Lectures on Chern-Weil Theory and Witten Deformations

LECTURES ON CHERN-WE11 THEORY AND

WITTEN DEFORMATIONS

Nankai Tracts in Mathematics - Vol. 4

LECTURES ON CHERN-WEN THEORY AND

WITTEN DEFORMATIONS

Weiping Zhang Nankai Institute of Mathematics Tionjin, f R China

World Scientific ewJersey*London *Singapore *Hang Kong

Published by

World Scientific Publishing Co. Pte. Ltd. P 0 Box 128, Faner Road, Singapore 912805 USA ujj5ce: Suite IB, 1060 Main Street, River Edge, NJ 07661

UK ofice; 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

LECTURES ON CHERN-WEIL THEORY AND WITTEN DEFORMATIONS Copyright 0 2001 by World Scientific Publishing Co. Re. Ltd.

All rights reserved. This book, or parts thereoJ may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-4685-4 ISBN 981-02-4686-2 (pbk)

Printed in Singapore.

Dedicated t o my teachers Jean-Michel Bismut and Shiing-Shen Chern

Preface

These lecture notes are based on the notes of a graduate course of differential geometry I taught a t the Nankai Institute of Mathematics. It consists of two parts: the first geometric part contains an introduction t o the geometric theory of characteristic classes due t o Shiing-shen Chern and Andre Weil, as well as a proof of the Gauss-Bonnet-Chern theorem based on the Mathai-Quillen construction of Thom forms; while the second part, which is analytic in nature, contains analytic proofs of the Poincark-Hopf index formula as well as the Morse inequalities based on deformations introduced by Edward Witten. We hope this book can serve as a text book to cover materials not generally contained in an introductory course in differential geometry. With this reason, we have not tried hard to make this book being completely self-contained. However, we will give detailed references when (possibly) nonstandard results will be quoted. On the other hand, we have tried to make each chapter in the text to be relatively independent from the other chapters. As a result, we will list the references of each chapter a t the end of that chapter. We will work in smooth (i.e. C") category throughout this book. I would like t o thank Dr. Huitao Feng for taking the preliminary notes for my lectures. Part of these notes were prepared during a short visit t o the Institute of Mathematics of Fudan University in May, 2000, and during my visit t o the Department of Mathematics of MIT for the Spring semester of 2001. I would like t o thank Professor Jiaxing Hong of F'udan University for kind hospitality. I am also grateful to Professors Richard Melrose and Gang Tian for arranging my visit t o MIT, and to MIT for financial support. vii

viii

Preface

Finally, I would like to thank the Ministry of Education and the National Natural Science Foundation of China for their support during the writing of this book.

Contents

Preface

vii

Chapter 1 Chern-Weil Theory for Characteristic Classes 1.1 Review of the de Rham Cohomology Theory . . . . . . . . . . 1.2 Connections on Vector Bundles . . . . . . . . . . . . . . . . . . 1.3 The Curvature of a Connection . . . . . . . . . . . . . . . . . . 1.4 Chern-Weil Theorem . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Characteristic Forms, Classes and Numbers . . . . . . . . . . . 1.6 SomeExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Chern Forms and Classes . . . . . . . . . . . . . . . . . 1.6.2 Pontrjagin Classes for Real Vector Bundles . . . . . . . 1.6.3 Hirzebruch’s L-class and A-class . . . . . . . . . . . . . 1.6.4 K-groups and the Chern Character . . . . . . . . . . . . 1.6.5 The Chern-Simons Transgressed Form . . . . . . . . . . 1.7 Bott Vanishing Theorem for Foliations . . . . . . . . . . . . . . 1.7.1 Foliations and the Bott Vanishing Theorem . . . . . . . 1.7.2 Adiabatic Limit and the Bott Connection . . . . . . . . 1.8 Chern-Weil Theory in Odd Dimension . . . . . . . . . . . . . . 1.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 4 6 8 10 10 11 12 14 16 17 18 20 22 26

Chapter 2 Bott and Duistermaat-Heckman Formulas 2.1 Berline-Vergne Localization Formula . . . . . . . . . . . . . . . 2.2 Bott Residue Formula . . . . . . . . . . . . . . . . . . . . . . . 2.3 Duistermaat-Heckman Formula . . . . . . . . . . . . . . . . . . 2.4 Bott’s Original Idea . . . . . . . . . . . . . . . . . . . . . . . .

29 29 35 37 38

A

ix

X

2.5 References

Contents

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

39

Chapter 3 Gauss-Bonnet-Chern Theorem 3.1 A Toy Model and the Berezin Integral . . . . . . . . . . . . . . 3.2 Mathai-Quillen’s Thorn Form . . . . . . . . . . . . . . . . . . . 3.3 A Transgression Formula . . . . . . . . . . . . . . . . . . . . . 3.4 Proof of the Gauss-Bonnet-Chern Theorem . . . . . . . . . . . 3.5 Some Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Chern’s Original Proof . . . . . . . . . . . . . . . . . . . . . . . 3.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 43 46 47 50 51 54

Chapter 4 PoincarB-Hopf Index Formula: an Analytic Proof 4.1 Review of Hodge Theorem . . . . . . . . . . . . . . . . . . . . . 4.2 Poincark-Hopf Index Formula . . . . . . . . . . . . . . . . . . . 4.3 Clifford Actions and the Witten Deformation . . . . . . . . . . 4.4 An Estimate Outside of UpEzero(V~Up. . . . . . . . . . . . . . . 4.5 Harmonic Oscillators on Euclidean Spaces . . . . . . . . . . . . 4.6 A Proof of the Poincark-Hopf Index Formula . . . . . . . . . . 4.7 Some Estimates for DT.i’S, 2 5 i 5 4 . . . . . . . . . . . . . . . 4.8 An Alternate Analytic Proof . . . . . . . . . . . . . . . . . . . 4.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 57 60 61 63 64 67 69 73 74

Chapter 5 Morse Inequalities: an Analytic Proof 5.1 Review of Morse Inequalities . . . . . . . . . . . . . . . . . . . 5.2 Witten Deformation . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Hodge Theorem for (R*(M).d ~ f . ). . . . . . . . . . . . . . . . 5.4 Behaviour of U T j Near the Critical Points of f . . . . . . . . . 5.5 Proof of Morse Inequalities . . . . . . . . . . . . . . . . . . . . 5.6 Proof of Proposition 5.5 . . . . . . . . . . . . . . . . . . . . . . 5.7 Some Remarks and Comments . . . . . . . . . . . . . . . . . . 5.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 77 78 79 81 83 88 89

Chapter 6 Thom-Smale and Witten Complexes 93 6.1 The Thorn-Smale Complex . . . . . . . . . . . . . . . . . . . . 93 6.2 The de Rham Map for Thom-Smale Complexes . . . . . . . . . 95 6.3 Witten’s Instanton Complex and the Map eT . . . . . . . . . . 97 6.4 The Map P,. TeT . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.5 An Analytic Proof of Theorem 6.4 . . . . . . . . . . . . . . . . 102

Contents

6.6 References

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

xi

102

Chapter 7 A t i y a h Theorem on Kervaire Semi-characteristic105 7.1 Kervaire Semi-characteristic . . . . . . . . . . . . . . . . . . . . 106 7.2 Atiyah’s Original Proof . . . . . . . . . . . . . . . . . . . . . . 107 7.3 A proof via Witten Deformation . . . . . . . . . . . . . . . . . 108 7.4 A Generic Counting Formula for k ( M ) . . . . . . . . . . . . . . 112 7.5 Non-multiplicativity of k ( M ) . . . . . . . . . . . . . . . . . . . 113 7.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Index

117

Chapter 1

Chern-Weil Theory for Characteristic Classes

The theory of characteristic classes of vector bundles over smooth manifolds plays important roles in topology and geometry. The book of Milnor and Stasheff [MS] contains a beautiful introduction to the topological aspects of this theory. This chapter contains an introduction to the geometric aspects of this theory, which was developed by Shiing-shen Chern and Andre Weil.

1.1

Review of the de Rham Cohomology Theory

This section contains a brief review of the de Rham cohomology theory. For more details, we recommend the standard book of Bott and Tu [BoT]. Let M be a smooth closed manifold. Let T M (resp. T * M ) denote the tangent (resp. cotangent) vector bundle of M . We denote by A*(T*M)the (complex) exterior algebra bundle of T * M , and

the space of smooth sections of A * ( T * M ) .In particular, for any integer p such that 0 5 p 5 dim M , we denote by

the space of smooth p-forms over M . Let

d : R*(M)

-

1

R*(M)

Chern- Weal Theory for Characteristic Classes

2

denote the exterior differential operator. Then d maps a pform t o a ( p + 1)form. Furthermore, there holds the following important formula,

d2 = 0.

(1.1)

We adopt the convention that both R-l(M) and RdimM+l(M)are spaces of zero. From (l.l),one finds that for any integer p such that 0 5 p 5 dim M , one has

d R p ( M ) c kerdlnp+1(M), which leads to the definition of the de Rham complex as well as its associated cohomology: de Rham cohomology.

Definition 1.1 The de Rham complex (R*(M),d ) is the complex defined by

O-RO(M)

-5R'(M) 5.. . -41, Rd'"M(M)

--+ 0.

Definition 1.2 For any integer p such that 0 5 p 5 d i m M , the p t h de Rham cohomology of M (with complex coefficient) is defined by

IqR(M;C ) =

ker dlC2p(M)

dRp- ( M ) '

The (total) de Rham cohomology of M is then defined as dim M

H & ( M ; c)=

@ H:R(M; c). p=o

F'rom the definition of the de Rham cohomology, one sees that any closed differential form w on M , that is, any element w E R*(M) such that dw = 0, determines a cohomology class [w] E H&(M, C ) . Moreover, two closed differential forms w , w' on M determine the same cohomology class if and only if there exists a differential form r] such that w - w' = dr]. If w , w' are two closed differential forms on M and a is a constant function on M , then one verifies easily that the following identities in H& ( M ; C ) hold,

[awl = a [ w ] ,

[w

+ w'] = [w] + [w'].

Connections o n Vector Bundles

3

Moreover, for any two differential forms rl, 7’ on M , one verifies that (W

+ d q ) A (w’+ dv’)

=w A

+

+

w’ d (7 A w’ rl A drl’

+ (-l)degww A rl’) .

Thus the cohomology class [wAw’]depends only on [w] and [w’]. We denote it by [w][w’] and call it the product of [w] and [w’]. If wl’is a third closed differential form on M , then one can verify that

+

+

([w] [w’]) * [w”] = [w] . [w”] [w’] . [W”]. From the above discussion, one sees that the de Rham cohomology of

M carries a natural ring structure. The importance of the de Rham cohomology lies in the de Rham theorem which we state as follows, and which we refer to the book [BoT]for a proof.

Theorem 1.3 If M is a smooth closed orientable manifold, then for any integer p such that 0 5 p 5 dim M, (i) dim H & ( M ; C) < +m; (ii) H & ( M ; C ) is canonically isomorphic t o H&,,(M; C ) , the p - t h singular cohomology of M .

1.2

Connections on Vector Bundles

We still refer to the book [BoT] for the basic theory of vector bundles over smooth manifolds. Let E --+ M be a smooth complex vector bundle over a smooth compact manifold M . We denote by W ( M ;E ) the space of smooth sections of the tensor product vector bundle R * ( T * M )@ E obtained from R*(T*M)and

E. f l * ( M ;E ) := r ( R * ( T * M )8 E ) . A connection on E may be thought of, in some sense, as an extension of the exterior differential operator d t o include the coefficient E. Definition 1.4 A connection V E on E is a C-linear operator V E : r ( E ) + fll(M; E ) such that for any f E C”(M), X E r ( E ) ,the following

Chern- Weal Theory for Characterastic Classes

4

Leibniz rule holds,

V E ( f X ) = (df)X + fVEX. The existence of a connection on a vector bundle can be proved easily by using the method of partitions of unity. Certainly, there are a lot of connections on a vector bundle if one does not impose further conditions (In fact they form an infinite dimensional affine space). In many cases it is important in geometry to find and study connections verifying various specific geometric conditions. If X E r ( T M )is a smooth section of T M , then a connection VE induces canonically a map

05 : r(E)-+

r(E)

via the contraction between TIM and T * M . We call it the covariant derivative of VE along X. Just like the exterior differential operator d, a connection V E can be extended canonically to a map, which we still denote by V E ,

VE : R*(M;E )

-

R*+I(M;E )

such that for any w E R*(M),X E I'(E),

V E : wX 1.3

H

(dw)X

+ ( - l ) d e g w ~A V E X .

The Curvature of a Connection

The importance of the concept of a connection lies in its curvature.

Definition 1.5 The curvature RE of a connection V Eis defined by

RE = VEoVE : r ( E )-+ R2(M;E ) , which, for brevity, we will write RE = (VE)'. The following property of curvature is of critical importance.

(1.2)

The Curvature of a Connection

5

Proposition 1.6 The curvature RE is Cm(M)-linear. That is, for any f E C m ( M ) and X E I'(E), one has

R E ( f X )= f R E X . Proof. By using (1.1) and (1.2) , one deduces that

+

R E ( f X )= V E( ( d f ) X f V E X )

= (-l)degdfdfA

V E X + df A V E X + f (VE)' x =f

REX.

0

Let End(E) denote the vector bundle over M formed by the fiberwise endomorphisms of E . From Proposition 1.6, one sees that RE may be thought of as an element of I'(End(E)) with coefficients in R2(M). In other words,

RE E R2(M;End(E)). To give a more precise formula, if X , Y E r ( T M ) are two smooth sections of T M , then RE(X,Y ) is an element in r ( E n d ( E ) ) given by

R E ( X , Y )= VgVF

-

VFVg - VL,,],

(1.3)

where [ X , Y ]E I ' ( T M ) is the Lie bracket of X and Y defined by the formula that for any f E C m ( M ) ,

[ X ,Y ] f= X ( Y f )- Y ( Xf) E c - ( M ) . Finally, in view of the composition of the endomorphisms, one sees that for any integer k 2 0, k

*k

( R E ) = REo...oRE: r ( E )

-

R2'"(M;E )

is a well-defined element lying in OZk( M ;End(E ) ) ,

Chern- Weil Theory for Characteristic Classes

6

1.4

Chern-Weil Theorem

We continue the discussion in the above section. For any smooth section A of the bundle of endomorphisms, End(E), the fiberwise trace of A forms a smooth function on M . We denote this function by tr[A]. This further induces the map tr : R*(M;End(E))

-

R*(M)

such that for any w E R*(M) and A E r ( E n d ( E ) ) , t r : wA H wtr[A]. We still call it the function of trace. We also extend the Lie bracket operation on End(E) t o R*(M;End(E)) as follows: if w , q E R*(M) and A, B E r ( E n d ( E ) ) , then we use the convention that [wA,qB] = (wA)(qB) - ( - l ) ( d e g w ) ( d e g q ) ( ~ B ) ( ~ A )(1.4) . The following vanishing result is then obvious.

Lemma 1.7 For any A, B E R*(M;End(E)), the trace of [A, B ] vanishes. Lemma 1.8 If V E is a connection o n E , then f o r any A E R*(M;End(E)), one has dtr[A] = t r [[VE,A]].

Proof. First of all, if

(1.5)

vEis another connection on E , then from the

Leibniz rule in the definition of the connection, one verifies that

V E-

eEE R'(M; End(E)).

Thus by Lemma 1.7 one has t r [[VE-

vE,A]]

= 0.

That is t o say, the right hand side of (1.5) does not depend on the choice of VE. On the other hand, it is clear that the operations in the right hand side of (1.5) are local. Thus for any J: E M one can choose a sufficiently small open

Chern- Wed Theorem

7

neighborhood U, of x such that EIu, is a trivial vector bundle. Then one can take a trivial connection on E~,-J,for which (1.5) holds automatically. By combining the above independence and local properties, one sees directly that (1.5) holds on the whole manifold M . 0 Let

f (z) = a0

+ a12 + . . . + a,xn + . . .

be a power series in one variable. Let RE be the curvature of a connection V Eon E . The trace of

f ( R E ) = a0 + a@

+ ..

*

+a, ( R E ) ,

+ ..

*

is an element in a*( M ) . We can now state a form of the Chern-Weil theorem (cf. [ C ] ) as follows.

Theorem 1.9 (i) The form tr[f ( R E ) ]i s closed. That is, d t r [f ( P )= ]0;

vE

(ii) If is another connection o n E and is a differential form w E R*(M) such that tr [f ( R E ) ]- tr [f

EE its curvature,

I)”(

then there

= dw.

F‘rooJ (i) F’rom Lemma 1.8 one verifies directly that dtr

[f ( R E ) ]= tr “vE,f(RE)l]

as for any integer Ic 2 0 one has the obvious Bianchi identity

[VE, ( R E ) k ] = [VE, (VE)2k]= 0. (ii) For any t E [0,1], let Vf be the deformed connection on E given by

V f = (1- t ) V E +ti?.

(1.8)

8

Chern- Weil Theory for Characteristic Classes

Then Vf is a connection on E such that over,

* vE dt

=

-

Of = V E and Vf

=

oE.More-

V E E R1(M;End(E)).

Let R f , t E [O, 11, denote the curvature of V f . We study the change of t r [ f ( R f ) ] when t changes in [0, 11. Let f'(z) be the power series obtained from the derivative of f(z)with respect to z. We deduce that

-dt r [ f ( R F ) ] dt

= t r [--&' dR," (Rf)]

=tr

[- (vf)2

f'

(RF)]

dt

where the last equality follows from the Bianchi identity (1.7). Combining with Lemma 1.8, one then gets d -tr dt

[f (RF)] = d t r

from which one gets

This completes the proof of the part (ii).

1.5

0

Characteristic Forms, Classes and Numbers

By Theorem 1.9(i), t r [ f ( e R E ) ] is a closed differential form which determines a cohomology class [ t r [ f ( e R " ) ] ] E H&.(M; C ) . While (1.6) says that this class does not depend on the choice of the connection V E

Definition 1.10 (i) We call the differential form t r [ f ( g R E ) ] the characteristic form of E associated t o V E and f , and denote it by f (E,V E ) .

Characteristic Forms, Classes and Numbers

9

(ii) We call the cohomology class [tr[f(q R E ) ] ]the characteristic class of E associated to f , and denote it by f ( E ) .

Thus, a characteristic form is a differential form representative of the corresponding characteristic class. We also call a product of characteristic forms (classes) a (new) characteristic form (class). We now assume M is oriented, so that one can integrate differential forms on M . Let E l , . . . ,E k be k complex vector bundles over M , and VE1,. . . , VEk the connections on them respectively. Given k power series fl,. . . ,fk, one can then form the characteristic form

Lemma 1.11 T h e number defined by

=

s,

{fl

(E1,vE1) ' * * f k(Ek,VEk)}max

(1.11)

does n o t independent o n the choices of the connections VEa,1 5 i 5 k. Proof. Without loss of generality we assume that is another connection on El. By Theorem 1.9(ii) there is a differential form w on M such that

vEl

fl

( E l ,VE1) - fl ( E l ,

vE1)dw. =

One then uses Theorem 1.9(i) and the Stokes formula to deduce that fl

(El,VE')

"'fk

(Ek,VEk)

Chern- Weal Theory for Characteristic Classes

10

from which the lemma follows easily. The number defined in (1.11) is called the characteristic number associated to the characteristic class fl(E1) . . . fk(Ek), and is denoted by (fl(El)'*'fk(Ek),

1.6

[MI).

Some Examples

In this section we describe some well-known characteristic classes appearing in many places in geometry and topology. 1.6.1

Chern F o m s and Classes

Let V E be a connection on a complex vector bundle E over a smooth manifold M , and RE the curvature of V E . The (total) Chern form, denoted by c ( E , V E ) ,associated to V E is defined by (1.12) where I is the identity endomorphism of E . Since

in view of the following power series expansion formulas for log(1 exp(z), log(l+z) = z - 22 2 and

+ . . . + (-l)n+lZn n

+...

+ x) and

Some Examples

11

one sees that c ( E , V E )is a characteristic form in the sense of Definition 1.10. The associated characteristic class, denoted by c ( E ) , is called the (total) Chern class of E. By (1.12) it is clear that one has the decomposition of the (total) Chern form that

c ( E , V E )=l+ci(E,VE)+...+Ck(E,VE)+... with each C;

(E, V E )E C12i(M).

We call ci(E,V E )the i-th Chern form associated to V E ,and its associated cohomology class, denoted by ci(E),the i-th Chern class of E. Now if one rewrites (1.13) in the form that

+

-RE)) log (det (I7

= tr [log ( I +

(1.14)

then from the above power series expansion formulas for log(1 +x), one deduces that for any integer k >_ 0, tr[(RE)k]can be written as a linear combination of various products of ci(E,VE)'s. This establishes the fundamental importance of Chern classes in the theory of characteristic classes of complex vector bundles.

1.6.2

Pontrjagin Classes for Real Vector Bundles

Let now E be a real vector bundle over M , and V Ebe a connection on E.* Let RE be the curvature of V E . One sees easily that one can proceed in exactly the same way as in Sections 1.2-1.5 for real vector bundles with connections. Moreover, the Chern-Weil theorem can be formulated and proved in exactly the same way as in Theorem 1.9. Now similar to the Chern forms for complex vector bundles, we define the (total) Pontrjagin form associated to V E by p ( E ,0") = det

( (5)') "') (I -

(1.15)

*The definition of a connection on a real vector bundle is the same as that for a complex vector bundle in Definition 1.4, by simply replaxing 'C-linear' there t o R-linear.

Chern- Weal Theory f o r Characteristic Classes

12

The associated characteristic class p ( E ) is called the (total) Pontrjagin class. Clearly, p ( E ,V E )admits a decomposition

p ( E , V E )=l+pl(E,VE)+...+pk(E,VE) +.*. with each pi ( E ,V E )E n4i(IM).

We call p i ( E , V E )the i-th Pontrjagin form associated t o V E ,and call the associated class pi(E) the i-th Pontrjagin class of E. The discussion a t the end of Subsection 1.6.1 also applies here t o show the fundamental importance of Pontrjagin classes in the theory of characteristic classes of real vector bundles. Finally, if we denote by E @ C the complexification of E , then one has the following intimate relation between the Pontrjagin classes of E and the Chern classes of E @ C ,which is usually taken as the definition of Pontrjagin classes: for any integer i 2 0, p , ( E ) = ( - q i C z i ( E 63 C ) .

1.6.3

(1.16)

Hirzebruch's L-class and A-class

In this subsection we discuss some characteristic classes which are especially important when defined for the tangent bundle of a manifold. These classes were first defined by Hirzebruch (cf. [HI). We start with the L-class, which is associated to the function X

L(z) = tanh (x)* Let V T Mbe a connection on the tangent vector bundle TIM of a smooth closed manifold M . Let RTMbe the curvature of V T M . , by L(TM, V T M )is, defined by The L-form associated to V T Mdenoted L ( T M , V T M ) - det

((

TRTM

tanh G R T "

)) '")

E

n*(M).

(1.17)

Its associated cohomology class, called the L-class of T M , is denbted by

L(TM).

Some Examples

13

As a very special case, if dim M = 4, one has 1

{ L ( T M ,VTM)}max= : p i ( T M ,V T M .)

(1.18)

The importance of the L-class lies in the Hirzebruch Signature theorem (cf. [HI) which says that when M is oriented, then the L-genus of M , denoted by L ( M ) and defined by

L ( M ) := ( L ( T M ) [MI) , =

/

L ( T M ,V T M ,)

M

equals to the Signature+ of M. In particular, L ( M ) is an integer. The integrality of characteristic numbers such as L ( M ) is highly nontrivial. For one more example, we consider the A-class, which is associated to the function h

A(x) h

=

x/2 sinh (2/2)

Let M be as before a smooth compact oriented manifold. Let V T Mbe a connection on T M . Let RTMbe the curvature of V T M . We define

A^(TM,V

) --det

((

fR

G

E

TM

O*(M),

(1.19)

sinh g R T M

and denote the associated cohomology class by A^(TM). As a special case, when dim M = 4, one has

{ A ( T M ,V T M ) } m a x

= --1 pl

24

( T M ,v T M ) .

(1.20)

We define the &genus of M , denoted by A^(M),by

A ( M ) = ( A ^ ( T M ) ,[ M I ) =

/

A ( T M , V T M .)

M

tThe Signature of a manifold is defined as follows: if d i m M = 4m for some integer m, then there is a natural symmetric quadratic form H d g ( M ;R)x Hdg(A4.R) + R defined by ( [ u ][,u ’ ] )+ w A u’. Then the Signature of M is defined t o be the signature of this quadratJ%rm. If dim M is not divisible by 4,then define its Signature to be zero.

14

Chern- Weil Theory for Characteristic Classes

From (1.18) and (1.20), one sees that if d i m M = 4, then

L ( M ) = -8A^(M).

(1.21)

Now by a theorem of Bore1 and Hirzebruch [BH], one knows that if M is spin,$then A^(M) is still an integer. Moreover, when d i m M 3 4 mod 8, Atiyah and Hirzebruch [AH11 refined this result by showing that A^(M)is an even integer. Combining the later with (1.21), one recovers the famous Rokhlin theorem which says that the Signature of a smooth closed spin four manifold is divisible by 16. The proofs in [AH11 and [BH] are purely topological and are indirect. The natural attempt t o search for a more reasonable and direct explanation of these integrality results lead t o the discovery of the celebrated Atiyah-Singer index theorem [AS]. We recommend the two excellent books of Berline-Getzler-Vergne [BGV] and Lawson-Michelsohn [LM] t o the interested reader who wants to know more about index theory (The reader who knows Chinese can also consult the book of Yu [q). On the other hand, there is a higher dimensional generalization of the above mentioned Rokhlin theorem due t o Ochanine [O], which states that the Signature of a smooth closed 8k+4 dimensional spin manifold is divisible by 16. We refer to the article of Liu [Liu] for a modern proof of this result. This proof involves elliptic genus and in particular a “miraculous cancellation” formula which generalizes (1.21) t o arbitrarily dimensions and which in dimension 12 was first discovered by the physists Alvarez-GaumB and Witten [AGW]. 1.6.4 K-groups and t h e C h e r n Character We now back to the case of complex vector bundles. Still, let E be a complex vector bundle over a compact smooth manifold M . Let V E be a (C-linear) connection on E and let RE denote its curvature. The Chern character form associated t o V E is defined by

[

YR’)] ch ( E , V E )= tr exp ( -

E fleVen(M).

(1.22)

The associated cohomology class, denoted by ch(E), is called the Chern $We refer to the book of Lawson-Michelsohn [LM] for more details about spin manifolds.

Some Examples

15

character of E. The importance of the Chern character lies in its intimate relationships with the K-group of M . Recall that if E , F are two complex vector bundles over M , then one can form the Whitney direct sum of E and F , denoted by E @ F , which is the vector bundle over M such that each fiber ( E @ F ) , at z E M is the direct sum Ex@ Fx of the fibers E, and F,. From (1.22), it is clear that if E and F are two complex vector bundles over M , then

ch(E @ F ) = ch(E) + ch(F)

E

H g n ( M ;C ) .

(1.23)

Denote by Vect(M) the set of all complex vector bundles over M , then under the Whitney direct sum operation, Vect(M) becomes a semi-abelian group. We now introduce an equivalence relation '-' in Vect(M) as follows: two vector bundles E and F are equivalent to each other, if there exists a vector bundle G over M such that E @ G is isomorphic to F @ G. The quotient of Vect(M) by this equivalence relation, Vect(M)/ N , is still a semi-abelian group. Following Atiyah and Hirzebruch [AH2], we define the K-group of M , denoted by K ( M ) ,t o be Vect(M)/ -, with the group structure canonically induced from the above semi-abelian group structure. Then by (1.23), one deduces easily that the Chern character can be extended naturally to a homomorphism ch : K ( M )

-

H r ( M ;C ) .

The importance of this homomorphism lies in the following result due to Atiyah and Hirzebruch [AH2], which says that if one ignores the torsion elements in K ( M ) ,then the induced homomorphism

is actually an isomorphism. On the other hand, the integrality results in Subsection 1.6.3 can be generalized t o allow complex vector bundles as coefficients. For example, one has that for any complex vector bundle E over an even dimensional

Chern- Weil Theory for Characteristic Classes

16

oriented spin closed manifold M , the characteristic number (A^(TM)ch(W, [MI) is an integer (cf. [AHl]). Of course, all these integrality results are special cases of the AtiyahSinger index theorem [AS].

1.6.5

The Chern-Simons h n s g r e s s e d Form

We now take a further look at the formula (l.lO), which we rewrite as follows, tr

[f ( R E ) ]- tr [f ( X E ) ]

= -d

/’

tr

0

[* dVE

f’(I??)] dt.

(1.24)

The transgressed term (1.25) appearing in the right hand side is usually called a Chern-Simons term. In many interesting cases, it is a closed form and thus induces a cohomology class in H,& (M; C). A typical example is when both BE and are flat connections, that is, when both RE and kE equal to zero. We now examine another typical case, where E = T M , the tangent bundle of a smooth compact oriented three dimensional manifold M . 5 Recall the standard result due originally to Stiefel (cf. [S]) that for a smooth compact oriented three manifold M , the tangent bundle T M is topologically trivial. Thus one can choose a fixed global basis el, e2, e3 of T M . Then every section X E r ( T M ) can be written as

oE

where fl, fz, f 3 are smooth functions on M . Let dTM denote the connection on T M defined by dT M ( f l e l

+ fze2 +

f3e3)

= dfl

. el

+ dfz . ez + df3 .

e3.

§As we have noted, although (1.10) is proved for complex vector bundles, the same strategy works without change also for the real vector bundles.

Bott Vanishing Theorem for Foliations

17

Then any connection V T Mon T M can be written as

V T M= dTM + A wifh

A

E

fll(M;End(TM)).

For any t E [0,1], set

VTM = dTM

+ tA.

Take f(x) = - x 2 . By dimensional reason, the left hand side of (1.24) vanishes. Thus, the form in (1.25) is closed and one deduces that

dVTM

dt =

-I'

tr [A(-2) ( d T M

+ t A ) 2 ]dt

= 2 I ' t r [tAA dTMA+t 2 A A AA A ] dt

3

2 AAdTMA+-AAAAA, 3 which is precisely (up t o a rescaling) the Chern-Simons form [CS] appearing recently in so many places in topology, geometry as well as in mathematical physics (cf. for example, the paper of Witten [W] on the Jones polynomial of knots).

1.7 Bott Vanishing Theorem for Foliations

As an application of the Chern-Weil theory, we discuss a vanishing theorem on foliations due to Bott. We recommend the interested reader t o Volume 3 of Bott's Collected Papers [Bo] for further developments arising frbm this simple and beautiful result. We will work on real vector bundles in this section.

Chern- Weil Theory for Characteristic Classes

18

1.7.1

Foliations and the Bott Vanishing Theorem

Let M be a closed manifold and T M its tangent vector bundle. Let F c T M be a sub-vector bundle of T M . We say F is an integrable subbundle of T M if for any two smooth sections X, Y E r ( F ) of F , their Lie bracket is also a section of F , that is,

[X,YI E r ( F ) .

(1.26)

If such an integrable subbundle F c T M exists on M I then we call M a foliation (or a foliated space) foliated by F . We now assume that M is a foliation which is foliated by an integrable subbundle F of T M . Let T M / F be the quotient vector bundle of T M by

F. Let p i , ( T M I F ) ,. . ., pi, ( T M I F ) be Ic Pontrjagin classes of T M / F . We can now state the Bott vanishing theorem as follows. Theorem 1.12

If i l + + . + i k > (dim M

p i , ( T M / F ) ...pi,( T M / F ) = 0

- dim F ) / 2 , then

in

H:$"."+ik)(M;R)

(1.27)

ProoJ To simplify the exposition, we take a Riemannian metric g T M on TM.7 Then T M admits an orthogonal decomposition

TM=FCBF' such that F and F' are orthogonal t o each other with respect to gTM. Moreover, T M / F can be identified with F'. Let V T Mbe the Levi-Civita connection on T M associated t o g T M . Let g F , g F L be the metrics on F , F' induced from g T M . Let p , p' denote the orthogonal projection from T M to F , F' respectively. Set

V F =pVTMp,

V F L =p

l VTMpl.

Then one verifies easily that V F ,V F Lare connections on F , F' respectively. Moreover, they preserve g F , g F L respectively. It is clear that to prove (1.27) one needs only to show that there is a smooth form w E R * ( M )such that when il+...+ik > (dimM-dimF)/2, p i , ( F ' , V F L > . ' . p i k (F',VFL>= d w .

(1.28)

TFor the basics of Riemannian geometry, see the book of Chern-Chen-Lam [CCL]

Bott Vanishing Theorem for Foliations

Following Bott, we will construct a new connection that

19

vF1on F'

such (1.29)

+

when i l . . . + ik > (dim M - dim F)/2. The Bott connection on F L can be defined as follows.

vFL

Definition 1.13 For any X E I'(TM), U E r(F'), (i) If X E I'(F),we define

v g u = pqx,U ] ; (ii) If

X

E I'(FL), set

v $ l U = O$lU.

The part (ii) is not essential. The importance of the part (i) lies in the following result of Bott. Let EF' denote the curvature of

oFL

Lemma 1.14 For any X,Y E I ' ( F ) , one has

P ( X ,Y)= 0. Prooj Let 2

E

r ( F * ) be any smooth section of F'-. By (1.3) and (i)

above, EFl

(X, y ) z = 951q' 2 - QFl o g 2 - vFL [X,YIZ

= O$'p'[Y,Z]

-OF'pqX,z]-pL[[X,Y],Z]

P[Y, 211 - P l [Y, P[Z,XI1 = 0, -P'-[X, where the last equality follows from (1.26) and the Jacobi identity. This completes the proof of Lemma 1.14. 0 Let F'>* denote the dual bundle of F'-

Chern- Weal Theory for Characteristic Classes

20

From Lemma 1.14, one sees easily that

fiFL E r ( F L Y *A) R* ( M ;End (F')) Thus, for any integer j with 1 5 j 5 k, (1.30) From (1.30), one deduces that

il

Since dim F J - = dim M -dim F , one sees directly from (1.31) that when . . . ik > (dim M - dimF)/2, formula (1.29) holds. From (1.29) and the Chern-Weil theorem, one gets (1.28). The proof of Theorem 1.12 is thus completed. 0

+ +

1.7.2

Adiabatic Limit and the Bott Connection

One may argue that from the geometric point of view, the connection VF' is also a natural connection on F'. In fact, by passing g T M t o its adiabatic limit, one sees that the underlying limit of VF' and the Bott connection are ultimately related. To be more precise, for any E > 0, let g T M be the metric on T M defined by

oF'

Let V T M >be€ the Levi-Civita connection of gTM+. Let VF+(resp. VFL>€) be the restriction of V T M +to F (resp. F*). We will examine the behavior of VF'+ as E 0. The process of taking the limit E -+ 0 is called taking the adiabatic limit. The standard formula for Levi-Civita connection (cf. [CCL]) implies that for any X E r(Fj,U , V E r ( F L ) , --f

= ([x,U],V)--(X,VT,~U+V~~V)-;(x,[u,v] 1

2

(1.32)

Bott Vanishing Theorem for Foliations

Let qF"'i* be the connection on F' any sections U , v E r p ) ,

which is dual t o

21

oFL.That is, for

d (U,V)= (""'7.7, V) + ( U , vFLi*V) .

(1.33)

Set WFL

Let

-

V F " ' , + - GF"'

(1.34)

VF"'be the naturally induced connection on F L defined by

Then one verifies easily that VF' preserves gF1 The following result is taken from [LiuZ].

Theorem 1.15 For any smooth section

X

E r ( F ) , one has,

(1.36) Proof. For any

X

E

r ( F ) , U , V E r ( F ' ) , by (1.33) and (1.34) one has

= - (U, VSMV - VFMX)- (

x)- ( V y U, x)

= - ( V y f V,

-

V y u- V p - X v) , + X ( U , V)

(V,V y v ) - ( V y u ,v)+ X(U, V).

(1.37) Note that the last three terms cancel. So (1.36) follows directly from (1.32), (1.35) and (1.37). 0

Remark 1.16 If for any X E r ( F ) , wFL(X) = 0 , then one says that ( M ,F , g F L )admits a Riemannian foliation structure (cf. [TI).

Chern- w e d Theory for Characteristic Classes

22

1.8

Chern-Weil Theory in Odd Dimension

The theory of characteristic forms and classes we have discussed in the previous sections are mainly even dimensional. In this section, we will describe an odd dimensional analogue of this theory. Let M be a smooth closed manifold. Let g be a smooth map from A4 to the general linear group G L ( N ,C ) with N > 0 a positive integer:

-

g :M

G L ( N ,C ) .

(1.38)

Let C N I denote ~ the trivial complex vector bundle of rank N over M . Then the above element g can also be viewed as a section of Aut(CNIM). Let d denote a trivial connection on C N I ~Then . one gets a natural element g-ldg E R'

( M End ; (c"l11.1)) .

(1.39)

If n is a positive even integer, one verifies that

On the other hand, from the equality gg-' = 1, one deduces that (1.41)

dg-1 = -9-'(dg)g-1.

From (1.40) and (1.41) one deduces that if n is a positive odd integer, then

= -ntr

[(g-ldg) ] n+l

= 0.

(1.42)

The following lemma shows that the cohomology class determined by the closed form tr[(g-ldg)"] does not depend on smooth deformations of g : M 4 G L ( N ,C ) . L e m m a 1.17 If gt : M 4 G L ( N ,C) depends smoothly on t E [0,1], then for any positive odd integer n, the following identity holds,

Chern- Weil Theory in Odd Dimension

23

Proof. By an analogue of (1.41), one deduces that

One also verifies that

d (gtldgt)

2

=d

( g t l d g t ) g t ' d g t - g,ldgtd

( g t l d g t ) = 0,

from which one deduces that for any positive even integer k,

d (g;ldgt)k = 0.

(1.45)

From (1.40), (1.44) and (1.45), one verifies that

= ntr

[

[ii;'dgt,g;'g]

= ntr

[

[gclcigt,g;l* at

"-'I

(gt'dgt)

(

at

The proof of Lemma 1.17 is completed. 0

Corollary 1.18 Iff, g : A4 -+ G L ( N , C ) are two smooth m a p s from M t o the general linear group GL(N, C), t h e n f o r a n y positive odd integer n, there exists w, E W - ' ( M ) such that the following transgression formula holds,

Chern- Weil Theory for Characteristic Classes

24

Proof. We consider the direct sum of two trivial complex vector bundles

, 1 with the trivial connection induced from t h a t on C N . We equip C2N For any u E [0, $1, let h(u): A4 -+GL(2N,C) be defined by cosu

sinu

)(

:) (

cosu

-sinu

sinu

cosu

Clearly,

h(O)=(

:)

and

h(:)=(

fg

0l ) .

Thus, h(u)provides a smooth deformation between two sections (fg, 1) and (f,9 ) in r ( A u t ( C Z NIM)). By applying Lemma 1.17 t o h(u),one gets (1.46). 0

Corollary 1.19 Let g E r(Aut(CNl,)). If d' is another trivial connection on CNI,, then f o r any positive odd integer n, there exists w, E Rn-l(M) such that the following transgression formula holds,

[

tr (g-ldg),]

[

= tr (g-l~l'g)~]idw,.

(1.47)

Proof. Clearly, there exists A E I'(Aut(CNIM)) such that

d' = A-' . d . A.

(1.48)

From (1.48), one deduces that

(1.49)

Chem- Weil Theory in Odd Dimension

25

From (1.41), (1.49) and Lemma 1.17, one sees that for any positive odd integer n, there exists wn E Qn-l(M) such that

[

[

tr (g-ld'g)n] = tr (Ap1 ((AgA-l)-'

d (AgA-I)) A)"]

+ tr [ ( ~ - ' d g ) ~+] t r

[(A-'c~A)~]- dw,

= tr [(AdA-')"I

= tr

[(~ - l d g ) ~-]dw,,

which is exactly (1.47). 0

Remark 1.20 By Lemma 1.17 and Corollary 1.19, one sees that the cohomology class determined by tr[(g-ldg)"] depends only on the homotopy GL(N,C). class of g : M ---f

Let n be a positive odd integer, we call the closed n-form

the n-th Chern form associated to g, d and denote it by G ( g , d ) . The associated cohomology class will be called the n-th Chern class associated to the homotopy class [g] of g. We denote this class by cn([g]). We define the odd Chern character form associated to g and d by (1.50) Let ch([g]) denote the associated cohomology class which we call the odd Chern character associated to [g]. For any two f , g : A4 4 GL(N,C), by Corollary 1.18, one has the following additive property, ch([fgl) = ch([fI)

+ ch"g1)

in

Hd"gd(M).

The following integrality result partly explains the choice of coefficients in (1.50): if M is an odd dimensional closed oriented spin manifold, E a

Chern- Wed Theory for Characteristic Classes

26

complex vector bundle over M and g : M -+ G L ( N ,C ) a smooth map from M to the general linear group GL(N ,C ) ,then

is an integer. We refer to Baum-Douglas [BD] and Getzler [GI for the index theoretic interpretations of this integrality result.

1.9

References

[AGW] L. Alvarez-GaumB and E. Witten, Gravitation anomalies. NUC. Phys., B 243 (1983), 269-330.

[AH11 M. F. Atiyah and F. Hirzebruch, Ftiemann-Roch theorems for differentiable manifolds. Bull. Amer. Math. SOC.65 (1959), 276-281. [AH21 M. F. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces. Proc. Symp. Pure Math. Vol. 3. pp. 7-38. Amer. Math. SOC., 1961. [AS] M. F. Atiyah and I. M. Singer, The index of elliptic operators on compact manifolds. Bull. Amer. Math. SOC.69 (1963), 422-433.

[BD] P. Baum and R. G. Douglas, K-homology and index theory. in Proc. Sympos. Pure and Appl. Math., Vol. 38, pp. 117-173, Amer. Math. SOC. Providence, 1982. (BGV] N. Berline, E. Getzler and M. Vergne, Heat Kernels and Dimc Operators. Grundlagen der Math. Wissenschften Vol. 298. Springer-Verlag, 1991.

[BH] A. Bore1 and F. Hirzebruch, Characteristic classes and homogeneous spaces 111. Amer. J. Math. 82 (1960), 491-504. (Bo] R. Bott, Collected Papers Volume 3: Foliations. Birkhauser, 1995.

References

27

[BoT] R. Bott and L. Tu, Differential Forms in Algebraic Topology. Graduate Text in Math. Vol. 82, Springer-Verlag, 1982. [C]S. S. Chern, Geometry of characteristic classes. Appendix in the Second

Edition of Complex Manifolds without Potential Theory. Springer-Verlag, 1979. [CCL] S. S. Chern, W . H. Chen and K. S. Lam, Lectures o n Differential Geometry. Series on Univ. Math. Vol. 1. World Scientific, 1999. [CS] S. S. Chern and J. Simons, Characteristic forms and geometric invariants. Ann. of Math. 99 (1974), 48-69. [GI E. Getzler, The odd Chern character in cyclic homology and spectral flow. Topology 32 (1993), 489-507. [HI F. Hirzebruch, Topological Methods in Algebraic Geometry. Grundlagen der Math. Wissenschften Vol. 131. Springer-Verlag, 1966. [LM] H. B. Lawson and M.-L. Michelsohn, Spin Geometry. Princeton Univ. Press, 1989. [Liu] K. Liu, Modular invariance and characteristic numbers. Commun. Math. Phys. 174 (1995), 29-42. [LiuZ] K. Liu and W. Zhang, Adiabatic limits and foliations. The Milgram Festschrifl. Eds. A. Adem et. al., Contemp. Math., To appear. [MS] J. Milnor and J. Stasheff, Characteristic Classes. Annals of Math. Studies Vol. 76. Princeton Univ. Press, 1974. [O] S. Ochanine, Signature modulo 16, invariants de Kervaire gknkralisks et nombres charactkristiques dans la K-thkorie rkel. Supplkment au Bull. SOC. Math. France, 109 (1981), mkmoire no 5.

[S] N. Steenrod, The Topology of Fibre Bundles. Princetion Univ. Press, 1951.

28

Chern- W e d Theory for Characteristic Classes

[TI Ph. Tondeur, Geometry of Foliations. Birkhauser Verlag, Basel, 1997. [W] E. Witten, Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121 (1989), 351-399.

[YlY. Yu, Index Theorem and Heat Equation Method. (in Chinese) Shanghai Sci. & Tech. Press, 1996.

Chapter 2

Bott and Duistermaat-Heckman Formulas

In Chapter One we have defined characteristic classes and numbers in terms of curvatures of connections on vector bundles. A natural question is how to compute these characteristic numbers. In this chapter we will discuss a localization formula due to Bott [Bo] which shows that for a compact manifold admitting a compact Lie group action, the calculation of characteristic numbers on this manifold can be reduced to the fixed point set of the group action. The philosophy of localizing a computation on a manifold t o that on the fixed point set of certain group actions on that manifold has a wide range of implications in topology and geometry. The Duistermaat-Heckman formula [DH] in symplectic geometry is another important example for this. It turns out that the Bott localization formula and the DuistermaatHeckman formula can be put into the unified framework of the equivariant cohomology theory. In this chapter, we will first prove an equivariant localization formula due t o Berline-Vergne [BV] and Atiyah-Bott [AB], then show how the Bott and Duistermaat-Heckman formulas can be deduced from it.

2.1

Berline-Vergne Localization Formula

Let M be an even dimensional smooth closed oriented manifold. We assume that M admits an S1-action. Let g T M be a Riemannian metric on T M , the tangent vector bundle of 29

Bott and Duistermaat-Heckman Formulas

30

M . Without loss of generality we assume that g T M is S1-invariant.* The $-action on M induces an action on Cm(M) such that for any f E C " ( M ) , x E M and g E S1, ( 9 . f)(x) = f(q). Let t E Lie(S1) be a generator of the Lie algebra of S1. Then t induces canonically a vector field K in the following manner: for any f E Cm(M) and x E M ,

Since the S1-action preserves g T M , K is a Killing vector field on M . It induces a skew-adjoint homomorphism from T M to T M by X H V Z M K , where V T Mis the Levi-Civita connection associated to g T M . That is, for any X , Y E I ' ( T M ) , one has

(Vg'K, Y ) + ( V F M K X , ) = 0.

(2.1)

Proof of (2.1). Let L K denote the Lie derivative of K on r ( T M ) . Since the S1-action preserves g T M , L K also preserves g T M . That is, for any XI Y E I ' ( T M ) , one has

K ( X ,y ) = ( L K X Iy ) + (x,C K y )

=

( V g M x- v g M K , Y )+ ( V y Y - V F M K I X ) = K ( X ,Y ) - ( V g M K ,Y ) - ( V F M K X ,

),

from which (2.1) follows. 0 The Lie derivative C K on I ' ( T M ) induces canonically an action on R*(M) which we still denote by LK and call it the Lie derivative of K on R*(M). The following Cartan homotopy formula on R*(M) is well-known,

CK = diK f i K d ,

(2.2)

where i~ : R* ( M ) 4 R*-'( M ) is the interior multiplication induced by the contraction of K . *In fact, given any metric on T M , one can integrate it over S1 to get an S1-invariant one.

Berline- Vergne Localization Formula

31

Proof of (2.2). First of all, for any f E C " ( M ) , one verifies directly that LK f

=

+ iKd) f = Kf.

(di,

Secondly, since LK commutes with the exterior differential d, one has LKdf

= dLK f = diKdf =

(diK

+ iKd)df.

Since locally every one form can be written in the form df for some f E

C" ( M ), one sees that (2.2) also verifies for all one forms on M . Finally, since both sides of (2.2) verify the Leibnize rule, from the above two facts one sees by induction that (2.2) holds for all forms on M . 0

Let

be the subspace of LK-invariant forms. Set dK = d

+ ZK

: R*(M) -i R*(M).

One verifies easily that d& = diK

+ iKd = LK

Thus dK preserves R>(M) and

The corresponding cohomology group

is called the S' equivariant cohomology of M . Consider now any element w E R*(M). w e say w is dK-closed if dKw = 0. By (2.4), a dK-closed form is LK-invariant. The equivariant localization formula due to Berline-Vergne [BV] (see also Atiyah-Bott [AB]) shows that the integration of a dK-closed differential form over M can be localized to the zero set of the Killing vector field K . For simplicity, we will only prove this formula for the special case where the zero set of K is discrete.

Bott and Duistermaat-Heckman Formulas

32

We start with the simplest case.

Proposition 2.1 If K has n o zeros o n M , then f o r any w E R*(M) which as dK-closed, one has JM w = 0. Proof. We use a method due t o Bismut [Bi2]. Let 9 E R1(M) be the one form on M such that for any X E r ( T M ) ,

ix9 = (XIK ) . Since C K preserves g T M , one verifies easily that

+

From (2.4), one then sees that (d i K ) e is dK-closed. The following lemma is due to Bismut [Bia].

Lemma 2.2 For any T 2 0 , one has

Proof.Since d ~ is0 dK-closed, one verifies directly that

from which one verifies directly, as dKw = 0, that

1,

wexp ( - T d ~ 0 )-

= 0. 0

Since

Berline- Vergne Localization Formula

33

one sees that

=

dim M / 2

JM

T a(do)')

.

(2.6)

Now as K has no zeros on M , IKI has a positive lower bound 6 > 0 on M . One then sees easily that when T -+ +m, the right hand side of (2.6) is of exponential decay. Combining this fact with Lemma 2.2, one gets Proposition 2.1. 0 We now assume that the zero set of K, which we denote by zero(K), is discrete. By using the exponential map a t every p E zero(K), one can assume that for every point p in the zero set of K , there is a sufficiently small open neighborhood U p of p and an oriented coordinate system (x' , . . . ,x2') with 1 = dim M such that on Up one has

g T M = (dz')'

+ .. + ( d x 2 1 ) 2

and

with each X i Set

# 0 for 1 5 i 5 1. X ( p ) = X I . . . x1.

We now state the Berline-Vergne localization formula [BV] in this case as follows.

Theorem 2.3 If the zero set of K is discrete, t h e n f o r a n y dK-closed differentia1 form w E R*(M), one has

where wIO]E C m ( M ) is the 0-th degree component of w .

Bott and D u i s t e n a a t - H e c k m a n F o n d a s

34

Pro05 By (2.5), one has

(2.8) Since K has no zeros on M \ U p E ~ e r o ( ~ ) one U p rcan proceed as in the proof of Proposition 2.1 to show that

s

w exp ( - T d K B )

4

M\UpEzero(K)

0 as T

4

(2.9)

+co.

Now on each Up, one verifies directly that

8 = x1 (z2dX1 - d d X 2 ) + . . . + xl (X21dX21-1 - X21-1d221) . Thus,

+ .. .+ ~

de = -2 (xldX1dX2

~

d

~. ~

~

- ~ (2.10)

Also, one verifies directly that, on Up,

JKI2= A;

((x')' + ( x ~ ) ~+)... + A;

(

+ (x",')

.

(2.11)

For any integer i such that 0 5 i I 21 = d i m M , let w[Zl E Ri(M) denote the corresponding component of w , then one verifies directly that for any p E zero(K),

Now we make the rescaling change of the coordinate system

x = ( 2 1 , . . . ,2 1 )

-+ @X

= (@X1,.

. . ,f i x , , ) .

By (2.10) and (2.11), one finds that if 0 5 i 5 1 - 1, then

-0

as T-++co.

(2.12)

d

~

~

Bott Residue Formula

35

On the other hand, if i = 1, then one computes that

-

(274-

w lo] (0) A1 . . . A1

as T--++oo.

(2.13)

From (2.8), (2.9), (2.12) and (2.13), one gets (2.7). 0

We refer to [BV], [Bi2] and [BGV, Chap. 71 for the general case where the zero set of K may not be discrete. 2.2

Bott Residue Formula

We make the same assumptions as in the previous section. In particular, we still assume that the zero set of the Killing vector field K is discrete. Let RTM be the curvature of the Levi-Civita connection V T M . Let 21, . . . ,ik be k positive even integers. For any p E zero(K) and 1 5 j 5 k, set

AiJ ( p ) =

A t + ...

f

Afj.

One can state a version of the Bott residue formula [Bo],which reduces the computation of characteristic numbers of T M to quantities on zero(K), as follows.

Theorem 2.4 If i l

+ . . . + ik = 1, then the following identity holds,

(2.14)

Bott and Duistermaat-Heckman Formulas

36

Moreover, if

il

+ . . . + i k < 1,

c

then Xi' ( p ) . . . X i k ( p ) 11111

= 0.

(2.15)

" W f

pEzero( K )

Proof. Clearly, the interior multiplication i~ can be extended canonically to an action on R*(M;End(TM)). Also, since both V T Mand K are S1-invariant, one sees directly that

[ V T MC,K ] = 0 ,

[ZK,L K ] = 0.

(2.16)

Moreover, one verifies directly that

(VTM+ i

+

~= RTM ) ~ [ V T Mi , ~ ]

+LK,

(2.17)

Ro(M;End(TM)).

(2.18)

= R~~ -I-C K

= VTMK E

From (2.16) and (2.17), one gets the following Bianchi type formula

[VTM+ Z K , RTM + L K ]

= 0.

(2.19)

From Lemma 1.8 and formulas (2.18), (2.19), one sees that for any integer h,

+ iK)tr [(RTM + L K ) h ] = tr [ [VTM + i K , ( R ~ M+ L K ) ' ] ] = 0. This means that each tr[(RTM+ L K ) ~ ~1 ]5, j _< k, is dK-closed. Their (d

product is thus also dK-closed. One can then apply Theorem 2.3 to get /Mtr [(RTM)$']. . .tr [(RTM)"]

37

Duistermaat-Heckman Formula

Now by (2.18) and the explicit expression of K given in the previous section, one sees that

( L K ( ~= ) )-diag ~ {A:, A,: . . . , A t , A t } . Thus, for each 1 5 j 5 k,

Theorem 2.4 then follows from (2.20) and (2.21). 0 The generalization of Theorem 2.4 in the case where the zero set of K may not be discrete was first proved by Baum and Cheeger in [BC].

2.3

Duistermaat-Heckman Formula

In this section, we further assume that M is a symplectic manifold with the symplectic form given by w E R2(M). We assume the S1-action preserves w . Moreover, we assume that the S1-action on ( M ,w ) is Harniltonian. That is, there exists a smooth function p E C m ( M ) such that dp = i K w .

(2.22)

The Liouville form of the symplectic manifold ( M , w ) is given by w'

m.

We still assume that the zero set of K is discrete. The Duistermaat-Heckman formula [DH] can be stated as follows.

Theorem 2.5 The following identity holds,

Pro08 From (2.22), one finds (d Thus, one sees that exp(&ip

+ i ~ ) ( -w p ) = 0. - &iw)

is also dK-closed. One can then

Bott and Duistermaat-Heckman Formulas

38

apply Theorem 2.3 to get

from which (2.23) follows easily. 0 The generalization of Theorem 2.5 in the case where the zero set of K may not be discrete is also due to Duistermaat-Heckman [DH]. It was proposed by Witten that a formal application of the DuistermaatHeckman formula t o the free loop space of a compact spin manifold can lead to a heuristic proof of the index theorem for the canonical Dirac operator on that spin manifold. Witten’s idea was exposed in a talk of Atiyah [A] which in turn inspired Bismut [Bill t o give a probabilistic proof of the index theorem for Dirac operators. The paper [Bi2] contains, among other things, the family generalizations of this circle of ideas. 2.4

Bott’s Original Idea

Bott’s original proof of Theorem 2.4 in [Bo] uses the idea of trangression, and is thus different from the one we presented above. Here we give a brief description of Bott’s idea by re-proving Proposition 2.1. Thus let w be a dK-closed form on M , with K has no zeros on M . Let 0 E nl(M) be the one form on M such that for any X E I’(TM), ixe = K). Since

(x,

and K is nowhere zero on M ,

1 E R*(M) dK0

is well-defined. From the fact that d g 8

= 0,

one then verifies directly that

W = d K ( =B) ,A w from which Proposition 2.1 follows directly from the Stokes formula.

References

2.5

39

References

[A] M. F. Atiyah, Circular symmetry and stationary-phase approximation, Colloquium in honour of Laurent Schwartz, Vol. 2, AstLrique 131 (1985), 43-59. [AB] M. F. Atiyah and It. Bott, The moment map and equivariant cohomology. Topology 23 (1984), 1-28. [BC]P. Baum and J. Cheeger, Infinitesimal isometries and Pontryagin numbers. Topology 8 (1969), 173-193. [BGV] N. Berline, E. Getzler and M. Vergne, Heat Kernels and Dirac Operators. Grundlagen der Math. Wissenschften Vol. 298. Springer-Verlag, 1991. [BV] N. Berline and M. Vergne, Z6ros d’un champ de vecteurs et classes characthristiques 6quivariantes. Duke Math. J . 50 (1983), 539-549. [Bill J.-M. Bismut, The Atiyah-Singer theorems: a probabilistic approach.

J. Funct. Anal. 57 (1984), 56-99. [Bia] J.-M. Bismut, Localization formulas, superconnections, and the index theorem for families. Commun. Math. Phys. 103 (1986)’ 127-166. [Bo] R. Bott, Vector fields and characteristic numbers. Michigan Math. J. 14 (1967)’ 231-244. [DH] J . J . Duistermaat and G. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space. Invent. Math. 69 (1982), 259-268. Addendum, 72 (1983), 153-158.

Chapter 3

Gauss-Bonnet-Chern Theorem

In this chapter we will present Mathai-Quillen's proof [MQ] of the GaussBonnet-Chern theorem [Cl], which expresses the Euler characteristic of a closed oriented Riemannian manifold as an integral of the Pfaffian of the curvature of the associated Levi-Civita connection. The proof is based on the explicit geometric constructions of Thom forms given in [MQ], while the basic idea behind is the same as in [Cl]: transgression. We will first construct the Thom form of Mathai-Quillen and then use it to prove the Gauss-Bonnet-Chern theorem. We will work with real coefficients in this chapter.

3.1

A Toy Model and the Berezin Integral

We start with the simplest situation. Let E be an oriented Euclidean vector space of dimension n, which we view as a vector bundle over a point. Let x = (d, . . . ,x n ) be an oriented Euclidean coordinate system of E. Set

u(x)= e-lx12/2&1

A

.. . A &".

(3.1)

Then one verifies easily that

We now reinterpret (3.2) in terms of the language of Berezin integral. Let R*(E)be the exterior algebra of E. 41

Gauss-Bonnet-Chern Theorem

42

The Berezin integral of an oriented Euclidean space E is a linear map J B : A * ( E )+ R

defined by (3.3) which means that if e l , . . . , en is an oriented orthonormal basis of E and ael A . . . A en is the component of w of degree n, then JBw=a.

We lift A* ( E ) as a vector bundle over El and denote by R* (ElA* ( E ) ) the space of smooth sections of R*(E) over E. Then we can and we will extend the Berezin integral to R*(E,A * ( E ) )such that rB

]

rB

:QA

,8 E R* (ElA * ( E ) )H a!

]

,8 E R*(E),

(3.4)

with Q E R*(E), ,8 E r ( A * ( E ) ) . We now consider the identity map E .+ E t o be an element of Ro(E, E ) , with its exterior differential d x E R1(E,E ) . The following result gives the Berezin integral interpretation of the differential form U defined in (3.1).

Proposition 3.1 The following identity in R*(E) holds, (3.5)

Proof. As x = x'el

+ . + .+ P e n , one verifies directly that

- (-1)"("+')/'(-l)"

from which (3.5) follows. 0

/

B

(dx' A e l ) A . . . A (dx" A en)

Mathai-Quillen’s Thorn Form

43

Finally, let E be an oriented Euclidean vector bundle of rank n over a manifold M . Then by an obvious fiberwise extension, we can extend the Berezin integral defined above to define a map

J’

B

: R * ( M , A * ( E )4 ) R*(M)

in the way similar to that in (3.4). We still call it a Berezin integral. Let V E be a Euclidean connection on E (that is, V E preserves the metric on E ) , then it extends naturally to an action V on R * ( M , R * ( E ) ) . The following property is important for the applications in the next section.

Proposition 3.2. For any a holds,

E

R * ( M , A * ( E ) ) ,the following identity

dJ’Ba=J’BVa.

(3.7)

Proof. Let e l , . . . , en be an oriented orthonormal basis of E . Without loss of generality, we can assume that a = we1 A . . . A en with w E R*(M). Now since V Epreserves the Euclidean metric on E , one verifies directly that v a = ( d w ) e l A . . . A en

+ ( - l ) d e gAwV ~( e 1 A . . +A en)

= (&)el A

.

s .

A

en,

from which (3.7) follows. 0 3.2

Mathai-Quillen’s Thom Form

In this section, we construct Mathai-Quillen’s Thorn form by using the Berezin integral. Let M be an oriented closed manifold and p : E + A4 an oriented Euclidean vector bundle of rank n. Let V E be a Euclidean connection on E. Then V E lifts to a Euclidean connection on p*E and thus also to a derivation V on R * ( E ,A*(p*E)).

44

Gauss-Bonnet-Chern Theorem

On the other hand, for any s E r ( E , p * E ) ,the interior multiplication 2, on A * ( p * E )extends naturally t o a derivation on R*(E, A * ( p * E ) ) . We will apply Proposition 3.2 t o the triple ( E , p * E ,V). Since the interior multiplications decrease the degrees in A* (p*E),from Proposition 3.2 one gets

d S BQ =SB(V

+ is)Q,

(3.8)

for any Q E R*(E,A*(p*E)) and s E r ( E , p * E ) . Also, we identify so(E),the subset of End(E) consisting of skew-adjoint elements, with A 2 ( E )by the map

A

E

so(E) H x ( A e i , ej)ei A e j .

(3.9)

i<j

We now consider the following elements in the algebra R*(E, A*(p*E)):

(1) the tautological section x E Ro(E,p*E)= T ( E , p * E ) ; (2) the elements IxI2 E Ro(E) and Vx E R1(E,A1(p*E)); (3) the element p*RE E R2(E,A2(p*E))which is the pull back by p of the curvature RE = (VE)2 E R2(M,A2(E)),where we have used the identification (3.9). The following result is of critical importance.

Lemma 3.3 Let

(V

+

2,)

A = 0.

Proof. By Leibniz's rule, we have

v (1x12) = -22,vx. By the definition of the curvature, we have

~ ( V X=) V2x = (p*RE)x = ixp*RE,

(3.10)

Mathai-Quillen's Thom Form

45

while by the Bianchi identity we have

Vp*RE= 0. Combining all these with the obvious fact that

Z,/X~~

= 0, we get (3.10). 0

Following Mathai and Quillen [MQ], we now define a form U on E by

(3.11) The following result shows that U is a Thom form* for E .

Proposition 3.4 The form U is a closed n-form on E . Furthermore, one has the following formula for the fiberwise integration,

(&)n/2

J,,, u = 1.

(3.12)

Prooj Since 2

A E @Oi(E,Ai(p*E)), i=O

one gets n

e W AE

@Oi ( E , A i ( p * E ) ) . i=O

By (3.8), (3.10) and (3.11), one verifies easily that U is a closed n-form on E . To verify (3.12), one simply restricts to each fiber, on which one can apply directly (3.2) and (3.5). 0 Mathai and Quillen [MQ] originally obtained their Thom form by computing the Chern character of Quillen's superconnection [Q] associated t o *We refer to the book of Bott-Tu [BoT] for the topological significance of the Thom forms and the associated classes.

Gauss-Bonnet-Chern

46

Theorem

spin vector bundles. The Berezin integral formalism here is adapted from [BGV, Sec. 1.61 and [BZ, Sec. 31.

3.3

A Transgression Formula

The Thom form U defined in the last section depends on the choices of the Euclidean metric and the connection V Eof E . However, one can show, by using certain transgression formula, that the cohomology class it determines (that is, the associated Thom class) does not depend on these metrics and connections. For the proof of the Gauss-Bonnet-Chern theorem, here we only consider a special case of this transgression formula, i. e., the case where the metric on E being rescaled. This is equivalent t o change x to t x in A for t > 0:

At

=

-+ tVx - p xRE. t21X12

2

(3.13)

Let Ut be the Thom form corresponding to At as being defined in (3.11)

Proposition 3.5 W e have the transgression formula (3.14)

Proof. One verifies from (3.13) that dAt

--

dt

-

t1xI2

+ vx = (V +ti,) x.

(3.15)

On the other hand, (3.10) now takes the form

(V + ti,)At From (3.15), (3.16), one deduces that

and hence, in view of (3.8),

= 0.

(3.16)

Proof of the

Gauss-Bonnet-Chern

Theorem

47

(3.17)

3.4 Proof of the Gauss-Bonnet-Chern Theorem We now assume that E is the tangent bundle T M of M and V T Mis the Levi-Civita connection associated t o the metric g T M on T M . Let RTMbe the curvature of V T M . We also assume that n, which now equals to the dimension of M , is an even integer. Let v E r ( T M ) be a vector field on M . By (3.11) and Proposition 3.4, the pull-back v*U is a closed differential form of degree n on M given by the formula

In particular, if we take v = 0, the zero section, we get the so-called Euler

form

1

B

(-i)n/2Pf

( R T M ) := (-i)n/2

exp ( R T M )

.

(3.19)

The following result shows that the cohomology class associated to (-1)"/2Pf(RTM) does not depend on the choice of g T M . We call this class the Euler class of T M .

Proposition 3.6 If y M is another metric o n T M , TTMis the Levi-Civita , there connection associated to F Mand RTMis the curvature of T T M then exists a differential form w E !?-l(M) such that

Pf ( R T M ) Pf

( E T M )= dw.

(3.20)

PmoJ For any u E [0,1], let g z M be the metric on T M defined by g,TM = ug T M

Let Vf"

VZ*.

+ (1 - u)?".

be the Levi-Civita connection of g T M and RTM the curvature of

Gauss-Bonnet-Chern Theorem

48

By using Proposition 3.2, the Bianchi identity and (3.19), one deduces that

=d

SB

exp ( R z M ),

from which one gets

Pf(RTM)-Pf(ETM) = d / ' /

dVTM d u exp

(RTM)d u ,

0

which completes the proof of Proposition 3.6. 0 Let x ( M ) denote the Euler characteristic of M . We can now state the Gauss-Bonnet-Chern theorem [Cl] as follows. Theorem 3.7 T h e following identity holds,

x(W=

(

- 1 y 2

J-

Pf ( R T M ).

(3.21)

Proof. Let V be a transversal section of T M . That is, V is a tangent vector field on M such that the zero set of V, denoted by zero(V), is discrete and nondegenerate. The later means that for any p E zero(V), there is an oriented coordinate system y = ( y ' , . . . ,y") on a sufficiently small open neighborhood Up of p such that near p ,

V(Y)

= YA

+ 0 (lY12)

9

(3.22)

where A is an n x n matrix not depending on y verifying that det(A) # 0.

(3.23)

Proof of the Gauss-Bonnet-Chern Theorem

49

The existence of such a transversal section is an elementary result in differential topology. In order to prove (3.21), observe that by (3.18), (3.19) and Proposition 3.5, one has that for any t > 0, (-1)n/2

/

Pf ( R T M )

M

= (-1)n/2

/

M

sB (- (F

+ tVTMV - RTM

exp

For any p E zero(V), from (3.22) one sees easily that one can modify the coordinate system slightly so that (3.22) becomes V(Y) = YA.

(3.25)

Moreover, by Proposition 3.6, we can well assume that on Up the metric g T M is of the form g T M = (dy')2

+ . . . + (dy")' .

With these simplifying assumptions we can rewrite (3.24) as

(-1)n/2

/

Pf ( R T M )

M

/

= (-I)"/'

/".xp(-(T+tdV))

pEzero(V)

/

/ B exp

M\UpEZero(V)UP

(- (T +

tvTMV - RTM

(3.26) Since IVI > 0 has a positive lower bound on M \ UpEzero(V)Up, one sees easily that as t + +m, one has

s

/"eq

M\Up~eero(~)up

(- (y

+ t V T M V - RTM

Gauss-Bonnet-Chern

50

Theorem

On the other hand, for any zero p of V, one verifies directly that, as t

-+

+m,

/ / (- (v+ B

=

= tn det(A)

-

sgn(det(A))

A2

exp

UP

td(yA)))

lp (- (v)) dy' A . . . A dyn

exp

/ (- (q)) 1

det(A)(dyl A . . . A dyn

exp

R"

= ( 2 ~ ) ~ / ~ sdet gn (A)). (

(3.28)

Now recall the classical Poincark-Hopf index formula (cf. [BoT, Theorem 11.251) which says that

X(M)=

c

sgn(det(Ap)).

(3.29)

pEzero( V )

By (3.24) and (3.26)-(3.29), one gets (3.21). 0

3.5

Some Remarks

Remark 3.8 To see more closely the relationship between the above proof and Chern's original proof in [Cl], one integrates both sides of the transgression formula (3.14) to get for E = T M and any T > 0 that

(2) lT/' "I2 d

( ~ e - dt. ~ ~ )

(3.30)

Chern’s Original Proof

51

Now if one restricts (3.30) t o the unit sphere bundle SM of TM, one verifies directly that when T -+ +m,

Thus, when restricted to the unit sphere bundle S M , one has

(

;)n’2

p * P f ( RT M ) -

(i-)”’” dJdtw /”(xePdt)d t , -

(3.31)

which looks of exactly the same form as Chern’s transgression formula in [Cl]. We leave it to the interested reader to show that they are in fact the same one.

Remark 3.9 There is also a heat kernel proof of the Gauss-Bonnet-Chern theorem due to Patodi [PI, see the books [BGV] and [y1 for more details. On the other hand, there is an analytic proof of the PoincarQ-Hopf index formula (3.29) suggested by Witten [W]. We will present such a proof in the next chapter. 3.6

Chern’s Original Proof

In this section we describe Chern’s simple and elegant original proof of (3.31), from which Theorem 3.7 follows from the PoincarQ-Hopf index formula and also the Stokes formula. Here, instead of using the arguments in [Cl],we adopt a simplified version due t o Chern himself [C2]. We make the same assumptions and use the same notation as in previous sections. Recall that S M is the unit sphere bundle of the tangent bundle p : T M -+ M . Thus, x forms a unit length section over SM of p’TM. We denote this section by en. Let e l , . . . , e,-l be the (locally defined) sections of p’TM over SM so that e l , . . . ,en-l, en forms an oriented orthonormal basis of p’TM over S M . For any integer i, j such that 1 5 i, j 5 n, let w i j be the (locally defined) one form on S M define by n

(3.32) j=1

Gauss-Bonnet-Chern Theorem

52

Let

denote the (locally defined) two form on S M . From (3.32), one verifies easily that n

flij = d w i j -

Wik

A Wkj,

(3.33)

k=l

from which one finds the following (local) version of the Bianchi identity, n

n

k=l

k=l

(3.34)

For any 1 integers a1,.. . , a ~let , E ~ ~ .be. 1 . or ~ ~-1 if ~ 1 , . .. ,a1 is an even or odd permutation of 1,.. . ,1 respectively. Otherwise we define it to be zero. Following Chern [C2, (4)], for any integer k such that 0 5 k 5 f - 1, we define

We also define Q - 1 = Q n p = 0. One verifies directly that each of the @ k ’ s and Q k ’ S does not depend on the choice of e l , . . . ,e n - l , and thus is a globally well-defined differential form on S M . Thus, d @ k is also a globally well-defined differential form on S M , which, by (3.35), is given by n- 1

d@k = k

C

al,”’,a,-l=l

.

~ c x 1 . . . a ~ - l d f l a 1 a z‘AflCX2k-1CX2k ~.

Awa2,+1nA.. *~wa*-ln

Chern's Original Proof

53

Ad~cxzk+lnA * . . A wan-ln.

(3.36)

By (3.32), one finds easily that

w,,

= 0.

Now we substitute d R a l a 2 and dwa2k+lnin (3.36) by the right hand sides of (3.33) and (3.34) respectively. Since d@k is a global form on S M , we see that the terms involving (locally defined) we@,with 1 5 a, ,B 5 n- 1, should cancel each 0ther.t In summary, from (3.36) one deduces that

(3.37) Following [C2, (9) and (lo)], we define

k=l

1 . 3 . .. ( n - 2k - 1) . 2 k . Ic!

@k

(3.38)

and

By (3.37), one then gets as in [C2, (ll)]the following transgression formula on S M ,

-dH

= 0,

(3.40)

which is exactly (3.31). 0

Remark 3.10 The historical importance of Chern's proof, besides introducing the idea of transgression, is that it was the first time in history that the intrinsically defined sphere bundle was used to solve an important problem in geometry. *In fact, for any p E S M , one can find e l , . . . , e,-1 1 1. a , /3 5 TI - 1 (cf. [CCL, Theorem 4.1.21).

near p so that w,p(p) = 0 for

54

Gauss-Bonnet-Chen Theorem

Remark 3.11 Although the Mathai-Quillen formalism provides a reasonable interpretation of Chern’s transgression formula through the geometric construction of Thom forms, it is still mysterious how Chern constructed the forms ak’s and Q k ’ s , especially under the form appeared in [Cl]. Remark 3.12 There is also a generalization of the Gauss-Bonnet-Chern theorem to Finsler manifolds, see the paper of Bao and Chern [BC].

3.7

References

[BC] D. Bao and S. S. Chern, A note on the Gauss-Bonnet theorem for Finsler spaces. Ann. of Math. 143 (1996) 233-252. [BGV] N. Berline, E. Getzler and M. Vergne, Heat Kernels and Dirac Operators, Grund. der Math. Wiss. 298, Springer-Verlag, Berlin-HeidelbergNew York, 1992. [BZ] J.-M. Bismut and W. Zhang, An extension of a theorem by Cheeger and Muller. Aste‘risque Tom. 205, SOC.Math. France, 1992. [BoT] R. Bott and L. Tu, Dzfferential Forms in Algebraic Topology, GTM 82,Springer-Verlag, Berlin-Heidelberg-New York, 1982. [Cl] S. S. Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds. Ann. of Math. 45 (1944), 747-752. [C2] S. S. Chern, On the curvatura integra in a Riemannian manifold. Ann. of Math. 46 (1945), 674-684. [CCL] S. S. Chern, W. H. Chen and K. S. Lam, Lectures on Dzfferential Geometry. Series on Univ. Math. Vol. 1. World Scientific, 1999. [MQ]V. Mathai and D. Quillen, Superconnections, Thom classes and equivariant differential forms. Topology 25 (1986), 85-110.

[PI V. K. Patodi, Curvature and eigenforms of the Laplace operator. J.

References

55

Dzff. Geom. 5 (1971), 251-283. [Q] D. Quillen, Superconnections and the Chern character. (1985), 89-95. [W] E. Witten, Supersymmetry and Morse theory. J. Difl (1982), 661-692.

Topology 24

Geom. 17

[Y] Y. Yu, Index Theorem and Heat Equation Method. (in Chinese) Shanghai Sci. & Tech. Press, 1996.

Chapter 4

Poincare-Hopf Index Formula: an Analytic Proof

We have seen in the previous chapter that the PoincarB-Hopf index formula (3.29) plays an important role in the proof of the Gauss-Bonnet-Chern theorem. In this chapter, following an idea of Edward Witten [Wi], we will present a purely analytic proof of this classical result. The strategy of Witten’s proof is very simple. One starts with the analytic interpretation of the Euler characteristic obtained from the Hodge theorem, and deforms the involved elliptic operators by the vector field in question. In this process, one finds that the proof can be localized t o sufficiently small neighborhoods of the zero set of the vector field. A further investigation on these small neighborhoods will then complete the proof. In this chapter, we will first review the Hodge theorem and the consequent analytic interpretation of the Euler characteristic. We then introduce Witten’s deformation and show how it leads t o a proof of the PoincarBHopf index formula. We work with real coefficients in this and next chapters.

4.1 Review of Hodge Theorem Let M be an n-dimensional closed oriented manifold. Recall that the de Rham cohomology of M has been defined in Section 1.1.* The theorem of Hodge provides an analytic realization of this cohomology group. To begin with, let g T M be a metric on T M . Then the Hodge star *Although in Section 1.1we worked with complex coefficients. The same strategy works for real coefficients parallelly. 57

PoincarGHopf Index Formula: a n Analytic Proof

58

operator

* : A*(T*M)-+ Rn-*(T*M) can be defined as follows: if e l , . . . ,en is an oriented orthonormal basis of

TM and e l , . . . ,en is the corresponding dual basis in T * M determined by gTMl then for any integer k between 1 and n,

* : el A . . . ~e~

H

e"+l A . . . ~ e " .

(4.1)

It can be verified easily that the operator * is well-defined. The following properties of the Hodge star operator are easy t o verify. (i) For any integer k, ** = ( - l ) n k + k: @ ( M ) -+ Rk(M); (ii) For any integer k and any a , P E Rk(M),a A *P = P A *a; (iii) a A *a = 0 if and only if a = 0. From these properties, one can define an inner product (., .) on R*(M) as follows: for any a , P E R*(M),

Recall that d : R*(M) -+ R*(M) is the exterior differential operator on

M. Definition 4.1 Let d* : R*(M) ---f R*(M) be the operator defined by

a

E

R"M)

w (-l)nk+n++l

* d * a E &+l(M).

(4.3)

From (4.3) and Property (i) above, one verifies that for any a E Rk(M),

p E R2"+'(M), d(aA

*P) = (da)A *,B + ( - l ) k aA d * P = ( d a ) A *,B -

A

*d*,B,

from which one gets ( d a , P ) = (a,d*P).

That is, d* is the formal adjoint of d.

(4.4)

59

Review of Hodge Theorem

Definition 4.2 The de Rham-Hodge operator D associated to gTM is defined by

D

=d

+ d* : R*(M) +R*(M).

Set

i odd

i even

Let

be the restrictions of D to Revenlodd(M) respectively. Clearly, Dodd is the formal adjoint of D,,,,. Let

be the Laplacian of D. Then 0 preserves each Rk(M),0 5 k 5 n. We can now state the Hodge decomposition theorem as follows.

Theorem 4.3 The following decomposition formula f o r Cl*(M) holds,

R*(M) = ker 0 @ Im 0. We refer to the books of de Rham [de] and Warner [W] for a proof of this result. When restricted to each Rk(M)with 0 5 k 5 n, one further has

where we use the notation that for any integrer i such that 0 5 i 5 n, 0,= O l n . ( ~ ) ,d, = dln.(M) and d: = d*ln.(M).

Corollary 4.4 For any integer k such that 0 5 k 5 n, one has the identification ker O k

21

H t R ( M ;R).

PoincarC-Hopf Index Formula: a n Analytic Proof

60

Proof: First, if w E ker n k , then by (4.4) and (4.5), one has (dw,dw)

+ (d*w, d*w) = ((d'd + dd*)

W,W )

= 0,

(4-7)

which implies that dw = 0. Furthermore, if w , w' E kerOk such that w - w' = dw" for some w f t E a'-', then by (4.6) one sees that w = w'. That is, each element in ker Ok determines a unique element in H t R ( M ;R). On the other hand, if dw = 0, then by (4.6), one deduces easily that there is the unique decomposition w = w' dw" with w' E ker 01,and w~~ 0 2 " - 1 ( M ) ,which determines an element in ker a k . Combining the above discussions one completes the proof of Corollary 4.4. 0

+

Now by (4.5)-(4.7) one also deduces that ker 0 = ker (d

+ d*) c Q * ( M ) .

(4.8)

From (4.8) and Corollary 4.4 one gets the following analytic interpretation of the Euler characteristic x ( M ) of M : n

x ( M ) :=

C(-l)i dim H&(M; R) i=O

= ind

(Deve,= d + d* : QeVen(M) Qodd(M)), ---f

(4.9)

which by definition equals to dim (ker D,,,,)- dim (ker Dodd)

4.2

.

PoincarB-Hopf Index Formula

For convenience we here restate the PoincarBHopf index formula. Let V be a transversal section of T M . Then the zero set of V, denoted by zero(V), is discrete and for any p E zero(V), there is a sufficiently small neighborhood Up of p and an oriented coordinate system y = ( y l , ... . , y n ) such that on U p , V(Y) = YAP

(4.10)

Clifford Actions and the Witten Deformation

61

for some constant matrix A, such that det A,

# 0,

and that the Up’swith p E zero(V) are disjoint with each other. The PoincarC-Hopf index formula (cf. [BoT, Theorem 11.251) can be stated as follows.

Theorem 4.5 The following identity holds, (4.11)

4.3 Clifford Actions and the Witten Deformation For any e E T M , let e* E T * M corresponds to e via g T M . That is, for any X E I ’ ( T M ) , ( e * , X )= ( e , X ) . Let c(e), 2(e) be the Clifford operators acting on the exterior algebra bundle A * ( T * M )defined by c(e) = e* A -ie,

A

c(e) = e* A +i,,

(4.12)

where e*A and i, are the standard notation for exterior and interior multiplications. If e, e‘ E T M , one has

+

c(e)c(e’) c(e’)c(e) = -2(e, el),

+

2(e)E(e’) 2(e’)2(e) = 2(e,e’),

+

c(e)Z(e’) E(e’)c(e) = 0.

(4.13)

Let V T Mbe the Levi-Civita connection associated to the metric g T M . Then it induces canonically a Euclidean connection V A * ( T * M on)A* ( T * M ) . Let e l , . . . ,en be an oriented orthonormal basis of T M . Let e l , . . . ,en be the corresponding dual basis of T * M with respect to g T M . Since V T Mis torsion free, one verifies directly that n

(4.14)

PoincarGHopf Index Formula: a n Analytic Proof

62

€+om (4.14), one deduces that for any a, P E R*(M),

=

,/

d (Q. A *p) = 0.

Thus, -C:=, ieiVei A * ( T * M ) is a formal adjoint of d. Since we have seen that d* is a formal adjoint of d, by the uniqueness of the formal adjoint operators, one gets n

d* = -

2% '

v,;A * ( T * M ) : G * ( M )--+ R * ( M ) .

(4.15)

i=l

From (4.12), (4.14) and (4.15), one gets n

d

+ d* = C ~ ( e i ) V t ~ * :(R*(M) ~*~)

4

Q*(M).

(4.16)

i=l

Now let V E r ( T M ) .Following Witten [Wi], for any T E R, set

DT

=d

+ d* + TE(V): R*(M) -+

R*(M).

(4.17)

Then DT is a (formally) self-adjoint first order elliptic differential operator. Clearly, LIT exchanges Reven(M)and Clodd(M). Let DT,even/oddbe the respectively. Then &,odd is the formal restrictions of DT on flevenlodd(M) adjoint of D T , , ~ ~ ~ . By a standard fact for elliptic operators, and by (4.9), one has that for any T E R, ind

= ind D,,

=x(M).

The following Bochner type formula for D$ is crucial.

(4.18)

An Estimate Outside of u ~ ~ ~ ~ ~ ~ ( V ) U ~ 63

Proposition 4.6 For any T E R,the following identity holds, n

D$ = D2

+ T x c ( e i ) E ( V x M V )+ T21VI2.

(4.19)

i=l

Proof. From (4.13), (4.16) and (4.17), one deduces that

= D2

+T

n

+

c ( e i ) Z ( V z M V ) T21VI2. i= 1

4.4

An Estimate Outside of U p E e e r o ( V ) U p

Let (1 . ( ( 0 denote the 0-th Sobolev norm on R*(M) induced by the inner product (4.2). Denote by Ho(M)the corresponding Sobolev space. Let V E r ( T M ) be as in Section 4.2. The main result of this section can be stated as follows.

Proposition 4.7 There exist constants C > 0 , To > 0 such that for any section s E R*(M) with Supp(s) c M \ U p E s e r o ( ~ ) and U p T 2 To, one has ll&sllo

2 CJTllSllO.

(4.20)

Proof. Since V is nowhere zero on M \ upEzero(v)Up, there is a.constant C1 > 0 such that on M \ UpEZero(v)Up,

IVI2 2 c1.

(4.21)

Poincard-Hopf Index Formula: an Analytic Proof

64

Rom (4.19) and (4.21), one sees that there exists a constant such that

C2

>0

for any s E R*(M) with Supp(s) c M \ U p E x e r o ( ~ ) U p . Formula (4.20) then follows easily. 0 4.5

Harmonic Oscillators on Euclidean Spaces

Proposition 4.7 indicates that the proof of the Poincark-Hopf index formula can be 'localized' in some sense to small neighborhoods of zero(V). Without loss of generality, we assume that on each Up,the metric g T M is of the form

+ . .. + ( d y y .

gTM = ( d y y

Then each Up may be viewed as an open neighborhood of the n-dimensional Euclidean space En. In this section we investigate the Witten deformation in this Eulidean space for the vector field V = y A with det A # 0. Let ei = 1 5 i 5 n, be an oriented orthonormal basis of En. Equation (4.19) can be written explicitly here as

&,

=-c($) n

2

- TTr

[m]+ T 2(yAA*,Y )

i=l

(4.23)

The operator n

KT=-c($)

2

- TTr

[m] + T 2( y A A * , y )

(4.24)

k l

is a rescaled harmonic oscillator. By standard results concerning harmonic oscillators (cf. [GJ, Theorem 1.5.1]), one knows that when T > 0, KT is a

Harmonic Oscillators o n Euclidean Spaces

65

nonnegative elliptic operator with ker KT being of dimension one and being generated by

(4.25) Furthermore, the nonzero eigenvalues of KT are all greater than CT for some fixed constant C > 0. On the other hand, one has the following algebraic result due t o S. P. Novikov (cf. [S]).

Lemma 4.8 The linear operator

L

= Tr

[m] +1 c(ei)E(eiA)

(4.26)

i=l acting on A*(E;) is nonnegative. Moreover, dim(ker L ) = 1 with ker L c Aeven(EA)if det A > 0, while ker L c Aodd(EE)if det A < 0.

Prooj We write

with U E O(n). Also, let W E SO(n) be such that

JA*A = Wdiag(s1,. . . , s n } W * , where diag(s1,. . . , s n } denotes the diagonal matrix with each si i 5 n. Then one deduces that

Tr

[m] = 2%

> 0, 1 5

(4.27)

i=l

and n

n

C c(ei)E(eiA) = Cc(ei)?(eiUWdiag{sl,. . . , i-1

Now write

i= 1

sn}W*).

(4.28)

PoincarCHopf Index Formula: an Analytic Proof

66

From (4.28), one gets n

n.

n

=

C s j c (ejw*U*)Z ( e j w * > .

(4.29)

j=1

Set fj = e j W * , 1 5 j 5 n. They form another oriented orthonormal basis of En. From (4.26), (4.27) and (4.29), one finds n

L=

si (1

+ c ( f i U * )Z(fi)).

(4.30)

i=l

Now for any integer j such that 1 5 j 5 n, set

Then by (4.13) one verifies easily that each qj, 1 5 j 5 n, is self-adjoint and that q; = 1. Thus the lowest eigenvalue of each q j , 1 5 j 5 n, is -1. This proves that the operator L in (4.30) is a nonnegative operator. On the other hand, by (4.13) one also verifies qiqj = qjqi for 1 5 i,j 5 n. Moreover, one verifies that Z ( f j ) q j = - q j Z ( f j ) , while Z ( f j ) q i = qiZ((fj) when i # j. From these two facts one deduces easily, via induction, that

+

dim {x E A* (Ei) : (1 q j ) x = 0 for 1 5 j 5 n } =

dim A* (E:) 2"

= 1.

Moreover, let p E h*(E*,)denote one of the unit sections of ker L , then one

has

(4.31)

A Proof of the Poincart-Hopf Index Formula

67

Now it is easy to see that

From (4.31) and (4.32), one sees that p E Aeven/Odd det(U) = f l . This completes the proof of Lemma 4.8. 0

(EA) if and only if

Combining Lemma 4.8 with the properties of the (rescaled) harmonic oscillator K T , one gets (Compare with [S, Corollary 2.221)

Proposition 4.9 For any T > 0 , the operator

-2 i= 1

n

($)2

+ T x c ( e i ) Z ( e i A ) + T 2 (yAA*,y) i=l

acting on F(A*(EA)) is nonnegative. Its kernel is of dimension one and is generated b y (4.33)

Moreover, all the nonzero eigenvalues of this operator are greater than CT for some fixed constant C > 0 .

4.6 A Proof of the PoincarB-Hopf Index Formula We will use a simplified version of the analytic techniques developed by Bismut and Lebeau (cf. [BL, Chap. 91) t o prove (4.11). Without loss of generality we assume that each U p ,p E zero(V), is an open ball around p with radius 4a. Let y : R + [0,1] be a smooth function such that y(z) = 1 if IzI 5 a , and that y(z) = 0 if IzI 2 2a. For any p E zero(V), T > 0, set (4.34)

68

Poincari-Hopf Index Formula: a n Analytic Proof

(4.35) Then p , , ~ E n * ( M ) is a section of unit length with compact support contained in Up. Let ET denote the direct sum of the vector spaces generated by p P , ~ ' s . Then ET admits a Zz-graded decomposition

(resp. ET,odd)is the direct sum of the vector spaces generated where by those P,,T'S with det(A,) > 0 (resp. det(A,) < 0). Let E; be the orthogonal complement to ET in Ho(M). Then Ho(M) admits an orthogonal splitting

Ho(M)= ET CB E+.

(4.36)

Let p ~p k, denote the orthogonal projections from Ho(M)t o ET, E$ respectively. Following Bismut and Lebeau [BL, Chap. 91, we decompose the Witten deformation operator DT according to the splitting (4.36). That is, we define

(4.37) Let H1(M) denote the first Sobolev space with respect t o a (fixed) first Sobolev norm on O * ( M ) . We now state a crucial result which will be proved in the next section. Proposition 4.10 There exists a constant TO> 0 such that (a) f o r any T 2 TOand 0 5 u 5 1, the operator

is Fredholm; (ii) the operator D T , :~E$ n H1(M)-+ E$ is invertible.

Some Estimates for D T , I~s, 2

5a 54

69

Proof of the Poincard-Hopf formula (4.11). By (4.18), Proposition 4.10 and the homotopy invariance of the index of F’redholm operators, one deduces that for T 2 TO,

x ( M ) = ind (Dr: Cleven(M)-+ Clodd(M)) = ind

( D r ( 0 ): Cleven(M) -+ Clodd(M))

sgn(det(A,)).

= p&ero(V)

4.7

Some Estimates for D T , ~ ’ s2, 5 z

54

In this section, we prove Proposition 4.10. The proof will be based on certain estimates for the decomposed operators DT,~’s,2 5 i 5 4. These estimates are much simpler than the corresponding estimates in [BL, Chap. 91. We first state the estimates for

&,2

and D T , as ~ follows.

Proposition 4.11 There exists constant TO > 0 such that for any s E E; f~ H1(M),s’ E ET and T 2 TO,one has

(4.39)

Proof. It is easy to see that DT,3 is the formal adjoint of D T , ~ .Thus one needs only to prove the first estimate in (4.39). Since each p , , ~ , p E zero(V), has support in U p ,by (4.35) and Proposition 4.9 one deduces that for any s E E$ n H1(M),

70

Poincari-Hopf Index Formula: an Analytic Proof

Now since y equals to one in an open neighborhood around zero(V), d y vanishes on this open neighborhood. Thus by (4.40), one deduces easily that there exist constants TO> 0, C1 > 0 , C2 > 0 such that when T 2 TO, for any s E Ejk n H1(M),

~ ~ D T , z5sCIT”’~ ~ ~ o ex~(-CzT)llsllo,

(4.41)

from which the first inequality in (4.39) follows easily. 17

By Proposition 4.11, one sees that both D T , ~and D T , ~are compact operatorst mapping from H1(M) to Ho(M). Thus one gets the first part of Proposition 4.10. To get the second part of Proposition 4.10, one needs only t o show that there exist constants TO> 0, C, > 0 such that for any T 2 TOand s E E+ n H1(M), IIDT,4sllO 2 c3llsllO-

Now since for s E E+ n H1(M) one has

DTS = DT,ZS+ DT,QS, by Proposition 4.11, one needs only to show that for some constant C4 > 0 , IlDTsllo

2

~411s110,

when T > 0 is large enough.

Proposition 4.12 There exist constants TO> 0 and C > 0 such that for any s E Ek n H1(M) and T 2 TO, IlDTSllO

2 CJTllSllO.

tIn fact, they are finite rank linear bounded operators.

(4.42)

Some Estimates for D T , 's~, 2 5 i 5 4

71

Proof. For any 0 < b 5 4a, we denote U p ( b ) ,p E zero(V), the open ball around p with radius b. Following Bismut and Lebeau [BL, Chap. 91, we will prove Proposition 4.12 in the following three steps: (i) Step 1. We assume Supp(s) c ~ ~ , , , , , ( v ) U ~ ( 4 a ) ; (ii) Step 2. We assume Supp(s) c M \ Up,,e,o(~)Up(2a); (iii) Step 3. We prove the general case. We now start to prove Proposition 4.12 step by step. Step 1. We suppose Supp(s) E UpE,,,,(v)Up(4a). Then we can assume as well that we are in a union of Euclidean spaces Ep's containing Up's, p E zero(V), and can thus apply the results in Section 4.5. Thus, for any T > 0, p E zero(V), set

And for any section s verifying Supp(s) E UpEzero(V)Up(4a), set

pks =

Pb,T

1.

(&TI s> d'Ep.

(4.44)

pEzero(V)

Then p k is the orthogonal projection from @pEzero(~)Ho(Ep) to the finite dimensional vector space generated by P ; , ~ ,p E zero(V). Since ~ T =S0, we can rewrite p$s as

pks =

f;,T p€zero(V)

1 EP

((l-y(IY1)) ( ~ ) n i 4 J l r i P t o / e x P

As y equals to one near each p , by (4.45) there exists C:, when T 2 1, IIP;.sll;

c 5

I -11s11~.

0

By Proposition 4.9 and (4.44), we know that

'L'p~s)duE~*

>

(4.45) 0 such that

(4.46)

72

Poincark-Hopf Index Formula: a n Analytic Proof

Moreover, by (4.46) and Proposition 4.9, there exist constants C7 > 0 such that llDTSllo -

11 DT(S-&s)IIo

2

C6Tlls-&sllo

c 6

> 0,

2

C6T

2 ~ 1 l " l l ;- GflIIsII;, from which one sees directly that there exists TI

> 0 such that for any

T 2 Ti, (4.47) Step 2. Since now Supp(s) c M \ UpEsero(~)Up(2u), one can proceed as in the proof of Proposition 4.7 to find constants T2 > 0 and Cs > 0, such that for any T 2 T2,

ll&4lo 2 C s ~ l l s l l o .

(4.48)

Step 3. Let ;i; E C m ( M ) be defined such that on each Up,p E zero(V), ;i;(Y)= T(lYl/2), and that ;i;IM\UpEzero(")Up(4a)= 0Now for any s E E; n H1(M),one verifies easily that each

7s E E$ n H1(M). Thus, by the results in Steps 1 and 2, one deduces that there exists such that for any T 2 Ti T2,

+

llDTsll0

2

Cg

>0

1

5 (11(1- ?)DTSllO + II?DTSllO)

2 ClOJTllSllO

- C911S110,

where Clo = m i n { a / 2 , C8/2}, which completes the proof of Proposition 4.12. 0 The proof of Proposition 4.10 is thus also completed. KI

An Alternate Analytic Proof

4.8

73

An Alternate Analytic Proof

We observe that by Proposition 4.7, in the simplest case where V has no zeros on M , the PoincarB-Hopf formula is a direct consequence of Witten's deformation. On the other hand, a paper of Atiyah [A] contains another analytic proof of this simple statement, which we recall as follows. Let V be a nowhere zero vector field on M . Without loss of generality we assume that IVl = 1 on M . From (4.13) and (4.16), one verifies directly that

From (4.9), (4.49) and the homotopy invariance of the index for elliptic differential operators, one deduces that

x ( M ) = ind (d + d* : Oeven(M)-+ Oodd(M)) = ind (Z(V) ( d

-

(d

+ d * ) Z(V) : Oodd(M)+ Oeven(M))

+ d * ) + Z(V)

n

c(ei)Z(VzMV) : Oodd(M)-+ Oeven(M) a=

= ind (d

1

+ d* : Oodd(M)-+ Oeven(M))

from which one gets the desired equality x ( M ) = 0. In [Z], an alternate analytic proof of the PoincarB-Hopf index formula is given by extending the above idea of Atiyah to manifolds with boundary. The proof in [Z] works also for the case where the (isolated) zeros.of the vector field may be degenerate, i.e., the vector field in question may not be a transversal section of T M . This is different from Witten's proof presented in this section.

74

4.9

Poincard-Hopf Index Formula: a n Analytic Proof

References

[A] M. F. Atiyah, Vector fields on manifolds. Arbeitsgemeinschaft fur Forschung des Landes Nordrhein- Westfalen, Dusseldorf 1969, 200 (1970), 7-24.

[BL] J.-M. Bismut and G. Lebeau, Complex immersions and Quillen metrics. Publ. Math. IHES. 74 (1991), 1-297. [BoT] R. Bott and L. Tu, Differential Forms in Algebraic Topology, GTM 82, Springer-Verlag, Berlin-Heidelberg-New York, 1982. [de] G. de Rham, Differentiable Manifolds. Springer-Verlag, 1984. [GJ] J . Glimm and A. Jaffe, Quantum Physics. Springer, 1987.

[S] M. Shubin, Novikov inequalities for vector fields. The Gelfand Mathematical Seminar, 1993-1995. Birkhauser, Boston 1996, pp. 243-274. [W] F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups. GTM 94, Springer-Verlag, Berlin-Heidelberg-New York, 1983. [Wi] E. Witten, Supersymmetry and Morse theory. J. Diff. Geom. 17 (1982), 661-692.

[Z] W. Zhang, rpinvariants and the Poincark-Hopf index formula. in Geometry and Topology of Submanifolds A ' . Eds. W. H. Chen et. al., pp. 336-345, World Scientific, Singapore, 2000.

Chapter 5

Morse Inequalities: an Analytic Proof

In this chapter, we present Witten’s analytic proof of Morse inequalities by refining some of the arguments in Chapter 4. Witten’s original paper [Wl] has been very influential in various aspects in topology, geometry and mathematical physics. We will mention some of them in Section 5.7. We recommend the book of Milnor [Mi] for a beautiful account of some of the classical aspects of Morse theory. As in Chapter 4, we will work with real coefficients in this chapter.

5.1

Review of Morse Inequalities

Let M be an n-dimensional closed oriented manifold. Let f E C m ( M ) be a smooth function on M . A point z E M is called a critical point of f if

df(z)

= 0.

z E M is a critical point of f , then we say z is nondegenerate if the Hessian o f f a t z is non-singular, i.e.,

If

det(Hessf(z)) # 0.

It is easy to verify that every nondegenerate critical point z E M of f is isolated, that is, there is no other critical point of f in a sufficiently small open neighborhood of E M . A smooth function on M is called a Morse function if all the critical points of this function are nondegenerate. It is well-known (cf. [Mi]) that 75

Morse Inequalities: a n Analytic Proof

76

there always exists a Morse function on M . Clearly, a Morse function on a closed manifold has only a finite number of critical points. From now on we assume f is a Morse function on M . The following Morse lemma (cf. [Mi]) is important in many aspects of the theory of Morse functions.

Lemma 5.1 For any critical point x E M of the Morse function f , there is a n open neighborhood U, of x and a n oriented coordinate system y = ( y l , . . . ,y") such that o n U,, one has

f (y) = f (x)-

1

(y')2

- ...- f

(,.,(.))

2

+ -1 ( y " f ( x ) + l ) 2 + . . . + 1 ( f ) 2 . 2

(5.1) We call the integer n f ( x )the Morse index of f a t x . Also, for later use, we assume that for any two different critical points x, y E M of f ,

uxn u,= 0.

Now for any integer i such that 0 5 i 5 n, let Pi denote the i-th Betti number dim H & ( M ; R). Let mi denote the number of critical points x E M o f f such that n f ( x )= i. The Morse inequalities, for which an analytic proof will be given in this chapter, can be stated as follows.

Theorem 5.2 (i) Weak Morse inequalities: For any integer i such that 0I i 5 n, one has

Pi I mi.

(5.2)

(ii) Strong Morse inequalities: For any integer i such that 0 5 i 5 n, one has

Pi - Pi-1 + - . -+ (-I)',&

I mi - mi-1

+ . . . + (-1)'mO.

(5.3)

Moreover, Pn

-@"-I+ . . . + ( - I )

"Po

=mn-mn-1

+ a * . + ( -

1Inrno.

(5.4)

Clearly, (5.2) is a consequence of (5.3).* *In fact, one can apply (5.3) twice to i and i - 1 respectively, and then take sum to get (5.2).

W i t t e n Deformation

77

We refer to the book [Mi] for a topological proof of this result. In the rest of this chapter, we will present an analytic proof of it by following an idea of Witten [Wl]. 5.2

Witten Deformation

Recall from Section 1.1 the definition of the de Rham complex

(R*(M),d): O-Ro(M)

-Lfll(M) --% ... --% fldimM(M)-+ 0.

Given the Morse function f , inspired by considerations in physics, Witten [Wl] suggested to deform the exterior differential operator d as follows: for any T E R, set

dTf = e-Tf deTf.

(5.5)

Since d2 = 0, from (5.5) one has 2 dTf

= 0.

(5.6)

Thus, one can deform the de Rham complex (R*(M),d)to the complex (fl*(M),dTf) defined by

(R*(M),dTf): O+Ro(M)

dTr fll(M) dTr * . . dTr fldimM(M)+0.

Let

be the corresponding cohomology, with the Z-grading given by n

HFf,dR(';R)

= @H$f,dR(M;R), i=O

where for each integer i such that 0 5 i 5 n,

The first important result for the Witten deformation is as follows.

78

Morse Inequalities: an Analytic Proof

Proposition 5.3 For any integer i such that 0 5 i _< n, dimff$f,dR(hf;R) = d i m f f & ( M ; R ) .

Proof. For any a E O i ( M ) such that da! = 0, one verifies that d T f ( e - T f a ) = 0, while for any

p E Oi-'(hf), one has ePTfd,B = d T f ( e - T f p ) .

Thus, the map a!

E

R'(M)

H

e - T f a E o'(M)

induces a well-defined homomorphism from H&(M; R)to Hkf,dR(M;R ) . Similarly, one sees easily that the map

a E R'(M) H e T f a E R ~ ( M ) induces a well-defined homomorphism from H & ( M ; R) t o H$f,dR(M; R). It is now easy to verify that these two induced homomorphisms on cohomologies are in fact isomorphisms each of which is the inverse of the other one. 0 5.3

Hodge Theorem for ( O * ( M )d, ~ f )

Let gTM be a metric on T M . Recall that the Hodge theorem for the de Rham complex (R*(M),d ) has been reviewed in Section 4.1. Since T E R, by (4.4) one deduces that for any a, /3 E R*(M), (&fa,,B) = ( e - T f deTf a, 0) = ( a ,e T f d*e-Tf

p) .

Thus,

d>f := eTfd*e-Tf is the formal adjoint of d T f . Recall that D = d d*. For any T 2 0, set

+

DTf = dTf

+ d>f ,

(5.7)

Behaviour of O T f Near the Critical Points off

79

By (5.5) and (5.7), one sees that OTf preserves each fli(M), 0 5 i 5 n. Moreover, one can well establish the Hodge theorem for the complex ( f l * ( M )dTf), , a consequence of which implies that for any integer i such that 0 5 i 5 n, dim (kerOTfIni(M)) = dimH$f,dR(M;R) = d i m H & ( M ; R ) ,

(5.10)

where the last equality follows from Proposition 5.3. From (5.10), one sees that t o obtain the information about the pi's, one may take T -+ +m and study the behaviour of OTf under the limit.

Behaviour of O T ~Near the Critical Points o f f

5.4

Without loss of generality, we assume that on the open neighborhood Ux of a critical point x E M of f , with the coordinate system y = (y', . . . ,y") which are defined in Section 5.1, one has (5.11)

From (4.14)-(4.16), (5.5), (5.7) and (5.8), one verifies directly that dTf

=d

+ Tdf A,

dFf = d*

+ Tidf

and

where we identify df with its corresponding element in I'(TM) determined by gTM. Clearly, (5.12) is a special case of the deformation (4.17) in Section 4.3. However, the deformation operator in (5.12) has the advantage that the square of it preserves the Z-grading of R*(M), while the square of the deformation operator in (4.17) only preserves the Z2-grading of R*(M), in general. Now by the Morse lemma 5.1, one verifies that on each U,, one has df(z) = -pldpl - . . . - ynf(x)dynf(X)+ ynf(z)+ldynf(x)+l +

1 .

*

+.yndyn. (5.13)

&, 1 5 i 5 n, be the oriented orthonormal basis of T U X .

Let ei = Y

80

Morse Inequakties: an Analytic Proof

By (5.11)-(5.13) and the Bochner type formula (4.19), one deduces that on each U x ,

c($) n

OTf = -

2

- nT

+ T21yI2

i= 1

nf( I )

+T

C (1- c(ei)E(ei))+ T i= 1

n

+

(1 c(ei)E(ei)) i=nf (x)+1

(5.14) It is easy to verify that the linear operator

c

9

n f (2)

ieief A +

i=l

el A iei

i=nj (r)+l

is nonnegative, with the kernel being one dimensional and generated by

dyl A . . . A dynf (z). One then gets the folIowing 2-graded refinement of Proposition 4.9 in the current situation.

Proposition 5.4 For any T > 0 , the operator

acting on r(R*(E:)) is nonnegative. Its kernel is of dimension one and is generated by exp

(7)

-TlY12

. dy'

A

... A d y n f ( x )

Proof of Morse Inequalities

81

Moreover, all the nonzero eigenvalues of this operator are greater than CT C > 0.

f o r some fixed constant

5.5

Proof of Morse Inequalities

Recall that in the proof of the Poincarb-Hopf index formula in Section 4.6, we have used the deformation (4.38) t o reduce the proof t o a finite dimensional situation. However, if we would apply this deformation t o the operator D T ~now, we would see that the Laplacians of the deformed operators only preserve the &-grading of fl*(M),not the required 2-grading nature. Thus, one should deal with more refined arguments. Following Witten [Wl], we will instead prove the following result, from which the Morse inequalities will follow.

Proposition 5.5 For any c > 0 , there exists To > 0 such that when T 2 TO, the number of eigenvalues in [0,c] of O ~ f I n i ( 0~ 5 ) ,i 5 n, equals to mi. Proposition 5.5 will be proved in the next section. We now prove the Morse inequalities by using Proposition 5.5. For any integer i such that 0 5 i 5 n, let

F$ !i

c fl*(M)

denote the mi dimensional vector space generated by the eigenspaces of O ~ f l n . ( associated ~) with eigenvalues in [0,c ] . Since

and

d+fOTf = %fd?f one sees that d ~ (resp. f

=d&fdTfd&f,

Gf) maps each Fg$

t o Fg$+l (resp. FTlf,i--l). lo 4

Thus, one has the following finite dimensional subcomplex of (fl* ( M ) ,d T f ) :

(Fpp, dT,) : O-F[""] Tf,O

- -dTf

[O,c] dTf

FTfJ

...

dTf

F$$

+0.

(5.15)

Moreover, one can prove a Hodge decomposition theorem for this finite dimensional complex (or one can just apply the restriction of the Hodge

a2

Morse Inequalities: a n Analytic Proof

decomposition theorem for ( n * ( M )d, ~ f t o) this finite dimensional complex). In particular, for any integer i such that 0 5 i 5 n,

equals to dim(keroTfInicM,), which in turn equals t o /3i by (5.10). By Proposition 5.5, this implies the weak Morse inequalities. To prove the strong Morse inequalities, we examine the following decompositions obtained from the complex (5.15): for any integer i such that

Osisn,

&f[FEzyl) (Im

ker dTf IF[O,CI = dim

(Im

+ dim

dTflFgz\-l) +

(Im

dTf IFgzi)

’

(5.16) From Proposition 5.5 and (5.16), one deduces easily that for any integer i such that 0 5 i 5 n,

from which the strong Morse inequalities follows. In particular, we see that when i = n, the equality (5.4) holds. 0 Clearly, equality (5.4) is a special case of the Poincark-Hopf index formula proved in Chapter 4.

P T O O f Of PTOpOSZtiOn

5.6

5.5

83

Proof of Proposition 5.5

We will proceed as in Sections 4.6 and 4.7, which in turn rely on techniques developed in [BL, Chap. 91, to prove Proposition 5.5. As in (4.34) and (4.35), in view of Proposition 5.4, for any T > 0 and critical point z E M o f f , set

Then p , , ~E P f ( 2 ) ( M )is of unit length with compact support contained in U,. Let ET denote the direct sum of the vector spaces generated by ~ , , T ' s , where z runs through the set of critical points o f f . Let E$ be the orthogonal complement t o ET in Ho(M). Then Ho(M) admits an orthogonal splitting

Ho(M)= ET CB E+.

(5.18)

Let p ~p: , denote the orthogonal projection operators from Ho(M)t o E T , E$ respectively. As in (4.37), we decompose the Witten deformed operator DTf by setting

As in Section 4.7, the estimates summarized in the following proposition are crucial. Proposition 5.6 (i) For a n y T > 0, (5.20)

Morse Inequalities: a n Analytic Proof

84

(ii) There exists constant TI > 0 , such that for any s E E; n H1(M), s' E ET and T 2 TI, one has

IIDT,3S'IIO

Ils'llo. IT '

(iii) There exist T2 > 0 and C > 0 such that for any T 2 T2,

(5.21) s E

Ejk

fl H1( M ) and

ProoJ (i) Let zero(df) denote the set of critical points o f f . Then for any s E Ho(M),one verifies directly that

By (5.17) it is clear that for any x E zero(@),

has compact support in U,. Thus, (5.20) follows. (ii) This is a special case of Proposition 4.11. (iii) This is a special case of Proposition 4.12. The proof of Proposition 5.6 is completed. 17

Remark 5.7 Similarly, one can show that the operator D T , in ~ Section 4.6 is also a zero operator. We did not make this explicit since this fact was not used there. Now for any positive constant c > 0, let ET(c)denote the direct sum of eigenspaces of D T ~ associated with the eigenvalues lying in [-c, c]. Clearly, ET(c)is a finite dimensional subspace of Ho(M). Let &(c) denote the orthogonal projection operator from Ho(M) to

ET (4.

Proof of Proposition 5.5

Lemma 5.8 There exist C1 (7 E ET,

85

> 0 , T3 > 0 such that for

IIPT(C)O

-410

a n y T 2 T3 and any

C1 I rll.llo.

(5.25)

Proof. Let 6 = {A E C : ) A ] = c} be the counter-clockwise oriented circle. By Proposition 5.6, one deduces that for any X E 6, T 2 TI T2 and s E H1(M),

+

Il(A

-

D T f ) 410 L

1

2 IIAPTS - DT,2P+SllO

By (5.26), one sees that there exist T4 > TI for any T 2 T4 and s E H'(M),

Il(A - DTf) sllo 2 Thus, for any T >_

T4

+ T2 and C2 > 0 such that

~2llSllO~

(5.27)

and X E 6, X -D T ~ : H'(M)

+ Ho(M)

is invertible.

Thus, the resolvent (A - D T ~ ) - ' is well-defined. By the basic spectral theorem in operator theory (cf. [D]), one has PT(C)U

1

(5.28)

- u = ___

Now one verifies directly by Proposition 5.6(i) that

(0 - DTf1-l

- A-')

CT

= X - l (A - D T ~ ) - 'D ~ , s u .

(5.29)

From Proposition 5.6(ii) and (5.27), one then deduces that for any T 2

T4 and u E E T ,

From (5.28)-(5.30), one gets (5.25).

0

Morse Inequalities: a n Analytic Proof

86

Remark 5.9 Though there have been used complex numbers in the above proof (by which one needs to complexify the spaces and extend the operators accordingly, though this was not stated explicitly in the proof), one can well stay in the real coefficient category by working with the real part of the right hand side of (5.28). We leave these t o the interested reader.

Proof of Proposition 5.5. By applying Lemma 5.8 t o the ,o~,T'sfor E zero(df), one sees easily that when T is large enough, the P ~ ( c ) p , , ~ ' s for z E zero(@) are linearly independent. Thus, there exists T5 > 0 such IC

that when T

2 T5, dim ET(c)2 dim ET.

(5.31)

Now if dim ET(c)> dim E T , then there should exist a nonzero element s E ET(c)such that s is perpendicular to PT(c)ET.That is, ('7

(5.32)

PT(c)P2,T)HO(M) =

for any z E zero(@). From (5.23) and (5.32), one deduces that

+

c

('1

P2,T - 'T(')PZ>T)HO(M)

'T(')PZ,T'

(5.33)

zEzero(df)

By (5.33) and Lemma 5.8, there exists

C3

> 0 such that when T 2 T5,

c 3

IlPTSllO

Thus, there exists a constant

C4

Ir l l 4 o .

(5.34)

> 0 such that when T > 0 is large enough,

llPilSll0 2 llsllo - IlPTsllo

L

C4ll~llO.

(5.35)

87

Proof of Proposition 5.5

From (5.35) and Proposition 5.6, one sees that when T enough,

> 0 is

large

~ ~ 4 ~ l II ~IIDTfP+s((o l l o = IIDTfS - ~ T f P T s l l o

I IIDTfSllO

1

+ TllSllO,

from which one gets

1

IIDTfSllO

2 C C 4 ~ l l ~ ll OT;llsllo.

Clearly, when T > 0 is large enough, this contradicts with the assumption that s is a nonzero element in ET(c). Thus, one has n

dim ET(c)= dim ET =

mi.

(5.36)

i=O

Moreover, ET(c) is generated by P ~ ( c ) p , , ~for ' s all z E zero(@). Now in order to prove Proposition 5.5, for any integer i such that 0 5 i 5 n, denote by Qi the orthogonal projection operator from Ho(M) onto the L2-completion space of na(M). Since O T ~preserves the Z-grading of f l * ( M ) ,one sees that for any eigenvector s of D T ~associated with an eigenvalue p E [-c, c ] , OTf QiS = QinTfS = p2Qis.

That is, Qis E Ri(M) is an eigenvector of O T ~ associated with eigenvalue P2.

Thus, in order to prove Proposition 5.5, one needs only t o show that when T > 0 is large enough, dim Q ~ E T ( c=) mi.

(5.37)

To prove (5.37), one uses Lemma 5.8 t o see that for any z E zero(#), (5.38)

Morse Inequalities: a n Analytic Proof

88

From (5.38), one sees that when T > 0 is large enough, theforms Q n f ( ” ) P ~ ( c ) p , ,z~ ,E zero(@), are linearly independent. Thus, for each integer a between 0 and n, dimQ2ET(c) 2 m,.

(5.39)

On the other hand, by (5.36) one has n

n

x d i m Q i E T ( c ) 5 dim&(C) = x m i . i=O

(5.40)

i=O

From (5.39) and (5.40), one gets (5.37). The proof of Proposition 5.5 is completed. 0

Remark 5.10 Since the constant c > 0 in Proposition 5.5 can be chosen arbitrarily small, one sees that when T --+ +m, the eigenvalues in [O,c]of U T converge ~ t o zero. 5.7

Some Remarks and Comments

1). Witten’s original paper [Wl] was very influential in 1980’s. Many rigorous accounts of the analytic proof of the Morse inequalities appeared right after the appearance of [Wl]. Here we only mention the paper by Helffer-Sjostrand [HS] which was based on semi-classical analysis and the paper by Bismut [B]where a proof by heat equation methods was developed. The later also contains an analytic treatment of Bott-Morse inequalities which hold when the critical points are only nondegenerate in the sense of Bott [Boll. 2). Witten further suggested in [Wl] that under some generic conditions, from the complex (F$;”],d~f) defined in (5.15) one can even recover the Thom-Smale complex (cf. [L]) associated to the Morse function f. Witten’s this idea, which was proved rigorously in [HS] (Compare also with [BZ2, Sect. 6]), has a tremendous influence on the subsequent developments. For example, it is one of the sources for Floer’s conception [F] of Floer homology (cf. [Bo2] for a nice informal account on these). In another direction, Bismut and Zhang [BZl] used these ideas to give a heat kernel proof, as well as an extension t o the case of general flat vector bundles, of the theorems of Cheeger [C] and Muller ([Mull, [Mu2])on relations between

References

89

the Ray-Singer analytic torsion [RS] and the Reidemeister torsion. Most recently, a far reaching generalization of the main results in [BZl] and [BZ2] to the case of fibrations has been obtained by Bismut and Goette, see [BGl] and [BG2] for more details.

3). In a subsequent paper [W2],Witten also proposed certain holomorphic Morse inequalities for circle actions on Kahler manifolds. These holomorphic Morse inequalities were first proved rigorously by Mathai and Wu [MW] by a heat equation method for the case where the fixed point set of the circle action consists of isolated points. The paper [WZ] contains a proof by using the analytic arguments similar to what in this chapter. It also covers the case where the fixed point set of the circle action may be non-isolated. 4). The analytic localization methods described in Chapters 4 and 5, with necessary technical refinements if needed, are very useful for a wide range of problems in index theory (cf. [BL]). We hope to have shown that the basic ideas involved are in fact very simple.

5.8

References

[B] J.-M. Bismut, The Witten complex and degenerate Morse inequalities. J. Dig. Geom. 23 (1986), 207-240. [BGl] J.-M. Bismut and S. Goette, Formes de torsion analytique en thkorie de de Rham et fonctions de Morse. C. R. Acad. Sci. Paris, Skrie I, 330 (2000), 479-484. [BG2] J.-M. Bismut and S. Goette, Families torsion and Morse functions. Prkpublication 2000-59, Mathkmatique, Univ. Paris-Sud, Orsay. To appear in Aste'risque.

[BL] J.-M. Bismut and G. Lebeau, Complex immersions and Quillen metrics. Publ. Math. IHES. 74 (1991), 1-297. [BZl] J.-M. Bismut and W. Zhang, An extension of a theorem by Cheeger and Muller. Astkrisque Tom. 205, SOC.Math. France, 1992.

90

Morse Inequalities: a n Analytic Proof

[BZ2] J.-M. Bismut and W. Zhang, Milnor and Ray-Singer metrics on the equivariant determinant of a flat vector bundle. Geom. Funct. Anal. 4 (1994), 136-212. [Boll R. Bott, Nondegenerate critical manifolds. Ann. of Math. 60 (1954), 248-261. [Bo2] R. Bott, Morse theory indomitable. Publ. Math. IHBS. 68 (1989), 99-114.

[C] J. Cheeger, Analytic torsion and the heat equation. Ann. of Math. 109 (1979), 259-322. [D] R. G. Douglas, Banach Algebra Techniques in Operator Theory. Academic Press, New York, 1972.

[F]A. Floer, An instanton invariant for three-manifolds. Commun. Math. Phys. 118 (1988), 215-240. [HS] B. Helffer and J. Sjostrand, Puits multiples en mhcanique semi-classique IV: Edude du complexe de Witten. Commun. P. D.E. 10 (1985), 245-340. [L] F. Laudenbach, On the Thom-Smale complex. Appendix to [BZI].

[MW] V. Mathai and S. Wu, Equivariant holomorphic Morse inequalities I: a heat kernel proof. J. Diff. Geom. 46 (1997), 78-98. [Mi] J. Milnor, Morse Theory. Princeton Univ. Press, 1963.

[Mull W. Muller, Analytic torsion and R-torsion for Riemannian manifolds. Adv. in Math. 28 (1978), 233-305. [Mu21 W. Muller, Analytic torsion and R-torsion for unimodular representations. 3.Amer. Math. Soc. 6 (1993), 721-753

[RS] D. B. Ray and I. M. Singer, R-torsion and the Laplacian on Riemannian manifolds. Adv. in Math. 7 (1971), 145-210.

References

91

[Wl] E. Witten, Supersymmetry and Morse theory. J. Diff. Geom. 17 (1982), 661-692. [W2]E. Witten, Holomorphic Morse inequalities. Algebraic and Differential Topology, Teubner-Text Math., 70,ed. G. Rassia, Teubner, Leipzig, (1984), pp. 318-333. [WZ] S. Wu and W. Zhang, Equivariant holomorphic Morse inequalities 111: non-isolated fixed points. Geom. Funct. Anal. 8 (1998), 149-178.

Chapter 6

Thorn-Srnale and Witten Complexes

In the previous chapter, we presented an analytic proof of Morse inequalities by using the Witten deformation of the de Rham complex. We also pointed out in Section 5.7 that in his seminal paper [Wi], Witten further suggested that the Thom-Smale complex associated to generic Morse functions can also be recovered from his deformation, and that Witten's this suggestion was first realized rigorously by Helffer and Sjostrand [HS] by using semiclassical approximation methods. In this chapter we will examine this point of view of Witten by adapting a simpler treatment appearing first in the paper of Bismut and Zhang [BZ2]. Since this chapter is closely related to the previous one, we will make the same assumptions and use the same notation as in the last chapter.

6.1

The Thom-Smale Complex

Let f E C " ( M ) be a Morse function on an n-dimensional closed oriented manifold M . Let g T M be a metric on T M , and let

Of= (df)* E r ( T M ) be the corresponding gradient vector field of f . Then the following differential equation defines a group of diffeomorphisms ( $ t ) t E ~ of M : dy =

dt

-v f (y). 93

T h o m - S m a l e and Witten Complezes

94

If

3:

E zero(Vf), set

The cells W " ( x ) and W s ( x )will be called the unstable and stable cells at x respectively. We assume that the vector field V f verifies the Smale transversality conditions [S]. Namely, we suppose that for any IC, y E zero(Vf) with x # y, W " ( x )and Ws(y) intersect transversally. In particular, if n f ( y ) = nf(x)- 1, then W"(x)fl Ws(y) consists of a finite set I'(x,y) of integral curves y of the vector field -V f , with y-oo = x and y+m = y, along which W u ( x )and Ws(y) intersect transversally. By [S, Theorem A], given a Morse function f , there always exists a metric g T M on TM such that V f verifies the transversality conditions. We fix an orientation on each W " ( x ) ,x E zero(Vf). Let 2, y E zero(Vf) with n f ( y ) = nf(x)- 1. Take y E r ( x ,y). Then the tangent space TyWu(y)is orthogonal to the tangent space TyWs(y)and is oriented. For any t E (--00, +-00), the orthogonal space T$W"(y) to TYtWS(y)in TYtM carries a natural orientation, which is induced from the orientation on TyWu(y). On the other hand, also for t E (--00,+00), the orthogonal space TJtW"(z)to -Vf(yt) in TYtW"(x)can be oriented in such a way that s is an oriented basis of T4tW"(z)if (-Vf(yt),s) is an oriented basis of

TYt W" (x). Since W u ( x )and Ws(y) are transversal along y,for any t E (-00, +m), T$W"(y) and T4tW"(x)can be identified, and thus one can compare the induced orientations on them. Set nY(x,y) = 1 if the orientations are the same,

= -1

if the orientations differ.

The de R h a m Map for Thom-Smale Complexes

95

If x E zero(Vf), let [W"(z)] be the real line generated by W u ( x ) .Set

@

C*(W")=

[W"(3:)11

xEzero(Vf)

@

Ci(W")=

[WU(x)].

sE.ero(Vf)

nr(x)=i

If x E zero(Vf), set

c c

a w u ( x )=

(6.5)

ny(x,y)WY(d.

yEr(x,y)

uEZero(Vf)

nf(Y)=nf (x1-1

Then a maps Ci(W") to Ci- 1 (W"). The following basic result is due to Thorn [TI and Smale [S].

Theorem 6.1 (C*( W " ) ,a) is a chain complex. Moreover, we have a canonical identification between its Z-graded homology group H,(C,(W"), a) and the Z-graded singular homology group H , ( M ) .

If 3: E zero(Vf), let [W"(z)]* be the line dual to [ W " ( x ) ]Let . (C*(W"), 5) be the complex which is dual to (C*(W"), a). For any integer i such that 0 5 i 5 n, we have the identity

Ci(W")=

@ [W"(.)]*. zEzero(Vf)

nr(x)=i

Then by Theorem 6.1, one has the identification of the Z-graded cohomology spaces

H* ( C * ( W " ) , g )21 H,*,,,(M). 6.2

(6.7)

The de Rham Map for Thom-Smale Complexes

We now assume that for any x E zero(Vf), there exists a sufficiently small open neighborhood U, of x and a coordinate system y = (y', ...,)!y on Ux such that on U x ,

f(y) = f ( x ) -

1

2 (yl) - * . . -

;

( ? p X ) )

2

1

2

+ -2 ( y n , ( x ) + ' ) + . . . +

1

(y")2

,

Thom-Smale and Witten Complexes

96

Certainly we can assume that for any x, y E zero(Vf)with x

# y, UxnU, =

0. We still assume that V f verifies the Smale transversality conditions. By the Morse lemma 5.1 and by [S], given a Morse function f , there always exists a metric g T M on T M verifying the above conditions. We now state a result of Laudenbach [L, Prop. 21 which improves an old result of Rosenberg [R]. P r o p o s i t i o n 6.2 (i) If x E zero(Vf), then the closure r ( x ) is an n f ( z ) dimensional submanifold of M with conical singularities; (ii) wu(x) \ Wu(x) is stratified by unstable manifolds of critical points of index strictly less than nf(x). We refer to the original paper [L] for the proof of Proposition 6.2. By Part (i) of Proposition 6.2,one sees that one can integrate smooth forms over WU(x)'s,x E zero(Vf). If x E zero(Vf ), then the line [W'"(x)]has a canonical non-zero section Wu(x). Let W'(x)* E [Wu(x)]*be dual to W u ( z )so that

(Wu(x),Wu(x)*)= 1.

If

Q

E

Q * ( M ) ,then the integral WU(X)*

lies in [Wu(x)]*.Clearly, if nr(z)= i.

Q

E

O ' ( M ) , then JEu(z) a is non-zero only if

Definition 6.3 Let P, be the map Q

E

R*(M)

-

(6.9) zEzero(Vf)

Theorem 6.4 (Laudenbach, cf. [BZl, Theorem 2.91) The map P, is a Z-graded quasi-isomorphism between the de Rham complex (Cl*(M),d ) and

Witten's Instanton Complex and the Map eT

97

the dual Thom-Smale complex ( C * ( W " )g), , which provides the canonical identification of the cohomology groups of both complexes. The following particular formula, which shows that P, is actually a chain homomorphism, follows in fact easily from (6.5), Proposition 6.2 and the Stokes formula,*

P,d

-

= aP,.

(6.10)

What Witten [Wi] suggested is that Theorem 6.4 can be recovered from the deformations (5.5) and (5.7) by letting T -+ +m. This was first realized by Helffer-Sjostrand [HS] by using semi-classical approximation methods. In the next sections we will present a treatment which is adapted from [BZ2, Section 61.

6.3 Witten's Instanton Complex and the Map eT Let To > 0 be such that Proposition 5.5 holds for c = 1 and any T 2 TO. From now on we always assume that T 2 TO. Recall from Section 5.5 that for any integer i such that 0 5 i 5 n,

is the mi dimensional vector space generated by the eigenspaces of O T I~n i ( M ) associated t o the eigenvalues lying in [0,1], and that one has the finite dimensional subcomplex (5.15) of (R*(M),d T f ) ,

We call (FF;'], dTf)the Witten instanton complex associated t o T f . Now we equip C*(W")with a metric such that for any 5 , y E zero(V f ) with x # y, W"(x)* and W " ( y ) * are orthogonal t o each other, and that

( W " ( x ) *W.(z)*) , =1 for each x E zero(Vf ) . ~ R*(M) Recall that for any 2 E zero(Vf) and T 2 TO,the section p z , E has been defined in (5.17). *Compare with [L, Proposition 61.

98

Thom-Smale and Witten Complexes

Definition 6.5 Let JT be the linear map from C*(Wu)into R*(M) such that for any x E zero(Vf) and T 2 To,

Clearly, JT is an isometry from C*(W")into R*(M),which preserves the Z-gradings. Let P;"] denote the orthogonal projection from R*(M)on F$)ll. Clearly,

P$"] is exactly the orthogonal projection operator PT(c)defined in Section ;! is easily seen to be the same as &(c) 5.6 with c = 1, as the space ]F with c = 1 there. Definition 6.6 Let eT : C*(W")4 'F ;!

be given by

eT = P$JlJT.

(6.13)

The following result, which refines Lemma 5.8 significantly, is taken from [BZl, Theorem 8.81 and [BZ2, Theorem 6.71.

Theorem 6.7 There exists c

> 0 such that as T

-+

+oo, f o r any s E

c*(WU), (eT - J T ) s = 0 (e-CT) ))s110 uniformly o n M .

(6.14)

In particular, eT is an isomorphism. ProoJ Let 6 = {A E C : 1x1 = 1) be the counter-clockwise oriented circ1e.t Then we can write as in (5.28) that for any x E zero(Vf) and T > 0 large enough, (eT - J T ) W " ( x ) * = P T[O' 11Px,T - Pz,T

(6.15) tCompare with Remark 5.9 on the use of complex coefficients.

Witten’s Instanton Complex and the Map eT

99

For any p 2 0, let 11 . l i p denote the p t h Sobolev norm on R*(M). From Proposition 5.4 and the definition of p z , ~ one , sees that on a (fixed) sufficiently small open neighborhood of 2, one has

DTfPx,T = 0.

(6.16)

By (5.17) and (6.16), for any positive integer p , there is cp that as T 4 +co,

> 0 such (6.17)

llDTfPx,Tllp = 0 ( e - c p T ) . C1

Take q 2 1. Since D is a first order elliptic operator, there exist C s E R*(M),then

> 0,

> 0 and C2 > 0 such that if llsllq

I c1 (IIDsIlq-1 + 11~110~

(6.18)

I CTq([[(A - DTf)sIlq-l iIISIIO) ,

where the last inequality follows from an induction argument. On the other hand, by (5.27) one deduces easily that there exists C‘ > 0 such that for X E S, s E R*(M) and T > 0 large enough, [[(A - DTf)-lSllo 5 By (6.18) and (6.19), there exists C” enough, - D T ~ ) - ~ s II I , CTq(11S1lq-i

(6.19)

C’IISllO.

> 0 such that if T > 0 is large

+ C’llsllo)

I C”Tqllsllq-i.

By (6.17) and (6.20), there exists cq > 0 such that when T enough,

Il(X - D ~ f ) - ~ & f p ~=, 0 ~ (e-cqT) l / ~ , uniformly on X

E

(6.20)

> 0 is large 6.

(6.21)

Using (6.21) and Sobolev’s inequality (cf. [W, Corollary 6.22(b)]), we see that there exists c > 0 such that

[(A - ~

o

T f ) - l ~ T f p ~5, T l( e - c T ) ,

uniformly on M .

(6.22)

By (6.15) and (6.22), (6.14) holds for any s = Wu(x)*with z E zero(Vf). It then clearly holds for any s E C*(Wu).

Thom-Smale and Witten Complexes

100

Since JT is an isometry from C*(W")into R*(M),from (6.14) one sees easily that eT is an isomorphism when T > 0 is large enough. 0

6.4

The Map P,,TeT

Recall from (6.10) that the de Rham map

is a chain homomorphism between the complexes. From (5.5), (5.15) and (6.23), one verifies easily that if T enough, then the map

P m ,: ~ F]'!;

-

> 0 is large

c*(W")

defined by

poo,T: a H pooeTfa

(6.24)

is also a chain homomorphism of complexes. That is, when acting on F!];', one has

Definition 6.8 Let 3 E End(C*(W")) which, for z E zero(Vf), acts on [W"(z)]* by multiplication by f (z). Let N E End(C*(W")) which acts on C i ( W " ) 0, 5 i 5 n, by multiplication by i. The following result is taken from [BZ2, Theorem 6.111.

Theorem 6.9 There exists c > 0 such that as T

--f

+a, (6.26)

I n particular, P-,T is an isomorphism for T > 0 large enough. Proof. Take z E zero(Vf), s = W"(z)*.

101

By (6.23) and (6.24), we get

P , , T ~ T ~=

C

eTf(Y)Wu(y)*

(6.27)

eT(f-f(Y))eT s.

uEzero(Vf)

nf(Y)=nf ( X I

Clearly, for any y E zero(Vf), one has

f-f(y) 5 0

on

W(Y).

(6.28)

Since by Proposition 6.2(i) the W ( y ) ' s are compact manifolds with conical singularities, by Theorem 6.7 and (6.28), we see that if y E zero(Vf) with nf ( y) = nf(z), then

for some c > 0. Since the support of JTS is included in U,, using (5.17), (6.8), (6.12) and (6.29), we find that

Take now y E zero(Vf). By Proposition 6.2(ii) we know that m"(y)\W"(y) is a union of certain W"(y'), with nf(y') < n f ( y ) . Thus we find that for y E zero(Vf) with y # z and n f ( y ) = nf(z), then 5

6 w-(Y).

From (5.17), (6.12) and (6.31), we deduce that there is c' y E zero(Vf) with y # z and n f ( y ) = nf(z), then

JTS = 0 (e-c'T)

(6.31)

> 0 such that if

on W(y).

Using (6.28) and (6.32), we see that if y E zero(Vf) with y nf(Y) = nf(x),then

(6.32)

#

z and

(6.33) From (6.27), (6.29), (6.30) and (6.33), one gets (6.26) easily. The proof of Theorem 6.9 is completed. 0

Thorn-Smale and Witten Complexes

102

We refer to the paper of Bismut and Goette [BL] for a geneneralization of the results in above sections t o the case of fibrations. 6.5

An Analytic Proof of Theorem 6.4

Recall from (6.25) that when T

> 0 is large enough,

is a chain homomorphism. Thus it induces a homomorphism on cohomology groups

On the other hand, by Theorem 6.9, when T > 0 is large enough, P w , ~ is an isomorphism. Thus by (6.25) again one sees that

is also a chain homomorphism, which induces a homomorphism on cohomology groups

(PGt.>

H

: H*

-

(C*(W"),g)

H* ( F ! j ' l , d ~ f ) .

Clearly, Pz,T and (P&)H are inverse to each other. Thus P o o ,in~ duces a canonical isomorphism between H * ( C * ( W " ) 5) , and H * ( F g f l ,d ~ f ) which clearly preserves the Z-gradings. Theorem 6.4 then follows easily from Proposition 5.3 and from (6.24). 0

6.6

References

[BG] J.-M. Bismut and S. Goette, Families torsion and Morse functions. PrCpublication 2000-59, MathCmatique, Univ. Paris-Sud, Orsay. To appear in Astkrisque. [BZl] J.-M. Bismut and W. Zhang, An extension of a theorem by Cheeger and Muller. Aste'risque Tom. 205, SOC.Math. France, 1992.

References

103

[BZ2] J.-M. Bismut and W. Zhang, Milnor and Ray-Singer metrics on the equivariant determinant of a flat vector bundle. Geom. Funct. Anal. 4 (1994), 136-212. [HS]B. Helffer and J. Sjostrand, Puits multiples en mkcanique semi-classique IV: Edude du complexe de Witten. Commun. P. D. E. 10 (1985), 245-340. [L] F. Laudenbach, On the Thom-Smale complex. Appendix to [BZI].

[S] S. Smale, On gradient dynamical systems. Ann. of Math. 74 (1961), 199-206. [R] H. Rosenberg, A generalization of Morse-Smale inequalities. Amer. Math. SOC.70 (1964), 422-427.

Bull.

[TI R. Thom, Sur une partition en cellules associke B une fonction sur une variktk. C. R. Acad. Sci. Paris, Se'rie A, 228 (1949), 973-975. [W]F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups. GTM 94, Springer-Verlag, Berlin-Heidelberg-New York, 1983. [Wi] E. Witten, Supersymmetry and Morse theory. J. Diff. Geom. 17 (1982), 661-692.

Chapter 7

Atiyah Theorem on Kervaire Serni-characteristic

Recall that in Chapter 4 we have proved the Poincark-Hopf index formula (4.11) by making use of the deformation introduced by Witten [W]. Now if one changes V to -V in (4.11), one sees easily that the left hand side does not change, while the right hand side will change by a factor (-l)dimM. As a consequence, if dim M is odd, the Euler characteristic x ( M ) vanishes. On the other hand, a theorem due t o Hopf (cf. [S]) states that if a closed manifold M has vanishing Euler characteristic, then there exists a nowhere zero vector field on M.Thus, there always exists a nowhere zero vector field on an odd dimensional closed manifold. In this chapter we will discuss the following result due t o Atiyah [A], which considers the possibility of the existence of two linearly independent vector fields on 4q 1 dimensional manifolds.*

+

Atiyah vanishing theorem If there exist two linearly independent vector fields on a 4q 1 dimensional oriented closed manifold, then the Kervaire semi-characteristic of this manifold vanishes.

+

In this chapter, we will show that this result can also be proved by using the Witten type deformations similar to what has been used in Chapter 4. We will start with the definition of the Kervaire semi-characteristic.

*Dupont [D] has proved that there always exist three linearly independent vector fields on a 49 3 dimensional oriented closed manifold. This generalizes the classical three dimensional result of Stiefel.

+

105

Atiyah Theorem

106

7.1

on

Kervaire Semi-characteristic

Kervaire Semi-characteristic

+

Let M be a 4q 1 dimensional smooth closed oriented manifold. By definition, the Kervaire semi-characteristic of M , denoted by k ( M ) ,is an element in 2 2 defined by 29

k(M)z x d i m H & ( M ; R )

mod 2 2 .

(7.1)

i=O

One may think of k ( M ) as a mod 2 analogue of the Euler characteristic on odd dimensional manifolds. In particular, it admits an analytic interpretation via the Hodge decomposition theorem. We first describe this analytic interpretation, which is due t o Atiyah and Singer [AS], as follows. Take a metric g T M on TM. Let e l , . . . ,e4q+1 be a (local) oriented orthonormal basis of T M . We will use the same notation for Clifford actions and so on as in Chapter 4.

Definition 7.1 Let Dsig be the Signature operator defined by

Dsig = c(e1). . . c(eqq+l)( d + d * ) : aeven(M)-+ aeVen(M). A

A

(7.2)

Clearly, the operator Dsi,, is a well-defined first order elliptic differential operator. Moreover, by using (4.13) and (4.16), one verifies directly that Dsig is skew-adjoint. That is, for any s ,s’ E aeven(M),

(Dsigs,s’) = - (s,Dsigs’).

(7.3)

On the other hand, by Corollary 4.4, which is a consequence of the Hodge decomposition theorem, and by (4.8), one has 2q

dim (ker Dsig) =

dim

( M ;R).

(7.4)

i=O

Now for any skew-adjoint elliptic differential operator D , following Atiyah and Singer [AS], one can define an element in 2 2 , which is called the mod 2 index of D, as follows, ind2 D = dim (ker 0 )

mod 22.

(7.5)

Atiyah’s Original Proof

107

Furthermore, Atiyah and Singer showed that this mod 2 index is a homotopy invariant. That is, if D ( u ) , 0 5 u 5 1, is a smooth family of skew-adjoint elliptic differential operators on a closed manifold, thent indz D(1) = inda D(0)

in

Z2.

(7.6)

From (7.1), (7.4) and (7.5), one can write Atiyah-Singer’s analytic interpretation of the Kervaire semi-characteristic of M as follows,

k ( M ) = ind2 Dsig. 7.2

(7.7)

Atiyah’s Original Proof

Let V1, V2 E I’(TM) be two smooth vector fields on M . We assume that they are linearly independent over M . That is, for any z E M , Vl(z) and Vz(z) are linearly independent in T,M. Following Atiyah [A], we now show that under this situation, one has k ( M ) = 0 in Z2. Without loss of generality, we take a metric g T M such that for any 2 E M , Vl(2) and V2(z) are orthogonal to each other, and that Vl(z) and Vz(2) are of norm one. Following [A], we construct the following differential operator

?This follows from the easy fact that if a finite dimensional Euclidean space admits a skew-adjoint automorphism, then it is of even dimension.

Atiyah Theorem on Kervaire Semi-characteristic

108

From (7.10), one sees that D' is a first order elliptic differential operator. On the other hand, by (4.13) and (7.8), one verifies directly that D' is skew-adjoint. Thus, by using (7.10) again, one sees that for any u E [0,1],

D(u) = (1 - u)Dsig+ uD'

(7.11)

is elliptic and skew-adjoint. By (7.11) and the homotopy invariance of the mod 2 index, one then gets ind2 Dsig = ind2 D'.

(7.12)

Now by our assumption on gTM and by (7.8), one verifies directly that

Z(Vl)Z(V2), which preserves R""""(M), commutes with D'. Thus, i?(Vl)i?(V(v~) preserves the kernel of D'. On the other hand, one checks that

(Z(v1)Z(vz))2 = -1.

(7.13)

forms a comlpex struction on ker D', which implies By (7.13), Z(V1)Z(Vz) that dim (ker D')

=0

mod 22.

(7.14)

F'rom (7.7), (7.12) and (7.14), one gets the vanishing property of k ( M ) . 0

Remark 7.2 Conversely, Atiyah [A] and Atiyah-Dupont [AD] have shown that for a 4q 1 dimensional oriented closed manifold M , if both k ( M ) and the 4q-th Stiefel-Whitney class of T M vanish,$ then there exist two linearly independent vector fields on M .

+

7.3 A proof via Witten Deformation In this section, we present an alternate proof of the Atiyah vanishing theorem by adapting the deformation idea of Witten. We first give an alternate analytic interpretation of the Kervaire semicharacteristic k ( M ) . Let g T M be chosen as in the previous section. In this section, we denote by V = Vl and X = Vz. $See [MS] for a definition of the Stiefel-Whitney class of vector bundles.

A proof via W i t t e n Deformation

109

Definition 7.3 ([Zl]) Let DV : Oeven(M)+ Oeven(M)be the operator d&ned by 1

DV = - ( E ( V )( d + d * ) - ( d + d*) E ( V ) ) . 2

(7.15)

Clearly, DV is skew-adjoint. On the other hand, by using (4.13) and (4.16), one verifies directly that

from which one knows that D V is also an elliptic differential operator of order one. The following result, which is taken from [Zl], shows that the mod 2 index of DV gives an alternate analytic interpretation of k ( M ) .

Theorem 7.4 T h e follouring identity in

22

holds,

indz DV = Ic(M).

(7.17)

Prooj Let D” : Oeven(M)--f Oeven(M)be the elliptic differential operator defined by 49+ 1

Since V E I’(TM) is of norm one over M , one sees that for any integer i such that 1 5 i <_ 4q 1,

+

(V,v y v >= 0. Thus, by (4.13),

+

E ( V ) E ( V ~ M V ) E ( v y V ) Z ( V ) = 0.

(7.19)

From (4.13), (7.3) and (7.19), one verifies easily that D” is also skewadjoint. Thus, by the homotopy invariance property of the mod 2 index, one has indz D” = indz Dsig.

(7.20)

Atiyah Theorem on Kervaire Semi-characteristic

110

On the other hand, by (4.13), (7.2), (7.16) and (7.18), one verifies directly that

1

49+1

c(ei)Z(VzM i=l

= ker D v .

(7.21)

From (7.5) and (7.21), one gets, indz D" = indz Dv.

(7.22)

From (7.7), (7.20) and (7.22), one gets (7.15). 0 Next, we introduce a deformation of D v by using the second vector field

X. Definition 7.5 ([Z2]) For any T E R, let DV,T : fleven(M)+ Reven( M ) be the operator defined by

DV,T = Dv

+ TE(V)Z(X).

(7.23)

Remark 7.6 Since V and X are orthogonal to each other, by (4.13), (7.15) and (7.23), one can also write DV,T as 1

DV,T = 5 (qv)(d

+ d* + E(X))- (d + d* + Tz;(X))E(V)).

(7.24)

In view of (4.17), one may regard DV,T as a Witten type deformation of

Dv. Clearly, DV,T is elliptic and skew-adjoint. By Theorem 7.4 and the homotopy invariance property of the.mod 2 index, one gets that for any T E R, indz DV,T = indz DV = Ic(M).

(7.25)

A proof via Witten Deformation

111

We will prove the vanishing of k ( M ) by studying the behaviour of ker DV,T as T --+ 03. We first establish a Bochner type formula for -D$,T. Proposition 7.7 The following identity holds, 4q+ 1

-D$,T

= -D$

+ T C ( c ( e i ) E ( V z M X )- (Vz'X,

V )c(ei)E(V))

i=l

+T2/ X I 2 .

(7.26)

Proof. By (4.13), (7.16) and (7.23), one can rewrite DV,T as 4q+l

c ( e i ) 2 ( V E M V )+ T E ( X )

i=l

From (4.13), (7.15) and (7.27), one deduces that

= -D$+T((d+d*)E(X)+E(X)(d+d*))

From (7.28) and by proceeding as in the proof of (4.19), one gets (7.26). 0

We can now prove the Atiyah vanishing theorem as follows. Since 1x1 = 1 over M by our assumption, one sees easily that there exists To > 0 such that when T 2 To, 4q+ 1

T

C ( c ( e i ) Z ( V g M X -) ( V g M X ,V )c(ei)E(V))+ T2(XI2> 0 .

(7.29)

i=l

On the other hand, since DV is skew-adjoint, -D$ is a nonnegative operator. Combining this fact with (7.26) and (7.29), one sees that when

112

Atiyah Theorem on Kervaire Semi-chamcteristic

T 2 To,-D$,T is a positive operator, which implies that ker DV,T = (0).

(7.30)

From (7.25), (7.30) and the definition of the mod 2 index, one gets the vanishing property of k ( M ) . 0

7.4

A Generic Counting Formula for k ( M )

The proof in the previous section has the advantage that it also leads to a generic counting formula for k ( M ) in a way similar to what the PoincarQHopf formula is for the Euler characteristic. To state this counting formula, we recall that on the given 4q 1 dimensional smooth oriented closed manifold M , by the result of Hopf mentioned in the beginning of this chapter, there always exists a nowhere zero vector field V of M . Let [V]denote the one dimensional vector bundle generated by V . We consider the quotient bundle T M / [ V ] ,which is a 4q dimensional vector bundle over M . Take a transversal section X of T M / [ V ] ,which always exists by elementary result in differential topology. Since the rank of T M / [ V ]is 4q, and M is of dimension 4q 1, one knows that the zero set of X I denoted by zero(X), consists of disjoint one dimensional closed submanifolds (i.e., circles) in M . Let T M / [ V ]be equipped with a Euclidean metric. Take one circle F in zero(X). At any point y E F , the transversal , is the restiction of section X induces an automorphism of T y M / [ V y ]which T M / [ V ]a t y. By the linear algebra result Lemma 4.8, one can determine a one dimensional linear subspace in A* ( ( T y M / [ V y ] ) *Moreover, ). these linear subspaces form a real line bundle, denoted by o p ( X ) , over F . Clearly, as a topological line bundle over F , OF(X)does not depend on the Euclidean metric on T M / [ V ] . We define a mod 2 index, denoted by ind2(X, F ) , on F by

+

+

indz ( X ,F ) = 1 if OF(X)is orientable over F

(7.31)

Non-multiplicativity of k ( M )

113

and ind2 (X,F ) = 0 if OF(X)is nonorientable over F.

(7.32)

We can now state the generic counting formula for k ( M ) ,which is taken from [Z2], as follows.

Theorem 7.8 The following identity in

k(M)=

Z2

holds,

indz(X,F).

(7.33)

FEzero(X)

The basic strategy of the proof of Theorem 7.8 is the same as that of the proof we presented in Chapter 4 for the PoincarB-Hopf formula: one first apply the Bochner type formula (7.26) to localize everything to a sufficiently small open neighborhood of zero(X), and then completing the proof by making use of properties of harmonic oscillators in this small neighborhood. One notable difference is that since here zero(X) consists of circles instead of isolated points, the analysis of harmonic oscillators will lie in the normal spaces to zero(X) in T M , instead of in whole tangent spaces. We refer to the article [Z2] for more details.

Remark 7.9 It is interesting that while the Euler characteristic can be computed by counting isolated zero points of vector fields (cf. (4.11)), here the Kervaire semi-characteristic is computed by counting circles. 7.5

Non-multiplicativity of k ( M )

To conclude this chapter, we apply Theorem 7.8 to give an analytic proof of a non-multiplicativity result of Atiyah and Singer [AS] on the Kervaire semi-characteristic. We use the same assumptions and notation as in the previous section. We further assume in this section that H 1 ( M ;Z2), the first singular cohomology of M with 2 2 coefficient, is nonzero. Take a nonzero element a E H 1( M ;Z2). Let

-

xTT, : Ma be the double covering determined by

-+ M CY.

114

Atiyah Theorem on Kervaire Semi-characteristic

Let w l q ( T M )E H4q(M;Z2) be the 4q-th Stiefel-Whitney class of the tangent vector bundle of M . The non-multiplicativity theorem of Atiyah and Singer can be stated as follows.

Theorem 7.10 The following identity in

k

(za) ( a =

'

22

holds,

(7.34)

w4q(TM)i

Proof. Recall that V is a nowhere zero vector field on M and X is a transversal section of T M / [ V ] .Let = 7r:V and 2 = 7r:X be the pullback vector fields of V and X on respectively. Then 2 is a transversal section of ~ i i ; i , / [ V ] . Clearly, the zero set zero(2) of 2 is exactly 7r;'(zero(X)). Let La be the real line bundle over M which is determined by a. That is, L, is the (unique) line bundle over M such that wl(L,) E H 1 ( M ;Zz), the first Stiefel-Whitney class of L a , equals to a. For any connected component F , which is a circle, in zero(X), there occur two possibilities for 7r;' ( F ) : (i) If L a l is ~ orientable, then 7r;'(F) consists of two disjoint circles F1 and F2. Moreover, the restrictions of the pull-back line bundle - . z ( o ~ ( X ) ) on Fl and F2 have the same orientability. In summary, in this case one has

c

E,

(7.35) (ii) If L,[F is non-orientable, then

7r;'(F)

is connected and

is a double covering between circles. In this case, 7r:(o~(X)) is orientable over 7r; ( F ), and we get indz (2,7ro1F)= 1.

(7.36)

From (7.35), (7.36) and by using Theorem 7.8, one gets immediately that k ( z a ) equals to the number of connected components of zero(X) on which the restriction of the line bundle La is non-orientable. Now since by elementary obstruction theory (cf. [MS]), [zero(X)] E

References

115

H4q(M,Z2) is dual t o w ~ ~ ( T Mone ) , finds finally that

(h3i-)

(w1 ( L a I F ) 7

[F])= ( a .W4q(TM),[MI)

FEzero(X)

which is exactly (7.34). The proof of Theorem 7.10 is completed. 0 As was pointed out by Atiyah and Singer in [AS], Theorem 7.8 shows that the Kervaire semi-characteristic is a subtle invariant which, t o be different with respect t o the Euler characteristic, does not admit a direct differential geometric interpretation.

7.6

References

[A] M. F. Atiyah, Vector fields on manifolds. Arbeitsgemeinschaft fur Forschung des Landes Nordrhein- Westfalen, Dusseldorf 1969, 200 (1970), 7-24. [AD] M. F. Atiyah and J. L. Dupont, Vector fields with finite singularities. Acta Math. 128 (1972), 1-40. [AS] M. F. Atiyah and I. M. Singer, The index of elliptic operators: V. Ann. of Math. 93 (1971), 139-149. [D] J. L. Dupont, K-theory obstructions to the existence of vector fields. Acta Math. 133 (1974), 67-80. [MS] J. Milnor and J. Stasheff, Characteristic Classes. Annals of Math. Studies Vol. 76. Princeton Univ. Press, 1974. [S] N. Steenrod, The Topology of Fibre Bundles. Princetion Univ. Press, 1951.

[W] E. Witten, Supersymmetry and Morse theory. J. (1982), 661-692.

Diff.Geom. 17

[Zl] W. Zhang, Analytic and topological invariants associated to nowhere

116

Atiyah Theorem on Kervaire Semi-characteristic

zero vector fields. Pacific J. Math. 187 (1999), 379-398.

[ZZ] W. Zhang, A counting formula for the Kervaire semi-characteristic. Topology 39 (2000), 643-655.

Index

adiabatic limit, 20

foliation, 18

Berezin integral, 42 Bianchi identity, 7 Bott connection, 19 Bott vanishing theorem, 18

Hodge decomposition theorem, 59 Hodge star operator, 57 Kervaire semi-characteristic, 106 Killing vector field, 30

Cartan homotopy formula, 30 characteristic class, 9 characteristic form, 8 characteristic number, 10 Chern class, 11, 25 Chern form, 10, 25 Chern character, 14 Chern-Simons form, 17 Chern-Weil theorem, 7 connection, 3 covariant derivative, 4 critical point, 75 curvature, 4

Lie bracket, 5 Liouville form, 37 Morse function, 75 Morse inequalities, 76 Morse lemma, 76

odd Chern character, 25 Pontrjagin class, 12 Pontrjagin form, 11 Riemannian foliation, 21

de Rham cohomology, 2 de Rham complex, 2 de Rham theorem, 3 de Rham-Hodge operator, 59

Signature operator, 106 Smale transversality, 94 Thom class, 46 Thom form, 45

equivariant cohomology, 31 Euler class, 47 Euler form, 47

Witten deformation, 62, 77 Witten instanton complex, 97 117