows us to define a Cartan map
lying this map to p, gives'us a closed two-form ve E
Theorem 9.5.1 For
r:w+x,
y([(m,f)l=*(m) (9.10) where [(m,f)] denotes the equivalence class of (m, f ) . This makes W into a fiber bundle over X with typical fiber F. In the language of the topologists,
lei # 0
suficiently small, v, is symplectic.
Proof. Let w E R2(W) be the form obtained by applying the Cartan map (9.12) to (pr2)*3. Then if 2d := 2p + 2q = dim W, 2p := dim X ,
we have V:
2q := dim F
+
= equqA ( y * ~ ) O(eqf ~ l)
and it is easy to see that the first term on the right is nowhere zero. For applications of this minimal coupling form to elementary particle physics see [Stl]or [GS]. For applications to representation theory, see [GLS]. For applications to topology see [GLSW].
9.6 and mapping
P(w)
which is called the minimal coupling form with coupling constant e.
Minimal Coupling
Let M be a G-manifold on which G acts freely with quotient space X = MIG and with projection n:M-+X sending each m E M into its G-orbit. We. think of this as a principal Gbundle. Let F be another G-manifold. The diagonal action of G on M x F is free, so we can form the quotient
117
Symplectic Reduction
Let M be a G-manifold on which G acts freely and let 3 = w - q5 be an equivariant symplectic form with moment map q5 : M -+ g*. We will show that q5 is a subdersion, i.e. that
118
Chapter 9. Equivariant Symplectic Forms
9.6 Symplectic Reduction
is surjective. Indeed, if not, the components of 4 with respect to a basis tl,.. . ,tnof g would have to be functionally dependent at p. Put another way, some non-trivial linear combination of
This implies that there exists a two-form vo on Xo satisfying (9.13). To prove that uo is syrnplectic we use the identity L~
(ddE)P= 0
< # 0 E g. But
. . . L,,
( w )= ~ ( d ! / n ! ) ~A~ddl - ~A . - . A dC$n
at all rn E Mo
(9.14)
where dim M = 2d. (We leave the proof of this identity as ao easy exercise.) In fact, dim Xo = d i m M - n - n = 2(d-n), so to prove that vo is symplectic, it suffices to show that v,d-" is nowhere vanishing, which is the same as proving that (r;uo)d-n is nowhere vanishing. Since 7r;vO = i(;w,this is the same as proving that (z(;u)~-" vanishes nowhere on Mo. This is the same as showing that the right hand side of (9.14) does not vanish at any point of Mo. But the left hand side of (9.14) vanishes nowhere on M, since the are everywhere independent and w is symplectic. The operation of passing from
must vanish, i.e. for some
119
dq,E = --L.GMW
<.
where CM is the vector field on M corresponding to Since w is symplectic, we conclude that (tM),= 0, contradicting the assertion that the action of G is free. Since q5 is G-equivariant the level set
is known as syrnplectic reduction or ~arsden-weinsteinreduction. For a more detailed treatment see [GS] section 26. If the group G is abelian, there is nothing sacrosanct about the zem level set of the. moment map. For every a E g* the submanifold
is a G-invariant submanifold of M . Let
We have the inclusion io : Mo -+ M
and the projection
is G-invariant. Let
ro : Mo -,xo.
Theorem 9.6.1 Marsden-Wein~tein form, vo on Xo with the pmperty
X a := MJG
here exists a unique symplectic
and let
z
a
a
7ra:Ma+Xa
the inclusions and projections. Then the proof of Theorem 9.6.1 goes over unchanged to prove
Proof. The equivariant form z:i;r E n G ( M o )is dc-closed. Since 2:1$ = 0 by the definition of Mo, we have
Theorem 9.6.2 Marsden-Weinstein There exists a unique symplectic form, va on Xa with the property
Since dw = 0 we have
If the moment map C$ is proper, the level set Ma is compact, and hence so is its quotient space X a . In particular, its symplectic volume
where x l , . .. ,xn is the dual basis to our basis &, .. . ,tnof g. Thus
-
in other words iGw is basic with respect to the fibration Mo
xo.
is well defined and is a smooth function of a. In the next section we will prove the following theorem of Duistermaat and Heckman: b
Theorem 9.6.3 The symplectic volume (9.18) is a polynomial as a function of a.
120
Chapter 9. Equivariant Symplectic Forms
9.7
The ~uistermaat-~eckman Theorem
9.8 The Cohomology Ring of Reduced Spaces
121
Fkom (9.17), (9.21), and (9.24) we obtain
Let X := M/G
v]
and, s:M+X be the canonical projection. If G is abelian, the coadjoint action is trivial. Hence the fact that the moment map q5 : M --, g* is G-equivariant means that it is invariant. Thus it factors through s , i.e. there is a smooth map
such that
OX.
(9.19)
Since 4 is a submersion, so is $ and hence, by (9.19),
Let c denote the cohomology class of v - $r,uP,and let denote the cohcmology class of the curvature form ,ui. Letting [v,] denote the cohomology class of v,, we conclude that
In other words, [u,] "varies linearly with a". Since X, is compact and oriented, the embedding j,, : Xa-+ X defines a homology class [Xa] E H2(d-n) (X1 Z, in the integer homology group of dimension 2(d - n). This homology class depends smoothly on a, and being an integer class is thus independent of a. So let us fix an a, in the image of 4. Then
SO
We can now prove the Duistermaat-Heckman theorem. The integral (9.18) can be interpreted topologically as the pairing of the constant homology class [X,,] with the cohomology class
Let denote the inclusion, so that we have the commutative diagram
1 (C
( d - n)!
+~
w l ) ~ - ~
so the value is clearly a polynomial of degree d - n:
9.8 Now let us equip M with a connection and consider the associated Cartan map k ( M ) --,f l ( X ) . (9.22) Under this map, the equivariant symplectic form
gets mapped to
- *rpr where the pr are the curvature forms of the connection, and where v is the unique form on X with the property that
The Cohomology Ring of Reduced Spaces
Let (M, w ) be a symplectic manifold of dimension 2d, G a compact connected Lie group, r : G + Diff(M, w ) a Hamiltonian action of G on M with moment map 4 : M -+ g" and Z = $-'(0) the zero level set of the moment map. If 0 is a regular value of 4, then the action of G on Z is locally free, and the reduced space X = Z/G is a symplectic orbifo!d of dimension 2(d-n) where n = dim G. Let us assume C) is that $ is proper. Then X is compact, so its cohomology ring, H*(X, finite dimensional and satisfies Poincare duality. In the early 1980's, Kirwan, [Ki], showed how to compute the Betti numbers of X, using Morse theoretic techniques. Recently, quite a bit of progress has been made on the much more difficult problem of understanding the cohomology ring structure of X. Some relevant papers are [Wi], [Ka], and [JK]. See also the survey paper, [Dl. In particular, Jeffrey and Kirwan have found a general formula
[w,
122
Chapter 9. Equivariant Symplectic Forms
for pairiigs of cohomology classes on X which enables them, in principle, to determine the ring structure of X when the ambient space, M is compact. (In practice, the problem of decoding the ring structure from their formula is non-trivial.) The purpose of this section is to point out that a good deal of inform* tion about the cohomology ring structure of X can be extracted from the Duistermaat-Heckmann theorem. The version of this theorem that we will use is a slight sharpening of the version we proved in the last section: Let G = Tn be the standard n-dimensional torus, and for e E g' close to zero, set 2, = $-I(!) and Xe = Zt/G. (9.28)
ii *
9.8 The Cohomology Ring of Reduced Spaces
i
and hence, by Poincare duality, all relations of the form
I
Specifically,
123
Theorem 9.8.1 A s a dzffemntiable manifold X =: Xo = Xc and
where vtop denotes the top order homogeneous part of v and where Q = a@&. Clearly this last expression vanishes for all 171 = d - n if and only if QvtOp= 0. From this one obtains:
where [pel is the cohomology class of the symplectic form on Xt, b] = [pol is the cohomology class of symplectic form on X, and c = (cl, ...,c,,) is the Chern class of the fibration Z + X.
Theorem 9.8.2 If cl, . . ., c, generate H * ( X ,C), then H * ( X ,C) is isomorphic as an abstract ring to
Let u(q =
/xt
exp([irt]) =
/
X
~ P ( W +C e i s )
where Q(xl,. . .,I,) E ann(vtop) if and only i ; f ~ ( & ,.- . , &)vtop(l) = 0. (9.30)
be the symplectic volume of Xe. From (9.30) it follows that v(e) is a polynomial of degree d - n and that
In particular, if la1 = d - n
Thus the coefficients of degree d - n - k of the polynomial v(l) determine the cohomology pairings (9.31) and, in particular, the leading coefficients determine the cohomology pairings (9.32). The identities (9.31) and (9.32) are more or less well known, but they haven't been used very much as a tool for computing the ring structure of H D ( XC). , Possibly this is because they give no information if the fibration Z --+ X is trivial, which is frequently the case. On the other hand, if the c, generate the cohomology ring, one can read off from (9.32) all multiplicative relations of the form
In this section we discuss tko examples of applications of this theorem in the first of which X is the generic flag variety and in the second X is the toric variety associated with a simplicial fan. The cohomology rings of the flag manifolds were determined by Bore1 in his classic paper, [Bo], and for toric varieties by Danilov in [Dan]. It is interesting to note that the structure of H 8 ( X ,C ) in both these examples is given by the general recipe (6). Let us return to the case of an arbitrary (M, u).Let p be a fixed point of the G = Tn action on IM, and let 01,. . . , a d be the weights of the isotropy representation of G on the tangent space at p. We will call p an extremal fixed point if there exists a v E g such that
(See [GLS], Section 3.) If p has this property, we will prove in Section 9.8.3 that if is a regular value of # close to $ ( p ) then Xc is a toric variety, and we will show how to compute its moment polytope in terms of the a,. This has the following useful corollary. Suppose that M is compact and the fixed point set of G is finite. By the convexity theorem ([At] or [GS]) the image, A = 4 ( M ) , is a convex polytope. Let AOC A denote the set of regular values of #. The connected components, Al, . . . , AN of A0 are themselves open convex polytopes. By the Duistermaat-Heckrnann theorem, the diffeotype of the reduced space, Xe is constant as varies over each A,. In particular, the cohomology ring of this reduced space depends only on A,.
<
<
Theorem 9.8.3 Suppose the closure of A, contains a vertex of A. Then its associated reduced space is toric va'ety.
124
Chapter 9. Equivariant Symplectic Forms 9.8 The Cohomology Ring of Reduced Spaces
Thus for a Ai whose closure contains a vertex, we can compute the cohomology ring of its reduced space by theorem 9.8.6 below. In general, there will he many connected components of AO which don't have this property. This brings up the following interesting question: How does the cohomology ring of the reduced space, X e , change, as passes through a common (n - 1)dimensional face of two adjaceq Ai's? For some recent results concerning this question see [TW].
<
9.8.1
Flag Manifolds
Let K be a compact semi-simple Lie group and T its Cartan subgroup. The adjoint action of T on the lie algebra k has a T-equivariant splitting
Here t is the Lie algebra of T and tL is its orthocomplement with respect to the Killing form. Using the right action of T on K we get the fibration
mapping an element of K into its right T coset. This gives us a fibration of K over the flag variety KIT. The splitting (9.34) gives us a splitting of the tangent space to K at e into vertical and horizontal components, which can be extended via the left action of K to the whole of K, giving an intrinsic K x T invariant connection, 0, on the bundle (9.35). By minimal coupling, cf. Section 9.5, one gets a presymplectic form
125
Proposition 9.8.1 The reduced space of K x (t;)O with respect to T at C is isomorphic as a Hamiltonian K space to Ot. *
Proof. At the point (e,e) E K x (t;)O minimal coupling satisfies
the symplectic form defined by
But the expression on the right is precisely the Kirillov-Kostant form evaluated on z, y E k l t at & E Ot. Let c be the Chern class of the T fibration K + KIT. By definition, this is a t-valued cohomology class of degree two which we can write as
where
Hiis the standard Weyl basis oft.
Proposition 9.8.2 c l , . . . ,c, genemte the cohomology ring H 8 ( K / T C , ). For a proof of this see Borel [Bo~]. As a corollary, we see that the identities (9.32) completely determine the cohornology ring structure of KIT. To make these identities more explicit, we will use the following result, cf. [BGV] p.232, giving a formula for the symplectic volume of a coadjoint orbit, and hence an explicit formula for the left hand sides of (9.31) and (9.32): Proposition 9.8.3 The symplectic volume, v(C) of the coadjoint orbit, Ot is given by the formula 1
4 0 , pr2)
(9.36)
on K x t* where prz : K x t* + t * denotes projection onto the second factor. The presymplectic form given by (9.36) is symplectic on the set
where t; denotes the positive Weyl chamber and t: its interior. Letting K x T act trivially on the second factor in (9.38) gives a Hamiltonian action of K x T on K x (t:)'. The T-moment map for this action is (the restriction of) prz. For e E (t;)O let Or denote the coadjoint orbit of K through C. The stabilizer of e in K is T, so Or KIT
-
a s a K-homogeneous space. The reduction of k x (t;)O at !(with respect to T) is also KIT as a homogeneous K space. We claim
' r]:
v(t) = -
(C,a) men+ where a ranges over the positive roots, where (e, a ) is the inner product of C and a with respect to the form induced on t* by the negative of the Killing form and where the constant y i s given by
Notice that the right hand side of (9.40) is a polynomial in C of degree dimK = 2p + n where p denotes the number of positive roots. Plugging (9.40) into (9.32) gives
d
- n & required, as 2d = dimK + n ,
We will say a few words about how these results are related to the theorem of Borel which we mentioned above. Borel's theorem says that the cohomology ring of KIT is isomorphic to the ring
126
Chapter 9. Equivariant Symplectic Forms
I
C[xl,. . .,x,JW being the ideal generated by the Weyl group invariant polynomials of degree greater than zero. To deduce this from Theorem 1.2, one has to show that if Q(x1,. . . ,x,) is a homogeneous Weyl group invariant polynomial of degree greater than zero and v(e) is the function (12) then
9.8 The Cohomology Ring of Reduced Spaces
If one wants Xa to be a manifold rather than merely an orbifold, condition 3 should be replaced by the stronger 3*. The Pi(p), i = 1,.. . ,n, form a basis of L'. Let d denote the number of (n - 1)-dimensional faces of A. Condition 2 implies that these faces can be defined by equations of the form
Notice, however, that if a is a simple root and a,: t + t is reflection through the hyperplane a([) = 0, then a, maps all of the positive roots except a into themselves and maps a onto -a. Hence
where the vector belongs to L. The vector ui can be normalized by requiring it to be a primitive element of L, which then determines it up to sign. The sign can be fixed by requiring that A be contained in the half space
and since Q is Weyl group invariant,
These normalizations determine the u, and A, uniquely. Conversely, A is given as the intersection of the half spaces (9.44). Define the map T:R~-+v, T:~,++u, (9.45)
In particular, Q (g)v(e) vanishes on the hyperplane a(!)
= 0, so the monG mial a(!) divides Q (&) v(e). Since every positive root is Weyl group conjugate to a simple root, it follows that, for every root a , a(!) divides Q (&) v(P). Hence v(e) itself divides Q (&) v(e), and this is impossible unless Q (&) 44 = 0. Remark: For this argument we are indebted to David Vogan.
9.8.2
Delzant Spaces
We will refer to the toric varieties in this section as Delzant spaces siflce we will be thinking of them as symplectic manifolds (or orbifolds) rather than as complex projective varieties. Let us briefly review the definition of these spaces as given by Delzant [Dell. Let V be a real n-dimensional vector space and L c V and n-dimensional lattice, with dud space V* and dual lattice L*. Let T be the torus T = V / L . Given a convex polytope, A c V* satisfying certain axioms listed below, we will associate a symplectic orbifold, Xa of dimension 2n to A. We will equip XAwith a Hamiltonian T action so that A is the image its moment map. By the uniqueness theorem of Delzant this characterizes Xa up to isomorphism. The assumptions about A are the following: 1. Exactly n edges of A meet at every vertex. 2. The edges meeting at any vertex p point in rational directions (relative to L*), i.e. lie along half rays of the form
1
where {el, . .. ,ed} is the standard basis of Rd. Since
I
we get an induced map
I
.We have the standard linear action, p, of the toms td on Cd given by
.
i
where
I
I \
s = (xl,. . .',xd) E T~ and r = (21,. . . ,zd) E cd.
This action preserves the syrnplectic form
1
dzJ A dZj
-i
.
3
and is a Hamiltonian action with moment map
. 3. The h ( p ) , i = 1 , . . . , n , form a a basis of V*
127
Let G be the kernel of the map rI (cf. (9.46)). Restricting p to G gives a Hamiltonian action of G on Cd with moment map I
128
Chapter 9. Equivariant Syrnplectic Forms
where each a, is the weight of G on the one-dimensional subspace of Cd spanned by e, i = 1,. . . ,d. Set
where the A, are the same a s those entering into the description (9.44) of A. Let ZA be the X lwel set of the map 4
zx = { r :
9.8 The Cohomology Ring of Reduced Spaces
'1
which is the line bundle associated with the weight
I
lu12a;= A).
We refer to [Dell or [Gull for the following: Proposition 9.8.4 If A satisfies conditions 1-3 the action of G on ZA is locally free. If, zn addition, 3* holds, then this action is free.
i
We now define
C 6.q. But
so this line bundle is the trivial bundle and hence its Chern class is zero. . For the following see [Dan] section 11:
Proposition 9.8.7 The Chem classes c, genemte the cohomology ring H*( x a , c )If we now let X vary slightly about a fixed value, the symplectic volume varies, and we can apply (9.32). Once again, to apply this formula we need an alternative computation of the symplectic volume: Here we have a strikingly simple description, for whose proof we refer to [Gull section 6: Theorem 9.8.4 The symplectzc volume of XA is the Euclidean volume of the polytope A.
X* = Zx/G. By the proposition, this is an orbifold, and is a manifold if 3* holds. Since G is the kernel of ll,the group T acts on X* in Hamiltonian fashion. We refer again to [Del] or [Gull for a proof of
129
1
So if we let v(X) denote this volume, we can write (9.32) as
Proposition 9.8.5 The image of the moment map for the T action on X p is the polytope A. for any multi-index /3 with (PI = n . (Note that since v(X) is a polynomial of degree n the left hand side of (9.51) is a constant.) We will briefly describe the tie-in between this result and the result of Danilov which we alluded to in the introduction of this section. Let g be the Lie algebra of G and L the inclusion of g into Rn. From (9.46) one gets an . exact sequence
The aim of this subsection is to give a description of the cohomology ring of X* along the lines indicated in the introduction to this section. To this end we need to determine the Chern class of the fibration
Since Zx is a principal G-bundle, a one-dimensional representation of G gives rise to a line bundle over X*. Let L, be the line bundle associated to the representation of G on the one-dimensional subspace of Cd spanned by ei (with weight a,). Let c, be the corresponding Chern class.
o+~*~(R~)*CV*+O. For 1 and A' in Rd let A and A' be the polytopes defined by
Proposition 9.8.6 The Rd-valued cohomology class
!,
Proposition 9.8.8 A and A' are congruent if and only if X - A' E ker L" i n which case A1=A+v
takes values i n g, the Lie algebra of G, and is the Chern class of the fibration zx-,X*. Proof. We content ourselves with proving the first of these two assertions. .. ,K ~E )(Rd)* lies in the annihilator space, We must show that if K = (q,. go of g then C nici = 0. It is enough to prove this when the 6 1 , . . . ,~d are integers. But !hen nit, is the Chern class of the line bundle
We leave as an exercise the following result:
1
where X
- X1 = a'v.
J3om Proposition 9.8.8 one gets n first order partial differential equations , satisfied by v(A) namely
130
Chapter 9. Equivariant Symplectic Fonns
where w, = z'v, and v;, i = 1,.. . , n , is a basis of V; and from these equations one gets n homogeneous generators of degree two of the ideal, ann(v). However, in addition to these generators, there are also some generators of higher order. Namely by (9.44) one can associate with each of the standard basis vectors, e,, of Rd the (n - 1)-dimensional face of A defined by
Denoting this face by Fi it is not hard to show that for a multi-index, I = (il,. . . ,i k ) ,1 5 il < iZ < . . . < ik 5 d,
if and only if F,, n .. . n F,, = 0. We claim (but will not attempt to prove here): Theorem 9.8.5 ann(vtOp) is generated by the n genemtors of degree two associated with the n equations (9.52) and the genemtors of degree 2111 associated with the equations (9.53). Combining this with Theorem 9.8.2 one ends up with the theorem of Danilov: Theorem 9.8.6 H*(X*, C) is a ring with d generators (each of degree 2) satisfpng the relations (9.52) and (9.53).
9.8.3
Reduction: The Linear Case
Let V be a complex d-dimensional vector space equipped with a'positive definite Hermitian form, H and let w be the syrnplectic form w = ImH. Let G be an r-dimensional torus and p a unitary representation of G on V. We can ignore the linear structure on V and consider p as a Hamiltonian action. In this section we discuss what reduced spaces look like in this linear setting. Let el,. . .,e d be an orthonormal basis of V with the property that the one-dimensional subspaces spanned by the ei are G-invariant. Each such subspace has an associated weight, call it a,. This basis gives us a coordinate system (zl,. . . ,zd) on V.
'
9.8 ~ h e ' ~ o h o m o Ring 1 0 ~ of ~ Reduced Spaces
The action p is said to be quasibe if, for all p E V the stabilizer group, Gp is either of dimension greater than zero or is trivial. It turns out that for linear actions quasifree is equivalent to the condition that Gp be connected for every p E G, cf. [GPS]. We also refer to [GPS] for the following Proposition 9.8.11 p is quasi-free if and only if every r-element subset of {al,... ,ad) which spans g* is also a set of genemtors for the weight lattice of G. It is also clear from (9.54) that the image of q5 is the cone
For the following more refined result see [GLS], Proposition 3.28, Proposition 9.8.12 If a,, , . . . ,aim span a proper subspace of g* every point on the cone YP~.,; YP 2 0)
{C
is a critical value of the moment map (9.54). Conversely every critical value lies on one of these cones. Now let A be a regular value of the moment map. If the moment map is proper, the level set zx = ,$-'(A) is a compact submanifold of V. If the action is also quasi-free, the action of
G on Z A is free, and the reduced space
is a compact symplectic manifold. We will prove that this manifold is a
Delzant space (of the type considered in the preceding section): Let Rd+ denote the positive orthant
and let 7:~d+rJ*
Proposition 9.8.9 The moment map for the action p is
be the linear map sending the ith standard basis element into &. Let
We will henceforth assume that the action of G is effective. (This implies that the weights a, span g*.) For the following see [GLS] section 3:
T h e o r e m 9.8.7 X x is a Delzant space and its moment polytope is AA.
Proposition 9.8.10 The moment map (9.54) is proper zf and only if there exists a vector v E g for which ai(v) > 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<, 011) = -([v7ql, [<,El) + ([v,EI,IC,vl) = -([vyvl, [C,JI) - ([u,El, [v, CI). Thus ( 1 ~ ~[vvC1) ~ 3 , + ([v7v17IC,El) + ([v7Q7 [E7v1)= 0. We conclude that
exp : g --+ G which is G-equivariant for the adjoint action of G on g and the conjugation action of G on itself, This map is a diffeomorphism in a neighborhood of the origin. we can form the composite
where A denotes the alternating sum over all permutations of v, E, q, C. Let G be a Lie group with Lie algebra g and suppose that the adjoint representation of G leaves ( , ) invariant (which is automatic for the connected component of G). So q E (Ag'lG.
136
Chapter 9. Equivariant Symplectic Forms
Let 8 and 3 E R1(G, g) denote the left and right Maurer-Cartan forms. If L, and R, denote right and left multiplication by g E G, then the values of B and 8 a t g are given by
Og := dLg-l : TGg + TG,,
9.10 Group Valued Moment Maps
<
where ER is the right invariant vector field corresponding to (and so is the infinitesimal generator of left multiplication by expte) and tL is the left invariant vector field (corresponding to right multiplication). Now
I
8, := dRg-l: TG, -,TG,.
L(
31) - (3, [<, 81) + (3, [8,El) = 3(Et[3,8], = 6(<,d3) = 6d(8,().
In any faithful matrix representation 6 = a-Ida,
-
6' = da . a-'.
At any a E G we thus have
-
e,
= A& (8,).
Also (or directly from the definitions) for any fixed b E G,
137
f I i
Similarly,
I
Thus
L ( < L ) ([@, ~ >01) = -6d(@,E).
t
1 L(
+ 3,<).
(9.67)
RRcall from Section 2.1 that our notational convention in this book is to let LC denote the interior product with respect to the infinitesimal generator of the one parameter group ap(-tE), in out case the conjugation action of this one parameter group. Hence we can write (9.67) as
Here Ab denotes the conjugation action of b on G, so A: denotes pull-back via this action on forms, and Adb denotes the adjoint action of b on g. Note that Ada is the derivative of Ab at the point e which is fixed by Ab. The Maurer-Cartan equations say that
1
L ~ = X --d(B
2
+8 , ~ ) .
The map of g + R(G) given by
i
In particular, the three form 1
x := E(o,
1 - -[o,01) = --to, 12 10,01)
I
is G-equivariant. Indeed,
is bi-invariant. It is also closed since
1 x = --(0, 6
dB)
We may therefore define the equivariant three-form
by the Maurer-Cartan equations and hence At any a E G we have which vanishes by (9.62): In other words dx = 0. It is called the canonical three form of G - canonical relative to the choice of ( , ). We can extend this to an equivariantly closed three form on G relative to the adjoint action as follows: For any E E g, let [G denote the vector field which is the infinitesimal generator of conjugation by a p e . So
hence L(&) (0
+ 8)(a) = ( ~ d , ' - Ad,)
E.
(9.70)
In particular, LC
'
(0
+ 3, <) (a) = -L(
(6
+ 8, <) (a) = - (Ad, E - Ad,-1
and conclude that dGxG = 0.
E, 6) = O
138
Chapter 9. Equivariant Syrnplectic Forms 9.10 Group Valued Moment Maps
139
9.10.2 The Exponential Map
i
For each s E R let : g --,G
I
be defined by exps(rl) = exp(srl) where exp : g -+ G denotes the exponential map. Thus exp,(q) is given as the unique solution of the differential equation with initial conditions
If A, denotes the conjugation action of a E G on G, and Ad, denotes the adjoint action of a on g, then we have the equivariance condition
Consider the two form on g defined by
For E E g let v~ = Eg denotes the vector field on g corresponding to the adjoint action of G. We claim that T satisfies the following three properties:
i
! l
.
$1'
- --
exp:(Q. [Q. Q])ds-
-- a1 exp*([Q. Q1.8)
-- 1 = -- exp*([Q,@I,@) 12
I
We now prove (9.77). We first recall our notation. For any E E g, ER denotes the vector field on G which is the infinitesimal generator of left multiplication by exp tJ, Thus ER is the right invariant vector field corresponding to E and
/ Let EL denote the tangent at t = 0 to right multiplication by exptt, so EL is the left invariant vector field corresponding to E and
To prove the first of these equations, observe that A, = L,Ril where La denotes left multiplication by a E G and % denotes right multiplication. Also, 8 is invariant under right multiplication. Hence, by (9.73),
If we set a = exptt in (9.73), and differentiate with respect to t at t = 0 we get, for 71 E 9, 4 ~ ~ P , ( v=E <) R ( ~ X977) P - < L ( ~37). P If we apply 8 to both sides we get
Also, we may think of flow
6E
g as a constant vector field on g generating the
rlH71+tE. The invariance of ( , ) then proves (9.75). To prove (9.76) we use the MaurerCartan equations for 3. We have
Define
4,,, : s -- G,
4,,,(0 := exp,(q
+ C) exp_,(q)..
Thus exp,(71+ t<) = 4,,,(tE) exp,(q). Then L ( E ) ~ ~ P :=' &(&xp..
a
expsq)(E) = z43,q(tE)lt=~.
140
Chapter 9. Equivariant Symplectic Forms
9.10 Group Valued Moment Maps
On the other hand, the difIerential equation satisfied by the exponential at q+t
a
~ P , ( v+ tt)-lzexp,(7
<
We now apply (9.78) to both terms, to the integrand for all 0 5 s 1 and for the second term with s = 1. The second term becomes, when evaluated at q E g, 1 1 -(exp*B,, 5 - Adexp?t) = 5expw(8- Q , ~ ( I I ) . 2 The integral becomes
+ t<) = q + t<
which translates into
Let us differentiate this identity with respect to t and set t = 0. The right hand side gives <. Applying Leibniz's rule, we get a sum of two terms, the first coming from the t dependence in d(tt)-l gives
j
-
i
exp,(q)-l
& (q5,,,(t[)
exp,(q))
a
as ( ~ ( texp: ) Bexp,(rl)) .
Expanding the partial derivative with respect to s into two terms by Leibniz's rule again gives two terms, one of which cancels the preceding and we are left with
a
Adexp.(,)-l
(L(OWP:
8) = 5.
Applying A&xp,(,) and integrating from 0 to s gives
.
l'(<> a ~
I
~
(
(The d in the last expression serves to remind us that we are to identify elements of g with constant vector fields.) Adding the two terms gives (9.77).
9.10.3 GValued Moment Maps on Hamiltonian G-Manifolds Now let M be a Hamiltonian G-manifold with symplectic form w and with moment map q5 : M g*. Composing 4 with the identification of g' with g provided by ( , ), we obtain a map cP : M -+ g which may then be composed with the exponential map. That is, we may consider the map
v :M We can now prove (9.77): We have
d) - I %(<> ~ ~ X P~: 3 ) ~~s
= (exp*8, <) - d(. ,t ) .
j
The second, coming from the t dependence in the term gives
141
-, G,
.
v := expo*.
(9.80)
Clearly v is an equivariant map for the conjugation action of G on itself. Define the two form E on M by
The form Z is invariant under G since w and T are. To compute dz we use the fact that d u = 0 and (9.76). We get
Let [G denote the vector field on M corresponding to .$ E g, so that the moment map property of @ says that
Using (9.77) we obtain
142
Chapter 9. Equivariant Syrnplectic Forms
or
1
L((M)Z=-uf(9+s,[), 2
9.10 Group Valued Moment Maps
of the moment map. But, by another basic property of the moment map, (cf. [GS] page 184 again) this implies that EM()) = OWe have thus proved that
(9.83)
V[Eg-
We can combine equations (9.82) and (9.83) a s follows: Consider 2 as an element of R&(M). Then dGZ = -v*xG, (9.84)
143
ker 2, = {EM(z), C E ker (Ad,(,)
i
+ 1))
(9.86)
So far we have not made use of the non-degeneracy of w . Let us now use this property in computing the kernel of E. Suppose that v E TM, satisfies
if @(x)is a regular point of the exponential map. This leads Alekseev, Mallcin and Meinrenken [AMM] to make the following definition:
This means that
Definition 9.10.1 A G-manifold M together with a G-invariant two form S and a G-equivariant smooth map v : M -+ G is called a q-Hamiltonian G-space if (9.84) and (9.86) hold, i.e. i f L(V)W,= -L(v) (@*T),.
dGE = -U'XG
The right hand side is an element of T'M, which vanishes on all elements of ker d@,. So v E (ker d@,)Wz= (ker d4,)Wz, . the symplectic orthogonal complement with respect to w, of ker d@,. But one of the basic properties of the moment map (cf. [GS] p.184) is that this orthogonal complement consists precisely of the evaluation of vector fields coming from the G-action. In other words,
ker S, = {JM (x)I
and
E E ker (A&(,)
+ 1)).
We can summarize the preceding discussion (and read it backwards for the converse) as Proposition 9.10.1 Let M be a Hamiltonian G-space with moment map considered as a map : M -+ g. Suppose that the image of @ consists of regular points for the exponential map. Then
for some J E g. So, by (9.83), we conclude that
(a*expa(@+ 8, J)),, = 0. Now for any 8 E g, d Q ( 7 ~ = ) 729 and d exp g = n ~ Hence, . by (9.70), if we take the interior product of the preceding equation with ~ M ( Xwe ) get
(n. (~d;;,)
-A ~ ~ ( ~ = ) )0.E )
and
2 := w + 9*Y
give M the structure of a q-Hamiltonian G-space. Conversely, suppose M is a q-Hamiltonian G-space such that v(M) lies in expU where U w a nezghborhood of the origin for which the exponential map is a diffeomorphism. Then := exp-' ov and the preceding equation make M into a Hamiltonian G-space.
Since this must hold for all 17 E g, we conclude that
e ker (Ad:(,)
)
- 1 = ker Adv(,) - 1) @ ker (Adu(,)
(
+ 1) .
(9.85)
Conversely, if J lies in this kernel, then (L(
<
9.10.4 Conjugacy Classes Let C be a conjugacy class of G. We will show that C carries a canonical invariant two-form which makes C into a q-Hamiltonian G-space relative to the canonical embedding v : C -+ G of C as a submanifold of G. We first make a preliminary remark. If a and b are elements of G, then &b = b if and only if Aba = a. Hence, for E g,
The tangent spwe to C at any a E C consists of the vectors
144
Chapter 9. Equivariant Symplectic Forms 9.11 Bibliographical Notes for Chapter 9
145
We define t
This is well-defined since if vG(a) = 0, then Ada q
.
above remark. It follows from (9.70) that 1 E ( < G ( ~ ) , v G (= ~ )5 ) ((Ad;'
,
= q = ~ d , ' 7 by the
1
- Ada) V , C ) = I ~ ( q ~ ( a ) ) ( @ + La , E ) .
In computing p;v*x at u we can drop the Ad, occurring to the left in this expression by the invariance of ( , ). Thus
*
Expanding out the terms and using the invariance of ( , ) completes the verification. We have thus established (9.84). To establish (9.86), suppose that EG(a) E ker 5,. This means that Ad,
Thus
1 L ( ~ )=Z-vL(B +e,<). 2 So to verify (9.84) we must show that
Then pa(" . exptt) = (u exp tE. u-')wu-'
(uexp t<- u-')-'
so Since Ad,
<,Ad,
7) = (Ad, Ada 5,Adu 8)= (Ada E , 7)
we see that
1 = Z(Ada 6, B),
p:S hence
E E ker (
But for E
9.11
For this, consider the map
(~d,,,-l
or
1
1
+ z(Ada 0, el), 4 a left-invariant three-form. It will suffice to show that this equals -p;v*x. For this observe that at any u E G we have = d p ; ~= --(Ada[8, 8],8)
E - Ad,-I
< = 0,
~ d -i 1) = ker (Ad,, +1) @ ker (Ad,
-1)
ker (Ad, -1) we have &(a) = 0. Thus (9.86)holds.
Bibliographical Notes for Chapter 9
1. An action of G on a symplectic manifold, M, is called Hamiltonian if there exists an equivariant moment map, C#J : M 4 g*, having the properties described in Section 9.1. A necessary condition for an action of G on M to be Hamiltonian is that the symplectic form, w,be G invariant; however this is far from sufficient. A number of sufficient conditions for a G action to be Hamiltonian are described in [GS], Section 26. For instance if G is compact (as we have been assuming in this monograph) a G action on M is Hamiltonian if either M is compact, or H2(M,R) = 0 or G is semi-simple. 2. Berline and Vergne are, as far as we know, the first persons to make the observation that a G-action on M is Hamiltonian if and only if w is the "form part" of an equivariant symplectic form. This obsemtion plays an essential role in their beautiful proof of the Duistermaat- Heckrnann theorem in [BV]. (We will describe this proof in Section 10.9.) 3. The classification of homogeneous symplectic manifolds in terms of coadjoint orbits is due to Kostant [Kol]; however, the quantum version of this result was, in some sense, first observed by Kiriilov. Namely, Kirillov proved that if G is a connected unipotent Lie group there is a one-one correspondence between irreducible unitary representations of G and coadjoint orbits. This result was subsequently extended by Kostant [Ko~], Kostant-Auslander [AK], Sternberg [Stl], Duflo [Du] et "1. (e.g., [Zi], [Li]) to other classes of Lie groups as well. The arsenal of techniques which are used for associating unitary representations to coadjoint orbits are known collectively as "geometric quantization theory" (see [Wo]). To a large extent these techniques are due to Kirillov . [Ki],Kostant [Ko2] and Souriau [So].
146
f
Chapter 9. Equivariant Symplectic Forms
t
4. One important example of minimal coupling is the following: Let Y be a manifold and ?r : P --, Y a principal G-bundle. Let X = T * Y , and let M be the fiber product of X and P (as fiber bundles over Y). M is
9.11 ~ibliogra~hical Notes for Chapter 9
147
o n g* defined by the formula:
a principal G bundle with base, X; and given a connection on P, one can pull it back to M to get a connection on M. For this connection the minimal coupling form (9.11)-(9.12) is symplectic for all c. (See [St2].) Moreover, Weinstein [Wl observed that there is a way of defining this minimal coupling form intrinsically without recourse to connections: the product, T'P x F, is a Hamiltonian G-manifold with respect to the diagonal action of G, and its symplectic reduction is symplectomorphic to W with its minimal coupling form.
(f : g* -+ R being an a r b i t m y continuous function) is pieceurise polynomial. The measure, p o ~ is , called the Duistermaat-Heckmann measure. Since it is compactly supported, its Fourier transform:
5. This example of minimal coupling is used in elementary particle physics to describe the "classical" motion of a subatomic particle in the presence of a Yang-Mis field: Suppose that, when the field is absent, this motion is described by a Hamiltonian, H : T*Y 4 R. In the presence of a Yang-Mills field (i.e., of a connection on the bundle, P -4 Y) the motion is described by the Hamiltonian, p'H, p being the fibering of M over T'Y. (See [St21 and [SU].)
is a Cm function; and the version of the Duistermaat-Beckmann theorem which we will describe in Chapter 10 is a formula for computing (9.28) at "generic pointsn, x E g.
9. The notion of a q-Hamiltonian G-manifold is an outgrowth of recent attempts to extend various theorems in equivariant symplectic geometry to the action of loop groups on infinite dimensional manifolds. A baiic theorem of Alexetiv-Malkin-Meinrenken asserts that there is an equivalence of categories between the category of (infinite dimensional) symplectic manifolds equipped with a Hamiltonian loop group action with proper moment maps, and the category of finite dimensional qHamiltonian G-manifolds.
6. Let 0 be a coadjoint orbit of the group, G, n a fixed base point in 0 and GPOthe stabilizer group of po. If G, is compact, there exists a neighborhood, U of po in g* such that, for every p E U,the coadjoint orbit through p can be reconstructed by a minimal coupling construction in which the base symplectic manifold is 0 and the fiber symplectic manifold is a coadjoint orbit of G,. For some implications of this fact for the representation theory of compact Lie groups see [GLS]. perspective we recommend the paper [GLSW] of Gotay, Lashof, Sniatycki and Weinstein. (They consider the problem which we discuss in Section 9.5, namely the problem of equipping a "twisted product"
10. A beautiful observation of Alexeev-Meinrenken is that there exists an intrinsic volume form on q-Hamiltonian G-manifolds. If dim M = 2d, one might regard wd as a candidate for such a volume form. However it is in general not non-vanishing. It can be converted into a non. vanishing form by dividing by fix, where xp is the character of the representation of G whose dominant weight is one-half the sum of the positive roots.
of two symplectic manifolds, F and X, with a symplectic structure, from a more general point of view than ours: They don't assume that F is a Hamiltonian G space and that W is of the form (M x F)/G).
11. An analogue of the equivariant three form in all dimensions has recently been constructed by Alexeev,Meinrenken and Woodward based on an earlier construction of Jeffrey ([Je]).
7. For an insightful discussion of minimal coupling from the topological
12. In their study of q-Hamiltonian G-spaces, Aleveev and Meinrenken have been led to consider an entirely new kind of equivariant cohomology in which ( R ( M ) B S ( ~ * )is) replaced ~ by ( R ( M ) @ J U ( ~where ) ) ~ U ( g ) is the universal enveloping algebra of g. (Recall that U(g) is a filtered algebra and the Poincar6-Birkhhoff-Witt theorem asserts that its associated graded algebra is S(g) which is Z S(g:) in the presence of an invariant scalar product.)
8. The Duistermaat-Heckmann theorem described here is one of several versions of Duistermaat-Heckmann (another one of which we will discuss in $10.9). A version of Duistermaat-Heckmann which is easily deducible from Theorem 9.6.3 is the following:
~ h k o r e m9.11.1 Let ( M , w ) be a compact 2d-dimensional Hamiltonzan G-manzfold with moment map, q5 : M 4 ga. Then the measure
I
Chapter 10
The Thom Class and Localization Our goal in this chapter is to construct, in a rather canonical way, the equivariant version of the Thom form, following the construction given by MathaLQuillen [MQ] in the c.ase of ordinary cohomology. We then give some important applications of this construction. As motivation, we briefly recall the properties of the classical Thom class, referring to Bott-Tu [BT] for details. Let Z be an oriented d-dimensional manifold, and let X be a compact oriented submanifold of codimension k. Then integration over X defines a linear function on Hd-k = H d - k ( Z )where H e ( Z ) denotes the de Rham cohomology groups of 2. On the other hand, Poincare duality asserts that the pairing
is non-degenerate, where H: = H ' ( Z ) ~denote the compactly supportedcohomology groups. In particular, there exists a unique cohomology class r ( X ) E Hk(Z)O such that
This class is called the Thom class associated with X, and any closed form TX representing this class is called a Thom form. Thus a Thom form is a closed form with the "reproducing property"
for all closed a E fld-"z).Clearly any closed form with this property is a Thom form.
150
Chapter 10. The Thom Class and Localization 10.1 Fiber Integration of Equivariant Forms
Although the Thom class is canonically given by Poincar6 duality, until relatively recently, the construction of a Thom form involved choices, and hence was not suitable in situations where a more or less canonical construction is necessary for functorial considerations. Furthermore, as we shall see by example at the end of this chapter, Poincar6 duality is not available in equivariant cohomology. Both of these problems were solved by the MathaiQuillen construction whose algebraic aspects we have described in Chapter 7. We will now translate this algebra into geometry.
151
We list some elementary but important properties of fiber integration:
dx* = xed.
.
(10.2)
Proof. For any p E R'-'(Y)~ and P E Rm-'(X)o we have, by two applications of (10.1) and two applications of Stokes,
10.1 Fiber Integration of Equivariant Forms Let K be a Lie group acting as proper smooth transformations on the oriented manifolds X and Y (preserving the orientations), and suppose that x:Y+X is a fibration which is K-equivariant. Let m = dimY and n = dimX so that
so the result follows from the uniqueness of x,. If I$ : Y -. Y and 1C, : X -+ X are proper orientation-preserving maps which are T-related in the sense that x o I$ = 1C, o n then it follows from the uniqueness of (10.1) that
Let R e ( ~ ) o denote the space of compactly-supported e-forms on Y with a similar notation for X. If !. 2 k , there is a map
called fiber integration where, for p E Re(Y)o, a,p, is uniquely characterized by
**P A p =
/X p
A x*p V ~ Rm-'(~)O. E
It is clear that x*p fs uniquely determined by this condition, since an form Y on X with the property that
(10.1)
(e- k)-
must vanish. Once we know that our condition determines x.p uniquely, it is sufficient (by partitions of unity) to prove the existence in a coordinate patch where (xl,. .. ,xn, t l , . . . ,tk) are coordinates on Y, where (xl,. ..,xn) are coordinates on X and ?r is given by x(x, t ) = x. Then if
where the remaining terms involve fewer than k wedge products of the dt'; we can easily check that
.
If I$t and $t are one-parameter groups which are x-related, and if v and w are the corresponding vector fields, then v and w are x-related in the sense that dT,(v(y)) = ~ ( 4 ~ 1 ) . The infinitesimal version of (10.3) then becomes
where L denotes Lie derivative. Let v be a vector field on Y and .w a vector field on X such that v and w are x-related. We claim that
Indeed, by (10.1), for any
P E Rm-efl(X)o
But a*pAp=O since it is an (m
satisfies (10.1).
+ 1)-form on an m-dimensional~manifold.Hence
152
Chapter 10. The Thom Class and Localization 10.1 Fiber Integration of Equivariant Forms
153
Hence
l,B
=
A n,(i(v)p)
so (10.7) holds. Finally, suppose that we have an e x x t square of maps
+
since p A a,p = 0, as it is an (n 1)-form on an n-dimensional manifold. Fkom (10.2), (10.4), and (10.5) we see that
Theorem 10.1.1 : R(Y)O
-+
n(x),
is a morphism of K* modules. In particular, we may consider the Cartan complex for both R(Y)o and R(X)o and we conclude that
such that nl is a fibration, the restriction of g to each fiber r;'(zl) is a diffeomorphism, and g and h are proper. Then, in generalization of (10.3), we have (10.8) (r1)*g8= h*r*.
Proof. For every y E Y we can lind neighborhoods U of y and W of r(y) such that the square
U Lx R ~k ~
Explicitly, if we regard an element p of the Cartan complex of R(Y)o as a polynomial map p : k -+ R(Y)o, then r.(p) is the polynomial map from k to Q(X), given by (T*P)( t ) = a*( ~ ( 6 ) ) . The fact that r, is a morphism of K* modules then implies that
+
W-Rn3
+
is commutative where i and j are open embeddings and prl is projection onto the first factor. Any form w supported in U can be written in these local coordinates as
For our next property of fiber integration, observe that left multiplication of forms makes R(X), into a module over R(X). Since a* : Q(X) -, R(Y) is a homomorphism (in fact an injection), this makes R(Y), into an Q(X)module as well. Then n, is a homomorphism of R(X)-modules:
and r.w = j*
( J tz1(z, t)dr'dtl
)
. . .dtk . '
An element of g-l(U) can be described as (z, t ) , z E la-'(W) and Indeed, for any a
E
Qm-r-e(X) and hence at
t
we have
154
Chapter 10. The Thom Class and Localization
A partition of unity argument then proves (10.8) for all forms of compact support. We conclude this section by pointing out the relation between fiber integration and Thom forms. Suppose that X is a compact submanifold of a manifold Z, and U is tubular neighborhood of X which we identify with an open neighborhood of X in the normal bundle N X of X. Let T E Rk(U)o be a closed form, where k is the codirnension of X. Then is a zero-form, i.e. a function on X. Suppose that a,7 = 1 (that is integrating over every fiber results in the value one). Then T is a Thom form. Indeed, since X is a deformation retract of U ,for any closed form a on 2,the restriction of a to U is cohomologous to a*i*a.Hence, if a is of degree d - k = d i m X,
10.2 The Equivariant Normal Bundle
155
Let M be the compact Lie group SO(k) and V := R k its standard module. Then P is a principal M-bundle over X. Because the scaiar product ( , ) on N is K-invariant, the map
K xP
i
-
P,
(a, (2, e) o (ax,(ael,. . . ,aek))
defines an action of K on P which covers its action on X. We extend this to an action of K on P x V by letting K act trivially on V. The map makes P x V into a principal M bundle over N ,and the actions of K and M on P x V commute. Thus the map (10.9) descends to give a K-equivariant diffeomorphism of ( P x V)/M with N. Again, if K is compact, we may put a K-invariant connection on P. We are thus in the situation of Section 4.6 where we have two commuting actions, and P x V is equipped with connection forms relative to the SO(k)-action. We conclude that the Cartan map gives isomorphisms
and KX :~ M X K ( P ) O ~ K ( X ) O . Finally, we caa consider the equivariant de Rham complex O M ~ K ( V ) O Since . K acts trivially on V, we have +
10.2
The Equivariant Normal Bundle
Let Z be an oriented &dimensional manifold on which K acts, and suppose that X is a compact (n = d - k)-dimensional. oriented submanifold which is invariant under the action of K . Let
RMXK(V)O= RM(V)O@ s(k*IK. In particular, we have an embedding Here is how we will construct our equivariant Thom form: First we will make a slight modification in the universal Thorn-Mathai-Quillen form u, given by (7.16) and (7.19) so as to make it of compact support instead of merely vanishing rapidly at infinity. Let us call this modified form vo. So uo E RM(V)o and hence
denote the inclusion map. Let N = N X denote the normal bundle of X in Z. By the equivariant tubular neighborhood theorem, there exists a K-invariant tubular neighborhood U of X in 2, and a K-equivariant diffeomorphism
Let prz denote the projection onto the second factor: such that yoio = i SO
where io : X + N is the embedding of X into N as the zero section. Suppose we put a K-invariant scalar product ( , ) on N, which we can always do if K is compact. Let l? be the oriented orthonormal frame bundle of N:
P ~ G: QMXK(V)
+
.
Q M ~ K (xP V).
We define (10.10) r := K N (pr;(uo @I 1)) E RK(N). We will show that T , SO defined, has all the desired o roper ties of an equivariant Thom form.
156
Chapter 10. The Thom Class and Localization 10.4 Verifyiig that r is a Thom Form
10.3
157
Modifying v
The form v given by (7.16) and (7.19) is of the form
Let 'us tirst prove this in the non-equivariant case, i.e. when K = {e). E X, and let Po := *-'(xo) be the fiber over x0 and No the fiber of N over xo. We have the fibrations
Here the are linear coordinates on V and the xi are linear coordinates on m = o(k), and the pr are polynomials in the x and do not depend on the u. Let B denote the open unit ball in V and consider the SO(V)-equivariant diffeomorphism
the unique connection on Po arising from the fact that it is a homogeneous space for M = SO(k), the Cartan map
Fix some point xo
and the element TO = KO
Then
(prl Y O ) .
It follows from the functoriality of our constructions that TO
vanishes to infinite order as llull 1 which is enough to kill any growth coming from denominators in p*du'. Thus if we extend p'v by setting it equal to zero outside the unit ball, we obtain a form uo of compact support which is dM-closed and whose total integral is one. -)
10.4 Verifying that
T
where jo : No -+ N is the inclusion of No as a fiber of N . So to prove (10.11) in the case of trivial K, it is enough to prove it for the case of a fibration over a point. Since our connection forms all lie in Po, in the bundle
is a Thom Form
We know that pr;(vo @ 1) is closed form in R M x K ( Px V)o and hence from the Cartan isomorphism that T given by (10:lO) is dK closed. We wish to show that Once we do this, notice that if we then identify N with the tubular neighborhood U of X, and consider T as a form on Z (by extending by zero) then [TI, the class of T is a Thom class. indeed, since X is a deformation retract of U , idv and i o s are homotopic. Hence if a is any closed form, its restriction to U is cohomologous to ~ * i * aHence .
Po x V + No
.
all the spaces { p ) x V are horizontal and all curvature forms vanish. Let rn = o(k) and let xl,.. . ,xr be the dual basis. In the Cartan model, an equivariant de Rham form on Po x V is a polynomial in x l , . . . ,xr with ordinary de Rham forms ori Po x V as coefficients. So it is a sum of terms of the form p = p1xr.
&, . . . ,tr be a basis of
By Cartan's procedure, we must replace the xa by the curvature forqs and then take the horizontal component. But the curvatures all vanish, so we are left with the "constant term" Po. We must take the horizontal component of Po which amounts to restricting to each x V which is identified with No via projection. So TO is just the "constant term" in the expansion of vo as a polynomial in x, under the identification of No with V via V = {p) x V
= JXi*a~n.r
=
by (10.11). So we must prove (10.11).
La
.*
= JOT
-4
Po x V -No.
(The choice of p is irrelevant, by invariance.) But the integral of this "constant term" is precisely the integral of vo which equals 1. This proves (10.11) for the case that K is trivial. Let us now prove (10.11) in the general case. We keep the notation xl,.. . ,xr for a basis of the dual space to m = o(k) so that
158
Chapter 10. The Thom Class and Localization 10.6 The Fiber Integral on Cohomology
and hence 7 = IEN (pr;(~O@ 1)) = (P~;(VOI)~.,~ P* where the fia are now the "equivariant curvature forms", that is the curvature forms for SO(k) but computed using the operator dK instead of d. But
where the pa are the non-equivariant curvature forms and Q E k* @ RO(p). Thus T=To+Tj @$ where ro is the non-equivariant Mathai-Quillen-Thom form with the rj 6 n k - z j (N)o and the pi E Sj(k*). Upon fiber integration, all the terms n,(rj@
p t ) = ~ r@p' j = 0 since the fiber degree is less that the fiber dimension.
10.5
The Thom Class and the Euler Class
10.6
.
159
The Fiber Integral on Cohomology
Let N -r X be a K-equivariant vector bundle of rank k over a compact is a cochain map, it induces a map on cohomology manifold X. Since .~r,
Theorem 10.6.1 The map (10.13) is an isomorphism. Proof. In the case that X is a point, this is the content of (7.21) (except that we used rapidly-vanishing forms instead of compact ones, but this clearly makes no difference). Fkom this it follows that the projection P x V --+ P induces an isomorphism on cohomology
where N = ( P x V)/SO(k) a s above. Then the square
Suppose that X has wen codimension in 2. Theorem 10.5.1 Let i : X Z be the inclusion map, kt r(X) be the Thom class of X and e(N) the Eder class of its normal bundle, N. Then -+
Proof. We have represented N as the quotient space N = (Px V)/M so we have the commutative diagram
induces a square
. X*
X-N where the horizontal arrows are inclusions and hence the commutative diagram H h X d px v) HXN)
-
Hhx~(P) * where the horizontal arrows are isomorphiims and the vertical arrows are pullbacks. Our construction of the Thom class gives T(X) a s the image via the top horizontal arrow of
So i*r(X) is the image by the bottom horizontal isomorphism of pr;[i*uo @ 11 and the result follows from (7.20).
H&X,(P
x V)o
,*
IP*
H G xt ~ ( p )
in which the vertical arrows and the bottom horizontal arrow are isomorphisms. Hence the top arrow is an isomorphism. An important corollary of this result is that the equivariant Thom class [TI E H k ( N ) o is uniquely characterized by the property
110.7
Push-Forward in General
If a : Y -+ X is a submersion, we have defined a, on compactly supported forms. One of its key properties is that its induced map on cohomology is a transpose to pullback:
160
Chapter 10. The Thom Class and Localization
In ordinary cohomology, where we have ~oincar6duality, we could take this as the definition of push-forward, and it would work for any smooth map f : Y + X, not necessarily a submersion. We would like to define pushforward in general for equivariant cohomology as well, and we would like it to have this transpose property. .But we can not use this property as a definition, since we do not have Poincark duality. Any map f : Y -+ X can be factored
where -Y : y - , y x X ,
T ( Y ) := ( Y , ~ ( Y ) )
10.8 Localization
161
is to express the integral of an equivariant de Rham form over M in terms of its Uresidues" at the fixed points of G. So let X be a connected component of the fixed point set M G , and suppose we have chosen an orientation on X . Let rr : U -4X be an invariant tubular neighborhood and r E Qc(U)o an equivariant Thom form. We know that
the Euler class of N , where N = N x is the normal bundle of X in M . We have observed that the restriction of any closed form p E Q G ( M ) to U is homologous to rr*2*p and hence
is the inclusion into the graph and since is projection onto the second factor. So we need to define the push forward in equivariant cohomology for the case of an equivariant inclusion
Suppose that p is compactly supported in U. We can interchange the roles of T and p in the above argument to get
and check that it has the desires adjoint property relative to pullback. Let n : U -, X be an invariant tubular neighborhood of X in Z and let [r]be its Thorn class. For any p E R K ( X ) odefine
By the property of the Thorn form, for any ,8E R K ( Z )we have
If p is closed, aBi'/3 is cohornologous to p and so if
if
by (10.14). the codimension of X is odd, e ( N ) = 0 and the formula says that Jx i*p = 0. The formula becomes more interesting in the case of even codimension when we can write it as
p is also closed,
Now G acts trivially on X, since X consists entirely of tixed points. Hence
which is the desired adjointness property. where k = 2m and
10.8
Localization
Let G be a compact Lie group acting on a ;ompact, oriented, d-dimensional manifold M preserving the orientation. The idea of the localization theorem
f, E
s ; ( ~ * )cyi~E ,R2'(X)
with dai = 0. Suppose that f,,, # 0 and write the expression for e ( N ) as
162
Chapter 10. The .Thorn Class and Localization 10.9 The Localization for Torus Actions
163
Formally, let us take the inverse of both sides so as to get For the case of a point, we have identified the equivariant Euier class as (2r)-" times the image under h of the PfafEan in s ( 0 ( 2 r n ) * ) ~ ~So (~~). where q - 1 is the largest integer 5 +dim X. Both sides of (10.16) do not make sense as they stand, but if we multiply the right hand side by f$ we get a well defined expression
fm = ( 2 ~ ) - ~ h ( P f a E ) .
'
(10.21)
Let T be a maximal torus of G with Lie algebra t. Any invariant is determined by its restriction to t'. So we would like to compute the image of h(Pf&) under the restriction map
x : S(g*) + S ( t * ) .
Since a is dc-closed, we see that P ( N ) E Rc(X) is dc closed, and we have
For this we use the weight decomposition of No under the isotropy action of the torus: We may identify No with C m in such a way that the isotropy representation of T is given by
If we replace p by r 8 P ( N )A ,u in (10.15) we get
This formula expresses an integral over U in terms of an integral over X. We can write it more suggestively as
The expression l / e ( N ) on the right hand side of this equation is interpreted as the sum (10.16). So the summands on the right of (10.20) are not elements of ~ ( 9 ' but ) ~expressions of the form
<
where E t and the eJ in the weight lattice of T. As this representation preserves the standard Hermitian inner product on C m , we may identify
with the restriction to t for the equivariant m-th Chern polynomial of the bundle No + xo which is just the product of the roots
We have proved In other words, they are elements of the ring S(g*)Glocalized at fm. On the other hand, the left hand side of (10.20) is an element of S(g*)G. So there are a lot of combinatorial cancellations implied by (10.20). The above localization formula (10.20) was proved under the assumption that f m # 0. We now derive some equivalent conditions for this to hold: Let x0 E X and let No be the fiber of the normal bundle over so. The isotropy representation of G on No preserves the orientation of No (inherited from that of M and X ) . Since G is compact, we may put a positive definite scalar product on No which is preserved by the isotropy action of G. So we get a group homomorphism
Theorem 10.8.1 The restriction of fm to t is the polynomial (2a)-"el . - . em. I n particular, f, # 0 zf and only if the weights of the isotropy representation of T on No are all non-zero. Notice that since X is connected, the isotropy representation of T on the fiber of the normal bundle is the same at all points. Also the conjugacy theorem for Cartan subgroups implies that the choice of T is irrelevant. Thus the hypothesis of the theorem is independent of the choice of zo and of T . The fact that none of the ti vanish means that uT = X . So our condition amounts to the assumption that X is a connected component of MT.
10.9 So we get a transpose map, h, from the ring of invariants to S(g*)G. If jo denotes the inclusion of No in N then
S( 0 ( 2 r n ) * ) ~ ~ ( ~ ~ )
The Localization for Torus Actions
In this section we will obtain a more global version of (10.20). In view of the fact that (10.20) only holds when M G = we can, without loss of generality assume that G = T, i.e., that G is an T-dimensional torus. To globalize (10.20) we recall a few elementary facts about torus actions on . Eompact manifolds (see Lemma 8.5.1):
164 . Chapter 10. T h e Thom Class and Localization
10.9 T h e Locatization for Torus Actions
Lemma: Let V be a vector space over R and p : S 1 --* G L ( V ) a representation of S1 on V which leaves no vectorfized except 0. Let
T h e o r e m 10.9.2 There exist a finite number of weights PI, with the pmperty Mat i f E E g satzsfies
the corresponding vector field,
Then there exists a unique decomposition:
..., P,
165
in 2;
on M is non-zero except at points of M ~ .
<
Proof. If E # ( ~ )= 0, has to be in the Lie algebra o f the stabilizer group, G,; therefore for t o have the property above it suffices that and positive integers, 0 < m l <
... < mk
<
such Mat A = miJ, on V, with
J j = -I. Corollary: V admits a canonical complex structure. In particular V is even dimensional and has a canonical orientation. Now let G be an r-dimensional torus. Letting exp : g -+ G be the exponential map, there is an exact sequence
o f groups, ZG being the group lattice o f G in g, and its dual, 22,being the weight lattice o f G in g'. W e will denote by g~ the set o f elements in g of the form 77 = < / m with E ZG and m E Z - 0. These are precisely the elements in g with the property that the subgroup of G genemted by q is a circle. Now let M be a compact, oriented G-manifold. W e will need below the following standard result.
<
T h e o r e m 10.9.1 (Finiteness t h e o r e m ) Only a finite number of subgroups, Gi, i = 1, .. ., e, can occur as stabilizer groups of points of M . Proof. For p E M , let G p be its stabilizer group, let X be the orbit o f G through p, and let N, be the normal space to X at p. By the equivariant tubular neighborhood theorem
( M ,X ) s ( N X ,X )
(10.22)
and N X 2 (G x Np)/Gp and from (10.22) and (10.23) one reads o f f L e m m a 10.9.1 There exists a neighborhood U of X i n M with the property that the groups whach occur as stabilizer groups 'of points of U are contained in G , and are identical with the groups that occur as stabilizer groups for the linear action of G, on N,. This reduces the proof o f the finiteness theorem t o the proof o f the analgous theorem for linear actions, (and for linear actions the proof is easy). Q From this finiteness theorem we get as a corollary:
except when gi = g. However, each o f the Lie algebras g, can be defined by a set o f equations o f the form, PI ( E ) = . .- Pk(<)= 0 with A's in 22. Corollary 10.9.1 Let ( be in gg and satisfy (lO.Z4). Then the circle group, S1,generated by E has the property
Let's describe one important consequence o f this fact: Consider the set (10.25)
{EEg,Pi(t)# O , i = 1 , . . . , ~ } .
T h e connected components of this set are called the action chambers o f g (vis a vis the action of G on M ) .
. Proposition 10.9.1 The connected components of M~ are of even codimension. Moreover, having fied an action chamber, one can assign to each of these components a canonical orientation. Proof. Let E E go be an element o f the given action chamber, and S' the subgroup o f G generated by <. T p n the connected components of MG are also connected components o f M S , and if p is a point on one o f these components and N, the normal space at p, the isotropy representation o f S1 on N, defines a canonical complex structure, and hence a canonical orientation, on N,. It's clear moreover, that this orientation doesn't depend on the choice of p but only on the choice o f the action chamber in which is contained. We will now state and prove the global version o f (10.20) which we alluded t o above. Let d = dim M and, for the moment, let G = S1.
<
T h e o r e m 10.9.3 (Localization t h e o r e m for circle actions) If p E R;,(M), k 2 d, is dsl-closed
the sum being over the connected componepts of M ~ ' i, x being the inclusion map of X into M and e x being the equivariant Euler class of the n o n a l bundle of X .
166
Chapter 10. The Thom Class and Localization
10.9 The Localization for Torus Actions
Let U = M - M ~ ' .Since S1 is one-dimensional it acts in a Proof. locally free fashion on U ; so Hg,(U) = H k ( U / S 1 )and, in particular, since dim r / / S 1 = d - 1, H i , ( U ) = 0 for k 2 d. Thus p = d s l v on U , for some v E R ~ ; ' ( u ) . Let Ux be a tubular neighborhood of X and px E C m ( U x ) o a SL-invariantfunction which is identically one in a neighborhood of X. Letting u' = v - C p x u , p = dS1ut Cpx
167
I f M~ is finite this localization theorem becomes particularly simple. For p E M G let ...,a,,, be the weights of the isotropy representation of G on T p . We showed in Section 10.8 that
.
+
so (10.28) reduces to
where px E O&(UX)Oand p x = p on a neighborhood of X. Thus
for all E E g satisfying and by (10.20)
ai,p(E)
#0
(10.30)
for all p E M and i = 1, ... ,m. Example. Let 3 = w 4 be an equivariant symplectic form with form part w E R 2 ( M ) G and moment map part 4 : M g* . Applying (10.29) to the form p = e x p 3 = 1 + 3 + -Cl2 +... (10.31) 2! one gets, for every ( satisfying'the conditions (10.30),
+
We'll now extend this result to torus actions. Recall that if K is a subtorus of G there is a natural map
-+
induced by the restriction map on forms. More explicitly, given an equivariant form in R G ( M ) , /J:g-+R(M),
Remark. Strictly speaking, one can't simply prove this identity by applying (10.29) to the form (10.31) since exp3 is not an equivariant de Rham form according to our definition. (It is an analytic function of J , but not a polynomial in J . ) However, it is easy to deduce (10.32) from (10.29) by applying (10.29) to each of the terms on the right hand side of (10.32) and summing. The formula (10.32) is the Duistemaat-Heckmann theorem. We will briefly describe how it's related to the Duistermaat-Heckmann theorem of Section 9.7: Let A be the image of 4. By a theorem of Atiyah and GuilleminSternberg, A is a convex polytope. Moreover, denoting by g,eg the set of regular values of 4, each connected component of A ng&, is an open convex subpolytope of A. Duistermaat-Heckmann define a measure on the Bore1 subsets, B, of g* by setting
one gets an equivariant form in R K ( M ) by restricting p to k, and the restriction, /I --, pik, induces the map (10.27) on cohomology. In particular, let E g~ and let K be the circle subgroup of G generated by E. If satisfies the conditions (10.24) MG = M K ; hence, applying (10.27) to plk and ( e x ) l k :
<
<
This identity holds for all 5 E g~ satisfying the conditions, Pi(5) # 0 , i = 1 , . . . q. However, g~ is dense in g. Moreover, the left hand side of (10.28)is a polynomial function of and the terms on the right are rational functions of <, so this identity holds, in fact, for all E g satisfying Pi(<) # 0 , i = 1 , . ..q, or finally, holds for all E g, providing one thinks of (10.28) as a formal identity in which the left hand .side is an element of the ring S ( g * ) and the summands on the right, elements of the ring
< <
<
,
for an appropriate f E S ( g * )- 0.
This measure is supported on A, and it is easy to show, using elementary measure theory, that it is,the product of a smooth function times Lebesgue measure on each of the components of A n g.,: Duistermaat and Heckmam prove, however, a much stronger result: they show that on each connected
168
Chapter 10. The Thom Class and Localization
component of A n grfeg,v is a polynomial function times Lebesgue measure, this polynomial function being the Duistermaat-Heckmann polynomial which we described in Section 9.7. Now the left-hand side of (10.32) can be rewritten in the form, e ( f * zdu(z) )
J
and, replacing
10.10 Bibliographical Notes for Chapter 10
169
4. In. an (unpublished) note, David Metzler describes an alternative way of defining the Thom class and push-forward operations in equivariant cohomology: Given a G-manifold, M, let E ( M ) be the complex of de Rham currents. This complex is a G* module; and if M is oriented, there is a natural inclusion
< by it, (10.32) becomes It is easy to see that this is a morphism of G* modules, hence it induces a map HG(O(M)) HG(E(M)) (10.35) +
This formula says that the Fourier t m n s f o n of the measure, v,is equal to the function on the right hand side at all points, E g, satisfying aj,,(<) # 0 for all j . By inverting this Fourier transform one can get a fairly explicit formula for v and hence for the Duistermaat-Heckmann polynomials, in terms of the weights cri,p. Recall,however, that the Duistermaat-Heckmann polynomials describe how the symplectic volume of the reduced symplectic manifold
<
Thus varies as one varies a in a fixed connected component of A n g,',. (10.32) enables one to express this variation of symplectic volume in terms of the linear actions of G in the tangent spaces to M at the fixed points.
10.10 Bibliographical Notes for Chapter 10 1. For a compact oriented G-manifold, M,.one has a natural bilinear map *)~ by the pairing of H G ( M ) into s ( ~ given
This pairing, however, can be highly singular. For instance, if the action of G on M is free the integral over M of a dG-closed form is zero, so the pairing is trivial. Therefore, one can't define push-forward operations in equivariant de Rham theory,simply by invoking Poincark duality as in the non-invariant case. 2. Some vestiges of Poincark duality do survive, however, in the equivariant setting. For instance, Ginzburg [Gi] proves that if M is a Hamil-
tonian G-manifold the pairing (10.34) is non-singular on cohomology. (Thus push-forward operations in equivariant cohomology can be defined by Poincare duality for maps between some G-manifolds.) 3. Our construction of the equivariant Thom form in Section 10.2 is, as we already noted in Chapter 7, due to Mathai and Quillen.
We claim: Theorem 10.10.1 (Metzler) The map (10.35) is bijective. Proof. By a well-known theorem of de Rham the map induced by j on ordinary cohomology
is bijective and hence by Theorem 6.7.1 the map (10.35) is bijective. de Rham observed that the isomorphism (10.36) gives one a natural way of defining push-forward operations in ordinary cohomology without explicitly invoking Poincar6 duality. The reason for this is that E is naturally a covariant functor with respect to mappings, i.e., if M and N are compact oriented manifolds, a map f : M -4 N induces a canonical push-forward on currents
and hence by (10.36) a map
f# : H ( M ) --+ H ( N ) . If M and N are G-manifolds and f is G-equivariant, it is easy to see that (10.37) is a G* morphism and, therefore, by the same argument as above, it induces a map on equivariant cohomology
which by (10.35) can be viewed as a map of HG(M) into Hc(N). In paticular, if M is a submanifold of N, and f is the inclusion map, the image under this map of the element 1 E H & ( M ) is the equivariant Thom class of M.
170
Chapter 10. The Thom Class and Localization
10.10 Bibliographicat Notes for Chapter 10
5. The push-forward operations in equivariant cohomology and the equivariant Thom class can also be defined by purely topological methods, not using de Warn theoretic techniques. (This is, for instance, the approach taken by Atiyah and Bott in !j2 of [AB].) 6. The localization theorem was proved independently by BerlineVergne and Atiyah-Bott [AB]. The proof that we gave of the localization theorem in this chapter is essentially that of Atiyah and Bott. Let us briefly describe the Berline-Vergne proof, confining o~rselvesto the case, G = S1and M~ finite: Let [ be the infinitesimal generator of the G action and let U be the complement of MG in M. It is easy to show that there exists a G-invariant one-form, 8, on U with the property, L ~= B 1. Consider the equivariant form
171
and hence by Stokes' theorem
*
On B;, a is cohomologous to the restriction of a to p, (viewed as an element, f,(x), of S(g*)) so there exists an equivariant form, /3 E nG(B:) satisfying a = f r ( ~+ )~GP.
[By
Hence v A a = f,(z)v+vAd~p = ~+(x)~-~G(~AP)+P.
If we integrate the right hand side over dB: the second term is zero, and by Stokes' theorem the third term is
x being the generator of S ( g 8 ) . Then
Fkom this one concludes that if a E RG(M) is dG-closed
which is of order O(e). Thus
Let p,, r = 1,. . .,N, be the fixed points of G a i ~ dlet 2 1 , . . . ,z, be a complex coordinate system centered at pr in which the G-action is the linear action e i t ~= ( e i A l , v t
Finally note that by (10.38)
a,- , . ,e t A n , * . t G )
and the vector field, 5, is Making the change of coordinates, One can assume without loss of generality that on the set
B: = { z , EA;,,
this integral becomes
lz12 < e2}
the form 8 is equal to where
Let U' = U
eo = '
J--r
- UB:. By (10.39)
Zk d i k
-&dZk
and S2"-' is the standard (2n - 1)-sphere. By Stokes' theorem
h
b(vAff)
*
172
Chapter 10. The Thom Class and Localization and the integral on the right is easily computed to be (27r)"; so for (10.43) we get (2~)~(n~k.r)-' and plugging this into (10.42) and letting e tend to zero we finally obtain the localization formula
Chapter 11 which is the same as the formula we derived in 310.9.
The Abstract Localization Theorem
7. The localization theorem is a relatively recent result; however, an important special case of this theorem was discovered by Bott in the mid-1960s. Bott's result is concerned with the computation of the characteristic numbers of a vector bundle E --,M (i.e., the integrals over M of the characteristic classes of E). Suppose that the circle group, G = S1, acts on M and that his action lifts to an action of G on E by vector bundle automorphisms. Then as we pointed out in Chapter 8, there is an equivariant Chern-Weil map
In this chapter we will examine the localization theorem from a more abstract perspective and explain why such a theorem "has to be true". As in Section 10.9 we will assume that the group G is a compact connected Abelian Lie group;'i.e., an n dimensiond torus. The main result of this chapter is a theorem of Bore1 and Hsiang which asserts that, for a compact Gmanifold, M , the restriction map, HG(M) -r H G ( ~ Gis) injective "modulo torsion". Ftom this we will deduce a theorem of Chang and Skjelbred which describes the image of this map when M is "equivariantly formal". For this we will need the equivariant versions of some standard results about de Rham c e homology and some elementary commutative algebra. We will go over these prerequisites in Sections 11.1-11.3.
(with K = U(n), O(n),...), and from this one can recover the usual Chern-Weil map by composing it with the restriction map
Applying the localization theorem to elements in the image of (10.44) one gets polynomial identities in the ring, S(g*) = C[z],and by setting 3: = 0 these become "localization" formulas for the usual characteristic numbers. These are the formulas discovered by Bott in the mid-sixties. (See [Bott])
11.1
Relative Equivariant de Rham Theory
Let M be a compact manifold and X a closed submanifold. Denote by R(M, X ) the space of de Rham forms whose restriction to X vanishes and by R(M - X), the space of compactly supported de Rham forms on M - X. Theorem 11.1.1 The inclusion map R(M - X), + R(M, X) induces an isomorphism on cohomology.
This we will deduce from the following:
.
Lemma 11.1.1 Let .rr : U -+ X be a tubular neighborhood of X in M and let i : X -+ U be inclusion. Then zfw E Rk(U) is closed and has the property i'w = 0, there exists a v E Rk--l(U) with the same property and additionally such that w = dv.
174
Chapter 11. The Abstract Localization Theorem
11.2 Mayer-Vietoris
-
Proof of lemma. X is a deformation retract of U , so the maps T* : R ( X ) --, R ( U ) and i* :R(U) R ( X ) induce isomorphisms on cohomology. Therefore, there exists a v E Rk-'(U) such that w = dv. Moreover, i'w = 0 = di*v,so i*u is closed. Thus if we replace v by v - x'i'v we get a form whose exterior derivative is w and whose restriction to X is zero.
175
Another result in equivariant de Rham theory which we will need below is the equivariant version of the standard Mayer-Victoris theorem. Let M be a G-manifold and Ul and Uz G-invariant open subsets of M. Fkom the short exact sequences
Corollary 11.1.1 The cohornology groups of the complex O(U,X ) are zero in all dimensions. Let's now prove the theorem: "
1. Surjectivity. Given w E R k ( M , X ) with dw = 0 we can find a v E Rk-l(U,X) with w = d v on U. Let p E Cm(U), be a compactly supported Cw function which is one on a neighborhood of X . Then w - d(pv) is in R Y M - X),. 2. Injectivity. Given w E R k ( M - X ) , and v E R k - l ( M , X ) with w = dv, there exists a tubular neighborhood, U , of X on which w is zero, and hence, on which v is closed. Thus there exists an a E w ~ - ~ ( XU), for which u = oh on U. Thus vl = v - dpa is in R(M - X ) , and w = dul.
Since R ( M , X ) is the kernel of the restriction i* : R ( M ) + R ( X ) , one has, by definition, a short exact sequence of complexes:
<
and
o
+
-
-
n(uln u,), n(u1),e n(u,), n(ulu u2), o -*
(11.5)
one gets the standard Mayer-Vietoris sequences in cohomology and in cchomology with compact supports. However, both these sequences are exad sequences of G* modules, so by Theorem 4.8.1 one also gets long exact sequences in equivariant cohomology and in equivariant cohomology with compact supports:
-.
H;-'(u~ n U2) + Hk(U1 U Uz) + Hh(U1)@ Hk(U2) + . . . - (11.6)
and
and hence a long exact sequence in cohomology
and hence, Theorem 11.1.1, a long exact sequence
Suppose now that M and X are G-manifolds. Then (11.1) is a short exact sequence of G* modules, and hence by Theorem 4.8.1 one gets a short exact sequence of equivariant de Rharn complexes
One also gets an inclusion map
which by Theorem 6.7.1 and Theorem 11.1.1 induces an isomorphism on cohomology. Thus there is a long exact sequence
Here we will review some standard facts about modules over commutative rings. (The materid below can be found in any standard text on commutative algebra, for instance [AM].) Let A be a finitely generated S(g8)-module and let IA be the annihilator ideal of A: I A = {f E S ( g f ), f A = 0) The support of A is the algebraic variety in g @ C associated with this ideal, i.e., supp A = { x E g 8 C , f ( x ) = 0 for all f E la} . (Here we are identifying S(g8) with the ring of polynomial functions on g@C.) If f E I A then by definition, f = 0 on suppA, and conversely, if f = 0 on supp A, then some power f N lies in IA (Hilbert's Nullstellensatz). The following is an easy exercise:
-
Lemma 11.3.1 If A B supp B supp A U supp C.
4
C is an ezact sequence of S(g*)-modules,
176
Chapter 11. The Abstract Localization Theorem
If A is a free S(g*) module the only f annihilating A is zero; so for a free module, supp A = g @ C . Given an arbitrary finitely generated S(g*) module A an element a E A is defined to be a torsion element if f a = 0 for some f # 0. The set of torsion elements is a submodule of A, and A is called a torsion module if this submodule is A itself, i.e., if every element is a torsion element. It is clear from the definition that A is a torsion module iff suppA is a proper subset of g 8 C. In the examples we will be considering in the next section, A will be a gmded S(g*)-module. Hence IA will be a graded ideal, i.e., it will be generated by homogeneous polynomials fl,...,fk. Thus supp A will be defined by the equations, f i = . . . = fr; = 0 , and hence will be a conic subvariety of g @ C: if x E supp A and X E C, Xx E supp A.
11.4
The Abstract Localization Theorem
The manifolds we will be considering below will be G-manifolds, M , with "finite topology" i.e., with both dimH(M) < oo and dimH(M), < co. In this case HG(M) and Hc(M), are, by Theorem 6.6.1, finitely generated S(g*)-modules, and hence the abstract results which we described in the previous section can be applied to them. The following lemma is a simple but useful criterion for bounding the supports of these modules.
Lemma 11.4.1 Let K be a closed subgroup of G and : M -, G/K a G-equzvariant map. Then supp HG(M) and supp HG(M), aTe contained in k@C.
11.4 The Abstract Localization Theorem
177
L e m m a 11.4.2 There &ts a G-invariant neighborhood, U , of p with Me pwperty that for every G-invariant neighborhood W of p, supp HG(UnW) C k@ C .
.
By combining this with a Mayer-Vietoris argument we will prove the following key theorem. Theorem 11.4.1 Let M be a compact G-manifold and X a closed G-invariant submanifold. Then the supports of the modules HG(M-X) and HG(MX),,are contained in the set (11.8)
U~@C K
the union being over subgroups K which o m r as isotropy groups of points of M-X. Proof. Let U be a G-invariant tubular neighborhood of X. It. s f i c e s to prove that the assertion above is true with HG(M - X.)and HG(M - X), replaced by HG(M - P ) and H G ( M-8),. Since M - U is compact, one can find G invariant open sets, Ui, i = 1,.. . ,N covering M - and equivariant maps, di : Ui-+ G/Ki, each Ki being an isotropy group of a point in M -X. Let V, = Ul U . .- U U,-l.We .will prove by induction that the supports of HG(V,) and HG(V,), are contained in the set (11.8). By (11.6) and (11.7) there are exact sequences of S(g') moduIes
and
Proof. From the maps A4
2 G/K
-+
pt.
one gets the inclusions of rings
Hc(UrIc'@HG(V,),
+
HG(K+I),
+
HG(U, n V,),
and for these sequences the end terms are supported in (11.8) by induction. Hence by Lemma 11.3.1 the middle terms are as well. 0
We will now describe some consequences of this theorem: However, by Equation (4.29), HG(G/K) = S(k*)K;so, as an S(g*)-module, H c ( M ) is effectively an S(k8)-module. That is, the ideal of functions, f E S(g*), with f = 0 on k @ C , annihilates it. Thus, suppHG(M) k @ C . As for HG(M),, it is naturally a module over the ring, HG(M); and is a module over S(g*) by the morphism of rings, S(g*) -, Hc(M). Thus it too is effectively an S(k*) module; and, as above, supp HG(M), C k @ C . One situation to which this lemma applies is the following. Let p be a point in M with isotropy group K, let X be the orbit of G through p and let U be a G invariant tubular neighborhood of X. Being a tubular neighborhood, it has a G-invariant projection onto X. Therefore, identifying X with G/K we get a G equivariant map, 4 : U -,G / K . Moreover, for any G invariant open subset, U',of U we can restrict 4 to U' anderegard it as a G-invariant map of U' onto GIK. Thus we have proved
Theorem 11.4.2 Let i cokernel of the map
:
X
-+
M be inclusion. Then the kernel and the
i* : HG(M) -+ HG(X)
are supported in the set (1 1.8).
Proof. From the exact sequence
one observes that ker is is a quotient module of the module HG(M-X), and cokeri* is a submodule of HG(M - X),.Thus the supports of both these modules are contained in (11.8).
178
Chapter 11. The Abstract Localization Theorem
11.5 The Chang-Skjelbred Theorem
Since M is compact there are just a finite number of distinct closed subgroups of G which can occur as isotropy groups. Let
{K,}, z = 1 , ... , f
11.5
(11.9)
be a list of these groups and let K be one of the groups on this list. Letting X = M K in Theorem 11.4.2 we get a result which will play a pivotal role in our proof of the Chang-Skjelbred theorem in the next section: Theorem 11.4.3 The kernel and wkernel of the restriction map
i* : H G ( M )--, H ~ ( M ~ )
.
179
The Chang-Skjelbred Theorem
Let us assume that M is equivariantly formal i.e., that H G ( M )is a free S(g*) module. Then, as we've just shown, the map
embeds H G ( M ) as a submodule of H c ( M G ) . Our goal in this section is to prove the following theorem of Chang-Skjelbred [CS] (see also Hsiang [Hs]). Theorem 11.5.1 The image of i* is the set
have supports i n the set
Let's consider in particular the case K = G. In this case
the intersection being over all wdimension-one subtori H of G and in being the inclusion of MG into M H .
i.e., H ~ ( M is~ a )free S(g*)-module and hence if a is a torsion element in H G ( M ) it must get mapped by i* into zero. On the other hand, by Theorem 11.4.3 the kernel of i* is a torsion module so we conclude
R e m a r k 11.5.1 It is clear that the intersectwn (11.1.2) is a finite intersection only involving the codimension-one subtori, H , occUning on the list (11.9).
Theorem 11.4.4 (The "abstract" localization theorem) The kernel of the mapping
Proof. The image of i" is obviously contained in (11.12) so it suffices to prove containment in the other direction: i.e., to prove that (11.12) is contained in the image of i*. Since i" embeds H G ( M ) into H G ( M ~we ) can regard H G ( M ) as being a submodule of H c ( M G ) . Let e l , . . . ,e K be a basis of H G ( M ) as a free module over S(g*). By Theorem 11.4.6 there exists a monomial p = al. . .O N , the a,'s being non-zero weights of G such that for every element, e, of H c ( M G ) , pe is in H G ( M ) . Hence pe can be written uniquely in the form f l e l f .' . + f ~ e ~ with fi E S ( g W ) .Dividing by p
is the module of torsion elements i n H G ( M ) . We pointed out in Chapter 6 that there is a spectral sequence which computes the equivariant cohomology of M and whose El term is H ( M ) 8 S ( g * ) . Following Goresky, Kottwitz and MacPherson [GKM] we will say that M is equivariantly formal if this spectral sequence collapses, in which case H G ( M )is isomorphic as an S(g*)-moduleto H ( M ) @ S ( g aand, ) in particular, is a free S(g*) module. Thus, as a corollary of Theorem 11.4.4 we get Theorem 11.4.5 If M is equivariantly formal the map (11.I 1) is injective. Let's come back again to the restriction map,' i : H G ( M ) --+ H c ( M K ) , K being one of the groups, (11.9). The cokernel of i* is supported in the set (11.10) so for every polynomial which vanishes on this set, some power of it annihilates coker i*. In particular since the summands, k, @ C, in (11.10) do not contain k, one can find a weight, cri E g*, of G which vanishes on k, 8 C , but not on k. By taking the product of these weights and then some power of these products we obtain:
Theorem 11.4.6 There exists a monomial p = a1 . - . C Y N , each a; being a weight of G which does not vanish on k , such that p is in the annihilator of cokeri*. In particular for ev2ry e E H G ( M K ) ,pe is i n the image of 'i : H G ( M )-. H ~ ( M ~ ) .
If f, and p have a common factor we can eliminate it to write
with g, and p, relatively prime. Hence we have proved
~ )be written uniquely i n the L e m m a 11.5.1 Every element of H ~ ( M can fonn
with fi E S ( g l ) and pi a product of a subset of the weights, a l , . . . ,C Y N ,and 1, containing no common factors.
pi and
180
Chapter 11. The Abstract Localization Theorem
Let us now suppose that e is in the image of H G ( M H ) .Then by theorem 11.4.6 there exist weights, {Pi),i = 1,. ,r of G such that ( p j ) l h# 0 and pl - . .pr e = fiel - .+fKeK We can assume without loss of generality that the &'s are contained among the a,%; and hence, in the "reduced r e p resentation" (11.13) of e, the p,'s occurring in the denominators are products of the pj's. Thus we have proved
+.
.
Theorem 11.5.2 If the element (11.13) of H G ( M ~ )is contained zn the image of H ~ ( M none ~ ) of the p, 's vanish on h. Suppose now that this element is contained in the intersection (11.12). Let p, = a,, . . .a,*and assume s > 0 i.e., p, # 1. Let h be the set {x E g , a,,(2)= 0). Since a,, is a weight, h, is the Lie algebra of a subtorus, H , of codimension-one, contradicting Theorem 11.5.2. Thus, for all i, p, = 1, and e = fiel . . . f K e K . Hence e is in HG(M).
+
11.6
11.7 Two Dimensional GManifolds
Some Consequences of Equivariant Formality
The assumption "HG(M) is a free S(g*)-module" has the following important consequence.
181
1. M G is finite.
'
2. The set of elements of G which act trivially on A4 is a closed codimension-one subgroup, H.
3. M is diffeomorphic to S2. Moreover, this diffeomorphism conjugates the action of G / H on M into the standard action of S1 on S2given by rotation about the z d. For the sake of completeness we will sketch a proof of this theorem: Equip
M with a G-invariant metric, and let p be a fixed point. The exponential map exp : T, -+ M
(11.14)
is surjective and intertwines the linear isotropy action of G on Tp with the action of G on M. Therefore, if an element of G acts trivially on Tp it also acts trivially on M. However, there exists a weight, 9,of G and a linear isomorphism, Tp -+ C, conjugating the linear isotropy representation with the linear action of G on C given by
Theorem 11.6.1 For every closed subgroup H of G each of the connected components of M~ contains a G-fized point.
(See, for instance, Section 10.9.) Thus the subgroup H of G with the Lie algebra { x ~ g , a ( x =) 0 )
. will assume that Proof. Let X be a connected component of M ~ We X n M~ = 0 and derive a contradiction. The fact that this intersection is empty implies that the G-equivariant Thom class rG(X) is sent to 0 by the restriction map Hc(M) -+ H G ( M ~ ) .But H c ( M ) is a free S(g*)-module, so it has no torsion. Therefore ~ G ( X=) 0. But under the map H c ( M ) + HH(M) induced by inclusion, rG(X) is sent into r~ ( X ) which must therefore be 0. But then the Euler class of X in M is ~ H ( N=) i * r ~ ( X= ) 0 which contradicts Theorem 8.1.
acts trivially on Tp (and hence acts trivially on M). Moreover, (11.14) maps a neighborhood of 0 diffeomorphicallyonto a neighborhood of p, and by (11.15) T: = (0). Hence p is an isolated fixed point. Thus we have proved the first two of the assertions above. To prove the third assertion let v be the infinitesimal generator of the action of the circle group G / H on M. By Assertion 2 the zeroes of this vector field are isolated, and from (11.15) it is easy to compute the index of v at p and show that it is one. Thus the Euler characteristic of M
11.7
Two Dimensional G-Manifolds
There is a standard action of SO(3) on S2 and R P ~ and an action of T Z on itself, and it is well known that these are basically the only actions of a compact connected Lie group on a two dimensional manifold. Rom this one easily deduces the following. Theorem 11.7.1 ,Let G be an n-dimensional torus acting non-trivially on an oriented, compact, connected two-manifold M. If MG is non-empty, this action has the following properties:
is just the cardinality of MG. Hence M has positive Euler characteristic and
has to be diffeomorphic to S2. Coming back to the invariant Riemann metric on M, by the Korn-Lichtenstein theorem (cf. [C]) this metric is conformally equivalent to the standard "round" metric on 9 and hence G / H acts on M as a one dimensional sub. group of the group of conformal transformations SL(2, C ) of S2. However, the connected compact one-dimensional subgroups of SL(2, C ) are all conjugate to each other; so, up to conjugacy, the action of G / H on M is the standard action of S' on S2 given by rotation about the z-axis.
182
.
Chapter 11. The Abstrmt Localization Theorem
We will now use the results above to compute the equivariant cohomology ring, H G ( M ) . Since H is a closed subgroup of G one has an inclusion map, h + g and hence a restriction map
Theorem 11.7.2 The *variant whomology ring H c ( M ) is the subring of S(g*)@ S ( g C )consisting of all pazrs
s
i.e., to check that
dim^^(^*)
+
= dim s k ( h * ) dim
s~-'(~*)
This, however, follows from the fact that the restriction map, (11.16), is onto and that its kernel is a . S k - ' ( g * ) , a being an element of g' - 0 which vanishes on h. 0
11.8
A Theorem of Goresky-Kottwitz-MacPherson
Let M be a compact G-manifold having the following three properties Proof. Since the ordinary cohomology of M is zero in odd dimensions, M is equivariantly formal by Theorem 6.5.3;so the map, H G ( M )-, H c ( M G ) , is injective, and embeds H G ( M ) into H G ( M G )as a subring. However, M G consists of two points, pl and pz; so H G ( M G )= S(g*)@S(g*),with one copy of S(g*) for each p,. Moreover the embedding of H G ( M ) into H G ( M ~is) given explicitly by
a) H G ( M ) is a free S(g8)-module. b) M~ is finite. c) For every p E MG the weights
of the isotropy representation of G on T p are pairwise linearly independent: i.e., for i # j , ai,, is not a linear multiple of ajVp.
i, being the inclusion map, i, : {p,) + M . Now note that since H acts trivially on M H $ ( M ) = H * ( M )@ S ( h * ),
The role of properiies a) and b) is clear. We will clarify the role of property c) by proving:
so the natural mapping of H Z ( M ) into H ; I ( M ) maps the element (11.19) into an element of the form
Theorem 11.8.1 Given properties a) and b) property c) is equivalent to: For every wdimension one subtorus H of G, dim M H 5 2. Proof. Let X be a connected component of M H of dimension greater than zero. By Theorem 11.6.1, X contains a G-fixed point, p. Moreover,
with a, E H ' ( M ) , i > 0 , and go,. . . ,ge in S ( h * ) . Thus Therefore, since H is of codimension one, its Lie algebra is equal to giving one the compatibility condition (11.18). Thus we have proved that H G ( M ) is contained in the subring of S(g*) @ S(g*) defined by (11.17) (11.18). To prove that it is equal to this ring we note that dim H , $ ~ ( M ) = dim ( H ' ( M ) @ s k ( g * )@ H ' ( M ) g =
s~-'(~'))
dim^^(^*) + dim^^-'(^*)
ai,, being one of the weights on the list (11.20), and hence TpX is the one dimensional (complex) subspace of TpM associated with this weight.
Remark 11.8.1 It is clear from this proof that dim M H = 2 iff the Lie algebra h of H is the algebm (11.21) for some i and p. Hence there are only a finite number of subtori
by Theorem 6.5.1; so it suffices to check that this dimension is the same as the dimension of the 2k-th component of the ring (11.17)-(11.18), viz. 2 dim s
~ ( -~dim * )s k ( h * )
with the property that dim M N k = 2, and if H is not one of the groups on the list, M~ = M ~ .
184
Chapter 11. The Abstract Localization Theorem
11.9 Bibliographical Notes for Chapter 11
Moreover, if H is one of these exceptional subtori, the ~ 0 ~ c 3 ~compoted nents C,,, of M ~ are ' two-spheres, and each of these two-spheres intersects MG in exactly two points (a "north polen and a "south pole"). For i fixed, the Cij's cannot intersect each other; however, for different i's, they can intersect at points of MG;and their intersection properties can be described by an "intersection graphn r whose edges are the Xij's and whose vertices are the points of M ~ (Two . vertices, p and q, of are joined by an edge, C, if C n M G = {p, q).) Moreover, for each C there is a unique Hi on the list (11.22) for which
11.9 ~ i b l i o ~ r a ~ h iNotes c a l for Chapter 11 ,
1. The main result of this chapter: that the restriction map, HG(M) -+ H G ( ~ G )is, injective modulo torsion, is due to Bore1 [Bo] and, with some refinements, to Hsiang [Hs] (see Chapter 3 of [Hs], in particular the comments at the bottom of page 39). 2. Let S be the polynomial ring, S(g*), and let S# be the quotient field of this ring, i-e., the set of all expressions
f f €S,g€S-0. -, 9
so the edges of I-' are labeled by the Hi's on this list. Since M G is finite
185
(11.27)
An equivalent form of Borel's result is the assertion:
3. Most of the material in § 11.4 is taken verbatim from Atiyah and Bott's article, [AB]. (In particular our Theorem 11.4.2 is Theorem 3.5 of [AB].)
and hence H G ( M ~= ) Maps(Vr, S(g*))
4. One can regard theorem 11.4.2 a s a sharpening of (11.28). In order
where Vi is the set of vertices of I?
to convert the map, HG(M) -+ H G ( M ~ )into , an isomorphism, one doesn't have to tensor with the ring of all quotients (11.27). It suffices to tensor with the ring of quotients
Theorem 11.8.2 [GKM] An element, p, of the ring
is in the image of the embedding
the ai's being weights of G for which the product, g, vanishes on the set (11.4.1). if and only if for every edge C of the intersection graph, compatibility condition Th~(v1)= ~ h ~ ( v 2 )
r, it satisfies the (11.25)
and vz being the vertices of C, h being the Lie algebra of the group (11.23), and (11.26) ~h : S(g*) 4 S(h*)
vl
5. We have already described, in the bibliographical notes at the end of chapter 6, some criteria for M to be equivariantly formal. Here are a few additional criteria:
(a) For every compact Gmanifold
being the restriction map. Proof. By Theorem 11.5.1 the image of i* is the intersection:
and by (11.18) the image of (iH,)* is the set of elements of Maps(Vr, S(g*)) satisfying the compatibility condition (11.25) at the vertices of r labelled by
--
-
and M is equivariantly formal if and only if chis inequality is an equality. (See [Hs], page 46.) (b) M is equivariantly formal iff the canonical restriction map
is onto ie., iff every cohomology class, c E Hi(M) is of the form
,
186
"
Chapter 11. The Abstract Localization Theorem
11.9 Bibliographical Notes for Chapter 11
-c being an equivariant cohomology class. For example suppose that every homology class [XIis representable by a G-invariant cycle, X, and hence, by Poincark duality, that every cohomology class, c, is the Thom class of a G-invariant cycle, X. Then one can take for the Fin (11.29) the equivariant Thom class of X. (Compare with item 3 in the bibliographical notes a t the end of Chapter 6.) (c) M is equivariantly formal if it possesses a G-invariant Bott-Morse function whose critical set is MG. Indeed if such a function exists, the unstable manifolds associated with its gradient flow (and, in the case of non-isolated fixed points, the G-invariant submanifolds of these manifolds associated with G-invariant cycles in MG) provide a basis for the homology of M. Hence M is equivariantly formal by criterion b). (d) In particular if M is a Hamiltonian G-manifold, it is equivariantly formal as a consequence of the fact that, for a generic E g, the <-component of the moment map is a Bott-Morse function whose critical set is M ~ .
<
(e) If M is equivariantly formal as a G-manifold, and K is a closed subgroup of GI M is equivariantly formal as a K-manifold. (This follows from the fact that the restriction map, Hc(M) --+ H(M), factors through HK (M).) 6. One important example of an equivariantly formal space is the compact Lie group, G, itself. For the left action of G, Hc(G) is trivial; however, for the adjoint action the restriction map
is surjective arid hence by criterion b) of item 5, G is equivariantly formal. (Here is a sketch of how to prove the surjectivity of (11.30) for G = U(n). For N large consider G and U(N) as commuting subgroups inside U(n + N); and let M be the Grassmannian
let E be the manifold of n-frames in Cn+N
and let T be the fibration of E over M with fiber, G. If k < N, then for every [p] E H ~ ( M ) r, p = dv for some v E Rk-'(E) by Proposition 2.5.2. Let u E M be the identity coset, and i : G --, E the inclusion of the fiber of E above u into E. Then i* d v = i*x*p = 0; so di'v = 0; i.e., i'v is closed, and hence represents a cohomology
187
class, [i'v] E Hk-'(G). By a theorem of Koszul, these "transgressive" cohomology classes generate H*(G). (See [Kol].)
-
+
Now notice that G acts on U(n N) by its adjoint action, and that this action induces an action of G on E which is intertwined by a with the natural left action of G on M. The latter leaves fixed the point, u; therefore the transgression construction which we outlined above can be carried out epuivariantly. Combining this observation with Koszul's result, the question of whether (11.30) is surjective can be reduced to the question of whether the map, HG(M) + H(M) is surjective. However, M is a Hamiltonian G-space; so this follows from Kirwan's theorem. (See item 4 in the bibliographical notes following Chapter 6.) Notice by the way that for z = 3, we've already proved the surjectivity of the map, H&(G) --, Ha(G)by exhibiting an explicit element of H&(G) whose image in Ha(G) is the generator of H1(G). (See 5 9.8.1.) An analogue of this "Alekseev-Malkin-Meinrenken form" in all dimensions has recently been constructed by Meinrenken and Woodward based on a construction of Jeffrey.) 7. An explicit reference for the Chang-Skjelbred theorem is: Chang, T. and Skjelbred, T., Ann. Math. 100, 307-321 (1974)page 313, lemma 2.3. Our treatment of this result in 511.5 is based on some unpublished notes of M. Brion and M. Vergne: "Sur le theorkme de localization en cohomologie &uivariantn. (Brion and Vergne show, by the way that the assumption we have been making throughout this section, "M compact", can be replaced by the much weaker assumption: "there exists a G-invariant embedding of M into a Euclidean space on which G acts linearly" .)
8. For another proof of the Chang-Skjelbred theorem within the setting of equivariant de Rham theory, using Morse theoretic techniques, see the paper of Tolman and Weitsman [TW]. Tolman and Weitsman prove that if M possesses a G-invariant Bott-Morse function whose critical set is MG, then the "Chang-Skjelbred generators" of Hc(M) can be explicitly represented by equivariant Thom classes associated with the unstable manifolds of the gradient flow of this function. (Compare with item 4d) above.) 9. The theorem of Goresky-Kottwitz-n/SacPhersonis actually stronger than the theorem we attribute to them in 8 11.8. In particular, in their paper "equivariant cohomology" means equivariant cohomology with coefficients in sheaves, and the result that we attribute to them is Section 11.8 is a specialization of their result to the constant sheaf C.
10. Let M be a compact Hamiltonian .G-space with symplectic form, w , and moment map, a. In addition suppose that MG is finite and
188
Chapter 11. The Abstract Localization Theorem O : MG --+ g* injective. In [GW]Ginzburg and Weinstein prove the following interesting result concerning "deformations" of the data ( w ,O ) .
-
Theorem 11.9.1 The data ( w ,O ) are determined up to deformation by the image of O : M G g* (this image being, by the hypothesis above, a finite set of points zn g*.) More explicitly let (w,, a , ) , c E R, be a family of G-invariant symplectic forms and moment maps 'depending smoothly on c with (wo,Oo) = (w, O ) . Suppose
Then for c small (w,, O,) is equivariantly diffeomorphic to (w, 9 ) .
Here is a sketch of their proof: Let G, be the equivariant symplectic form, wc - a,. (See 3 9.1.) Since M is equivariantly formal (by item 4, criterion d)) the restriction map, H G ( M ) + H G ( M G ) is injective and hence by (11.31) [GI = [G,]. Thus, by the "Moser trick" (see, for instance [GS])there is an equiv-
depending smoothly on c and equal to the identity for c = 0 such that
and @,of
=o.
Appendix
188
-
Chapter 11. The Abstract Localization Theorem <
3
9 : MG g* injective. In [GW] Ginzburg and Weinstein prove the following interesting result concerning "deformations" of the data (w,9 ) . Theorem 11.9.1 The data ( w ,9 ) are determaned up to deformation by the image of 9 : MG -4 g* (this image being, by the hypothesis above, a finite set of points an g*.) More q l i c i t l y k t (w,, a , ) , c E R, be a family of G-invariant symplectic forms and moment maps 'depending smoothly on c with (wo,9 0 ) = (w,9 ) . Suppose
Then for c small (w,, 9 , ) is equivariantly diffwmorphie to (w,9 ) .
Here is a sketch of their proof: Let G, be the equivariant symplectic form, w, - 9 , . (See § 9.1.) Since M is equivariantly formal (by item 4, criterion d)) the restriction map, H G ( M ) 4 H G ( M G )is injective and hence by (11.31) [GI = [G,] . Thus, by the "Moser trick" (see, for instance [GS])there is an equivariant difFeomorphism f c : M 4 M depending smoothly on c and equal to the identity for c = 0 such that
and
Appendix
Notions d'algsre diffhtielle; applicationa m groupes de Lie et am vari& OR opere m groupe de Lie Colloque deTopologie,CB.RM, Bmdles 15-27 (1950)
Soit A une alggbre (associative) sur un anneau commutatif K ayant un 616ment unit& Une structure gradu6e est d6finie par la donnCe de sous-espaces vectoriels AD (p =0, 1, ... ) .tels que l'espace wectoriel A soit somme directe des A'; un 616ment de ADest dit cc homogkne de degr6 p 12. On suppose de plus que le produit d'un 616ment de A' et d'un 616ment de Aq est un element de A M . On note a+ a l'automorphisme de A qui, a un 6lCment a E AD, associe 1'616ment (- 1)'a. Un endomorphisme B de la structure vectorielle de A est dit dc degrk r s'il applique AP dans.AHr pour chaque p. Parmi les endomorphismes, nous distinguerons les catkories suivantes : 1. On appelle de'rivation tout endomorphisme B de A, de dew6 pair, qui, vis-a-vis de la multiplication dans A,' jouit de la propri6t6 B (ab) = (Ba) b a (9b) . (1) .. 2. On appelle antidkrivation tout endomorphisme 6 de A, de deg& impair,qui jouit de la propri6td
+
+
Si en outre 6 est de degd 1 et si 66 =0, 6 s'appelle une diffkrentielle; on ddfinit alors, classiquement, l'alg lbre de cohomologie H ( A ) de A, relativement ti 6. C'est une al&bre gradu6e. Une d6rivation (resp. antiderivation) est nulle sur 1'616ment unit6 de A, s'il existe. Si 6 est une antidCrivation, 3 est une dkrivation; si 6, et
192
Henri Cartan Notions d'alghbre difT6rentielle
6, sont des antidkrivations,
Z2E, est une dkrivation. N f i nissons le crochet [Olr92] de deux endomorphismes 8, et Q2, comme d'habitude, par la formule
re,, e,j=e1e2 -e2e, .
hlors le crochet de deux derivations est une derivation; le crochet d'une derivation et d'une antiderivation-est une antiderivation. Une derivation, ou une antiderivation, est determince quand elle est connue su.r les sous-espaces A" et A', pourvu que l'alghbre A soit engendrke (au sens multiplicatif) par ses 616ments de degrC 0 et 1. Dans certains a s , on peut se donner arbitrairement les valeurs d'une derivation (ou d'une antiderivation) sur A', en lui donnant la valeur 0 sur A" :par exemple, lorsque A est l'algibre ezte'rieurc d'un module M (sur K) dont les ClCments sont de degre u n (I). Ezemple. - Soit a un module sur K, et soit A l'alghbre extkrieure du dual a' de ce module. Chaque Clement z de a dkfinit un endomorphisme i ( z ) de l'algbbre A, de degrC -1. appele cc produit int6rieur par z : c'est l'unique antide'rivation, nulle sur Ao=K, qui, sur A'= a', est kgale au ((produit scalaire definissant la dualit6 entre a et a' : i ( x ) - z ' = < z , x 1 > pour x E a et x'EA1; on a alors i (x). (x]' A - - - As,)' A = (- 1)"' <x, x ): .,'A. - ,i/zk1A-. Ax: (3) ))
))
-
l < k i r
(le signe A signifiant que le terme situe au-dessous doit Btre supprime) . L'opCrateur i (z) est de carre' nu1 : car i (z) i ( r ) est une dbrivation, kvidemment nulle sur A h t A', donc nulle prtout. Produit tensoriel d'algkbres gmdukes. - Soient A et B deux algbbres graduCes. Sur le produit tensoriel A@B de leurs espaces vectoriels, considdrons la loi multiplicative definie par (a8b) - (a'@b') = (- '' )1 ( a d )@ (bbl), si b est de deer6 q et a' de degrd p'. Ddfinissons sur C = A@B une structure graduPe, en appelant C' le sous-espace de C, somme directe des AP@Bqtels que p q = r. Alors C est munie d'une structure d'algibre gradue'e; cette algbbre graduee s'appelle le produit tensoriel des alghbres gradukes A et B. Le cas le plus intkressant est celui oir A et B ont un 616ment unite, les sous-algkbres A' et Be Btant isomorphes i l'an-
+
(I.)
Pour ce qui concerne les alghbres extbrieures en gbnbral, voir
BOL=BAXI, Algtbre, chap. III.
193
61&+
neau de base K. Dans ce cas, on identifiem toujours A i une sous-alghbre de A@ B, par l'application biunivoque a +a@ I de A dans AOB (on note 1 11k16ment unit6); de m&me, on identifiera B i une sous-algbbre de A8B. En outre, disignons par BC la somme directe des Bq pour 4 1; A 8 B est somme directe de la sous-alghbre A@B" (identifike Ci A) et de l'idhl A@B+. Cette dkcomposition directe definit un projecteur de A @ B sur A 8Be,donc une application linCaire de A 8B sur A ; cette application est compatible avec les structures multiplicative~;nous l'appellerons la projection canonique de AoB sur A. Elle identifie A B l'algbbre quotient de A 8 B par l'idhl A 8B+. On ddfinit de m&me la projection canonique de A 8 B sur B. Plaqons-nous toujours dans l'hypothbse oir A" Be =K. Soit donnee une application lindaire 0, de A dans C = ABB, de degrd pair, satisfaisant B la condition ( I ) , et une application linkaire 0, de B dans C, de m&medegrB, satisfaisant aussi A (1). I1 existe alors une dtrivation 0 de l'algkbre A 8 B, et une seule, qui se rCduise B 0, sur A et B 0, sur B; elle est dCfinie par 0(a@b)=B1(a) . b + a . 8,(b) (4) (le signe . designant la multiplication dans A 8 B) . De mbme, Btant donnkes une application linQire El de A dans C, de degrB impair, satisfaisant B (2), et une application linCaire Z2 de B dans C, de m&medegrB, satisfaisant aussi h (3), il existe une antide'rivation 6 de l'alghbre A@B, et une seule, qui se rCduise i 6, sur A et B Z2 sur B; elle estdefinie par E(a8b) = & ( a ) . b i d - Z , ( b ) . (5)
Pour simplifier l'exposition, on se bornera aux variCtks indkfiniment diffCrentiables. Les champs de vecteurs tangents que l'on considerera seront toujours supposes indifiniment diffkrentiables; de mGme, les formes diffkrentielles extirieures, de tous degrds, seront supposkes B coefficients indefiniment diffkrentiables. Les champs de vecteurs tangents constituent un module T sur l'anneau K des fonctions numkriques (indkfiniment diffkrentiables). Le module dual T1 est le module des formes diff6rentielles de degre' un. L'alghbre extkrieure A (T') du module T' est I'alghbre des formes diffkrentielles de tous degrBs (les fonctions, Clements de K, ne sont autres que les formes diffirentielles de degrk 0 ) . La diffkrentiation exthrieure, notCe d, est une cc diffkrentielle sur A(Tt), au sens du 1. Chaque 46ment z de T definit, outre le produit inte'ricur i ( z ) (qui opbre ))
194
Henri Cartan
Notions d'algkbre diikrentielle
sur A (TI) et est de degr6 -l), une transformation infinit6simale 0 (2) qui opkre aussi bien sur T que sur T' et A (TI); sur A (TI), c'est une de'rivation de d q r 6 0, qui est entierement cnnctkris6e par les deux conditions suivantes :
a(G) et dans son dual al(G) = A1(G) deux bases duales (zij et ( z : ) , on a
((
))
0 (z)d = d0 (z) (c'est-8-dire : 0 (z) commute avec d) ; (6)
0 (z) - f =i(z) - df pour toute fonction f E A'(Tt) .
<7)
Si z et y sont deux champs de vecteurs tangents, le champ de vecteurs 0(2) . y se note [z, y] ; cette notation se justifie parce que 0([x, y]) =fJ(z)4(y) -flS?l)Q(~).
(1)
En outre, sur l'alghbre diffkentielle .4 (TI), on a les relations O(z)i(y) = i ( y ) Q ( x ) + i ( [ x , ~ l ) ,
0 (x) = i (5) (1
+di (5)
(11) (111)
(formule qui, compte tenu de dd = 0, entraine la relation no6).
Soit G un groupe de Lie connese. Les champs de vecteurs tangents, invariants par les translations i gauche, f o m e n t un espace vecloriel a(G) sur le corps r6el; cet espace est en dualit6 avec l'espace vectoriel ar(G)' des formes diffbrentielles de degrb un, invariantes i gauche. L'alghbre ext6rieure A(G) de d ( G ) est I'alghbre (sur le corps rkel) des formes diffCrentielles de tous degrbs, invariantes i gauche. Les ClCments de degr6 0 (fonctions constantes) s'identifient aux scalaires (multiples de l'unitb) . L'algbbre differentielle A (G) a une alghbre de cohomplogie qui, lorsque G est compact, s'identifie k l'alghbre de cohomologie (rklle) de 1-espace G. Chaque 6lBment z de a(G) dCfinit un groupe B un paramMre d'automorphismes de G, qui ne sont autres que les translations h droite par un sous-groupe i un paramhtre de G. La transformation infinitesimale 0(z) de ce groupe opkre dans a(G); donc a(G) est stable pour le crochet [z, y]. L'espace a (G) , muni de la structure definie par ce crochet, est l'alglbre de Lie du groupe G. En outre, les 0 ( 2 ) ophrent dans A (G) , ainsi que les produits inlkrieurs i ( z ) . Sur l'alghbre diffbrentielle A (G), les ophrateurs d, i (x) et 0(x) satisfont aux relations ( I ) , (11) et (III) du parngraphe prCcbdent. Ici, on peut expliciter l'opbrateur diffkrentiel d de A(G) : dCsignons par e ( d ) la multiplication (? gauche) i par un 616ment z ' E A'(G) dans l'alghbre .4(G); alors, en prenant dans
195
.-
Cette formule a 6th donnke par Koszul dans sa thbse (I). AppliquQ aux Clkments de degd un de A(G), elle donne les tc Bquations de Maurer-Cartan )).
C'est une vari6tC 6, que nous supposerons indkfiniment diffhrentiable, et oh un groupe de Lie connexe G o@re de manibre que : lo L'application (P, s) -+ P - s de 6 X G dans 6 soit ind6finiment diffkrentiable; et (P . s) - t =P - (st) (ce qu'on exprime en disant que G opbre tc ? droite i *).; 2" G soit simplement transitif dans chaque classe d'6quivalence (fibre) ; . 3" L'espace CA (~.espacede base n ) quotient de 6 par la relation d'kquivalence definie par G, soit une vari6tk indBfiniment diffkrentiable; 4" Chaque point de 03 possbde un voisinage ouvert U tel que l'image r6ciproque de U dans 6 soit isomorphe (comme variCtC indkfiniment diffkrentiable) au produit U ' X G, la transformatiqn definie par un 616ment s de G 6tant alon (u, g) -+ (u,gs) On notera E l'alghbre des formes diffhrentielles (g coefficients indhfiniment diffbrentiables) de l'espace 6,munie de sa graduation et de l'op6rateur d de differentiation extdrieure. Tout vecteur x de I'alghbre de Lie' a(G) d6finit un champ de vecteurs tangents B 6 : en effet, chaque f i r e de & s'identifie G, d'une manihre bien dCterminke ?I une translation i gauche prbs du groupe G; donc un champ invariant ?I gauche, sur G, se tnnsporte sur chaque fibre d'une seule mnnihre. Ainsi, chaque element z de a ( G ) dhfinit, dans l'alg*bre E, un produit int6rieur i(z) et une transformation infinitesimale fl(x); et les relations (I), (11) et (111) du $ 2 sont satisfaites. D'ailleurs la transformation infinithimale 0 ( x ) n'est autre que celle du sous-groupe B un paramhtre de G (groupe d'op6rateurs h droite dans 6 ) d6fini par 1'616ment z de l'algbbre de Lie a(G) . Bull. Soc. mgth. d e France, 1950, pp. 65-127;voir formde (3.4).
(I)
p. 74.
Henri Cartan
196
Notions d'algebre diff6rentielle
L'alghbre B des formes diffgrentielles de l'espace de base U3 s'identifie B une sous-alg2bre de E, stable pour d, B savoir la sous-alghbre des 4l6ments annulhs par tous les opdrateurs i(z) et B(z) relatifs aux elements z de a(G). D'une maniere g6n6rale, soit E une alghbre diffkrentielle gradu6e oh operent des antidkrivations i(z) (de degd - 1 et de carre nul) et des d6rivations B(z) (de degd 0) correspondant aux 616ments z d'une algbbre de Lie a(G), de manihre B satisfaire i (I), (11) et (111). Nous dirons, pour abdger, que G opPre duns l'alglbre E. Cela Ctant, nous appellerons ClCrnents basiques de E les 6lCments annulks par tous les i(z) et les B(z) ; ils forment une sous-algbbre gradube B, stable pour d [en vertu de (III)]. On appelle 6lCments invariants de E les Bl6ment.s annul6s par les B(z) ; ils forment une sous-alghbre stable pour d, que nous noterons I,. Dans certains cas, I'homomorphisme canonique H(1,) + H(E) des alghbres de cohomologie de I, et de E est un isomorphisrne de la premiere sur la seconde. Il en est ainsi notamment dans les cas suivants : 1) E est de dimension finie, et l'algbbre de Lie a (G) est re'ductiue (i.e. : compode directe d'une algebre abClienne et d'une algebre semi-simple) ; 2) E est I'alghbre des formes diffkrentielles d'un espace fibr6 principal b dont le groupe G est compact. On a un homomorphisme canonique H (B) +H (I,). Un problbme important consiste 5 chercher des relations plus pr6cises entre les alggbres de cohomologie H (I,) et H (B) ; dans k cas 2) ci-dessus, ce sont respectivement les alebres de cohomologie de l'espace fibrC 6 et de son espace de base &.
5. COSWZSIONINFIKIT~SIMALE
DANS U N ESPACE F I B R ~PRINCIPAL
Une connexion infinithimale est dkfinie par la donnke, en chaque point P de l'espace fibrC 6, d'un projecteur Q, de l'espace tangent 21 & au point P, sur le sous-espace des vecteurs tangents B la fibre au point P, de manibre que : lo F, soit fonction indkfiniment diffkentiable du point P; 2" Les projecteurs cp, relatifs aux points d'une m&mefibre se transfoment les uns dans les autres par les op6rations d u groupe G. On peut prouver I'ezistence de telles connexions infinitksimales ('). De plus, la donn6e d'une connexion infinitksimale dans E revient B la donnCe d'une application lin6aire f du dual A1(G) de I'algebre de Lie a ( G ) , dans le sous-espace El (I)
*
T'oir la conf6rencr dc Ch. Ehrer;mann b ce Colloque.
-197
des elements de degrC un de l'algebre E, application qui satisfasse aux deux conditions suivantes : i(x) - f ( d ) = i ( z ) - 2 ) (scalaire de E, c'est-&-dire fonction constante sur &), (8) e ( t ) -f ( d ) = f ( e ( ~ )2) pour tout z E a (G) et tout z/ E A1(G). Supposons qu'on ait un autre espace fibr6 principal de mOme groupe G, et un G-homomorphisme de 6' dans 6 (c'est-&-direune application indkfiniment diffhrentiable de B dans 6, compatible avec les op6rations de G dans 6' et 6). Un tel homomorphisme d6finit d'une mani'ere Cvidente l'image r6ciproque d'une connexion infinithsimale sur 6 : c'est une connexion infinitbsimale sur 6'. On v6rifie aidment que l'application f' de A1(G) dans Eradbfinie par cette dernihre est compos6e de l'application f et de l'homomorphisme de E dans E' defini par l'application de l'espace 6' dans l'espace 6. Ce qui precede conduit ila notion abstraite de tc connexion al~6brique1, dans une alghbre differentielle E (avec 616mentunit6) dans laquelle opere un groupe G (au sens de la fin du s 4) : ce sera une application linCaire de A1(G) dans E' qui sntisfasse aux conditions (8). Soit alors f une telle connexion algkbrique. Supposons en outre que I'alghbre E satisfasse B la loi d'anticommutation v u = (- l)%v pour u de degr6 p et v de degrC q. Alors on peut prolonger f , d'une seule manihre, en un homomorphisme lmultiplicatif) de Z'algtbre A(G) duns l'alg2bre E, qui transforme 1'61Cment unit6 de A(G) dans 1'816ment unit6 de E. Notons encore f ce prolongement. On a alors, pour tout 616ment a E A(G),
I
i(z) - f ( a ) = f ( i ( z ) - a )
t
(80 - f ( a ) =f(fl(z) . a ) . Autrement dit, f est compatible avec les op6rateurs i(x) et B(z), qui operent dans A(G) et dans E. Mais, si z' E h l ( G ) , on n'a pas, en gkn6ra1, d(f ( z ' ) ) = f (dz') ; autrement dit, f n'est pas compatible avec les diff6rentielles de .4 ( 6 ) et de E. L'application z' --t d (f (z') ) - f (dz') de .il(G) dans Ez est ce qu'on appelle le tenseur de cowbure de la connexion. L'616ment d(f (z')) f (dz') n'est pas, en g6n6ra1, un 616ment basique de E; toutefois il est annuli par tous les produits inte'rieurs i (z) . Demonstration . i ( z ) d . ( f ( z ' ) ) =B(z). f ( d ) - d . ( i ( x ) . f ( 2 ) ) =f(Q(z).z')
Q(4
-
d'apres (8) rt, d'aprbs (87,
*
198
Henri Cartan
Notions d'algkbre diffkrentielle
i(z) -f(dz') = f ( i ( z ) -dz') =j(9(z) -2') - f(di(z) - 2 ' ) = f ( e ( ~ ) .z') 6. L ' A L G ~ RDE E WEXLD'UNE
ALG&BRE DE
.
LIE
Les considkrations pdc6dentes ont conduit Andrk Weil (dans u n travail non publi4) B associer k l'alg&bre de Lie a(G) une autre alghbre diffkrentielle, dont A(G) est un quotient, et que nous allons d4finir maintenant. Designons par S(G) l'alglbre symHrique du dual a'(G) de a (G). Si on prend une base (32)dans a' (G) , S(G) s'identifie B l'alghbre des polyndrnes par rapport aux lettres z: (commutant d e w B deux) . S (G) s'identifie aussi canoniquement B l'alghbre des formes multilinkaires symttriques sur l'espace vectoriel a (G) . On distinguera l'espace al(G) comme sous-espace A1(G) de A (G) , et comme sous-espace S' (G) de S (G) . On a un isomorphisme canonique h de A1(G) sur S1(G). On notera sou-
z'
I'ClCment h (9) , pour z' E A' (G) . Si on a une connexion algdbrique f de A1(G) dans une alghbre E comme ci-dessus, et qu'on prolonge f en un homomorphisme de A(G) dans E, on est amend B ddfinir une applivent
cation linkire f de S1(G) dans Ez,en posant =d(f(z')) --j(d&) . Pour que f conserve les deg~ks,on convient que les 616ments de S1(G) sont de degre' 2. Ceci conduit B graduer S (G) en convenant que les klkments de SP(G) (formes p-linkaires
-
symktriques sur a(G)) sont de degrk 2 p. L'application j se prolonge alors en un homomorphisme multiplicatif, de degrC 0, de l'alghbre (commutative) S ( G ) dans I'aIg6bre E.
-
On notera encore f l'homomorphisme prolongk. L'algPbre de Weil de I'alggbre de Lie a(G) sera, par definition, l'alghbre prad116e W(G) =A(G) 63 S(G) , produit tensoriel des algbbres p d u 6 e s A (G) et S (G) (cf. 5 1). Les homomorphismes j : A (G)--t E, et f : S (G) +E, dkfinissent un homomorphisme 7 de I'algbbre W (G) dans l'alghbre E, par la formule
-
f ( a 8 s> = f ( a > ~ f ( s .> L'homomorphisme. (multiplicatif) 7 est de deer6 0. On va definir, sur \V(G), d'une rnaniPre iinde'pendante de
199
l'algebre E et de la c o ~ ~ r ~ e z fi ,o des n opkrateurs i(z) , 0(z) et une diffbrentielle 6, de telle manihre que, pour toute connexion f dans une alghbre diffkrentielle graduke E dans laquelle o+re G (avec la loi d'anticommutation vu =(- l)pauv), l'homomorphisme f d6fini par j soit co~npatibleavec les opkrateurs i(z) , 8(z) et les op6rateun differentiels 6 (de W (G)) et d (de E) . Dtfinition de i(z) : i(z) est d6ja dkfini sous la sousalghbre A(G) de IV (G) = A (G) @ S (G) . Sur S (G) , convenons que 170p6rateur i(z) est nul. Sur W(G), i(z) sera l'unique antidkrivation qui prolonge i(x) sur A(G) et 0 sur S(G); on a i(z) i(z) =0, car i (z) i (z) est une derivation nulle sur A (G) et sur S (G) . Cela posk, on a, pour tout klkment w E W (G) ,
7 avec i (z) ) ,
i (z) .f(w) =f(i(z) .w ) (compatibilitk de
parce qu'il en est ainsi lorsque U Iest dans A (G) (relation (8') ) et lorsque w est dans S1(G) (les deux membres ktant alors nuls) . De'finition de B(z) : 0(z) est d6jB dkfini sur A(G) . On vn le ddfinir sur S1(G), puis on le prolongera en une dkrivat'Ion
z'
(de degrd 0) sur W (G) =A (G) 63 S (G) . Or soit E S1(G) , done d = h (z') ; on pose 6 (z) . i=h (6 (x) z') ; ceci dkfinit
-
.
0(z) sur S' (G) . 8(z) ktant alors prolong6 B F\-(G), on a bien, pour tout W E W(G), e ( ~ )f .( w )=j(e(2) - W ) (compatibilitk de j avec 0(z)), parce qu'il en est ainsi lorsque w est dans A (G) (relations (8') ) , et lorsque w est dans S1(G); en effet
--
-
-
0 (z)- f (P)=f (9 (z) .2') (vkrification imm6diate , grace B (8') ) . I1 y a intkr6t B dkcomposer l'op6rateur B(z) sur W (G) en la somme de deux op6rateurs partiels : e(z) =gA(z) +egi2) , oa 0,(z) est 4 p 1 B 0 (z) sur A (G) et nu1 sur S (G) , et 0,(z) est 6gal A O(z) sur S (G) et nu1 sur A (G) . DJfinition de Z'ope'mteur diffe'rcntiel 6 de \T (G). - La relation d(f (2) = f (d&> +j(z') conduit i poser (si l'on veut que f soit compatible avec les op6rateux-s diffkrentiels)
-
82'=dz'+Z'=dz'f
la(z')*.
(9)
200
Henri Cartan
Notions d'aigebre diffkrentielle
De meme la relation
--
--
-
-
' Chaque fois qu'on a une connexion algBbrique dans une algbbre diffkrentielle gmduee E oh o@re G [avec la loi d'anticommutation vu =(- 1)P q ~ on ~ ]obtient , un homomorphisme de l'alghbre de Weil W (G) dans l'alghbre E, compatible avec les graduations, les opkrateurs i(z) et B(z), et enfin compatible avec les optrateurs diffirentiels. Seuls, ce dernier point reste & verifier; or la relation
i(z)d -f(z')=B(z). f(z')=f(B(z).z')
-
conduit & poser
-
7
i(z).62'=8(~).2', ou, ce qui revient au m6me,
etant deux bases duales) . L'op6rateur 6 Btant ainsi dhfini sur A1(G) et S1(G), il se prolonge d'une seule maniBre en une antiderivation (not& encore 6) de W (G)=A (G) 8 S (G) . Son degre est 1. Les formules (9) et (10) permettent d'ailleurs d'expliciter 6 sur W (G)tout entier : Z=d*+ds+h, (11) o~ h prolonge I'application (d6j¬ee h) de A1(G) dans S1fG) en une antiderivation de W (G), nulle sur S(G) : ( (2,) et (2,')
+
-
-.
( e ( i ) d6sipe la multiplication par 1'816ment z' E S1(G)). Quant aux op6nteurs d, et d,, ils sont explicitds par les formules suivantes (dont la premiere dsulte de 13 formule (IV) du g 3) :
j(6w) =d(f (w)) a lieu si w est dans A1(G) ou dans S1(G), d'apds la manibre
meme dont 6 a BtB dbfini (cf. relations (9) et (10)); il en r6sulte qu'elle a lieu pour tout w E A (G) 8 S (G). C. Q . F. D. Cas particulier. - Prenons pour E l'algkbre A (G) , f Chnt l'application identique de A1(G) dans A1(G). Alors f n'est autre que la projection canonique (cf. fin du 5 1) de W(G) = A (G) @ S (G) w r A (G) ; cette projection canonique est compat.ible avec les opdrateurs differentiels 6 (de W (G)) et d =d, (de A (G) ) . Donc ltalgkbre differentielle A (G) s'identifie & un quotient de l'alghbre differentielle W (G) .
Remarque. - Si le groupe G est abe'lien, les op6rateux-s 6(z) sont nuls; alors la differentielle 6 de. W(G) se rMuit A l'antid6rivation h definie par la formule (12;. 7 . CLASSES CARACT~RISTIQCES D'UN
-
e (5.7 8. (z,)
ds =
(14)
k
Les formules (ll),(I?), (13), (14) explicitent complbement l'openteur E de \V (G). La relation analogme B (111) ( S 2 ) : ')(z) = i(x)E+Ei(z) a lieu sur \.\'(G). Xlle gsulte des relations suivilntes :
+
i ( z ] h hi(z) = O , 0, (2)= i ( z )dA d,i(z)
201
(15) (16) 0, (2)= i (2) . (17) On verifie que E est une diffgrentielle : B6 = 0 . I1 suffit. cie v6rifier que la derivation 66 est nulle sur A1(G) et S1(G). Ainsi, W(G) est une algkbre diffhentielle gndufe, munie d'op6nt.eur-s i ( z ) et 9(r) sat isfaisant nus conditions (I), (11). (111) du § 2.
+ , ds +.dsi (r) .
(R~ELLES)
ESPACE F I B R ~ P R I N C I P A L
Soit b un espnce fibre principal de groupe G, et soit E l'algebre des' formes differentielles de l'espace 6 . 11 existe une c o ~ ~ n e z i ofn, d'oh un homomorphisme 7 de l'algkbre de Weil W (GI dans E. Les ClBments basiques de W(G) sont transformes par 7 dans des Clements basiques de E, c'est-&-dire des Bldments de I'algbbre B des formes differentielles de I'espace de base (A. Or les Clements basiques de W (G) ne sont autres que les 616ments invariants de S (G) . Nous noterons I, (G) la sous-nlgbbre de ces SlPments invariants; elle s'identifie & l'alghbre des formes mnltilin6aires symetriques sur a(G) , invariantes par le groupe adjoint; on n'oubliera pas clue les elements de Ips(G) sont de degre' 3 p. Dnns la seconde conference, nous Ctudierons In structure de l'algkbre I,(G). La formule (14) montre ceci : pour qu'un element de S(G) soit un coc~clede \V(G) (c'est-&-dire pour clue le 6 de cet Clement soit nul), il faut et il suffit qu'il soit invuriant. Re\-enant A l'l~omomorphi~me f , il applique les Clements
Notions d'algebre diffbrentielle
de I,(G), qui sont des cocyles de W (G), dans des cocycles de B, c'est-H-dire des cocycles de l'espace de base. On les appelle les cocycles caracte'ristiques de la connezion; ils sont de degrks pairs. 11s forment une sous-alghbre du centre de B, appelke la sous-alghbre caradhristique de la connexion. Si on a un G-homomorphisme d'un espace fibre principal 6l (de g o u p e G) dans l'espace fibre 6, et qu'on envisage la connexion f' qu'il dkfinit (image rkciproque de la connexion !), l'homomorphisme 7 de W (G) dans E' est 6videmment compose de et de l'homomorphisme de E dans E' (dkfini par I'application de 6' dans 6).Donc l'homomorphisme Is (G) +B' est composk de Is (G) 4 B et de l'homomorphisme B +B'. Passons maintenant des cocycles de B H leurs classes de cohomologie, elements de l'alghbre de cohomologie H (B) . La wnnexion f dkfinit un hornomorphisme de Is(G) dans H(B), qui applique IPs(G) duns HW(B). Cet homomorphisme, introduit par A. Weil, joue un r61e fondamental; on verra (2confkrence) qu'il est indipendant du choiz de la connezion. C'est donc un invariant de la structure fibrie de l'espace 6. L'image de I,(G'I par cet homomorphisme est une sous-alghbre de I'alggbre de whomologie H(B) de l'espace de base, appelbe la sous-alg6bre caract6ristique de la structure fibrke; ses elements sont les classes caractkristiqws de la structure fibrke; elks sont de degr6s pairs. Si on a u11 G-homomorphisme d'un espace fibre principal & I , de m&me groupe G, dans l'espace E , l'homomorphisme H (B) +H (Bt) qu'il definit applique la sous-alghbre canct6ristique de H(B) sur la sous-alghbre canct6rktique de H (B') . 8. L'AI.G~BRE DE WEIL COMME ALGBBRE UNn'ERSELLE
On snit (') que si un espace fibre principal 6 , de groupe G, est tel que ses groupes d'homotopie ~ ~ (soient 6 ) nuls pour 0 ,< i ,
<
( I ) Voir par exemple STEE.\ROD, Annals of Math., G, 1944, pp. 294311, pour lc cas ou G est le group orthogonal.
203
H(B) -+ H(B1) est u~li\oqt~ernent cli.tern1inC Imr lit structt~re fibrCe de 6'. Un tel espnce 6 est [lit classijiccrtt poro la dit18c.rlsion N . Pnr esemple, si G est le groupe orthogonnl. on ronnait des espaces classifinnts pour des dimensions nrbit~xiremellt grandes (mais un meme cspnce n'esl [,as clnssifiant lmtlr toutes les dimensions). Leu13 Ihises sont cles gmss~nnnniennes (rkelles) . Revenant & I'alggbre de Keil 11-(G1 . on \oit cllt'elle se comporte, du point de vue Ilomolopique, conlrne itne olgbbre universelie pour les espaces fibrCz de prorlpe G , c'est-&-dire comme une algbbre de cochntnes d'un espnce fibr6 qui senit classifiant pour tous les espnces fibres de prouye G, cluelle que soit la dimension de leur espnce de base. L'algbbre Is(G) joue le r6le de l'alghbre des cochaines de I'espilce de base d'un tel espace fibre universel, nvec la pitrticularitf clue les 6lCments de I,(G) sont tous des cocycles. L'homomorpl~ismede W (G) dans E', d6fini par une connesion dnns l'algbbre C tles cochaines de I'espnce 6',joue le r81e que jonnit I'homon~orphisme E +E' defini par un G-homomorphisn~ed'un espace I,(G) +B' classifiant G dans l'espnce 6'; l'I~omomorpl~isrne joue le rale que jouait l'homornorphisme B +B'; enfin, 1.110momorphisme (unique) I, ( G ) +H (BO joite le r81e que jouait l'homomorphisme (uniquej H (B) --t H(B1). En fait, on verra, dnns In deusigme conf6rence ( 5 3 ,que si, G &ant cornpact (connese), I'espace & est clnssifinnt pollr la dimension N, alors Hm(R)est nu1 pour les 1 1 1 irrtpctirs ,< N , et I'hornomorphisme cnnonicpe I, (G'I --t H CB'I applique bztan ivoquement I1',(G) sur HZp(B pour 2 p \< N. Ceri donnern une preuve, a priori, du fnit que les espnceg de cohomolopie dec bases de deus espnces classifiiults pour In dimer~siorlK sont isomorphec pour torts Ies depres N.,
<
La transgression dans m groupe de Lie et dans nn espace fibi-6 principal Colloque deTopologie, CB.RM, Bmxelles 57- 71 (1950)
Les notations de la premiere conference (I) sont conservies. En particulier, I,(G) continue & designer l'algebre des 816ments invariants de S(G), c'est-&-dire des Blkments basiques de l'algebre de Weil W (G) A (G) @ S (G) . De plus, nous introduirons les notations suivantes : I,(G) pour l'alghbre des Blements invariants de A(G), et Iw(G) pour l'algkbre des 616ments. invariants de W (G) . Ce sont des algebres diffkrentielles gradukes (l'operateur diffkrentiel btant induit, pour I,(G), par celui de I'alghbre ambiante A (G) , et, pour I, (G) , par celui de I'alghbre ambiante I, (G) ) . En fait, l'opbrateur diffirentiel de I,(G) est nul, en vertu de la formule (IV) de la premiere conference ( 5 3). On notera HA(G) I'algebre de cohomologie de A (G) , qui, lorsque G est un groupe compact (connexe), s'identifie i l'alghbre de cohomologie reelle de l'espace compact G. Puisque les Clhments de I,(G) sont des cocycles, on a un homomorphisme canonique I,(G)+H,(G). I1 est bien connu que, lorsque l'algebre de Lie a(G) est re'ductive (c'est-&-dire composee directe d'une alghbre aulienne et d'une alghbre semisimple) , l'homomorphisme I, (G)+ HA(G) est une application biunivoque de I,(G) sur H,(G) . On en trouvera une demonstration dans la t h b e de Koszul ('). Ceci vaut notamment lorsque G est u n groupe compact.
En relation avec le caractere universe1 de l'algebre de U'eil (1" confkrence, 5 a), on a le thkoreme suivant : (I)
Page 15 de ce RecueiI.
Bull. Soc. Math. de France, 1950, pp. 63-127; voir It thhr6me 9.4 du chapitre IV. (')
La transgression dans un groupe de Lie
THEORBME 1. - L'algLbre de cohomologie de W(G) est triviale : Hm(W(G)) est nu1 pour tout entier m >/ 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.
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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.
<
Ln thkorie de Hirsch (I), mise n u point par Eoszt~l('), peut s'appliquer B l'alglrbre C et conduit aux rCsultats suivants . Supposons que G soit un proupe cornpact (connexe) ; ou que, E Ctant de dimension finie, I'alphbre a (G) soit re'dnrfive. Alors il existe une application linkaire r,, biunivoque (mnis non dkterminke de manilrre unique) de l'espnce vectoriel gra-
<
8. REDUCTION DU (') Comptes rendtrs, 227. 1948. p. 132s. (') Dans un travail non rnrorc pubtii..
<
GROUPE STRUCTUR.4L
Soit g un sous-groupe ferme, connexe, du groupe cprnpact 6 ;ou encore, dans un cadre purement algebrique, suppo-
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Henri Cartan
215
La transgression dans un groupe de Lie
sons que E soit de dimension finie, que l'algbbre de Lie MG) e t sa sous-algbbre a(g) scrent rkductives. On notem BG ce qui Btait not6 B nuparavant : BG sera la sous-alebre de E, form& des Plements annul& par les i(z) et les 0(z) relatifs aux Cldments z E a(G). De mGme, B, designera la sous-algbbre de E form6e des BlBments annul& par les i(z) et les B(z) relatifs a u x z f a(g) . On notera CG (resp. C,) la sous-alghbre des 816ments G-invariants de E B S (G) (resp. des 8lCments g-invariants de E B S (g) ) . L'homomorphisme nature1 de S (G)dans S(g) ddfinit un homomorphisme de EBS(G) dans E @ S(g) , qui applique CG dans C, et est compatible avec les diff8rentielles; d'oh un homomorphisme H(CG)+ H(C,), qui s'identifie i l'homomorphisme H(BG)+ H(B,) quand on identifie H(CG) B H(B,) et H(C,) B H (B,). Cela ktant, remontons B la definition d'un op6rateur cobord 8, dans l'espace vectoriel H (E) @ I,(G), obtenu h partir d'une application (s de H (E) O3 I,(G) dans C,, qui satisfasse aux conditions 1, 2 et 3 du 5 7. Cette application est determinee quand on connait sa restriction ?, au sous-espace H(E) 031. Composons (so avec I'application canonique de C , dans C,; on obtient une application de H(E)@1 dans C,, qui, prolongee B H (E) B I, (g) , donne une application satisfaisant aussi aux conditions 1,2 et 3. D'oh u n opdrateur cobord 6, dnns H(E) 81,(g), pour lequel I'espace de cohomologie de H (E)@Is(g) s'identifie i l'espace H (B,) . Finalement : l'application canonique H(E1 @Is(G)+ H(E) @Is(g) (d6finie par I'application I, (G)+ I, (g) ddduite de S (G)+ S(g)) associe i tout opCrateur cobord 8, de H (E)@I,(G) (obtenu par le procCdC du $ 7) un o*rateur cobord 6, dans H (E) @I, (g) , lui aussi obtenu par le proc6d6 du 7. Pour ddfinir 6, sur H (E) @ I ,on effectue successivement E, puis I'application canonique d e H (E) @I, (G) dans H (E) B I, (g) .
+,
))
+
Application. - Soit & u n espace fibre principal dont le groupe (3 est compact et connexe; soit g u n sous-groupe fermd, connexe, de 6 -Notons 6, I'espace quotient de 6 par G, et 03, l'espace quotient de 6 par g. L'applimtion canonique H(LRG)--t H ((A,) est donnCe par H(H(E) @Is(G))+ H(H(E)@Is(g)) . Lorsque g a mdme rang que G, l'application 1, (6)- l,(g) est biunivoque (voir ci-dessous). I1 en rhsulte t r h facilement que I'application H ( a G ) + H ((A,) est biunivoque. On retrouve ainsi, au moins dans le cas des espaces fibds inddfiniment diffkrentiables, un rCsultat de J. Leray ( I ) .
Nous allons appliquer la theorie prkckdente B la recherche de l'algbbre de cohomologie (rkelle) d'un espace homoghne Glg, lorsque G est un groupe compact connexe et g un sousp u p e fermk connexe de G . Cette question a dej; fait I'objet d'importants travaw de Samelson, Leray et Koszul ('). Nous nous placerons ici dans le cadre algebrique suivant : a(G) sera una algkbre de Lie rkductive, et a ( g ) une sous-algkbre rkductive duns a(G) (') . E sera alors l'nlgkbre A(G), dans laquelle ofirent les i(z) et les 0 (x) relatifs aux z E a(g) . La sous-algkbre B des a 616ments basiques de E s'identifie, lorsque G est un groupe compact connexe et g u n sous-groupe fermk connexe, B l'algebre des formes diffhrentielles de I'espace homogkne Glg, invariantes h mauche par G; et l'on sait que son algebre de cohomologie ?identifie B l'algkbre de cohomologie H(G/g) de l'espace homogi?ne. C'est pourquoi nous noterons desormais H (GIg) I'alghbre de cohomologie H (B) . D'autre part, l'alghbre de cohomologie H(G) de l'espace fibr8 G s'identifie canoniquement B la sous-algkbre I,(G) des elements G-invariants de A (G) . Le theoreme 4 est applicable, pnrce qu'il existe une connexion dnns. E = h(G) : une application linkire f de A1(g) clans A'(.GI, compatible avec les i ( z ) et les 0(x) relatifs nux z E a ( g ) . Une telle connexion est dkfinie par un projecteur de !'algkbre de Lie a(G) sur la sous-nlgbbre a ( g ) , projecteur qui soit compatible nvec les B(z) relatifs nux z E a ( g ) ; ou encore, par un sous-espnce vectoriel de a ( G ) , supplCmentaire de a ( g ) , et stable par les transformations infinitbsimales de a ( g ) . L'existence d'un tel sous-espnce r6sulte de I'hppoth6se suiunnt Incluelle 4( 9) est reductive dans a(G) . .Applicluons le theoreme 4 : H(G/g) s'identifie cnnoniquement i I'alghbre de cohomologie de la sous-algkbre C , des Clements g-invariants de I'algkbre A(G) €3S(g), munie de I'opdmteur A, explicit6 nu theoreme 4. Mais ici, non seulement la thkorie de Hirsch-Koszul est applicable comme au 5 7, mais (cf. la conference de Koszul) on peut astreindre l'application z (111 s 7 i Ptre comp;rtihle avec les structures rnultiplicatives. D'une faqon precise, I'nlg6bre H(E)@I,(G) envisagee au S 7 devient ici I,(G) @ I , ( g ) ; et va Ptre un isomorphisme de cette ((
,)
.
1)ibliographiques dans la thkr dc hoszu~;voir en I.rm4r aus Cornpfcs rendus, f . 228, 1949, p. 1902, e t t. !22!3; 1949, p. 280. + ( 9 Pnr~rc r l t r notion, \air lhke de Koszm, S 9. (')
RBfbrencrs
oot1.r les Sotrs de
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Henri Cartan
La transgression dans .un groupe de Lie
alghbre sur une sous-alebre de l'alggbre C,. Pour dCfinir F, on remarque que I, (G) est l'alggbre extCrieure du sous-espace P,(G) de ses 616ments primitifs; par suite sera dCtermin6 quand on le connaitra sur le sous-espace P,(G) 81 de 1, (G) 8 Is (g) . Pour dCfinir p sur oe sous-espace de manihre i satisfaire aux conditions 1, 2 et 3 du 5 7, on observe que tout Cltment de P,(G) est transgressif dans l'atglbre C, (voir ci-dessous), de sorte qu'on d6finira p en associant, tout 616ment a @ l de P,(G) 81, une cochalne de transgression dans l'alghbre C, des kl6ments g-invariants de A (G) @ S (g) . Dkmontrons que tout ClCment de P,(G) est transgressij dans C,, comme il a Ct6 annonc8. On va, pour cela, se servir I? nouveau de la transgression dans l'alghbre de Weil de a iG) : consid6rons I'homomorphisme canonique A de A(G)@S(G) dans A(G) 8s (g), dCfini par S (G)+ S(g). Il applique la sous-alghbre I, (G) des ClCments G-invariants de A (G) @ S (G)= W(G) dans la sous-alggbre C, des Clements g-invariants de A(G) BS(g); en outre, il est compatible avec les op6rateurs diffhrentiels A, et A, de I,(G) et C, respectivement (cf. la remarque de la fin du 6). Soit alors a un ClCment de P,(G) ; il est transgressif dans I,(G) ; si w est line cochaine de transgression de a @ 1 dans I, (G) , l ( w ) sera, dans C,, une cochafne de transgression pour a @ l , puisque sa diffkrentielle A,A(w) sera &ale B L'image, par l'application I,(G)--t I,(g), de & ( w ) . qui est un dl6ment de I,(G). Ceci prouve en outre que, pour obtenir une-transgression P,(G)+ I,(g) dans I'alggbre C,, il suffit de prendre une transgression PA(G)+ I, (G) (cf. s 3) , et de la composer avec I'homomorphisme canonique I, (G)+ I, (g) . RCsumons les rCsu1tats obtenus : TH~OREME 5. - Choisissons une transgression .;: PA(G)+ I,(G), et composo~zs-laavec l'homomorphisme canonique de Is(G) duns I,(g). On obtient une application lindaire de PA(Gj duns Is(g), qu'on prolonge en une antidkrivation de I, (G) B I, (g) , nulle sur I, (g) . Cette antidkrivation, dc degrP 1, est une diffirentielle 6 sur l'algtbre graduke I, (G) B ls( 9 ); l'algebre de cohornologie de I, (G) @ I, (9) , pour 8, est isomorphe & l'algdbre de cohomologie H(G/g) de I'espace homogtne G/g, par un isomorphisme compatible avcc lcs hotnomorphismes d u diagramme
+
H (1, (G)@ Is (9))
7
t
I \ 1/'
1s (9)
H (Gig)
217
(Dans ce diagramme, l'homomorphisme Is(g)+
H(IA(G) @Is(g))
est celui obtenu en considerant Is(g) comme sous-dglbre de 1, (GI 81, (g) , tandis que I'homomorphisme
H (IA(G) 8Is(.g) )--tIA(G! est celui obtenu en considerant IA(G) comme algkbre quotient de I, (G) 8I,(g) par I'idCal engendre par I,+ (g) .) Corollaire. - L'alggbre de cohomologie H(G/g) et les homomorphismes is(g)+ H (GI g)+ H (G) sont entihrement determinks par la connaissance de l'homomorphisme Is(G)-+ I, (g) , qui mnctCrise ainsi la cc position hornologique I) du sous-groupe g dans le groupe G- Signalons les deux cas extrgmes : lo Celui oh I'homomorphisme IS(G)+ I,(g) applique I, (G) sur I, (g) : c'est le cas, clnssique, oh g est non homologue 5 z6ro dans G; 2' Celui oh l'homomorphisme I, (G)+ I, (g) est hiunivocjue : c'est le cas oh les rangs r ( g ) et r(G) sont kgaux. Dans tous les a s , le theorgme 5 montre que I'alg6bre de cohomologie H(G/g) est justiciable de la thCorie de Iioszul concernant 1' I, homologie des S-algtbres (voir la confkrence de Koszul) : H(G/g). s'identifie B I ' lc algbbre d'homologie de la S-alghbre Is(g) I ) , S designant ici l'algbbre Is(G) , el la structure de S-alg6bre de I,(g) Ctnnt definie par I'homomorphisme de I,(Gi dans Is(g). I)
10. QIEI.QI.ESRESCLTATS CO.\CERlA.VT L A DES ESPACES HO~IOC.BSES
COHOMOLOCIE
Signnlons, s n s dernonstmtiou. une. &tie de rCsultats qui se dMuisent nssez fncilement de ce clui pr4c6de. Nous bcrirons, pour ahrCger, I(G) et I(g) nu lieu tle I,(G) et I,(g). On notera I+(G) I'idPn1 de I (G) form6 des ClCments de depre 0; dkfjnition analogue de I + ( g ) . Enfin, on Ccrira P(G\ su lieu de PA(G). Ecrivons la suite des homomorphismes canoniques
>
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~ ))
<
<
.
f'iThi-bri.n~t. C
.
I ~ I ~ I (l(s I ~ IS,~li~.sn\. . l t t n .
of .lJotl~..42. 1941. Satx V.
219
oricntds de dimension n dans l'espace numkrique de dimension N n ) . Posons Q (t, P) =idk
+
II
Alors le polyndme de Poincar6 de la ,wssmnnienne G,. si n et
N sont pairs :
n+N
- F+" (1 - tn) (I - tS) Q ( t , nT - 1) Q 1
N
- 1)
si n est pair et N impair :
+ NI- I) 1
Q (t,
N-1 si n kt N sont impairs : (1 + t"CS-1 )
I)
n+N ~ ( t T, Q(,&$)Q(~.~).
N -1
est :
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Index
,
G* algebra, 17 G* morphism, 18 G* module, 17 d G , 45, 55
connection elements, 24 connection forms, 23 curvature elements, 24 curvature forms, 24
basic cohomology, 26 basic elements, 26 basic subcomplex, 26, 54 Berezin integral, 78 Berezin integration, 77 Bianchi identity, 36, 38 Borel's theorem, 125
Delzant space$, 126 derivation, 14 derivation, even, 10 derivation, odd, 10 double complex, 61
canonical equivariant three-form, 135 Cartan model, 45 Cartan operator, 45, 55 Cartan structure equation, 36 Cartan structure equations, 38 Cartan's formula, 58 Cartan's theorem, 45 chain homotopy, 20 chain homotopy relative to a morphism, 22 Characteristic classes, 95 characteristic classes, 49 . characteristic homomorphism, 38, 48, 50 Chern classes, 97 Chern-Weil map, 38, 48 classifying bundle, 4 classifying space, 4 coadjoint orbits, 114 cohomology of flag varieties, 125 cohomology of toric varieties, 126 commutative superalgebra, 13 commutator, 13
Equivariant Characteristic Classes, 104 equivariant characteristic classes, 40, 50 equivariant Chern classes, 104 Equivariant Duistermaat-Heckman, 133 equivariant Euler class, 104 Equivariant Marsden-Weinstein, 132 equivariant Pontryagin classes, 104 equivariant symplectic form, 112, 132 Euler class, 98 Fermionic integral, 78 Fermionic integration, 77 flag manifolds, 124 Gaussian integrals, 79 Hamiltonian homogeneous spaces, 114 horizontal bundle, 23 horizontal form, 23 Kirillov-Kostant-Souriau theorem, 114
228
Index
Koszul complex, 33 Kuiper, 6 Lie derivative, 10 Lie superalgebra, 13 Lie superalgebra, 3, 14 locally free action, 23 Marsden-Weinstein reduction, 118 Marsden-Weinstein theorem, 118 Mathai-Quillen isomorphism, 42 minimal coupling form, 117 moment map, 112 Pfaffian, 97 Pontryagin classes, 97
structure constants, 9 summation convention, 9 super Fourier transform, 90 super Jacobi identity, 13 superalgebra, 12 supercommutator, 13 supervector space, 11 symplectic reduction, 119 tensor product of superalgebras, 15 The Weil algebra, 34 total Chern class, 101 total complex, 61 type (C), 24 universal Thom form, 86, 88
Quillen's law, 12
vector field corresponding to
splitting manifold, 107 splitting principle, 74, 106
Weil identities, 11 Weil model, 45
6, 10
