Seminar on Atiyah-Singer Index Theorem. (AM-57), Volume 57 9781400882045

Table of contents :
CONTENTS
PREFACE
CHAPTER I: STATEMENT OF THE THEOREM OUTLINE OF THE PROOF
§1. The index theorem
§2. The topological index
§3. The analytical index
§4. Appendix
CHAPTER II : REVIEW OF K-THEORY
§1. K(X) a finite CW-complex
§2. The Chern character
§3. The difference construction
§4. L-theory
§5. Products in L-theory
CHAPTER III : THE TOPOLOGICAL INDEX OF AN OPERATOR ASSOCIATED TO A G-STRUCTURE
CHAPTER IV: DIFFERENTIAL OPERATORS ON VECTOR BUNDLES
§1. Notation
§2. Jet bundles
§3. Differential operators and their symbols
§4. Hermitian bundles and adjoint operators
§5. Green's forms
§6. Some classical differential operators
§7. Whitney sums
§8. Tensor products
§9. Connections and covariant derivatives
§10. Spin structures and Dirac operators
CHAPTER V: ANALYTICAL INDICES OF SOME CONCRETE OPERATORS
§1. Review of Hodge theory
§2. The Euler Characteristic
§3. The Hirzebruch signature theorem
§4. Odd-dimensional manifolds
CHAPTER VI: REVIEW OF FUNCTIONAL ANALYSIS
CHAPTER VII: FREDHDIM OPERATORS
CHAPTER VIII: CHAINS OF HTLBERTIAN SPACES
§1. Chains
§2. Quadratic interpolation of pairs of hilbert spaces
§3. Quadratic interpolation of chains
§4. Scales and the chains (Z^n, V)
CHAPTER IX: THE DISCRETE SOBOLEV CHAIN OF A VECTOR BUNDLE
§1. The spaces C^k(ξ)
§2. The hilbert space H^0(ξ)
§3. The spaces H^k(ξ)
CHAPTER X: THE CONTINUOUS SOBOLEV CHAIN OF A VECTOR BUNDLE
§1. Continuous Sobolev chains
§2. The chains {H^k(T^n, V)}
§3. An extension theorem
§4. The Rellich, Sobolev, and restriction theorems
CHAPTER XI: THE SEELEY ALGEBRA
CHAPTER XII: HOMOTOPY INVARIANCE OF THE INDEX
CHAPTER XIII: WHITNEY SUMS
§1. Direct sums of chains of hilbertian spaces
§2. The Sobolev chain of a Whitney sum
§3. Behaviour of Smblk with respect to Whitney sums
§4. Behaviour of Intk and σ^k under Whitney sums
§5. Behaviour of the index under Whitney sums
CHAPTER XIV: TENSOR PRODUCTS
§1. Tensor products of chains of hilbertian spaces
§2. The Sobolev chain of a tensor product of bundles
§3. The # operation
§4. The property (S6) of the Seeley Algebra
§5. Multiplicativity of the index
CHAPTER XV: DEFINITION OF ia AND it ON K(M)
§1. Definition of the analytical index on K(B(M), S(M))
§2. Multiplicative properties of it
§3. Proof of Lemma 1
§4. Definition of it and ia on K(M)
§5. Summary of the properties of ia and it on K(M)
§6. Multiplicative properties of i on K(X )
§7. Direct check that ia = it in some special cases
CHAPTER XVI: CONSTRUCTION OF Intk
§1. The Fourier Transform
§2. Calderón-Zygnund operators
§3. Calderón-Zygmund operators for a compact manifold
§4. Calderón-Zygmund operators for vector bundles
§5. Definition and properties of Intr (ξ, η)
§6. An element of Into(S^1) with analytical index -1
§7. The topological index of the operator of §6
§8. Sign conventions
CHAPTER XVII: COBORDISM INVARIANCE OF THE ANALYTICAL INDEX
CHAPTER XVIII: BORDISM GROUPS OF BUNDLES
§1. Introductory remarks
§2. Computation of Ωk(X) ⊗ Q
§3. The bordism ring of bundles
CHAPTER XIX: THE INDEX THEOREM: APPLICATIONS
§1. Proof of the index theorem
§2. An alternative formulation of the index theorem
§3. The non-orientable case of Theorem 2
§4. The Riemann-Roch-Hirzebruch theorem
§5. Generalities on integrality theorems
§6. The integrality theorems
APPENDIX I : THE INDEX THEOREM FOR MANIFOLDS WITH BOUNDARY
§1. Ellipticity for manifolds with boundary
§2. The difference element [σ(d, b)]
§3. Comments on the proof
APPENDIX II : NON STABLE CHARACTERISTIC CLASSES AND THE TOPOLOGICAL INDEX OF CLASSICAL ELLIPTIC OPERATORS
§1. Characteristic classes
§2. τ-homomorphism
§3. The character of classical elliptic operators

Citation preview

Annals of Mathematics Studies Number 57

ANNALS OF MATHEMATICS STUDIES E dited by Robert C. Gunning, John C. M oore, and Marston Morse 1.

Algebraic Theory of Numbers, by

H erm ann W

3. Consistency of the Continuum Hypothesis, by

eyl

K u r t Go d el

11.

Introduction to Nonlinear Mechanics, bu N.

20.

Contributions to the Theory of Nonlinear Oscillations, Vol. I, edited by S.

21.

Functional Operators, Vol.

24.

Contributions to the Theory of Games, Vol. I, edited by H. W.

25.

Contributions to Fourier Analysis, edited by A. A. P. C a l d e r o n , and S. B o c h n e r

28.

Contributions to the Theory of Games, Vol. II, edited by H. W.

30.

Contributions to the Theory of Riemann Surfaces, edited by L.

33.

Contributions to the Theory of Partial Differential Equations, edited by L. n e r , and F. J o h n

34.

Automata Studies, edited by C. E.

38. 39. 40.

1,

by

J ohn

and N.

Kr ylo ff

B o g o l iu b o f f L

efsch etz

N eum a n n

von

and J.

Sh annon

Kuh n

W.

Zyg m u n d ,

M

and A. W.

T uck er

M.

M o rse,

T ra n sue,

Kuhn

Ah lfo rs

and A. W.

T uckj

et al. B ers,

S.

Boc

cC a rth y

Linear Ineaualities and Related Systems, edited bu H. W.

Kuhn

and A. W.

T uck er

Contributions to the Theory of Games, Vol. Ill, edited by M. D r e s h e r , A . W. T u c k e r and P. W o l f e Contributions to the Theory of Games, Vol. IV, edited by R. D u n c a n L u c e and A. W. T u ck er

41.

Contributions to the Theory of Nonlinear Oscillations, Vol. IV, edited by S.

42.

Lectures on Fourier Integrals, by S.

43.

Ramification Theoretic Methods in Algebraic Geometry, by S.

44.

Stationary Processes and Prediction Theory, by H.

45.

Contributions to the Theory of Nonlinear Oscillations, Vol. V, edited by L. S a l l e , and S. L e f s c h e t z

F

e fsc h etz

46.

Seminar on Transformation Groups, by A. Theory of Formal Systems, by R.

48.

Lectures on Modular Forms, by R. C.

49.

Composition Methods in Homotopy Groups of Spheres, by H.

50.

Cohomology Operations, lectures by N. E.

B o rel

A bhyankar

u rsten berg

47.

E

L

B o ch n er

C e s a r i,

J.

et al.

G u n n in g

St e en r o d ,

T oda

written and revised by D. B. A.

M il n o r

52. Advances in Game Theory, edited by M. 53. Flows on Homogeneous Spaces, by L.

D resh er,

A u sla n d er,

54. Elementary Differential Topology, by J. R. 55. Degrees of Unsolvability, by G. E. 56. Knot Groups, by L. P.

a­

Sm u lly a n

p s t e in

51. Morse Theory, by J. W.

L

L. L.

Sh a p l e y , Green ,

F.

and A. W. H

ahn,

M u n kres

Sacks

N e u w ir t h

57. Seminar on the Atiyah-Singer Index Theorem, by R. S. 58. Continuous Model Theory, by C. C.

C hang

and

H.

J.

P a l a is , K e is l e r

et al.

T u ck er

et al.

SEMINAR ON THE A T IY A H -S IN G E R INDEX THEOREM BY

Richard S. Palais

WITH

CONTRIBUTIONS

M . F . A T IY A H

BY

R. T . SE E L E Y

A . BO REL

W . SH IH

E. E. FLO Y D

R. SO LO VAY

PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS 1965

C O P YR IG H T ©

1 9 6 5 , B Y PR IN CETO N U N IV E R SIT Y PRESS A L L RIGH TS RESERVED l .c

.

card

: 6 5 -17 15 7

PR IN TED IN T H E U N IT E D STATES OF A M E R IC A

CONTENTS

PREFACE

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

CHAPTER:

STATEMENT OF THE THEOREM OUTLINE OF THE PROOF, by A. B o r e l....................................................

§ 1. §2. §3. §k .

The index th e o r e m ...................................................................................... The to p o lo g ic a l index ............................................................................. The a n a ly t ic a l index ............................................................................. Appendix...........................................................................................................

6 9

K(X) a f i n i t e CW -com plex..................................................................... The Chern c h a r a c t e r .................................................................................. The d iffe r e n c e co n stru ctio n ................................................................ L - t h e o r y ....................................................................................................... Products in L -th e o ry ..................................................................................

13 it

CHAPTER IV:

27

DIFFERENTIAL OPERATORS ON VECTOR BUNDLES, by R. S. P ala is 51

N o t a t i o n ....................................................................................................... J e t bundles ................................................................................................... D i f f e r e n t i a l op erato rs and t h e ir sym bols....................................... Hermitian bundles and a d jo in t o p e r a t o r s ....................................... G reen 's f o r m s .............................................................................................. Some c l a s s i c a l d i f f e r e n t i a l op erato rs ........................................... Whitney s u m s ............................................................................................... Tensor products .......................................................................................... Connections and co v a ria n t d e r iv a t iv e s ........................................... Spin str u c tu r e s and D irac op erato rs ...............................................

CHAPTER V:

15 18 20

THE TOPOLOGICAL INDEX OF AN OPERATOR ASSOCIATED TO A G-STRUCTURE, by R. S o lo v a y ...............................................

§ 1. §2. §3. § 14-.

1 k

13

CHAPTER I I I :

§ 1. §2. §3. § 14-. §5. §6. §7. §8. §9. §10.

1

REVIEW OF K-THEORY, by R. S o l o v a y ...........................................

CHAPTER I I : § 1. §2. §3. § 14-. §5.

ix

ANALYTICAL INDICES OF SOME CONCRETE OPERATORS, by M. Solovay ...............................................................................................

51 55 61

69 73 75 79 81 Qk

91 95

Review o f Hodge th e o r y ............................................................................. 95 The E u ler C h a r a c te r is tic ..................................................................... 96 The H irzebruch sig n a tu re t h e o r e m .................................................... 98 Odd-dimensional m a n i f o ld s ................................................................. 103

CHAPTER V I: CHAPTER V II:

REVIEW OF FUNCTIONAL ANALYSIS, by R. S. P a la is . . . . FREDHDIM OPERATORS, by R. S. P a l a i s .............................. v

107 119

CONTENTS

CHAPTER V I II: §1. §2. §3. § 14-.

. . .

125

C hains........................................................................................................... Q uadratic in te r p o la tio n o f p a ir s o f h ilb e r t spaces . . . Q uadratic in te r p o la tio n o f chains ............................................... S c a le s and the chains Z n , V ) } ...........................................

125 131 139 i in

CHAPTER IX: § 1. §2. §3.

THE DISCRETE SOBOLEV CHAIN OF A VECTOR BUNDLE, by R. S. P a l a is .............. ’ .................................................................. ;

The spaces Ck (£) ................................................................................. The h ilb e r t space H°U) ................................................................ The spaces Hk ( | ) ................................................................................. ...

CHAPTER X: § 1. §2. §3. § i.

CHAINS OF HTLBERTIAN SPACES, .by R. S. P a la is .

THE CONTINUOUS SOBOLEV CHAIN OF A VECTOR BUNDLE, by R. S. P a l a i s .........................................................................................

Continuous Sobolev c h a i n s ................................................................ ... The chains {H^T11, V ) } ........................................................................ An ex ten sio n t h e o r e m ............................................................................ The R e llic h , Sobolev, and r e s t r ic t i o n theorem s.........................

CHAPTER XI:

THE SEELEY ALGEBRA, by R. S. P a l a i s .................................. ...

U7 114-7 1 14-8 1 14-9 155 155 156 162 1 6b

175

CHAPTER X II: HOMOTOPY INVARIANCE OF THE INDEX, by R. S. P a la is . . , 185 CHAPTER X III: WHITNEY SUMS, by R. S. P a l a is ................................................... § 1. §2. §3. § t. §5.

D ir e c t sums o f chains o f h ilb e r t ia n sp a ce s.............................. ... The Sobolev chain o f a Whitney s u m ............................................., Behaviour o f Smbl^. w ith re sp e c t to Whitney sums . . . ., Behaviour o f In t^ and o^. under Whitney sums....................... . Behaviour o f the index under Whitney sums ..................................

CHAPTER XIV: § 1. §2. § 3. § t. §5.

Tensor products o f chains o f h ilb e r t ia n sp ace s......................... The Sobolev chain o f a tensor product o f b u n d les..................... The # o p e r a t i o n ................................................................................. ... The p ro p erty (S6) o f the S e ele y A l g e b r a .................................. ... M u l t i p li c a t iv i t y o f the index ............................................................

CHAPTER XV: § 1. §2. §3. §1*. §5. §6. §7.

TENSOR PRODUCTS, by R. S. P a l a i s ...........................................

DEFINITION OF

i & AND

it

191 192

193 193 19t 197 197 201 206 209 210

ON K(M) , by R. M. Solovay, 215

D e fin itio n o f the a n a ly t ic a l index on K(B(M), S(M)). . .. M u lt ip lic a t iv e p r o p e rtie s o f i^ ................................................... Proof o f Lemma 1 ..................................................................................... D e fin it io n o f i^ and i a on K(M).............................................. Summary o f the p r o p e rtie s o f i & and i t on K(M) . . ., M u lt ip lic a t iv e p r o p e rtie s o f i on K ( X ) ................................ .. D ir e c t check th a t i & = i^ in some s p e c ia l cases . . . ., vi

191

215 217 222 223 226

228 232

CONTENTS CHAPTER XVI:

CONSTRUCTION OF

I n t k ; b y R. S. P a la is

and R. T. S e e l e y .......................................................................................235 §1. §2. §3 . § 3+. §5 . §6.

The F o u r ie r T r a n s f o r m ........................................................................................235 C alderon-Zygnund o p e r a t o r s * . 2^3 Calderon-Zygm und o p e r a to r s f o r a compact m a n i f o l d .........................259 Calderon-Zygm und o p e r a to r s f o r v e c t o r b u n d l e s ..................................266 D e f i n i t i o n and p r o p e r t ie s o f I n t r (£ , r\) ............................................... 270 An elem ent o f I n t Q( S 1 ) w ith a n a l y t i c a l in d e x - 1 .........................272

§7. §8.

The t o p o l o g i c a l in d e x o f th e o p e r a to r o f §6 . . . . . . . . . 275 S ig n c o n v e n t i o n s ............................................................................................2 8 1

CHAPTER X V II:

COBORDISM INVARIANCE OF THE ANALYTICAL INDEX, b y R. S . P a l a is and R. T. S e e l e y ....................................................... 285

CHAPTER X V III: §1. §2 .

BORDISM GROUPS OF BUNDLES, b y E . E. F lo y d ............................ 303

In tr o d u c to r y re m a rk s.............................................................................................303 Com putation o f ftk (X) ® Q ............................................................................... 306

§ 3 . The bord ism r i n g o f b u n d l e s ...........................................................................307 CHAPTER XIX:

THE INDEX THEOREM:

APPLICATIONS, b y R. M. S o lo v a y . .

313

§1. §2.

P ro o f o f th e in d e x theorem ............................................................................... 313 An a l t e r n a t i v e fo r m u la tio n o f th e in d e x t h e o r e m ............................. 315

§3. § 3+.

The n o n - o r ie n ta b le ca se o f Theorem 2 318 The R iem ann-R och-H irzebruch t h e o r e m .........................................................32^

§ 5 . G e n e r a l i t i e s on i n t e g r a l i t y th eorem s........................................................ 326 §6. The i n t e g r a l i t y th eo rem s................................................................................... 329 APPENDIX I :

THE INDEX THEOREM FOR MANIFOLDS WITH BOUNDARY, by M. F . A t iy a h .................................................................................................... 337

§1. §2.

E l l i p t i c i t y f o r m a n ifo ld s w ith b o u n d a r y ............................................... 338 The d if f e r e n c e elem ent [ cr(d, b) ] .............................................................3^6

§3.

Comments on th e p r o o f ........................................................................................350

APPENDIX I I :

NON STABLE CHARACTERISTIC CLASSES AND THE TOPOLOGICAL INDEX OF CLASSICAL ELLIPTIC OPERATORS, by W. S h ih

C h a r a c te r is tic c la s s e s

. .

353

§1. §2 .

t - h o m o m o rp h ism .................................................................................. ....

. . . . . .

360

§3.

The c h a r a c te r o f c l a s s i c a l e l l i p t i c o p e r a to r s ............................

362

vii

353

PREFACE

The Index Theorem is a striking and central result in a rapidly developing field of research which may be described as the study of the re­ lation between analytic and topological invariants of a certain class of linear maps between sections of differentiable vector bundles (the class of integro-differential, or pseudo-differential operators).

The field is not

really new and has several classical results, for example the Hodge theory of harmonic forms and the Hirzebruch formulation of the Riemann-Roch theorem, both of which are in fact closely related to the Index Theorem.

Moreover,

since the Index Theorem there have been other notable results discovered by Atiyah, Bott, Singer and others.

To mention only two there is a mod 2 "in­

dex” defined for elliptic operators on real bundles with self adjoint sym­ bols, which in certain cases seems to be related to the Arf invariant of a manifold, and there has been a remarkable generalization of the Lefschetz fixed point formula, which already has had important applications to the theory of h-cobordism and to the study of the fixed points of periodic trans­ formations . This book consists mainly of slightly revised notes of a semi­ nar held at the Institute for Advanced Study in 1 9 6 3 -6k upon the initiative of A. Borel.

Exceptions are Chapters XVT and XVII and the Appendix by M.

Atiyah which were written somewhat later. Aside from going through the details of the proof of the Index Theorem, the major emphasis of the seminar was placed on developing the to­ pological and analytical machinery associated with integro-differential operators.

On the topological side the agreement was to assume a reasonable

degree of sophistication.

Thus, the basic facts concerning K-theory and

characteristic classes are reviewed rather than proved and the emphasis is on showing how, with these tools, elliptic operators give rise to cohomology

ix

PREFACE classes and on studying the properties of these classes. side, it was decided to start more or less ab i n i t i o .

On the analytical

The reason for this

somewhat unbalanced exposition is in part due to the predelictions of the organizers of the seminar, but also in part it is due to the fact that while most of the algebraic topology involved is covered in complete detail in easily accessible published papers, much of the analysis is quite recent, and the published versions often refer explicitly only to the case of trivial bundles over domains in Euclidean space. I would like to thank the many persons who attended the semi­ nar lectures and whose suggestions lead to a smoother presentation.

In par­

ticular the section on jet bundles in Chapter IV was considerably improved by A. Vasquez and L. Charlap, and a suggestion by M. Kneser led to a smoother version of the section on Fredholm operators (Chapter VII). I would also like to express my great appreciation to F. Browder, E. Nelson and E. Stein for their constant and invaluable advice while I was writing my part of these notes. Finally, a very careful reading of the entire manuscript by W. Shih led to the elimination of a great number of misprints and other errors. For this I am sure he has the readers thanks as well as mine.

Richard S. Palais Brandeis University January, 1 9 6 5

CHAPTER I STATEMENT OF THE THEOREM OUTLINE OF THE PROOF A. Borel As an introduction to the subject matter of this seminar, this lecture gives the statement and a rough description of the proof of the index theorem.

More details on the proofs of the results stated and on

the concepts discussed here will be found in the subsequent lectures. Manifolds are compact, smooth (i.e.,

C°°), orientable and o r i ­

e n t e d , consist of connected components of the same dimension, and, unless otherwise stated, have no boundary. be denoted at a point

Complex vector bundles will usually

bythe same letter as their total spaces. x of a bundle

E and

C°°(E)

tions of a smooth complex vector bundle

§1.

the space E

Let

on

X.

d:

C°°(E) -► C°°(F)

X

be a manifold,

A differential operator

of smooth cross sec­

over a manifold.

d

E, F

from

E

smooth complex vector bundles to

F

is a linear map

which is given locally by a matrix of ordinary (i.e.,

scalar) differential operators.

More precisely, let

definition of a local chart, x 1, ..., xn (n = dim X) C°°(U)

thespace ofcomplex valued

of

and

E

Fx .

F over Then

d

U

1

defines a map

< j < q) , where

C°°-functions on

which at each point

(x e U)) given by a matrix

< p;

denotes the fibre

The index theorem 1.

and

Ex

P^

be the domain of

local coordinates, U, and choose sections

x e U

C°°(U)P -► C°°(U)q

U

form a basis of

(p = dim Ex , q = dim Fx

P(x, d) = (P(x, d/dx.,, ..., e C°°(U) [ c^/dx^ , ..., 1

Ex

d / d x ^ . The order

(1 < i r

of

2 d

BOREL

§1

is the maximum of the degrees of the

The characteristic matrix

Q(x, 5)

taking the terms of degree partial derivatives vector

P^j

of

d

in the partial derivatives.

at

x

is obtained from

r, substituting indeterminates

d/dx^,

^ d x +-..,+ £n &X-n ,

and multiplying by

(-l)r ^2 .

there is then associated a

with complex coefficients.The system is elliptic if matrix is invertible for all

i ^ 0

and for all

cumstances, it is known that

ker d

and

coker

P

for the To each cotangent

p x q matrix

p =

q and

x e X.

by

Q(x, |)

ifthis

Under those cir­

d = C°°(F) /im d

are finite

dimensional; the difference of their dimensions is the index of

d.

In

order to distinguish it from another index, to be defined below, we shall call this the analytical index of (1)

d

and denote it by

ia (&)•

Thus

iQ cl(d) = dim ker d - dim coker d

It is known that

ia (d)

is invariant under deformations of

d, and this led

Gelfand to ask whether it could be expressed in terms of topological data. The index theorem provides a positive answer to that question. formulate it, a mixed rational cohomology class on the symbol of

d

ch d e H (X; Q ) , depending

(see section 2, below), is introduced.

^(X) be the Todd class or

In order to

the complexified tangentbundle of

Let moreover X.

It is

obtained by considering the product of formal power series -xf(l-e_V where the

x^

1 • (l- e V 1

(i = 1, ..., s > dim X/2)

,

are indeterminates, expressing it

as a formal sum of homogeneous polynomials, which are then symmetric in the 2

x^, the

writing these as polynomials in the elementary symmetric functions in 2 x^ 's, and then replacing the j-th symmetric function by the j-th

Pontrjagin class of

X

(j = 1, 2, ...).

The topological index i^Cd)

is

then defined as (2)

it(d) = (ch d *^(X))[X]

(n = dim X)

,

where the right hand side stands for the value of the n-dimensional component of

ch d. 3T(Yf)

on the fundamental cycle of

THEOREM (Atiyah-Singer) . on the manifold

X.

Then

Let

d

X.

We have then the

be an elliptic operator

ia (d) = it(d) .

§1

I: OUTLINE OF PROOF

3

This is not the most general form of the theorem. Singer have extended it:

Atiyah and

(a) to elliptic complexes, I.e., sequences of

operators whose spribols form an exact sequence, but this can easily be re­ duced to the case of one operator; (b) to a wider class of operators; this is quite important for the proof, and will be dealt with at length in this seminar; (c) to boundary value problems; this will probably not be touched upon here, for lack of time and material. M. Atiyah.]

[See however, Appendix I, by

A fourth generalization; which would include Grothendie.ck> s

version of the Riemann-Roch theorem over

C, is contemplated but, as far as

I know, has not yet been carried out. The proof falls naturally into two parts: which investigates the properties of cerned with

ia (d).

it(d) > and an analytical one, con­

The former one also gives the motivation for the latter

one, and here we shall summarize it first. give some more details on 2.

a topological one,

ch d. ch d.

The definition of

The Grothendieck group

Before doing that, however, we

K(X)

of

X

Let

X

be a finite CW-complex.

is the quotient of the free commutative

group generated by the isomorphism classes of complex vector bundles on by the subgroup generated by the elements E*

0

K(X)

is an exact sequence.

If a base point

is the kernel of the homomorphism

bundle

E

the dimension of

relative Grothendieck group means

X

with

Y

E - E ! - EM

E .

If

K(X, Y)

Y

x e X

K(X) -► Z

where

o -► E" -*■ E -►

has been chosen, then

which assigns to each

is a closed subcomplex of

is by definition

K(X/Y),

X

be again a manifold.

Let

B(X)

(resp.

with respect to some Riemannian metric, and It is known that, via cup product,

coefficients) is a free module over n = dim X,

n,

the inverse of

H*(X),

it: B(X) -+ X

where

X/Y

a

be

T*(X),

the natural pro­

H*(B(X), S(X)) (any ring of with a canonical generator

whence the existence of an isomorphism

cp*: H (B(X) , S(X)) -► H (X), by

the

S(X))

the unit ball (resp. unit sphere) bundle of the cotangent bundle

of degree

X,

pinched to a point, which is then taken as base point.

Let now

jection.

X

U

cp*: H*(B(X),

the Thom isomorphism, which decreases dimensions a o U(a e H*(X)).

d: C°°(E) -+ C°°(P)

Let on

§1

BOREL

k

X.

be a differential operator of order

It is well-known that the matrix

r

Q(x, t), which was defined above

using local coordinates, has in fact an intrinsic meaning and associates to each

£ e T*(X)x

a linear map of

of course smoothly on the bundles on symbol of

d.

B(X)

x, |,

Ex

Fx «

This linear map depends

whence a homomorphism

lifted from

E

and

Ellipticity means that

isomorphism.

into

F

via

*,

[.trp (d) ] e K(B(X), S(X))

**F

of

to be called the

&r (d), restricted to

In that case, one can associate to

difference element

a^(d) : jt*E

S(X),

is an

(jt*E, tc*F, ap (d)|S(X))

(see [2]).

Then

ch d

a

is de­

fined by (3)

ch d = ( - D n(n+1)/2 T#ch[ 0, a. € z ) , QJ

-L

QJ

the Fourier transform

=

-I-

1 ... sn n . We put a lso

1 ... 3 ^ n , denote by

7

f -+ f

where

1112 = Z I?,

and

f(|) =

(2*)-n/2 / f(x)ei(x'l)dx. The Riesz operator

Ra

$ “1 o (|/| g1 )a o $ f

is

course, the middle factor stands for the multiplication by A = Za aa (x)Ra

a finite linear combination efficients in the space

B°° = B°°(R n ) of

where, of (l/|l|)a .

To

of Riesz operators, with co­

C°°-functions all of whose partial

derivatives arebounded, we associate the symbol

a ( A) = Z aa (x) (|/| 11 )Q;,

which may be viewed as a function on R n x Sn_1 .

The symbol map

A -*> a (A)

is injective, and extends by continuity (in suitable topologies) to a 1-1 map of a certain class of operators of H“ ( R n )

H°°(R n )

to

(where

is the space of functions all of whose partial derivatives are

square integrable), to be called

B°° singular integral operators of order

zero, onto the space of functions on R n x Sn_1, are bounded.

A

B°° operator of order

A

is of order zero, and

of

1 +a ,

a

H°°( R n),

(a

A = iF_1 o (1

being the Laplacian) .

r

all of whose derivatives

is then a product o $

+ 1112) 1 The symbol

B = AAr ,

where

is the square root

cj (B)

is then

(A) •U | r . These operators, modulo operators of lower order, include the

differential operators in

B°°,

because, since

D = Z|a |_r aQ,(x)DQ: of order Da = jp-1 o (-i|)a © $ f

(Z aQ,(x)Ra) * (-iA0)p , where A,

andit can be shown that Let now

on

X

X. A linear map (1)

for

(2)

for

is of order

D =

is the "square root" of < r-2.

be a manifold, E, F smooth complex vector bundles

A: C°°(E) -► C°°(F) belongs to cp, y e C°°(X)

order

with coefficients

we may write

A Q = SF-1 ° || | © $ Ar - AP

r,

with disjoint

Intp (E, F)

if:

supports, cpAty

is of

< r-i; cp, \|r € C°°(X)

borhood

with supportsin some coordinate neigh­

U, over which sections of

section i, by a matrix

cpAt

E, F

are chosen as in

is given, modulooperators of order

(A^.) of> ®°° operators of order

r.

< r-l,

8

BOREL

The matrix

(ar (A.j_j))

the symbols is shown to have an intrinsic meaning,

and associates to every Ex

into

Fx ,

Intr (E, F) denoted class

A.

The operator

and if

cr^(A)

Ellp (E, F) . ch A

g € T*(X) - 0

whence a homomorphism

the symbol of

§3

A

(x e U)

cr^(A) : jt*E -► jc*F,

r

A e Ell (E, F), then, using

if it is in

a (A), the cohomology

are defined as in the case of

Moreover, it can be shown that

ker A, coker A

finite dimensional, so that

ia (A)

are true.

may now be viewed as a function on

Consequently

i&

S(X) ,

The set of such operators is

and the topological index

differential operators.

defined on

is elliptic of order

is an isomorphism.

If

a linear transformation of

are

is again defined, and that (ii), (iii) K(B(X), S(X)),

and the next steps of the proof consist in showing that it has the same pro­ perties as

i^, namely: 7.M u l t i p l i a a t i v i t y .

ellipticoperators on in the sense

X, Y

and

One X x Y

wants to know:

if

A, B, C,

such that [a(A)] • [o(B)J

of the pairing of §2, section 3, then ia (A) *

are =

[cr(C) ],

= ^a^ *

This will be proved by making use of a pairing of operators: (A,B) *-► A # B which is suggested by the construction underlying section 3 above. however, meets with some technical difficulties since eral an integro-differential operator.

A # B

This,

is not in gen­

It will nevertheless be a limit of

such operators and this will allow one to keep track of symbols and indices. 8.

If

X

is even-dimensional, and

V

a bundle on

X,

we

define V*'

v) ■

•V

»

where the right hand side stands for the analytical index of any operator with symbol belonging to

V •

X

.

In fact, using a connection on

the operator

D Q mentioned in k, we may find a differential

that symbol.

One then has to prove that (a) to (d)

with

i^

replaced by

1&.

V

operator with

in section k are

This is fairly standard, except,

and

true

however, for

(c), which is one of the main parts of the whole proof. The index theorem for even-dimensional manifolds then follows from t, 5 and 8.

In order to extend it to odd-dimensional manifolds, we

need the following statement:

§3

Is OUTLINE OP PROOF 9. There exists on the circle

9

an elliptic operator

from the trivial bundle to the trivial bundle 3 with

i^E^)

EQ,

= i^(EQ ) 7^

0,

which, together with the multiplicativity properties 3, 7, allows one to reduce the odd-dimensional case to the even-dimensional one by multiplying with

Eq .

10. every element of

The analytical index is an integer by definition. K(B(X), S(X))

Since

is the class of a symbol of an elliptic

operator in the Seeley algebra, the main theorem for these operators implie the COROLLARY.

Let

X

be a manifold, and

Then the topological index of

a

a € K(B(X), S(X)).

is an integer.

As we shall see, this yields all the integrality theorems per­ taining to the Todd genus or the

A-genus.

§h . Appendix The order of exposition in the sequel does not coincide with the one adopted in the previous outline.

In order to orient the reader, we

make here some comments on the contents of the different chapters. Chapters II to X give some background material both for the to pological and the analytical parts.

In conformity with the purpose of this

seminar, the treatment of the latter one is practically self-contained, omitting only proofs of some quite standard facts, while much is taken for granted on the topological side.

Chapter II reviews briefly K-theory,

Chern characters, and shows that

K(X, Y) may be defined as the set of equi

valence classes of suitable sequences of vector bundles.

Chapter III de­

scribes a method to compute the topological index when the bundles underly­ ing the differential operator and the tangent bundle to the base manifold are associated in a suitable way to a given principal bundle. Chapter IV introduces some basic material on differential oper ators: definition, symbols, the jet bundle exact sequence, adjoints, Green operators, some classical differential operators. Chapter V defines the differential operator leading to the in­ dex of a manifold and checks the index theorem in some special cases.

10

BOREL Chapters VI to X review some notions and results in functional

analysis:

in particular bounded operators On Banach spaces with finite di­

mensional kernel and cokernel (to be called Fredholm operators) the Sobolev ir spaces H , the Sobolev inequality, and Rellich's theorem. The Seeley algebra is introduced axiomatically in XI, by means of five conditions, the existence proof being postponed to XVI. XII, XIII, XIV, and part of XV Seeley algebra: invariance of

Chapters

are devoted to the main properties of the

regularity properties of elliptic operators (XI), homotopy ia (XII), (this is (i) of §3; property (ii) is essentially

built in the axioms, so that its validity is really part of the existence proof), behaviour under Whitney sums (XIII) ; and the multiplicativity pro­ perty 7 above, which is dealt with with the help of the sixth condition im­ posed on the Seeley algebra (XIV) . Chapter XV proves that both conditions (a), (b), (d) of t, and that that (c) also is true for

i

s

i^ and i i^

satisfy the condition

also verifies (c).

The proof

is much harder and is given in XVII.

Chapter XVI proves the existence of the Seeley algebra of integro-differential operators, with the properties postulated in XI and XIV. The method finally adopted in these Notes is different from that of [5 ], after which the above summary was patterned. convenient for a self-contained exposition.

This turned out to be more Chapter XVI also contains the

construction of an elliptic scalar operator on the circle with both indices equal to -1, comments on the sign conventions made in defining

i& and i^,

and historical remarks on singular integral operators. Chapter XVIII is topological, and gives the proof of the unique­ ness theorem in Section 5 of 1, using the main results of Thom's cobordism theory. How the results of Chapters XI to XVIII yield a proof of the index theorem is briefly recapitulated in Chapter XIX, which also contains an extension to the non-orientable case, further remarks on the theorem, and some of the main applications (the Riemann-Roch theorem, the Hirzebruch in­ dex theorem, various integrality theorems of algebraic topology).

I: OUTLINE OP PROOF Appendix I is devoted to the index theorem on manifolds with boundary.

It gives the precise statement, and discusses the notions under­

lying it; but the proof is only briefly sketched.

Finally, there is an ap­

pendix by Weishu Shih, pertaining to Chapter III, which treats in a more general setting, characteristic classes and the topological index of differ­ ential operators associated to G-structures.

REFERENCES [1 ] M. F. Atiyah, "The index of elliptic operators on compact manifolds," Sem. Bourbaki, Mai 1963, Exp. 2 5 3 . [2 ] M. F. Atiyah and F. Hirzebruch, "Analytic cycles on complex manifolds," Topology 1 (1 9 6 2 ), pp. 2 5 -^6 . [3 ]

M. F. Atiyah and I. M. Singer, "The index of elliptic operators on compact manifolds," Bull. A. M. S. 69 (1 963), pp. ^22-I4-3 3.

[k]

R. T. Seeley, "Singular integrals on compact manifolds," Amer. J. M. 81 (1959), PP. 658-690.

[5 ] R. T. Seeley, Integro-differential operators on vector bundles, Trans. A.M.S. 117 (1965), pp. 167-20^.

CHAPTER II REVIEW OF K-THEORY Robert Solovay

§1. group, {|},

K(X),

Let

X

be a finite CW-complex.

as follows.

The generators will be the equivalence classes,

of complex vector bundles over

X.

allows the dimensions of the fibers of ness components of

X.) (E)

there is a relation,

One defines an abelian

(If £

X

is not connected, one

to differ on different connected­

For each short exact

sequence of vectorbundles,

0

-

i*

-+ | -

{|} = U'} + {£"},

in

o

,

K(X) .

The tensor product of vector bundles makes tative ring:

{(•} • {rj} ={£ ® t]} . A mapping

the pull-back of vector bundles) a map Let dim({|})

x

be a point.

=dim £. If

X

Then

f:

K(X)

into a commu­

Y -► X induces

f ': K(X) -* K(Y) : dim: K({x}) = Z

is a space with basepoint

(through

f ‘(Q)) = {f*|}.

is defined by x, we put

K(X)

=

kernel (K(X) — K((x))) . If

(X, Y)

is a finite CW-pair, we let

tained by collapsing for

X/Y.

(If

Y

Y = 4,

The relative group,

to a point X/Y

K(X, Y) ,

(Y)

X/Y be thespace

ob­

which we take as the basepoint

is the disjoint union of is by definition

X

with a basepoint.)

K(X/Y) .

In [1 ], Atiyah and Hirzebruch have shown

that one can define

"cohomology groups" K t x , Y)

(i e Z 2)

which satisfy all the Eilenberg-Steenrod axioms (modified for except for the dimension axiom, and such that 13

Z 2-grading)

§1

SOLOVAY

^k

K°(X, Y) = K(X, Y) The exact sequence in cohomology reduces to an exact hexagon: K°(X, Y)

K°(X)

K'(Y)

K°(Y) 5

K (X)

K (X, Y)

(The construction of this cohomology theory uses the Bott periodicity theorem.) Moreover, the ring structure on commutative ring structure on

K(X, Y)

extends to an anti-

*

K (X, Y) :

K t x , Y) • KJ(X, Y) C Ki+J(X, Y)

§2. If

.

The Chern Character. X

is a topological space,

n j> 0 HJ(X, A) . Let

H**(X, A)

is the direct product

| be an n-dimensional complex vector bundle over

with total Chern class (rational coefficients) c( i) = 1 + c1(!) + ..*+ cR (g) Following Borel and Serre, one considers

c^(|)

as the

symmetric function in the indeterminants

x 1, ..., xn :

. i—

n

c( e) = II (1+x1)

.

i=1 Then by definition,

ch(|) e H**(X, Q)

is

i=i Of course, if

X

is finite dimensional,

H (X) = H *(X)

and

elementary

X

§3

CHAPTER II: The map

ch

K (X, Y) - H

B**(X, Y ; Q)

the

the

H* = IIi=o(2) ^

(X, Y; Q)

Z 2-grading: H

(where

15

extends to a ring homomorphism ch:

If we give

REVIEW OF K-THEORY

311(1

=tf®H

= Hj_=i (2)

>

tlien

cl1

compatible w ith

Z 2-gradings and commutes with the coboundary homomorphism, If

(X, Y)

is a finite CW-pair, then

K*(X, Y)

5.

is finitely

generated and ch 0 1q : K*(X, Y) 0 Q s H is an isomorphism. K (X)

* K (X, Y)

admits a natural graded module structure over

which is compatible, via

H*(X, Yj Q)

over

(X, Y; Q)

ch,

with the module structure of

H*(X; Q) .

The Bott periodicity theorem entails the following description of

K*(Sm ): The map

ch:

K* (Sm ) -►lf1(Sm ,Q)

the subgroup of integral elements,

§3.

is a monomorphism with image

if^S111, Z).

The Difference Construction. (After [2 ]).

3.1.

Suppose that

are vector bundles over

X

(X, Y)

is a finite CW-pair, that

E

P

and that cc:

E|Y s F |Y

is an isomorphism of the restrictions of

and

difference element d(E, F, a) € K(X, Y) is defined as follows: Let

A C X x I

be

X x {0 } U Y x I U X x (13

to

Y.

Then the

E

and

16

Thus

SOLOVAY X x I/A s S1(X/Y)

(S1(X)

§3

is the reduced suspension of

X.)

Put

AQ = X x {0} U Y x (o, 1) and

A 1 = X x {1} U Y x (0, 1)

and let

f^:

A^ -+ X

Define a bundle

% is

f*(E);

Let d(E,

|

over

on

F, a)

be the restriction of the projection map A

AQ n A 1

as follows: we identify

be theimage of

U)

on

AQ ,

f*(E)

|

with

is

X x I -* X.

fQ (F);

f*(E)

on

using

A 1,

a.

under the composition

K°(A) —►KtXx I, A) s K1(s’ (X/Y)) sK°(X, Y) 3.2.

The following lemma gives the principal properties of

difference construction. LEMMA 1: (i)

For

a map

f:

(X*, Y') -► (X, Y)

d (f*E ,f*F, f*a) = f (ii) (iii) (iv)

(v)

d(E, F, a)

Y = (jf

For

*d (E , F, cc)

we have

f ‘:

K(X, Y)

on

X

such that the isomorphism

(E © G) |Y s (F © G) |Y

defined on the whole of

E © G

with

F © G

X.

d(E © E 1, F © F ’, cc) = d(E, F, a )

+ d(E», F T,or1) .

d(P, E, a-1) = -d(E, P, a). If

G

is a vector bundle over

X

d(G ® E, G ® F, 1q ® a) = G (ix)

we have

if and only ifthere is

d(E, F, a) = o

extends to an isomorphism of

(viii)

K(X),

element

G

cc.

d(E, F, cc) = E - F.

.

a ©

(vii)

.

f *(d(E, F, of)) = E - F

a vector bundle

(vi)

have

depends only on the homotopy class of

For the natural map

The

we

Suppose that

cc:

E|Y s E ’|Y

and

then • d(E, F, cc) . a*: E'|Y ^ E" |Y.

Then d(E, E M), a 1 o cc) = d(E, E f, cc) + d(E»,E ,f, a'). PROOF:

(i)— (iv)

and (vi)— (viii)

are proved in [2 ].

CHAPTER II:

§3

REVIEW OF K-THEORY

17

PROOF of (v): First suppose that there is a bundle

G

on

X

and an isomorphism

a: extending

cc © 1^.

If

f:

E ®G sF ® G

(X, Y)

(X, X), then

d(E © G, F © G, cc © 1Q ) = f Jd(E © G, F © G,. a) = 0 (sinceK(X, X)

= o) . By (vi),

0 = d(E © G, F © G, a © 1Q ) =d(E, But

d(G, G, 1q|Y) = f'd(G, G, 1q) = 0.

Suppose now that

d(E, F, cc) = 0 .

assume

that

Eis the trivial bundle a:

and that

N

fibers of a bundle Let

P|Y

is large compared to F|Y

overX/Y;

F

E = X/Y x CN

if

cc

X x C^, ss Y X C N

©*G, a © 1^) ,

that

we may

cc is a trivialization

,

By means of

cc,

we identify the

The result of this collapsing process is

be

is the natural map, the map of

p*d(E, F, cc) = d(E, F, cc) = o.

We have

.

of high dimension such that

by (E © G, F

p: X -► X/Y

and let

G

dim X.

with one another.

1q |Y)

This establishes the sufficiency.

Select

(E, F, cc)

E © G s X x C N . Replacing

F, a) + d(G, G,

then

p*F = F. induced by cc.

E|{Y) “ F|(Y)

But

p !: K(X/Y) -► K(X, Y) is bijective, andthe natural map

K(X/Y)

F = E

is large compared to dim X > dim X/Y,

in

K(X/Y) .

Since

dim F

F = E = X/Y x C N .If we identify identity, then comes

1E |Y

F = p F

a:

with

E

E © G s X x C^,

then

E = p E

X).

cc

becomes

so that

To sum up:

cc © i^.

Thus

if

the

cc = p cc G

be­

is of

extends to an isomor­

E © G ^ F © G. PROOF of (ix):

By (v) and (vii) there is a bundle

and an isomorphism : E © E ’© F ^ E ’© E © F extending

is injective.

so that

is identified with

(which manifestly extends to

high dimension, and phism

F

K(X/Y)

cc © cc-1 © ip.

F

over

X

18

SOLOVAY One has a commutative diagram: a © a ’ © 1F E ® E’© F

E' © E m © F

©

a'a © E r © E" © F

of isomorphisms over over

X.

Put

Y

in which the vertical maps extend to isomorphisms

= d(E © E f © F, E* © E" © F, a © a 1 © 1p)

and

r\2 =

d(E’ © E © F, E' © E n © F, 1-g, © a fa © 1 ^) . From the diagram and (i), we get

ti1 = t]^ • But

r\2 = d(E, E " , a '

t]1 = d(E, E ', a) + d(E', E 11, a')

a )

for the same reasons.

by (v) and (vi), and

The proof of (ix) is complete.

L-Theory We are going to give a Grothendieck type definition of the rela­ tive groups

K(X, Y) .

Let

(X, Y)

be a finite CW-pair.

We define

(X, Y)

to be the set of all triples, E = (Eq, E,, a) such that

Eq

and

isomorphism of

E1

E Q |Y

are complex vector bundles over

with

E 1 |Y.

X,

and

a

is an

The addition of triples is defined using

the Whitney sum. A triple,

E,

of the form (Eq , E q , 1E q |Y)

is said to be elementary. image of the map

Note that if

{ S (X, X) —

Two triples are elementary triples

and

S (X, Y) Q,

E © P

are called equivalent if there

such that there is an isomorphism ^

E’© Q

(An isomorphism of the triples morphisms

is elementary, it lies in the

S (X, Y)) .

E, E 1 e P

E

E'

and

E"

is a pair of iso­

CHAPTER II: defined over

X,

REVIEW OP K-THEORY

such that, over

Y,

the diagram

E o ----1—

I

19

^ E1

,

I

E o ---- 2

- Ei

is commutative.) DEFINITION:

S 1(X, Y ) ,

L 1(X, Y)

is the set of equivalence classes of

under the equivalence relation just defined.

L 1(X, Y)

a priori, an abelian semigroup with respect to Whitney sum.

is,

We put

L 1(X) = L 1(X, Of) . L 1(X, Y)

is a contravariant functor of the pair

(X, Y ) . We

define a natural transformation L 1(X, Y) - K(X, Y)

X:

as follows: then X

X(

E €

If

S 1(X, Y)

[E ]) = d(EQ , E 1, a).

represents the element

[E]

of

L 1(X, Y ) ,

Using Lemma 1of Section 3, one checks

is well-defined and additive.

Moreover, if E 0 - Ei

Y = 0,

x([El)

that

is just

.

The main result of this paragraph is THEOREM 2.

The map

L 1(X, Y) — K(X, Y)

X:

is an

isomorphism. PROOF. s K(X/Y, (Y)) ,

We first show that

we may assume that

is surjective.

X

Y

So let where

dim Ex = dim E^.

Let

a:

CY))

j

s

a € K(X, {x}) . The image of

Y)

is a point:

L,(X, Y)«-------- V X / Y ,

’ K(X, Y)

Since K(X,

K(X/Y, (Y)) a

in

K(X)

may be written as

E - E’

Ex ^ E^. Then

E = (E, E», a) lies in

t*N, a ) . f duced by

f.

Let f:

(Cf. ( h ) . )

:

(B(X), S(X)) — (B(V), S(V)) be the map

in-

Then we put Qf = f* 7

+ In Lecture II, K-theory was discussed only for the category of finite CW-pairs. Therefore, strictly speaking, we should work with N-universal classifying space, Bq n , and then let N tend to infinity. (The spaces Bq ^ can be finite CW-complexes.) However, we shall continue to speak only of

Bq .

III:

§2 2.1+. on

31

OPERATORS ASSOCIATED TO G-STRUCTURES

Let

a e K(B(X), S(X))

be associated to the G-structure

X. We are going to show eventually how to find an explicit formula

chCD^)

for various concrete a.

for

The following theorem is a first step

in

that direction. THEOREM 1 . Suppose that map

f:

Let X

X

2f t .

be a manifold of dimension

has a G-structure with classifying

a e K(B(X), S(X))

X -► Bq. Let

ciated to the G-structure. constructed from M,

N,

be asso­

We suppose that

oc

and o' as discussed

is above.

We suppose also that the rational Euler class x(V*) e H*(Bq, Q)

is non-zero.

Then

ch(M) - ch(N) is divisible by (6 )

x(V ),

and

ch(D ) ■ f * * ( ( V “ V REMARK 1 : According to Borel,

sors

of zero.

(Cf. [1 ].)

Thus

ch(M)-ch(N) ) X(V*) '

.

the ring H*(Bq, Q)

has no divi­

of (6 ) iswell-defined.

the righthand side

REMARK 2 : The right hand sideof (6 ) depends only and not

on

the isomorphism PROOF of Theorem 1 .

o.

Therefore, the same

Let

y = d(jt*M, **N, a).

on

M

and

is true of Since

chCD^) .

ch and

the Thom isomorphism are natural, ch(Da) = (-1 )^ cp* " 1 ch(a) = (-1)* cp* " 1 ch(f! 7 ) = (-1 )* f**(cp* " 1 ch(r)) Thus it suffices to show that X ( f ) U cp* " 1 ch(7 ) = ch(M) - ch(N)

(7) Let

i:

diagram

(B(V ) , Q) -+ (B(V ), S(V ))

.

be the inclusion map.

N

Consider the

32

SOLOVAY

§2

! K(B(V*), S(V*))--

^ ------------ -

ch .

K(B(V), 0)

ch I ► H**(B(V), 0; Q) I **

**

H~"(B(V"), S ( T ) ; Q)

*

fl Since

ch

v;

----------------------- “

H *(X) = II p > o H^(X).) (1 ) and (2 ) of 1 .0 . (8 ) Since

--------------------

is natural, the upper square is commutative.

taG, Q)

The commutativity of the lower square follows from

Since the diagram is commutative,

q>*~ (ch(r)) U x(V ) = (it

y = d(jt* M,

a

(We recall that

)“ (ch(i*(r))

N, cr) ,

jt*

i, . * ~ ~ i ’(7 ) = it M - jt N (by IILemma 1 (iv))

so the right hand side of (8 ) is just

ch M

- ch N .

The proof is complete.

Review of the Borel-Hirzebruch Formalism (after [2 ]).

§3. 3.1.

Let Tr

be a torus of dimension

Tr = S 1 x •* * x S 1 Then the classifying space,

B^,

(r

r: times)

is the Cartesian product of

infinite-dimensional complex projective space

r copies

of

CP(00) :

Bt = CP(oo) x ••• x CP(00)

.

Therefore we have H*(Bt , Q) sQtx,,

xp ]

and H**(Bt , Q) ^ Q(x 1 , ..., xp ]

(= the ring of

formal power series) where

x^ e H 2 (B,p, Q) , 3 .2 .

mal torus of sifying spaces

G.

Now let

G

The inclusion map of

be a compact Lie group, and let T

in

G

T

determines a map of clas­

be a ma

III:

§3

33

OPERATORS ASSOCIATED TO G-STRUCTURES

BT

BG

(well-defined up to homotopy). The Weyl group,

W(G, T) ,

is the group of automorphisms of

which extend to inner automorphisms of homology ring,

G.

The Weyl group acts on the co­

H*(Bt , Q ) ,(since it consists of automorphisms of

We can now state Borel*s description of

T

T).

H (Bq , Q ) .

THEOREM 2 . The map P* :

H* (Bq , Q) — H* (Bt , Q)

is injective.

The image consists of those elements

in

which are invariant under the action of

H*(Bt , Q)

the Weyl group

of

G. (I.e.,

g •y = y

y

for all

g € W(G, T ) .) We shall frequently identify H*(5j, Q)

under

with its image in

p.

3.3for any torus

*

H (Bq , Q)

Let T,

S1

be the group of reals

mod 1: S1 =R/Z.

One has

canonical isomorphisms p:

Hom(T, S1) — H1 (T, Z)

v:

Hom(T, S1) — H2^ ,

and Z)

which we shall treat in the future as identifications .

ed as follows: Let * = h a. To describe

They may be describ

o € H 1(S1, Z)

Then

u,

and

be the canonical generator. 1 we identify temporarily the groups S

n(h) U(1)

via the map s - e 2*13 A homomorphism on

B,p.

h:

T -*■ S 1

(s € S 1)

gives rise to a principal

U(1)-bundle,

We put u(h) = c/lh) 3.^.

M

Let

a G-module, and

Eq

G

be a compact Lie group,

T

a maximal torus of

the universal principal G-bundle over

G,

Bq . Recall

3^

SOLOVAY

that

M = Eq Xq M

§3

is the vector bundle over

corresponding to

M.

We

shall now recall the Borel-Hirzebruch description of the characteristic classes of

M. CASE 1 : M

M

as a T-module.

dimension one:

is a complex G-module (of dimension

Then

M

is the direct sum of irreducible T-modules of

M = M 1 © ••• © Mn . On

M^,

t • m = e 2 iria)j ^ where M.

ok

e Hom(T, S 1).

n) . We consider

The

T

acts by

•m

(m € Mj)

,

are called the weights of the G-module

The set of weights does not depend on the particular decomposition of

M into irreducible T-modules (by the Jordan-Holder theorem). Now consider the 3 .3 ), and identify

as elements of

2

H (B,p,

Q)

H*(Bq, Q) with a subring ofH*(BT , Q)

(as discussed in in 3 .2 .

as

Then the following formulas are valid: n 1 + c 1 (M) + ••• + cn (M) =

(9)

~ = ch(M)

(1 0 )

Vy

[[ (1 i=l

+ a>±)

e®i

i =1 REMARK 1 : Formula (9 ) says that symmetric function in the weights of REMARK 2 : Let M.

X:

The classical character,

ch(x)(g) = Trace (x(g)).

G

On

T

M.

Aut^(M)

ch(X),

c^(M)is the ittl' elementary

describe the action of

G

on

is a complex valued function on

G:

we have the identity 2 jrico .(t)

ch(x)(t) = 2 ^

e

J= 1 which bears a close resemblance to (1 0 ). of

M

(on

T),

If we know the classical character

we can write down immediately the expression for

CASE 2 (2 f) : M

is a real G-module of dimension

Now, as a T-module, dimensional T-modules:

M

2n

splits into the direct sum of

ch(M) . (2 n+i). n

two-

§3

III:

OPERATORS ASSOCIATED TO G-STRUCTURES

(11)

M = M 1© •••

((11

')

M =

andT

acts trivially We give

narmal basis of

where ment

M 10

35

© Mn

••• © Mh 0 R 1

,

on R ). M a G-invariant metric. Then

Mj,

the action of

t e T

with respect to an ortho-

is described by a matrix

cos 2itcDj(t)

-sin 2jtooj(t)

sin 2jtcDj(t)

cos 2jra>j(t)

co. e Hom(T, S 1). The cd.'s are called the weights of M. The eleJ J cd^ . depends on the orientation of the basis in M j ;if we change the

orientation in

Mj ,

cd^ .

is replaced by -ok .

The Pontrjagin

•

classes of the real vector bundle,

M,

are given

by the formula n (12)

Thus 2

1 +* P 1(M) + ••• + pn (M) =

p^(M)

is the

[J (1 + i=i

i^*1 elementary symmetric function of the elements

2

•

•••> mn * CASE 3: In this weights of M,

M

is an oriented real G-module of dimension

case, the bundle

M

is oriented.

wenow choose orientations

ceives its given orientation(cf. ( 1 1 )).

in the

2n.

In defining the

IVL's

Then the Euler

so that

M

re­

class of M

is

given by n (13)

x(M)

=

n

®i

1=1

3.5.

PROPOSITION,

with weights G-module

M

(m|m € M} , (b)

Let

Mn

(a)

Let

n . has weights then

Xm = Xm,

Mn

be a complex

G-module

Then the complex conjugate

-a^, ..., -®n -

(If

M =

nfj” + m^ = m 1 + m 2, g*m = gii.)

be a complex G-module with weights

, ...,

36

SOLOVAY n .

be a real G-module

a^, ..., o)n

ofdimension

M ® C

as a complex G-module are

(if m = 2 n),

+ and using the fact that

0 ^(05^

...

(by (9 )), we get

a>m ) = ci (M)

00 V"

( 17 )

00.

111

2

Tk < ° l < S >»

• • • ’ cm(K) )

■

k=0

n

—

i=i

1-e

1

But (1 6 ) and (17) yield (1*0 . By naturality it suffices to prove the proposition when BSO (n) ’

M =R1

and

Then

r\ 0 C = M 0 C.

+ y^,

and possibly

t|= M... Let the weights be By Proposition 3.5(c),

0.

Since

x/(l-e"x )

appropriately interpreted, is

Here the

y^

are elements of

1 + P 1 + *** + Pg.

If

N

complex

M

and

N

G

k

y.

I] i= 1

---3 -y± 1 -e 1

if

(1 +yf) =

y±

1-e 1

are complex vector spaces, we let M

with

G-modules.

Iso(M, N) de­

N. V

a real G-module, and

M

Suppose that S(V*) — Iso(M, N)

Then if

P

is a principal G-bundle, with base

X,

(19)

E = P Xq M, a

F = P Xq N,

and

•X-

it:

T = P Xq V

,

gives rise to a map ap :

where

o/(l-e”°)

The proof is complete.

be a compact Lie group,

is a G-equivariant map.

then

1,

An Application to ch(D) .

cr:

space

has constant term

+ y 1,

Thus the proposition is valid in the universal example,

note the set of isomorphisms of Let

has weights

H2(B,p, Q) . However, by (12),

(Cf. the proof of (1*0 .)

§*.

Vn

be an oriented real vector space equipped with

We recall that there is an isomorphism .k (V) /T7~ *n-k/(V) s A

*:

A

defined as follows: A^(V)

has a natural inner product characterized by the identity

< e1 a If

e1, ..., en

is

... a ek ;

f1

det

* isdefined

(2 0 )


= det(a * *b)

*(e, * ... * e p

is even,

1

An (V) ^ R.

(a, b e Ak(V))

that

- ek+1 * .. . a en *2 = (-i)k (n-^

*2 = (-1)k have dimension

24,

and let

on

of V,

then

. on Ak (V)

which we shall always assume, then

(22) V

e )=

form an oriented orthonormalbasis

Prom this it ise'asy to deduce

Now let

e± , fj > |

by the identity

e^., ..., ep

(21)

dim V

> = det|
=

II

(e 71 - e71)

1=1 -ft H (B gQ(2n)^

where we identify and let

y 1, ..., yR

PROPOSITION 6.2.

be the weights of

Let

let

X

* H (BT),

V.

be a Riemannian manifold, and

„ D:

with a subgroup of

„ Cf°(A_(T (X)))

C°°(A+ (T (X)) -

be an elliptic operator whose symbol arises from auniversal construction.

Then it(D) = I] y± /tanh y± [X] i

where the Pontrjagin classes are viewed as the elementary symmetric function in the

p

y^ .

§6

SOLOVAY

k6

REMARK,

y/tanh y

is even and therefore is given by 2

QCy } •

a power series in

PROOF.

We put

M = A (V),

and

N = A_(V) ;

applying Theorem 3

and Proposition 6.1, we get

n ch(D) =

-J±

n

1=1

ch(D)^(X) =

y± z^r—

1

“ "yi - ey i J] e -v yi

i=l n

~yi

n

y-

-y.

1-e "7i

1-e yi

yi

= 11 y-( -____— y, i=1

(1-e

) (1 -e

-y± ) /

•

1)

We now use the identity

1 - U2

U '1 - U (1 -U)(1 -U_1)

(1 -U)(U-1 )

1 +U

U1 /2 + u - 1 /2

U-1

u 1 /2 - u - 1 /2

to get

ch(D)lSr (X) =

Thus by

lt(D) = ch(D)^(X) 2y^,

j] y1 ± 1

cosh y . / s — sinh y± /2

=

[|yi/tanh (yn-/2) ± 1 1

[X] = J], yj/tanh (y± /2) [X] .

the term in grading

2k

will be multiplied by

top-dimensional term is multiplied by

If we replace 2 . Thus the

2n , and we have

n 2n i t(D) =

J] 2yi/tanh y ± [X] i=1

or equivalently,

(2 3 )

=

REMARK.

II y ±/t a,nh y ± [X] i

The right hand side of (2 3 ) is just the

Hirzebruch L-genus of

X.

y±

§6

III:

OPERATORS ASSOCIATED TO G-STRUCTURES

^7

PROOF of Proposition 6.1. 1. a € A k (V),

Let

b € A^(W).

( 2k)

V

and

W

be even-dimensional vector spaces.

Let

Then the formula

V©¥

(a a b) =

*y (a) a *w (b)

follows easily from the definition of

*

(cf. 2 0 )).

A routine computation

now shows that

°v©w(a A b) = °V(a) A From this it follows that A+(V © W) = (A (V) 0 A (W)) © (A_(V) 0 A _(¥)) (25) A (V © 2.

Let

p:

by the inclusion, is a map

X:

W)

i,

E^

=

(A

(V) 0

B y -*■ Bq of

T

in

A

(¥)) ©

(A

(V) 0

A_(W ))

be the map of classifying spaces induced G.

Then by the definition of

p,

there

Eq. such that

X(x • t) = X.(x) • t

(for

x € Et

and

t e T)

and the diagram

is commutative. Let

Xx M1

M

be a G-module,

1^: E^, x M -*■ E^ x M M inducing

p

M1

Thus,

Since the map

it suffices in view of

p*t

M* = p*M. H*(Bq , Q) -*■ H*(Brp, Q)

2, to show

ch(A+ ( R 2n)~) - ch(A_( R 2n)~) = n (e i where R 2 n

The map.

passes to quotients to define a bundle map of

on base-space s.

3.

the underlying T-module.

is considered as a T-module.

-3L 1 - e 1)

is a monomorp

^8

§6

SOLOVAY k.

Recall that if

G

is a compact Lie group,

R(G)

free abelian group generated by the irreducible G-modules. G-module,

M,

An arbitrary

splits into the direct sum of irreducible G-modules:

M 1 © ••• 0 M n,

and one defines

duct of G-modules makes ring of

is the

R(G)

[M] € R(G)

as

[M^].

into a commutative ring:

M =

The tensor pro­ the representation

G. The map R(G) - H**(Bg , Q)

Ch:

defined by ch([M]) = ch(M) is a ring homomorphism. 5.

We return to the T-module

R

2n

.

This splits into irreduci­

ble T-modules:

R 2n = R 2 © ... ® R 2 Let

VI

be the

i

th.

copy of R

get the following equation in

2

considered as a T-module.

Prom (2 5 ) we

R(T):

[A+( R 2n) ] - [A _( R 2n)] =

n

(CA+ (V1 )] - tA_(V± ) ])

.

i=i Taking

ch

of both sides, n

(26)

Ch(A+ ( R 2n)~) - Ch(A_(R2n)~) =

n

(ch(A+(V1)~) - Ch(A_(V±)~)

.

i= 1

6.

The Euler class of

is

y^.

Thus Proposition

6.1 will

follow from (2 6 ) and the. following: LEMMA.

Let

V

vector bundle.

be an oriented 2 -dimension real vector bundle, Then if

y

is the Euler class of -y.

ch(A (V)) - ch(A_(V)) = e

1

y.

- e 1

V,

§6

III: PROOF.

By naturality, we may assume

R 2-bundle

ented

over

V

e1, e2

2 is the usual basis in R ,

*e l

= e 2>

*02

1- i ( e 1

a

A+

e2)

If y:

^9

is the universal ori­

b sq(2) *

If

Thus a basis for

is

OPERATORS ASSOCIATED TO G-STRUCTURES

*1

=

1 + i(e1 a e2)

is

and

= -e i 9

e1

a

and

e 2 ,

* (e 1 a

e1 + ie2 ;

e 2)

= 1

.

a basis for

A_

ie2 .

SO(2) -► S*

is

thestandard weight,

then

ysends the

matrix ^ cos 2*0

-sin 2*0 v

sin 2*0 into 0.

Then the

(Here

0

Let

[X]

€ S 1.) denote the1-dimensional

SO ( 2) -module s

[e^ + ie2]

cos 2*0

have weights

[i

+ i(e1

+ y

a e2)]

subspace spanned have weight

respectively.

0,

by x.

and the modules

The Lemma now results from

(1 0 ) .

REFERENCES [1]

A. Borel, Sur la cohomologie des espaces fibres principaux et des espaces homogbnes de groupes de Lie compacts, Ann. of Math., 57(1953), pp. 1 1 5 -2 0 7 .

[2]

A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces

I, Amer. J. Math., 8 0 (1 9 5 8 ), pp. t58-538.

CHAPTER IV DIFFERENTIAL OPERATORS ON VECTOR BUNDLES Richard S. Palais § 1. M ary,

will denote a paracompact

C°°(M,

C)

the subring gory of

Notation C°° manifold, possibly with bound­

the ring of complex valued

of real valued functions.

C°°complex vector

ally be denoted by

V(M)

bundles over

g, n, £

C°°

M.

Objects of

F*:

of

g . We denote by

C°° cross-sections of C°°( r\)

C°°(g)

is of course a

g

) . C“ (g)

linearly into

V(M)

and to

The fiber of

|

into

t\

g

F € Hom( g, tj)

F*(f)(x) = F(f(x))

at

x

which for each

CQ (g)

V(M)

modules, that 0“ (ti)

is a

F*

and maps

the induced map

x € M

maps

C°°(g)

the subspace of

C” (g)

(g, i\)

then

into L

C“ (g)

and

L( g

F*

consisting C” (g)

T -*• T

con­

dM.

C” (g) maps

L( g, n) tj )

V(M) x V(M)

whose fiber

of linear maps of

T(x) = T|gx e L(g, tj)

We

are subC“ (g)

into

V(M) which

gx

into

at

x

is

tj .

If

and the bundle structure of

is a natural equivalence of the functors

which we regard as an identification, i.e., 51

to

L(g, r\)

T e C°°(L(g, r\)) .

L(g, t|) “is uniquely characterized by the property that Then

tx

C~( n) .

the functor from

the bundle

the complex vector space T e Hom(g, t\)

module, that

is a module homomorphism, and that

We denote by assigns to

C°°(M, C)

to

the vector space

sisting of sections whose supports are compact and disjoint from C°°(g)

will

(an element of Hom( g, r\)

will denote the subspace of

of sections with compact support and

note that

R)

will usu­

C°° the additive functor from

g

defined by

C°° map of

C°°(M,

we will generally use the same symbol to

the category of complex vector spaces which assigns to C°°(g)

M,

will denote the additive cate­

denote a vector bundle and its total space. be denoted by

functions on

Horn

Horn = C°°L .

and

C°°L

52

PALAIS If

¥^ e V(M) bold

and

C°°(M, ¥)

If

and we will use the sym-

If

g e V(M)

is a C°°

its dual bun­

real vector bundle then

denote the dual real bundle. IfM = M 1 x •• • x

a bundle

7

vector space then

M x¥

interchangeably.

= L(-g, C M) . Similarly if

g*

will

is afinite dimensional complex

will denote the product bundle

C°°(¥m )

dle is 7*

¥

§1

over

M

f1 €

g1 e V(M^)

and

whose fiber at

x = (x-,..., xv)

K is

I

f 1 0 ••• 0 fk ,

we define

g1 0 ••• 0 gk

then

0

g.L

X1

a function on

is

••*0 g^

Xk

M, by

(f1 0 ••• 0 fk) (x) = f 1 (x1) 0 ••• 0 fk (xk) . The bundle structure of v i v ••• 0 g is characterized by the property that f 0 •••0 f e C& (|1 0 ••• 0 gk)

wherever

f^ € C°°(gi).

Ifg1 € V(M) i = 1 , .. ., k g1 0 •••

0 gk = A* (g1 0 *•* 0 gk),

agonal map,

x -*■ (x, ..., x) .

g1 0 ••• 0 gk = then

.

if

ments of

where

0 kg

lift fixed by all such

0 kg

into itself

. The set of ele­

ir is denoted by

Sk (g),

where the sum is over all

(1 , ..., k) . ¥e will also use of 0 kg 0 t] onto

•x M

•

the

k—

( k) There is a canonical projection S^ ’ e

Z *,

namely

•

by

{1 , ..., k}

it of

jt(v1 0 ••• 0 v^) = v ^ . ^ 0 • • • 0

Hom(0 kg, Sk (g)),

0 id

M x

M -►

a:

jr is a permutation of the set

g.

tions of

••♦ 0 gk e V(M)

g1 = g (i = 1 , ..., k) we write

If

symmetric tensor product of

S ^

define g1 0

we

ir induces a bundle morphism, also denoted by

characterized by

g1 0

S ^

to denote

k! permuta­

theprojection

Sk (g) 0 tj.

¥e note that the above tensor and symmetric tensor products are of course

also defined for Let 7

be

a

non-negative integer k bundle over

M

C°° real bundles.

C°° real bundle over

M.

For each

we defineLk (7 ,

to

be aC°°

is the space L k (7

whose fiber at x

(over R) maps of

(rx )k

into

gx

having the obvious bundle structure. bundle whose fiber at maps of iL

(7 v)k into

X

x g .

X

is If

7 x -► gx , defined byf(v)

g)

(we define ¥e define

the space SX

X

= f(v,..., v)

, g) of

Lk (7 , l)

and

complex vector k

L°(rx , lx ) = ix ),

Lg(7 x , !x )

f € Lk (7 „, g„)

g e V(M)

linear and

to be the sub­

of symmetric k-linear

there is an associated map i.e.,

f = f°A

where

is

§1 a

:

IV: k rx -► (?x)

DIFFERENTIAL OPERATORS ON VECTOR BUNDLES

is

diagonal map.

identified, it is the space of degree

k

then

7X

of

, ix )

into

The image of the map

P ^ ( r x , Sx ) | .

[If

e^

f f

1=1

= k,

xiei)

-*■ ( I

..., en

map is called polarization.

a = (c^ , .

P ^ ( 7 , |),

are the elements of

L^( 7 „, £„) S X X

P^ ( 7 X

with

It follows that

L g(?,

the bundle whose fiber at

characterized by the property that corresponds to the map

&x ) •

(w1 , ..., w^) -* v 1 (w1) ••• v^-(w^-)e

identified with the sub-bundle

of

£ g

then

corresponding element of

L^ ( 7 X, lx ),

of

Lk (7 , |) .

(v 0 ••• 0 v) 0 e

the inverse

Lk (7 , |),

0 e in ®^7X ® l x

Sk (7*) §> t

e

iY),

X

0 ^7 * 0 | with

Under this identification the sub-bundle

and

£x ] . The map

is

(v1 0 **• 0 v^)

Lg(7, |)

a )

i)is naturally equiva­

x

There is a canonical identification of

v € 7*

7x

is a basis for

X 1 1 ••• xn n)^a

a

ja

and the

is an isomorphism of

lent to

is easily

of homogeneous polynomial maps

where the sum is over all n-tuple of non-negative integers

f -*• f

v

consists of all maps of the form

(I

such that

53

is in

? x > lx ) •

in

0k7* §> I

is

In particular if

^ ( 7 *) 0 |

and the

namely

(w1, ..., wk) — v(w1) •**v(wk)e will be denoted by

S^(v) 0 e.

Note that we how havethree

naturally equivalent bundles

Sk (r*) ® I *» Lg(r, I) «p (k)(r, I) In general

we will find it convenient to "prefer"

Lg(r, I)

(some choice

has to be made) however, it will beconvenient to pass back and forth be­ tween these various ways of looking at the same thing and we shall often do so

tacitly. We denote the tangent bundle of

T(M)*,

the cotangent bundle of

bundle

T*(M)

pairs

(v, x)

by

T(M),its dual bundle

T*(M) andwe denoteby

T'(M)

the

with the zero section removed; so elements of

T'(M)

are

where

x e M

and

M, by

M

v

is a non-zero linear functional on the

^

PAIjAIS

tangent space to jt(v, x) = x. (v, x) T ’(M) number

M

Then

is

x, and the projection

**(g)

so

such that k

at

§1 «:

T'(M) -► M

is a vector bundle over

Hom(ir*|,

cr(v, x)

tj)

T'(M)

whose fiber at

consists of functions

is a linear map of

g„ X

a

For each real *)£ * Hom(ir g, * t]) by X

Smblk (g, n) = (a e Hom(ir*g, *%)|a(pv, x) = pkcr(v, x) If

k

with domain

t\'.

into

Smblk (g, i\) of

we define a linear sub space

is given by

p > 0}

if

is a non-negative integer then we define a subspace

Smbl^g, tj)

of

Smblk (g, tj) by Smbl£(t, n) = C” (P(k)(T*(M), L( 5 , t,))

a

or, to be precise, an element if and only if for each of degree

k

from

x e M

*

T (M)x

of

Hom(it*|, n %)

belongs to

Smbl^d, ti)

there is a homogeneous polynomial map

into

L( gx , t]x )

such that

f

X

a(v, x) = ^x (v ) •

There is a canonical isomorphism Xk : Hom(Lk(T(M), |), ^)

*

Smbl£(|, t|)

defined by *-k (F)(v, x)e = P(Sk (v) ® e) We leave

it to the reader to verify that

xk

is the composition

of the following sequence of canonical isomorphisms: Hom(Lk (T(M), ?), t,) - C"(L(Sk (T*(M)) ® 5 , r,)) ■» C” ((Sk (T*(M)) ® I)* ® 11) « C°°(Sk (T(M)) ® (I* ® n)) - C°°(P(k)(T*(M), L( |, t|)) The inverse inverse isomorphism to o h a,

Thus if

terized by

a € Smblk (g, n)

then

xk

.

will be denoted by

cf e Hom(L^(T(M), g), tj)

is charac­

c7(Sk (v) 0 e) = a ( v , x)e. Let

V

and W

be

of finite dimension and let thedifferential of

f at

respectively real and complex vector spaces f:

x

dfx (v) = lim l/t(f(x+tv) - f(x))

V -*• W

be a

is the element and

x -*■ dfx

C°° map. df

of

is a

Then for each L(V, W) C°° map

x € V

defined by df:

V -► L(V, W)

§2

IV:

DIFFERENTIAL OPERATORS ON VECTOR BUNDLES

called 'the first differential of L(V, L k ” 1 (V, W)) « Lk (V, ¥)

f . Making the canonical identification

we get inductively a sequence of

dkf = d(dk" 1 f) : V -+ Lk (V, W)

e = (e1, .. ., en)

dual base for

V*

is a base for

(a

V

f ..., a )

an I,

l 2 il

and

a of complex numbers tial operator then

e ^

a‘

xa = x 1 1 1

d^/dx-j

is the

Lg(V, W)

x 1, •••> xn

V)

the

then

is an "n-multi index", i.e., an n-tuple

of non-negative integers, we will put 1

and we let

dkf : V

... e. ) = --- ^ ------ (x) k dx. ... dx. 1 1 xk

1

a =

we have in fact

C°° coordinate system for

dkf ( e

If

C°° maps

and by a standard result of advanced calcu­

lus (the "equality of cross-derivatives") If

55

.

+ ••• + a

|a| = if

and

x = (x1, ..., xR)

cc! =

is an n-tuple

a

••• xn n dxR n .

and we write If

Da

for the differen­

e = (e1, ..., e )

is any n-tuple

|a|-tuple

(®ri) - 4 ^ ) where

( Lk (T(M)p; Sp) _ o is exact. PROOF. h - h' € hence

If

h, h' £ C°°(|*)

Z°(|*)so if

dk(h • f)

- dk (h'o f)

then it follows that its value

h(p)

f £ Zk_1(l)

at

satisfy then

h

If

then

X 1, ..., Xk £T(M)

depends on

and is clearly a linear

by reflexivity there is a unique element

h'(p)

• f - h'o f = (h-h1) » f £ Ik+1,

= dk(h-h') ° f) = o.

dk(h o f)(X1, ..., X k) p

h(p) =

function

h only through of h(p), hence

dkf(X1, ..., Xk) in

g^

such

that d£(h o f)(x1, Since

° f)

is symmetric and multilinear in

dpf € Lk (T(M) , g ).

that

X k) = h(p)(dkf(X1, .

X k))

X 1, .. ., X k

. it follows

The remaining statements of the lemma are now

trivial consequences of Lemmas 1 and 2 . q.e.d. Since into

Z^(g) C Zk_ p — p 1 (g),

Jk (l)p = C°°(|)/Zk (|)

Jk(l)p

into

Jk-1(?)

the canonical map

induces a linear map

with kernel

f •■+ J, (f)p k

jk (f)p M ’

of C°°(g) of

Zk_1 (5 )/zj^( I) . By Lemma 3 we have an

exact sequence 0 -*■ Lk(T(M)p , Ip) - Jk ( | ) p - Jk"’( e ) p - 0 where the injection fact that

i:

Lk s (T(M) p , |p ) —► Jk (6)_ p

is characterized by the

§2

IV:

DIFFERENTIAL OPERATORS ON VECTOR BUNDLES i(Sk (v) 0 e) = jk ( -gj- g^f)p

where

g

is any element of

and where that

f

C°°(M, C)

is any element of

Jk U ) p

j

satisfying

g(p) = o

such that

f(p) = e.

C°°(|)

is isomorphic to® k=Q L^(T(M)p , |p )

is a basis for Lg(T(M)p , |p)

59

. If

and

dg^ = v

If follows

e = (e^

..., eR)

T(M)_ then f -*■ [f ( e ^ )} .( is an isomorphism of P I I“Hi with ®| a |_m I , hence Jk U ) p is isomorphic to •

®|cc | < k ^p* Let f € C°°(|)

by

the form into a

jk (f)(p) = J*k (f)p-

J, (f)p k C°°

Up € M J k ( l ) p

Jk (|) =

it follows that there is at most one way to make

Jk (|)

Let

called the N

a half space in Let

cp: 0

tion of

N

|

M

such that if

Jk :

fh

f e C°°(£)

then

Jk (f) is a linear map of

and let

W

be a chart in

0.

Jk (i) e C°°(|)

k-jet extension map.

be either a finite dimensional vector space

V

over

Since every element of

for is of

vector bundle over

C°°(Jk (|)),

Jk (f) : M - J k (|) Jk (l)p

C°°(Jk (|)) and hence such that into

and define

V

or else

be a finite dimensional complex vector space M

and let

\|r:

^\0 ^ 0 x ¥

To prove that the above bundle

Jk (l)

be a trivializaexists we will

describe an associated trivialization

k Jk (?) I 0 « 0 x © L“ (V, W) « 0 x © m=0 |of|< k

W

It will be obvious that the trivializations so obtained are smoothly related Namely each into

W.

d111?^ = 0

f e C°°(|)

gives rise to a

By Taylor's Theorem if m = 0, 1, ... k.

p e N

C°° map then

f = t ° f ° cp-1 j, (f)p = 0 k

(See the proof of Lemma 1 .)

P =

if and

satisfies

vanishes in a neighborhood of

g(p) = 0 f(p) = e.

p

then clearly

j, (f) = 0 k p

Hence

Ji_(f*| O) K

0

T(p) e L( Jk (|

PROOF. ^ k ^ fi^p^

base

for Jk (|)

5ij

for x

near

M then

, tip).

is a base for

P* Let

such that

T € C°°(L(Jk (|), n)) = Hom(Jk (|), r\)

Necessity is clear.

p.Let

M

A necessary and sufficient

x »-► T(x) jk (f) be inC°°(Tj)

near

jk (f)p M>

Jk (|) | 0 ** Jk (t \ 6 ) .

T be a function on

condition that is that

is open in

induces an isomorphism

P

THEOREM 3. Let

and let

i(Sk (v) 0 e) =

satisfies

f € C°°(|)

is

Proved above.

THEOREM 2. If

so that

Jk(f)p ^ Jk_-, (f)p

Given

jk( ^ p

T (x )Jk (fj_)x = £ aij(x)g^(x)

p e M

5131(1 5161:106

hp € C°°(Jk U)*) §1 > •••> Ss

for each

so that

f e C°°(g).

choose ^ k ^ fi ^ hp (x)

be a local basis for

near

p.

Then

{f^}

in

C°°(g)

is a local (jk(fj)x ) =

i\

near

p

§3

61

IV: DIFFERENTIAL OPERATORS ON VECTOR BUNDLES

T(x) = 2 a1j(x)(h1 (x) ® 8j(x) then

is

L(Jk (|), n)

C“

near

and hence

near

T

p.

If

x*-» T(x)Jk (f1)x

is a smooth section of

is in

C ° ° ( t|)

Jk (t)* ® n

p. q.e.d.

§3.

Differential operators and their symbols DEFINITION. g to

from

A k-th order differential operator

t) is a linear map

such that, for each Df(p) = 0 C°°U)

into

g

p € M,

= 0

tip vanishes on

implies

f -+ Df(p)

(i.e., the linear map

the set of from

D: C°°(|) -*• C°°(t])

Zp(|)).

of

We denote

k-th order differential operators

to

t]

by

Diffk (£, ti) .

We now record some trivial consequences of the definition. First we note that a k-th order differential operator is local, i.e., if D e Diff^U, ti) and if and

Dfg

Diffk (|, C°°(M, C)

0.

agree on ti)

f 1, fg e C°°(|) agree on an open set Secondly

is a subspace of

then

Hom^,(C°°( |),C°°(ti))

it is a sub

D e Diffk (£, ti)and

will be called a universal k-th

order differential operator for

£

k-th order differential operator there is a unique

|

and in fact

A k-th order differential operator

U e Diffk (|, w)

of

Df1

T*D e Diffk (£, £) •

DEFINITION.

If

then

Diffk-1(|, ti) C_Diffk (£, tj) . Thirdly,

module. Finally we note that if

T e Hom(Ti, (;)

0

if for each

D e Diffk (|, ti)

T e Hom(w, ti) such that

U f€ Diffk (g, w ’)

D = T*U.

is a second universal k-th order operator

then by the usual uniqueness argument for universal objects there is

a unique isomorphism

T:

THEOREM 1.

w ~ w*

such that

U* = TU .

jk : C°°(|) - C ” (Jk (|))

k-th order differential operator for PROOF.

That

jk

is a universal g.

is a k-th order differential operator is just

the tautologous statement that if

J^^p = 0

then

Jf*)

= 0.

Given

62

PALAIS

D € Diff*k U , n) ZpU) into

the map

f -► Df(p)

§3

of

C°°(0

hence there is a unique linear map

into

vanishes on

of

C°°(|)/Z^(l) = Jk (l)p

T(p)

n

such that Df(p) = T(p)jlr(f)_. By Theorem 3 of Section 2 P K P T € Hom(Jk (|), ti) and by definition D = T*L . K q.e.d. COROLLARY 1 . The map ofHom(Jk (|), ti)

T -*■ T*jk

is an isomorphism

Diffk (S, n)

with

(as C°°(M, C)

modules). COROLLARY 2.

T -*■ T*

Hom( i , t})

is an isomorphism of

with DiffQ (|, n)• PROOF.

Since

Z°(|) = Cf € C°°(|) |f(p) = 0} there is a

identification of

J°(t)

with

|

through which jbecomes

natural

the identity

map. Since

differential operators are local it follows that if

D € Diff^d, r\) ential

@ is open in

and

operator denoted

= (Df) | O for each

by

f € C°°(i) .

M

there is a unique k-th order differ­

%| 0

D|a n

existsit is e € ix#

i(Sk (v) ® e),

where

i:

and

f

D(^y(g-g(x))kf)(x)

on

If

of Section 2

T'(M),

D e Diffk (| ,

we define a function

called the symbol of

it is

is the ’’inclusion" map

It follows that if

D e Diffk (£, ^)

such

^y(g-g(x))kf

is well-defined.

DEFINITION. a^(D)

f e C°°(|)

the k-jet of

Lk (T(M), |) -► Jk (|)

of the Jet Bundle Exact Sequence.

unique.

and infinitely many

f(x) = e, however given any suchg

Smbl£(S, n) - 0

Recall that since

M

.

is paracompact the additive cate­

is semi-simple (i.e., every exact sequence of vector bundles

splits) hence any additive functor from

category is exact.

V(M) to any additive

In particular applying the contravariant functor

L( , t|) : V(M) -► V(M)

to the Jet Bundle Exact Sequence (Theorem 1 of Sec­

tion 2) 0 -*■ Lk (T(M) , |)-i* Jk (|) - Jk_1 (I) - 0 we get an exact sequence in

V(M); namely .

*

0 - L(Jk_1 (|), ti) — L( Jk (|), T,)-n. L(Lk (T(M), |), T,) - 0 If now we apply the functor

C°°

.

to this sequence and recall that C°°L = Horn,

then using Corollary 1 above we get the exact sequence o where of

rk

—

(|, n ) - D i f f k U ,

= C°°(i ). Let

Hom(Jk (|), t|)such that

choose

g

e C°°(M, R) with

by Theorem l of Section 2

n) -L-. Hom(Lk (T(M), i ) , n) -

D e Diffk (£, ti)

and let

D = T*jk - Given dgx = v

and

v e

f € C°°(|)

T

0

bethe unique

T(M)X

and e e

with

f(x) = e.

i(Sk (v) 0 e) = Jk (^r( g-g(x))kf)

hence

element | Then

PALAIS

§3

rk (D)(Sk (v) ® e) = (C“ (i*)T)(Sk(v) ® e) = (i*T(x))(Sk (v) ® e) = T*(i(Sk (v) ® e)) = D(-jjr(g-g(x))kf) (x) = ok(D)(v, x)e If we now compose

.

with the isomorphism *k : Hom(Lk (T(M), |), r,)) ~ Smbl£( I, t,)

of Section 1 we get the exact sequence 0 - Diffk_1 (I, n) - Diffk(I, n) — —

^ — ► Snibl£( s, T)) - 0

.

Moreover, by the above calculation ck (D)(v, x)e = rk (D)(Sk (v) ® e) = xkrk(D)(v, x)e

Let us now paraphrase the information contained in the exactness of the Symbol Exact Sequence. symbol of some

First every element of

k-th order differential operator from

Smbl^U, n) £

to

is the

r\ and second­

ly two such operators have the same symbol if and only if they differ by a differential operator of order An element

a

of

is a linear isomorphism of £x D € Diff^fl, t\)

k-1. is called e l l i p t i c

Smbl^( |, n) tj

a(v, x)

for all

(v, x) €

T»(M), and

is called an elliptic k-th order

operator

if and only if

its symbol is elliptic.

with

if

We note that we can now assert the existence of an

elliptic operator corresponding to any elliptic symbol. Caution: a

If

D

is an elliptic

k-th order operator then

D

is

(k+l)st order operator but n o t an elliptic (k+l)st order operator, in

fact

*k+1(D) = 0 We now investigate what differential operators and their symbols

look like in coordinates. R n

and let

linear map

V D:

and

W

N

be either R n

or a closed half space in

be finite dimensional complex vector spaces.

C°°(N, V)

ential operator with

Let

C°°

A

C°°(N, W) is called a k-th order partialdiffer­ coefficients if it is of the form

§3

IV:

DIFFERENTIAL OPERATORS ON VECTOR BUNDLES

Df ‘

where

for each

into

n-multi-index

L(V, W) .

Daf(p)

belongs to where

X 1 9 '**> xn

a with

which implies

Diff^U, ti)

y € N,

Ac P ° f

I

I or | < k

|a|
v ) be

Then

°k P)(v,

y) (e) = jjr D(gkf)(y)

= jjr

]>

Aa (y) (D°gk) (y)e

I« I < k Clearly

Dagk (y) = 0

if

|a| < k

and

(Dagk)(y) = k!va

if

|a| = k,

hence : 0 ~ R n

such that if

positive

C°° function

p

on R n

is a char1: such that

any continuous, complex valued function

0.

compact subset of

Jf

is a Radon measure

M

there is a strictly

/ fdii = /(f°q>_1 (x))p(x)dx on

M

having as support a M

gives

M

and also one on

for the corresponding integrals.

dM

and we shall write

We note that if

f

M

with compact support then

We will denote by whose fiber at

x

Met (|)

|f| = 0 = > f = 0.

the complex fiber

bundle over

the obvious bundle structure. C°°(Met(£))

( , )^

are hermitian bundles over (which

namely if

An hermitian structure for

£

£ ,

with

is an element

and by an hermitian bundle we mean a complex vector bundle

with a specific choice

L( tj,|)

M

is the space of positive definite, hermitian symmetric,

conjugate bilinear forms (i.e., Hilbert space inner products) on

I

is a

m

continuous function on

of

for

p(x) = (det g ^ ( x ) ) 2 . We shall assume as

given a fixed such measure on and

f

M

For example, a Riemannian structure on

rise to such a measure, with

Jf M

69

M

of hermitian structure.

there is a natural map

*

If

T eL( £x , tix )

then

T*€ L( tjx , ix )

isdefined

r\

and

L(|, t\) into

of

is a real bundle isomorphism but is conjugate

£

linear)

by

(Te, e 1)

*

(e, T*e1)^.

This gives rise to a conjugate linear map, also denoted by * of Smb 1^.(1, r\) into Smbl^(r], |) defined by a (v, x) =■ a(v, x) . If a

|

is an hermitian bundle then for

C°° complex valued function

Clearly,

(f, g) -► (f, g) ^

and the support of of

f

and

g.

< f, g >

and

(f, g)g(x) = (f(x), g(x))^.

f

and

g

belong to the space

= JM (f, g)

is well defined. 00

C_(|) c

(f, g) Clearly

and the associated norm

C~(|) of

has compact < , >

If

I

and

D e Dif fk (|, r\)

tj are hermitian bundles over then

D * € D i f f k (ii, |)

is a -

< f, f >?

||f||^ -

DEFINITION. M

by

having compact support, then

prehilbert space structure for will be denoted by

M

is included in the intersection of the supports

In particular if |

on

we define

is hermitian symmetric and conjugate bilinear,

(f, g)

C°° cross-section of support and

(f, g) ^

f, g € C°°(£)

*

will be

5

70

PALAIS called a formal adjoint for < f, D*g >£

whenever

g € Cq (t])

PROOF.

Let

then

if

< Df, g >^ =

f e C” (g) and

S € Differ], i)

LEMMA 1 . If all

D

and if

g € C” (t]). Sg =

0 for

S = 0.

h € C°°(g) . We must show

Sh = 0,

and since e

is continuous it will suffice to prove that

Sh(p) = 0 if p

cp be a

subset ofM - dM

C°° function with support a compact

identically one in a neighborhood of and

since g

agrees

with

h

p

and let

g = cph.

in a neighborhood of

p,

dM.

Sh Let

which is

Then

Sg = 0

Sh(p) =Sg(p)

= 0. q.e.d.

LEMMA 2.

If

D € Diff^(|, t}) has a formal

adjoint

then it has a unique formal adjoint. PROOF T

Suppose

T

In particular taking

Sg

D

such that for each

n 0^

gives

||Sg|L =0hence 1

Diffk (g, tj)

D* e Differ)! O a , I I &a ) • D|

= 0

> g - < f, T'g > g = g = < f, Tg

for

T ’ are adjointsfor

- T T.By Lemma 1 it will suffice to show that

Now if

of

and

- < Df,

q.e.d.

{0

and that

is an open cover

hasa formal adjoint

Then clearly

D*|0 a n 0

D* | 0 a n 0

= 0 .

Sg = 0.

a Da = D| 0 a

so by Lemma 2

g

is a formal adjoint

= D* | 6 a n0 ^ •By

Corollary 3

of Theorem 1 of Section 3 there is a uniqueD € Differ], g) such that * # *#■ D |0 a = Da for all a and clearly D is a formal adjoint for D (by an obvious partition of unity argument) hence LEMMA 3.

If

D e Diffk (g, t\)

cp: 0 ~ ® C_Hn adjoint then LEMMA Ij-. space

and if for

of an atlas for M, D|

0

each chart hasa formal

D has a formal adjoint.

Let

M

be either

{x e R n |xn < 0} . Then

formal adjoint

D*

and

Rn

or theclosedhalf

D e Diffk (g, ti) = (-1)kcjk (D)* .

has a

IV: PROOF. space, the proof M = Rn.

We shall consider only the case where being essentially the same but

Let the measure on

that on dM = R n_1 a

are positive

M

be given by

t| = M x V 2

inner products by

T f:

where V^ ( , )^.

V 2 -*■ V 1 ,so

are If

alittle simpler when

L

= J^f(x)p(x)dx and let

We can assume

| = M x V1

finite dimensional Hilbert spaces with

T:

V1

V 2 is linear we denote its adjoint

(Tv, w) 2 = (v, T'w)^ The hermitian structure for

of positive operators on las

is a half

/^M f = / n _1 f(x)a(x)dx where p and R ^ C°° functions on M and R n_ respectively and dx are

r\are uniquely given by

and

M

be given by

the respective elements of Lebesgue measure. and

71

DIFFERENTIAL OPERATORS ON VECTOR BUNDLES

C°° maps

V1

(e, e')g = (A(x)e, e ,)1

and

A

V2

if

and

e, e ’ € nX .

Then if

by

T* = A(x) _1T'B(x) . Let

D:

of

e, e 1 € lx

and

^

into the spaces

(e, e ’)

its adjoint

T*:

C~(M, V 1) — C°°(M, V 2)

Df =

M

respectively according to the formu­

T e L(e X, Xr\ )

if

B

|

= (B(x)e, e ’)2

X -►X|

is given

be given by

Ca (D“f)

a, where the sum is over all n-multi-indices is a

C°° map of If

M

into

s ^m b M a or \ i i

D g =

^

where

(-l)Wi A _1D >'(pC^Bg)

x i =

G o k_,f, jk-ig)

dM Then if so

f

and

g

have supports disjoint from

< Df, g >^ = < f, D*g >£

hence

D*

is a formal adjoint for

Moreover the characteristic polynomial for

(-Ok

G(jk_.jf, ^k-1^ = 0

dM,

D

D.

is

£ A(x)‘1C^(x)B(x)va = (-1)k £ C*(x)va |a|=k la I=k

- 0 - l)k

*

I

a |=k which since that

Z|a |_k Ca (x)va

is the characteristic polynomial of

D

proves

*) = (-l)kak (D)*. q.e.d. We note the following Corollary of Proof.

Grk (|, t])

be the bundle over

of conjugate bilinear maps of

In the case dM = R n”1 x

M = {x e R n |xn < o},

whose fiber at into

x C.

let

is the space Then for each

§5

IV:

D e Diffjj.Cl, n)

DIFFERENTIAL OPERATORS ON VECTOR BUNDLES

there is a cross

< Df, g > n - < f,

whenever

f € C^(|)

and

section G €C (Gr^l, 11)

D*g > 5 =

73

such

that

G(jk. 1 f, J ^ g )

J dM

g € C^(rj) .

The main result of this section, which now follows trivially from Lemmas 2 , 3 and k and Theorem 3 of Section 3 THEOREM. M

If

then each

adjoint

|

r\ are hermitian bundles over

and

D e Diff^d, r\)

D*e Diff^d,

We note that

D

DEFINITION.

If 5

is

= D

has a unique formal

|) . Moreover and if D €

ak (D*) = (-1 )kak (D)*.

Differ), £)

then

(DD)* = d V .

D € Diff2k(|, |) 2k

isan hermitian bundle over

is called strongly elliptic of order (-1 )^ ^ ( D ) (v, x)

if and only if

operator on for all

COROLLARY.

If

in

is injective

| )

for all

D € Diff^d, t\)

elliptic of order

2k

for all

is a positive

( ( - 1 )ka 2 k (D) (v, x)e, e) | > 0

!x (i.e.,

e ^ 0

M then

(v, x) € T !(M) . then

D*D

is strongly

if and only if ^(D) (v,

x) :Sx -► r\x

(v, x) e T ’(M). Hence if

Tj have the same fiber dimension then elliptic if and only if

DD

is

D

£ and

is k-th order

2 k-th order strongly

elliptic.

§ 5.

Green* s forms In this section we again assume given strictly positive smooth

measures on M.

If

M

and on

D € Diff^d, t))

f € C“ (g)

and

dM

and that

|

and

then the formula

r\ arehermitian bundles over

< Df, g

= < f, D*g

g € C^d) have supports disjoint from

generally we should be able to express

< Df, g

£M

suggests that more

- < f, D*g

as an in­

tegral

of

some expression involving the jetsof f

theory

of

boundary value problems requires that thisbemade precise and we

shall now do so.

andg

where

over

dM.

The

PALAIS DEFINITION.

§5

We define a bundle

whose fiber at

x

I, ti)

over

SM

is the space of conjugate bilinear

maps of

jk~1(£)x x

into

section

G e C°°(Grk (i, t}))

order Green's form of

g

A C°° cross

will be called a t\

and

and will be called

D e Diffk (g, i\)

a Green's form for

k-th

if

< Df, g > n - < f, D*g > t = J G(Jk_,(f), j ^ g ) ) 9M for all f € C” (|) REMARK.

Let

is a Green's form for

and

g £ C“ (n) .

M = Cx £ R n |xn < o},

D

Diff2(g,

e

also

so

t\)

| = ti = M x C . Then if isG !,

G't^f, j,g) = G(J,f, J,g) + This

G

defined by

(fg)

.

shows thatin general Green's forms for operators are not

uniquely

determined. REMARK.

If

valued functions on

G e C°°(Grk (g, ti))

M

and

is a bilinear form in

similarly for let

{cp^}

{ ^ a },

j^.-j^g)*

be a locally finite

Ga e C°°(Grk (g| O that

G(jk_,(f), jk_i(g))

D e Diffk(|, t]) and

tj| ,

for

has a Green's form. D € Diffk (t, i\)

THEOREM.

Every

has a Green's form.

PROOF.

Immediate from the Lemma and the Corollary ofProof of

Lemma k of the preceding section.

§6.

Some classical differential operators a.

The Lie derivative Let

g be a complex vector bundle over

consists of all complex valued tensors at metry type.

Given a vector field

be the Lie derivative operator.

X

x

on

M

whose fiber at x

of a fixed variance and sym­ M

define

C°°(|) -► C°°(|)

Then bya basic property of

^x

to

we have

a>x (gf) = (xg)f + gdxf if

g € C°°(M, R)

that = v

f € C°°(|) .

and

^x e Diff^S, g) . Also if

It follows from Theorem 5 of Section 3 g € C°°(M, R)

vanishes at

x

and

dg^.

then tf-l^xMv, x)f(x) = dx (gf)(x) = v(X)f(x)

i.e.,

a1(^x )(v, x)

is scalar multiplication by

is elliptic if and only if b.

M

v(X).

is one dimensional and

X

It follows that dx never vanishes.

The exterior derivative Let

| = a (T*(M) ) ® C = ® L q

complex valued differential forms and let

A1 (T*(M)) ® C be the bundle of d:

C°°(g) -► C°°(|)

as usual de­

note exterior derivation. If

g

is a C°° real

valued function on M and

d(gca) = dg a cd + g By Theorem5 of Section

3 again

g(x)

then

= o and

dg^^. = v

A

cde

then

da)

we see that d e Diff1(g, |).

d(go))x = v a cdx

C°°(|)

Also if

76

§6

PALAIS a1(d)(v, x) = va 9

which proves that c.

left exterior multiplication by v.

The codifferential and Laplacian Maintaining the notation of (b) assume now in addition that

is an n-dimensional Riemannian manifold.

The Riemannian structure for

defines strictly positive smooth measures on hermitian structure on to

Af = 0.

hence

df = 0 ; Clearly if

and on

6 = d

*

dM, and also a

t

= (d+5 )2 .

a

Recall that

a

2

Note that 5

and a = d5 + 5d.

f € C°°(i) is called 0

it is called co-closed if 5f = f

M

5 , a first order operator and the Laplacian

a second order operator, by

closed if

M

| . We define two differential operators from

the codifferential

(d*)2 = (d2)* = 0

M

and harmonic if

is closed and co-closed it is harmonic.

If

f,

g ^ cjd) < Af, g >£ = < d5f, g >£ + < 5df, g >£ =< 5f, Sg >| Af = 0

In particular if

then taking

g = f gives

5f =

+ < df, dg > s . df = 0 ,i.e.,

conversely a harmonic form is closed and co-closed if its support is com­ pact and disjoint from

dM.

In particular on a compact manifold without

boundary every harmonic form is closed and co-closed. To give a more precise description of M

is oriented and we let

sure (i.e.,

forT (M) ).

Var i£t £s D i f f e r e n t i a b l e s .

and

we assume that

a

a> be the n-form describing the Riemannian mea­

cox= e1 a... aeR

normal base

s

where

(e^

..., en )

is any oriented ortho-

For details of what we sketch below Recall that the inner product in

pletely described by the properties that the

see de Rham'e |

A^(T*(M)x ) ® C

is com­

are mutually

orthogonal and (v,a...a

vp ,

V / ...A

Wp) 5 =

£

eCnMv,,

W)[(1))

•••

( v p> w f l ( p ) )

Jt where the sum is over all permutations of (1 ,...,

p ) . Equivalently, if

(e., ..., e )

is an orthonormal basis for T

then

(1 < p < n,

i1< i2
e = y tfA * g in M Let (-1)p

on

w be the automorphism of

Ap (T*(M)) ® C

|

which is multiplication by

so d(f a g) = df a g + w(f) a dg

We note that

[w(f) a d*g]R = -[f a d*w(g)ln hence if f, g [d(f a* g)]n =

[df a*

f, g € C*(s)

Stokes'

y [f ** g]n _.,

= y [d(f A* g)]n = < df, g > 5

dMdefining the

= G(e, e !)vx for x €

is the

(n-l)-form on

dM and

e, e 1 e J°(l)x = £x

then

by G

5

has degree -1, hence since

d

has degree

+1,

0.

Given =

7

d.

We note that has degree

and moreoverthat if

Riemannian measure and if we define G € C°°(Gr.j(£, |))

is a Green's form for

A

- < f, w * d * w g > i

M

that s = w*d*w

[e a * e ,]n_1

g]n - [f a * w*d*w(g)]n

formula gives

dM This proves

C°°(£) then

g]n - [f a d*w(g)]n

= [df a * If

€

v e T*(M))X

(-1)J + 1 (v,© j)

A • .•A e^

define a

...

a

iy : !x

e .

= (e, ive !)|* It suffices (since v -*■iv belonging to an orthonormal basis (e^ also assume that

e

and

(1 < p < n; i1 < ... < ip) the formula in this case.

e*

byiy (e1 a ...a

e )

Then we claim that(v A e,

e ’)^

is linear) to provethis for

v

..., en)

for

T

(M)

and we can

belong to the orthonormal basis {e. a ...a e. } for

£x>

Then since

the theorem of Section k that follows from Theorem k of

Sx

We leave it to the reader to check a1 (d) (v , x ) = v a

a1(5)(v, x) = -i . Since

Section 3 that

o2(a)(v, x )

=

it follows from a = d& + 5d -(v a i

it

+ i^v a ) .

78

§6

PAIAIS

A straightforward application of definitions shows that 2

= ]|v|| e, hence

hence

2

is scalar multiplication in g by -||v|| , 2 * is strongly elliptic. Since a = (d+8) = (d+8) (d+8), by the

a

ct2(a )(v

, x)

corollary at the end of Section k and

5

are of degree

+1

and

d + 5

-1

is elliptic-*

sub-bundles of

I

defined by

|° = 9 ^ A2p+1 (T*(M)) *® C

ge = ^

then

a differential operator from

+ 5.

d + 5

A2p(T*(M)) 0 maps

C°°(|e)

We haveseen that

[f A* g]n-l

I

d

d + 5 maps a p-form into e o Hence if g • and | are the

to

if

C

C°°(ge) into

C°°(g0)

f, g

and

C°°(g?),

and as

is still elliptic.

operator plays a basic role in the Index Theorem. d

Note that since

respectively

the sum of a (p-l)-form and a (p+1)-form.

formfor

v a i^e + i^v a e

This

We now compute a Green's

e C ” (|)

- < df, g > e - < f, 6g

>s

.

SM Complex conjugating(and noting

*g = *g)

- e - < eg,

f >5

.

in the latter equation and subtracting from the

former
5 -
5 = y

[f

a

*

g-g

a*

f]n _1

dM from which a

Green'sform for

d + 5

A (the prototype and the namesake of necessary to first replace

f

by

is clear. To get a Green's form for all Green's forms) it is now only

(d+5)f

and next replace

g

by (d+5)g

in the above equation and then add the two resulting equations. d.

The operator Now let

n = 2m.

Then g = ^

§

M

be a complex analytic manifold of real dimension

^ gP'^- where

holomorphic local coordinate

(gp 'q)x

z1, ..., zm

consists of forms which in at

x

have the form

Z a. . . . dz. A . . . a dz. a dz. 11 ** pJ1 *’*Jq 1 p

a ... a

dz. Jq

(i, < ... < ipj Ji < ... < jq) Elements of

g£>q

.

are calleddifferential forms of type (p, q) and clearly

§7

IV:

^>+q-r

DIFFERENTIAL OPERATORS ON VECTOR BUNDIES

= Ar (T*(M)) 0 C = |r .

Diffjd, l )

such that

if

zm

z1,

d = d + §, d2 = §2 = 0

arelocal

6

can he written in

then

and

df = Z1 ? i=1

C°°(|P,cl+1)

0

r\

r\.

r\ and

M,

| 0 r\

= (§f) 0

sections

over

(9

f € C°°(£)

holomorphic.

Then

g1,...,

then

F =

If

Riemannian structure and duced hermitian structure. the proof that

§ + t

fer the reader to

we call

(p, q)

with coeffi­

gq = Z a ^ lu

ajiPj_) ® kj

§

such that if 6

is a basis of

holomorphic

can be written unique­ §

exists we must holo­

a^. e C°°( (9 f C)

where

anc^ since

is

d ( a ^ f q) = a ^ d f ^

which proves

d| 0

we

is well defined

completes the proof of

with the stated properties.

r\ an hermitian structure.then The computation of the adjoint

If we give £ 0 t\ t

M

of

§

and We re­

Neue Topologische Methoden in dev Alge-

bvaisahen Geometvie (p. 117) for details.

§7.

Whitney sums Given

(C°°-complex) vector

their Whitney sum is the bundle (tj_)x

a

has an in­

is elliptic is similar to that in (c) above.

F. Hirzebruch's

is

is another basis of

Corollary 3 of Theorem 1 of Section 1

the existence of a unique

M

^5=0) oz

is holomorphic then

h1 , ..., h^.

Z^ df^ 0 g^ = Z^ d(Zq aj^Pj_) ® kj

and unique.

into

then (since

every F € C°°(| 0 r|)

0.

over

0

a...a

We note further that if

g e C°°( t|| (9)

In fact if

O then over

morphic sections over

f € C°°(|)

C°°(|-P,q-)

§ € Diff 1 (| 0 t), t 0 n)

and if

g.

dF = Z^ 5f 0 gq

and

Namely,

and

F = Z^ f^ 0 gq ; f^ € C°°( 11 (9) . Then if sucha

ly as

have

dz. a

ei ais^v ' x ^ei^

Similarly there is a linear isomorphism e

Diffk (ii , t)j) ~ Diffk (©1 e± , ©j hj), (D-yl — ®

11J

d±j J

where ( £ Given ©.

J

D ^ K © fi) - (®± D±1 fi,

D € Diff^(©^ g^, ©j tj. j)

.Diff, (|., n.) K

1

j

^

D ls f±)

.

the corresponding element

will be called the matrix of

of

D and where convenient

will be written as a matrix D 11

•••

D ls

D rl

...

D rs

V

-±® V V

We note that

so

). . D. . is

(v, x)))

k th order elliptic if and only if the matrix ((a,x (D. i j .)

defines an isomorphism of

with

©j(T^)x

for all

(v, x) e T'(M). There is a more general notion of ellipticity due to Douglis and Nirenberg that applies in this situation. DEFINITION.

©.

. D . . is called elliptic in the sense

of Douglis and Nirenberg if there exist non-negative integers

s.

i

and

t.

J

and such that for all

such that

D .. € Diff

(v, x) e T'(M)

i

. (g. j

the matrix

§8

81

IV: DIFFERENTIAL OPERATORS ON VECTOR BUNDLES

((081 -tj (Di 3 ) ( v * X)))

defines an isomorphism of (If

with

®j_ j Dpj

is

©j(iij)x .

order elliptic we can take

tj = 0) . This becomes a little clearer in the case i ^ j. and

In this case we say that

©^

y

= ©^ j

is d i a g o n a l

©^ ^

Then in this case

and only if each

€ Diff^U^, rj-^)

r = s

©^

is

s^ = k ,

and

= 0,

and we put

D^ =

is kth order elliptic if

kttL order elliptic, while

©^

is elliptic in the sense of Douglis and Nirenberg if and only if there exist non-negative integers order elliptic

§8.

k^(= s^- t^)

i = 1,

(|^, t^)

is

k^*1

r.

Tensor products. Let

and let

^

M

and

and l2

N

Then

(x, y)

U-|)x ® (i2)y

is

^

0 l2

f1 0 f2(x, ^

over

M

and

x Smbljdg, Tg)

C°° manifolds, at least one without boundary,

is the vector bundle over and whose bundle

e C00^ ) ,

the property that if (where

be

be (C°° complex) vector bundles over

spectively.

t).j

such that

=

t)2

over

into

N

and

M x N

N

re­

whose fiber at

structure ischaracterized by

fg € C°°(l2)

®

M

then

f1 ® f2 € C00^ ®

l2)

we are also given vector bundles we define a bilinear pairing of S m b l ^ ^ ,

Smbl^+fd 1 ® i 2 , >i1 ® n2),

(ap

a 2) — a., ® Og,

t^) de-

fined by a1 0 a2((v^ , v 2), (x, y)) = LEMMA. and

If

D1 e D i f f ^ ,

T e Diffk+J^(i1 0 i2,

n,),

x)

0 a2(v2,y)

i>2 e Diff^(l2,

0 t]2)

Choose with

Let

g1 € C°°(M, R ) g2(y) = 0

g-j (x) + g2(y). fi € c°°^i^

and Then

with

cr^Dg).

((v 1 ,v2), (x, y)) e T*(MxN) and let

with

g1(x) = o

and

(hg2)y = v2 . Define g(x, y) = o

f.(x) = e^

and

)

satisfies

T f f ^ f2) = L 1f10 D 2f2, then ak+i(T) = ^ ^ 0 PROOF.

tj 2

e±

(dg-|)x = v 1 and g € C°°(MxN, R)

dg^x ^

.

= (v1 , v2)

by

€ (li)x .

g2 e C°°(N, R) g(x, y) =

so choosing

82

§8

PAIAIS °k+£(T) (v1, v2) (e1 e2) 1 (k+£)

T(gk+£f1® f2) (x, y)

1^ , t ( (k+£)!

I

l_

4 * < - f 2) ( l , y)

m < k+£

® — 1—

lyD , ( # , ) ( * )

. Theorem 2], PROOF.

ak (Dk (a))(v, x)e = ak (a* • D k) (v, x)e =

a 0 (af^) ak (Dk) (v, x)e = a(Sk (v) 0 e) of

a,

just above, we have

by Theorem 5.

Since by the'definition

a^S^Cv) 0 e) = a(v, x)e

we get

ak ^ k ^ a^

=

q.e.d If

£and n are vectorbundles then recall [§1] there is a ('k') ~k • natural projection S v J of the bundle0 | 0 i\ onto thesub-bundle Sk (£) 0 t|,

namely symmetrization, defined by 3 x )ai( = S ^ ) (v 0 (... 0 (v 0 = S ^ ((v 0 ... 0 v) 0 and in thecanonical (v 0 ... 0 v) 0 e

x)

•••

e) ...) e) = (v 0 ... 0 v) 0 e

identification of S^(T*(M)) corresponds to

x )e

0|

with L g(T(M), £)

S^(v) 0 e. q.e.d

COROLLARY.

Given

v6

canonical isomorphism

and

v

Vm*

J (£) «

k ©

there is a m L (T(M), £)

m=o such that

jk (f) « {Dm (f))0 < m < k

.

§1 0 .

IV: DIFFERENTIAL OPERATORS ON VECTOR BUNDIES PROOF.

91

Immediate from the theorem and the definition of a kth

total differential.

§10.

Spin structures and Dirac operators We begin by recalling some basic facts about Cliffordalgebras

and spinor groups.

Details can be found in J. Milnor's mimeographed

notes "The Representation Rings

of

The Clifford algebra

An

(1963)

Some Classical Groups.11

is characterized to within unique

isomorphism by the following: 1)A Q

is a real algebra with unit;

2) R n C An and generates 3)

= -1 and e ^ j =

An ;

~ejej_

where

e1, ..., en is the

standard basis for R n . It follows that for

An so in particular

and

A“

1 1

(i^ = 0 or 1)

is a basis

Also An = A* ® A “, where

A*

are the sub spaces spanned by those elements of the above basis

A~A~ C A*

is a subalgebra,

of

i-n

... en )

dim An = 2n .

having an even and odd number of

ence

i2

{e1 e2

aQ + a1 -► aQ + a1en An _1

with

all v e Rn

of

pin(n) onto

and

equal one respectively.

A^A” = A “A*

aQ e A^_1, a1 e A " ^

where

u

of

An

form a subgroup

such thatp(u)v =

pin(n)

of

An

0(n). The inverse image of

by Spin(n), the n-dimensional spinor group. (for

n > 1) subgroup

Since

7t1(S0(n)) = Z 2

coveringgroup of If

£ A“ ,

Then

A*

and the correspond­ is an isomorphism

A* .

Those units for

i^

of A*

and

for

n > 2

S0(n)

‘for

p

and p

is in R n

uvu-1

is a homomorphism

S0(n) under

p

is denoted

Then Spin(n) is a connected

restricted to .Spin(n) has

kernel +1.

it follows that Spin(n) is the universal

n > 2.

n = 2 k is even then the complexified algebra

An 0 C

is

simple and hence has to within isomorphism a unique irreducible left module S n of

which we can choose to be some minimal left ideal in Sn

are called n-dimensional spinors.

Sn splits uniquely as a direct sum

A_ 0 C. Elements n Considered as an A* 0 C module

Sn = S* © S”

of two inequivalent ir­

reducible modules, elements of which are called positive and negative spinors

92

§1 0

PALAIS

respectively.

Moreover

have

vS* C S”

gard

Sn

and

vS~ C S*

ble.

If

u € Spin(n)

and S^

A^S^

C S* for

so in particular since R n C A “ v e R n . Since Spin(n) C A*

we

we can re­

as Spin(n) modules, and as such they are still irreduci­ v € Rn

and

then we note that for

s € sn

u(vs) =

(uvu_1)(us) = (p(u)v)(us). Now let structure for from

S0(n)

M

M

be an oriented Riemannian manifold.

is meant a reduction of the structural group of

We are given an indexed set

{ 6>x) is an open covering of

fields defined on

(9

and

M

and

..., X^)}

X^, ..., X^

are

C°° vector

forming an oriented crthonormal basis at each point.

And we are also given maps forx e Ox

T(M)

to Spin(n). Expressed in the language of coordinate bundles

this means the following. where

By a spin

n (9^ -► Spin(n) satisfying

g^:

gX|i(x) g^v (x) = g^v (x)

p(gXtJ(x))1j = (Xj(x),

x € &x n

for

X^(x)) or equivalently

such a spin structure we let

Sn (M)

y , and finally

Xj =

. Given

be the associated bundle with fiber Sn .

Thus the total space of

Sn (M) is the disjoint union of the sets x n O .

(x) Since

S*

Sn (M)

splits canonically as the Whitney sum of sub-bundles

S”(M) . if

and

S~

are Spin(n) invariant submodules of

There is a bilinear

w € T(M)X ,

x e 0^

pairing

and w =

Sn

T(M) x Sn (M) Sn (M) v^X^(x)

it follows that S*(M)

and

given as

follows:

then

w(x, s, \i) = (x, vs, |i) . The identity

u(vs) = (p(u)v)(us),

noted above, with

u = g^(x)

this definition is consistent with the identifications. it follows that we have induced bilinear maps T(M)

x S“(M) — S*(M). We now define

o(v,

x)s = vs

in maps

Smbl^(S+(M), S'(M)) Sn (M)x

C S”

R nS*

T(M) x S*(M) -+ S”(M)

a e Smbl^(Sn (M), Sn (M))

(where we have identified

mannian structure). By

Since

shows

T*(M)

with T(M)

and

by via the Rie­

the above we can consider a by restriction to be or Smbl^(S‘(M), S*(M)). If

isomorphically onto itself, in fact

the defining relations of the Clifford algebras.

v^O

a(v, x) Thus-

a

2

then

±- e -> dSx(w ) = with

v

xi it follows that Now let

f(x) = s

®x

1 V^ for

we can define

T(M) . x

D maps

C°°(S*)

into

C°°(S”)

and let

g e C°°(M, R) with g(x) = 0

v) •

Then if

s € Sn (M)x (w^, v) s

and

f e C°°(Sn (M))

hence

a1(d)(v, x)s = D(gf)(x) = ^ ( w 1 , v)wis = vs = a ( v , x)s i D hasthe required symbol

S*(M)

and vice versa.

T(M)

and

That

we have

restricts to covariant derivatives for

then v^(gf)(x) = dg(w^)s =

which proves that

and

associated with the Riemannian

w • • •> ¥n

v 1, ••*> vn

so that

S*(M)

but there seems to be no particular advantage involved

in taking it) . Let T(M)X

V

S (M)

and hence is elliptic.

CHAPTER V ANALYTICAL INDICES OF SOME CONCRETE OPERATORS Robert M. Solovay In § 1 , we review Hodge theory.

This will be used in §§2 , 3 to

check the index theorem in a number of concrete cases. a role in the proof of the index theorem in general.

This check will play In §^, we show that

the topological index of a differential operator on an odd dimensional mani­ fold is zero. § 1 . Review of Hodge theory 1 .0 .

mension 5 , and

n. *

Let X

be a closed oriented Riemannian manifold of di­

We shall recall the principalproperties of the operators introduced 1.1.

in Chapter IV, § 6 . = Ak (T*(X)) 0 C

Let

differential k-forms over

X.

a Hermitian structure on X.

(a,

p) j 1 .2 .

(1 ) If of

be the bundle of complex valued

The Riemannian structure on

X

gives rise to

| , and to a strictly positive smooth measure on

In this way, we get a Hermitian inner product on

note by

d,

C^U^)

which we de­

(a, 0 e C°°(|k)). There is a bundle isomorphism *: Ak (T*(X)) ® C a An~k (T*(X)) ® C

V

is the fibre of T*(X) over

x,

at the point

x

of

X, then, on the fibre

(1 )is the complexification of theisomorphism *!

discussed in III, § 6 . will also be denoted by

Ak (V) S An_k(V)

The induced isomorphism of *

C“ (S^)

with

C°°(|n_P)

Recall the following properties of the map 95

*:

96

SOLOVAY

(2)

(a,

(3) If (10

p) = / a a * p f

( a> p e C°°( |k)) ;

*2 - (-l)k(n-k)

on

(5)

= *P, 1.3.

Then

d

=0.

C°°( |k) .

n is even, (3) simplifies to *2 = (-l)k

p

§1

Let

d:

on

C°°(?k),

i.e.,

*

(dim X

even).

is real.

C°°( £k) -► C°°( |k+1)

be the exterior derivative.

DeRham's theorem asserts that the cohomology groups of H*(X; C),

with complex coefficients,

X

are canonically isomorphic to the

cohomology groups of the complex C°°(|0) - i * Let- 6:

... c“ (ik) - L * c°°(!k+1) — C“ (|k+1) — C"(5k)

...

be the adjoint of

d.

. When

X

is

even-dimensional, we have (6)

5 = -*d* Let

is formally

| = A(T# (X)) ® C = E£=q

self-adjoint, (i.e.,

A = (d+5)2 = ds + 5d. Then C°°(|lc).

We have,

(7) If

dcp = 0;

0, we say that

this:

e very

C"(|) - C“ (|) We put

is homogeneous of degree zero: a : C^fS^) “**

0,

X K.

has a

112

PAIAIS PROOF.

B y Asooli's theorem it will suffice to prove that the

V

implies |4(x)| < SupC ||k|| | k e K}, and since X U(x) - H ( j ) | < ||x - y|| the elements of B * are even equicontinuous on X X. q.e.d. clearly

H€ B *

THEOREM 12.

If

T e K(X, Y)

then

PROOF. {J^n )

X

and

Y

are Banach spaces and

T* € K(Y*, X*).

By the lemma if

)

is a sequence in _

has a subsequence which converges uniformly on

or equivalently (since

TB^-

B *

then

y

hence on

TB^,

T*4n (x) = ^n (Tx))T*£n

has a subsequence which con* verges uniformly on B^., i.e., which converges in X . But this means pre * * cisely that T B * is relatively compact in X . Y q.e.d. THEOREM 13.

(F. Riesz)

k e K(X, X)

and

Let

T = I - k.

1)

ker T = T”1(o)

2)

T(X)

X

be a Banach space,

Then

is finitedimensional;

is closed in

X

and coker T = X/T(X)

is finite

dimensional. PROOF. k(b)

Let

= b - T(b) = b

V = ker T.

for

relatively compact, hence

b e By,

Then

if

i.e.,

By

is the unit ball of

k(By) = By.

But,

k(By)

is locally compact and by Theorem k

V

V is

dim V
o ||Tw|| > e ||w|| for

will be a Banach space, (wn ) x €

and we can assume wn

But But

dim (X/T(X)) < then

T € K(X, Y) .

is a bounded subset of

T(X)

and hence by

Theorem 3 it is relatively compact. THEOREM 15. G

If

H

is a Hilbert space then the group

of invertible elements of

subset of PROOF.

L(H, H)

is a connected

L(H, H) . T € G

If

then

T*T

is a strictlypositive operator

on

H and hence has a strictly positive square root

so

U*U = A"1T*TA“1 = A ' V a '1 = I

and

U

A. Let

is unitary.

U = TA_1

Since the set of

strictly positive operators is clearly a convex subset of

G

arc in Gfrom

U. Let 1/i log

A

to

I, hence an arc from

be the inverse of the function Then

T

= UA

t -► e ^ of

[0, 2*)

to onto

there is an

{z e C|

1 /i log is a bounded Borel function on the spectrum of

U

|z| = 1 }.

so by the

B = 1 /i log U is a bounded self 2it it U = J e dP^ is the spectraldecompo-

functional calculus for normal operators adjoint operator (equivalently if sitlon of

U

then

is an arc in

G

B = J2n tdPt) .

from

I

to

Then

U = elB

and

t — eltB 0 < t < 1

U. q.e.d.

LEMMA.

If

H

is a Hilbert space and

is self adjoint then either eigenvalue of PROOF. ||Txn || -► IITII^. and then

Since

||y|| = IMI^.

IMI*,

or “M o ,

is an

T.

Choose a sequence (xn ) T

T € K(H, H)

on the unit sphere of

is completelycontinuous we can assume

H

with

Txn -► y

By Schwartz's inequality

||Ty|| = 11m ||T2xn || > llmfT2^ ,

xn ) = 11m ||Txn ||2 = ||T||2

and HT2y|| • ||y|| > (T2y, y) = ||Ty||2 > ||T|£ = ||T||2 ||y||2 > ||T2y|| • ||y|| so

2

2

(T y, y) = ||T y|| • ||y||

the scalar being and we define

and therefore

2

T y

is a scalar multiple of

(T2y, y) /(y, y) = ||T||^/||T||2 = ||T||2 . We can assume

x = y

+ ||T||^1Ty.

If

x = 0

then

y,

T/0

y is an eigenvector of

T

11 k

PALAIS

belonging to

the eigenvalue

vector of

belonging to the eigenvalue

T

— | | T w h i l e if

x / o

then

x

is an eigen­

||TH^. q.e.d.

THEOREM 1 6 .

Let

T e K(H, H)

be self adjoint.

H

be a Hilbert space and let

E X(T) = {x € X|Tx = Xx) .

Then the E X(T)

mutually orthogonal subspaces of their Hilbert space direct sum.

®|x|>s set

H

is

Moreover, the

In fact, if

is

let

are

H, and

E^(T), with the possible exception of finite dimensional.

R

A, e

Given

E Q (T), are

e > o

then

dimensional, hence the

{X € R|EX(T) ^ o ) of eigenvalues of

T

has

no limit point except possibly zero. PROOF.

If

= (x, Ty) = n(x, y) then

TV C V

and

x € E X(T)

and so

(x, y) = 0

S = T|V e K(V, V).

uous inverse

so the unit ball of V

V

dimensional.

is finite

(Tx, y) = K(Y1, Y 1)

y € E (T)

and

Let

(x, Ty) = 0,since

if

X ^

Since

llTlY1 ^

V = ®|x |> e E X(T) S

has a contin­

has compact closure and, by Theorem kf

Y = © x^0 E x,(T)>

TY C Y;

= 0,

If

X(x, y) = (Tx, y)

||Sv|| > e||v||

hence

hence

Y 6 Y

311(1 x € Y 1 .

TY1 C Y 1, so

and is self adjoint. Since clearly

value, by the lemma

\i.

then

TlY1 has

TlY1 = 0

so

Then

TlY1 €

no non zero eigen­ Y 1 = E Q (T) . q.e.d.

COROLLARY. of

T

if

(xn )

(i.e ., If

X

There is a sequence

H.

Moreover,

is the sequence of corresponding eigenvalues

Tfn = xnfn)

then

xn - o .

is a compact space we denote by X with the norm

C(X) the

algebra of con­

||f II*, =

|x € X ) . If

of

of eigenvectors

which form an orthonormal basis for

tinuous complex valued functions on Sup{|f(x)|

(fn )

A = sp(A) =

H

is a Hilbert space and

{X € C

|A - X

A € L(H, H),

then the spectrum

is not a unit of the algebra

L(H, H)) .

VI: REVIEW OF FUNCTIONAL ANALYSIS THEOREM 17.

If H

is a Hilbert space and

is self adjoint, then

R

subset of

sp(A)

A e L(H, H)

is a non empty compact

and there is a norm preserving algebra

isomorphism

f -► f(A)

of

generated by

L(H, H)

115

of

C(sp(A)) A and

onto the subalgebra

I,

which is uniquely

characterized by the properties that the identity map of

sp(A)

PROOF.

corresponds'to

A and that

f(A) = f(A)*.

A short, elegant, and completely elementary proof, due

to John von Neumann, can be found in Appendix I of S. Lang's "introduction to Differentiable Manifolds" (Interscience, 1 9 6 2 ). structs the isomorphism

f -► f(A)

f(A)

If f = g + ih

are self adjoint.

the definition fying

where

f

g,

h

xn ~*"°*

are real valued then

is clearly the unique extension satis­

then by the corollary of Theorem 1 6 ,

A € K(H, H)

(fn)

of eigenvectors of

sp(A) = (0 } U (\n )

Clearly,

f(A)(fn ) = f(xn )fn

A

is completely continuous then

Recall that if 6 £ C

is open and

into a complex Banach space X, then

Given

z e C

pz :

[0 , 0

then

of

z > 0into

Re

PROOF. x

equalities

log t -► -oo

is given

f

is

If

with

6 -► X

is a map of

called holomorphic in exists in

Re z > 0

p z(t) = ez log t p_z

f(A)

f(o) = 0 .

f:

l l m 2^ z (z-zQ)~1 (f(z) - f(zQ))

zQ € O

has

Afn =xn ? n >

A,

and the mapf -► f(A)

is completely continuous if and only if

for each

H

and it follows that

COROLLARY.If

since

and the resulting

f (A) = f (A) *.

a completeorthonormal basis

by

for real valued

f(A) = g(A) + ih(A)

If

and

Lang in fact only con­

V

6

if

X.

define t> 0

if

is continuous and sat-

= PzP z t P- = Pz -Moreover, if z -► p z |[0 , L]

map

C([o, L]).

z = x + iy,

as t -► 0

is a holomorphic

x > 0

then

it follows that

p_,_. z+ z = p„p„. z z andp- = zp_

are obvious. z

|pz(t) | = |ex log t | and pz

is continuous.

If for

Re z

The

> 0

we

11 6

PAIAIS

define

qz(t) = (log t)pz(t)

that by L ’Hospital's rule n

for

t > o

limt_^0

qz(o) = 0

and

(log t)n ex

^ = 0

a positive integer) just as above it follows that

moreover, an elementary calculation gives (ez lo®

h”1 (e*1 lo^ ^ - 1 - h log t)

h(-l-(log t)2 e^z+p^

log

then (recalling

qz

k-1

for

“ pz ^ ^

h_1(p2+h(t) - pz(t))

” qz ^

=

which by Taylor's theorem equals

0 < p < 1,

'*'og ^

and

is continuous,

so putting

x = Re z > 0,

lh_1(Pz+h(t) - p z(t)) - qz(t)| < |h|(l(log t)2 e ^x'lh ^ log *0 (log t)2

x > 0

is bounded on

[o, L]

tends uniformly on

and since

it follows that as h — 0,

[0, L]

to

qz(t).

q.e.d. THEOREM 18.

Let

A € L(H, H)

be a positive operator.

sp(A) £ [o, oo) and

pz

H

be a Hilbert space and let

and hence if

is as in the lemma then

a well-defined element of 1)

z € C

z —*■A z

(i.e.,

with Re z > 0

A z = P Z(A)

Re z > 0

A z+z’ = A ZA Z 1);

A z = (Az)*;

3)

If A

is completely continuous and

then

Az

If A

is strictly positive then so is

for

into

and is a semi-group homomorphism

2)

k)

is

L(H, H) . Moreover,

is a holomorphic map of

L(H, H)

Then

Re z > 0

is completely continuous;

x > 0.

Moreover

z -► A z

Ax

extends to a

(not necessarily continuous) function of Re z > 0 into

L(H, H)

whose value at a point iy of the

imaginary axis is a unitary operator such a way that

z -► A zv

Re z > 0 into H

for each

Ax+iy = A lyAx PROOF. if

f

If

is non-negative

if

sp(A)

in

is a continuous map of v e H.

Moreover,

x > 0.

f e C(sp(A)) on

Aiy,

then

f(A)

is positive if and only

(cf. S. Lang, l o o . o i t .

page 118) hence

V I : REVIEW OF FUNCTIONAL ANALYSIS in particular if we take

11 7

f = identity map of sp(A) we get

if and only ifsp(A) C [o,

«>) .

The existence of

A

is positive

A z as well

as properties

1), 2), and 3) now follow from the lemma together with Theorem 17 and its corollary.

If

t > 0

0 < t < 1

then

A = A^A1

More generally if again

A^

t > 0

V = im (A)

A

hence if then

v -► H

by

= ker(A1“iy).

H

(since

ker(A1“iy) = 0

im(A^)

If

is strictly positive so is

A^ = (At/n)n

A ^ v = A 1+^oo

But

A

is clearly positive.

where

n = [t] + 1

.

and so

A is.

is strictly positive, and hence injective,

is dense in

hence the closure of

so

A^ = (A*^2)* ( A ^ 2)

is strictly positive if Now if

A^-Y;

then

V = (ker A*)1 = (ker A)1).

if v =

Acd.

then

Define

Then im(A^) = im(A1+^ ) ,

is the orthogonal complement of ker(A1+^ ) *

A, and hence

A 2,

and

is dense in

im(Aiy)

is injective and H.

A 2 = (A1+iy) (A1”iy)

Now if

v =

Acd

e V

HA^vll2 = (A1+i^co, A 1+i^co)

hence

A^

=

(A1"1*

=

( A c d , Acd)

v =

with

Acd

cd)

e V.

In general given

||u-v || < e.

u €H

=

= ||v||2

A 1 + z cd

and

( A 2cd,

cd)

,

extends uniquely to a unitary map of lim A zv = lim z iy z iy

if

A 1+ 1 ^ c d ,

H onto itself.

= A 1+ e >

i y cD

Note that

= Aiyv

0we can find

v € V

Then

||Azu - A^ull < ||Az(u-v) || + ||Azv - A^vll + A ^ v - u ) || < (1 + ||Az||Je + ||Azv - A ^ V H Since

I|Az ||m
0

Re z > 0 it continues

Re z ! > 0, and in particular

Ax+iy =

Ai3rAx .

q.e.d.

CHAPTER VII FREDHDIM OPERATORS Richard S. Palais DEFINITION. element atorfrom

T

If

X

of

X

and

Y

L(X, Y)

to

Y

are Banach spaces, an

is called a Fredholm oper­

( o r simply an F-operator) if

1)

ker T = T -1(o)

is finite dimensional;

2)

coker T = Y/T(X)

is finite dimensional.

We denote the set of F-operators from F(X, Y) ind:

THEOREM 1 .

and

dim coker T

If

then

is canonical and suppose Then

T:

X © W -► Y

closed in

and

T(X)

Y

are Banach spaces and

is closed in

Indeed since S

Y,

vector spaces.

T

factors as

W

so [VI, Theorem 3]

X

Sh

where

Then

W T

such h: X -+

T(X) = S(X/ker T),

is a Banach space.

T(X) Define

is clearly continuous, linear,

Theorem 1 ] an isomorphism of Since

T e L(X, Y)

be a linear complement to

is closed in

X and

X © W, T(X)

Y as to­ = T(X)

is

Y. Then by [VI, Corollary 2 of Theorem 10]

dim coker

T e

T* e F(Y*, X*)

is injective, and since

T(x, w) = Tx + w.

and bijective hence [VI, pological

ind(T) = dim ker T - dim coker T.

T is injective. Let

dimW < » by

X

< ».

we can Y.

by

The first conclusion holds even for

X/ker T

in

by

ind(T*) = -ind(T).

PROOF. that

to Y

and define the index function

F(X, Y) -*■ Z

F(X, Y)

X

T and

dim ker(T*) =

dim ker T = dim coker(T*) and the other conclusions fol­

low.

q.e.d. 119

120

PAIAIS THEOREM 2. L(X, Y)

Let

T € F(X, Y) .

TS - Iy

S, S ’ € L(Y, X)

and

TS 1 - Iy € K(Y, Y) .

Conversely, if

S € L(Y, X)

T e F(X, Y) there

such that

K(X, X)

and

K(Y, Y)

T”1(o) C (ST)_1(o)

PROOF.

ST - Ix

and

T(X)

respectively) .

which is finite dimensional by

By the same theorem

(TS*) (Y)

is a closed subspace of

T(X) D (TS*)(Y)

of finite codimension and since

Theorem 7] that

be Banach spaces, T e

have finite rank (hence by [VI, Theorem 1^]

belong to

Y

Y

ST - Ix e K(X, X)

is in fact an

[VI, Theorem 13].

and

and suppose there exists

such that Then

X

it follows from [VI,

is closed and of finite codimension in

Y.

Hence,

T € F(X, Y ) . Conversely given that

X = T“1(o) © V

subspaces of X onto

Y

Y = T(X) © W

respectively.

T(X)and by [VI, Theorem 1]

(T|V)-1 Then

and

and

T e F(X, Y) it follows from [VI, Theorem 6]

to a continuous map

S:

ST - 1^ = -projection of

-projection of

Y

onto

¥

where Then

(T|V)"1 Y -► X

X

along

T

V

and V

are closed bijectively

is continuous.

by letting

onto

W

maps

T“1(o)

S

Extend

be zero on

along

V,

W.

TS - Iy =

T(X) . q.e.d.

COROLLARY 1 . then

T € F(X, Y)

and

k € K(X, Y)

T + k e F(X, Y) .

PROOF. compact operators.

If

Choose Then

S € L(Y, X)

so that

ST - Ix

S(T + k) - 1^ = (ST - I-^) + Sk

and TS - Iy and

are

(T + k)S - Iy

= (TS - Iy) + kSl are also compact operators by [VI, Theorem 11]. q.e.d. COROLLARY 2. PROOF.

Given

ST - Ix = k € K(X, X) T' € L(X, Y)

so

and

IISAll^ < 1

the algebra

and

F(X, Y)

is open in

T e F(X, Y)

then

and by [VI, Theorem 9 ] Let

S e L(Y, X)

TS - Iy = k» e K(Y, Y) .

llT-T'll^ < [jSH"1

L(X, X ) .

choose

L(X, Y) .

B

We will show that if

T'e F(X, Y) .

Ix - SA

such that

Let

A = T - T',

is an invertible element of

be its inverse and let

S’ =

BS.

Then

VII: FREDHDIM OPERATORS ST»

= ST -SA = k + (Ix - SA)

Bk.Similarly then

T'S

T'S" - Iy

Iy - AS

so

S'T!= BST*

has an inverse C

121 = Bk + Ix

in L(Y, Y)

orS ’T' - Ix = and ifSn = SC

= TS - AS = k ! + (Iy -

AS)so T TSn = T TSC = k'C + Iy

= k !C . Now Bk e K(X, X)

and

k'C e K(Y, Y)

or

by [VI, Theorem 11]

and the corollary follows. q.e.d. COROLLARY 3.

If

T1 € F(X, Y) and

X, Y

and

Z are Banach spaces,

T2 € F(Y,

Z) then

T = TgT., e

F(X, Z). PROOF. S1T1 - Ix , operators

Choose

k2 = T1S1 - Iy , and let

S =SjSg.

S1 e L(Y, X)and k3 = S2T2 Then

S1k^T-j + S1T1 = S1k^T1 + k1 + Ix

Iy and

ST = or

S2 e

L(Z, Y)

so that

k^ = T2S2~ Iz

^

=

arecompact

S1(S2T2)T1 = S ^ k ^ + Iy)Ti =

ST - Ix = S1k^T1 + k1 .

is a compact operator by [VI, Theorem 11],

Similarly

But S1k ^ + k1

TS - Ix = T2k2S2 + k^

which again is a compact operator. q.e.d. THEOREM 3.

Let

T e F(X, Y) and

X, Y S €

and Z

be Banach spaces

F(Y, Z) .Then

ind(ST) = ind(S) + ind(T) PROOF. is well defined.

o

We have just seen that

.

ST € F(X, Z)

so that

ind(ST)

From the exact sequence

— T“ 1 (o) -*• T- l s _1 (o) - L s _1 (o) n t(x) — o

we get

dim T-1S-1(o) = dim T_1(o) + dim(S_1(o) n T(X)) or 1)

dim ker (ST) = dim ker T + dim kerS - dim(S“1(o)/S~1(o)

n T(X)).

From the exact sequence 0 — SY/STX - Z/STX — Z/SY

0

we get 2)

dim coker (ST) = dim coker S + dim( SY/STX) . From theexact

0 — T(X) + S-1 (0) -*■ Y

sequence

SY/STX — 0

SY/STX 9sY/T(X) + S'1(0 ) =* (Y/T(X))/(T(X) + S"1(o)/T(X))

122

PALAIS

so 2 ’)

dim coker(ST) = dim coker S + dim coker T - dim(T(X) + S-1 (o)/TX)

Hence by 1) and 2 ’) 3)

ind(ST) = ind(S) + ind(T) + dim(TX + S~ 1 (o)/TX) - dim(S' 1 (o)/S"1 (o) n TX)

But 0 — S ' 1 (0 ) fl TX — S ~1 (o) — TX + S ' 1 (0 ) /TX — 0

is exact so that dim(S'1 (0 )/S"1 (0 ) n TX) = d±m(TX + S_1 (o)/TX) which with 3 ) completes the proof. q.e.d. THEOREM t.

If

ind: F(X, Y) of

= ind(T)

we can that map

Z

are Banach spaces then

is constant on each component

F(X, Y) .

PROOF. ind(S)

X and Y

Let if S

T € F(X, Y) . It will suffice to prove that

is sufficiently close to

choose closed subspaces

X = ker T © V

and

(v, w) \-* Sv + w

of

and

X

and

Y = T(X) © W. Given

S

e L(X,

© W into

W

Y.

If

S =T

tion, hence by [VI, Theorem1 ] an isomorphism lows by [VI, Theorem 9 ] that near

T,

S

maps

V

=dim W =

dim coker T. Since

write

S© Z © V

and then

disjoint from SV,

Y)

S

with

S near T;

Then

hence S(V)

Z © V,

so

dim ker S + dim Z = codim V = dim ker T

ind S =dim ker

for of V

S

Y, we

can

SZ is

Y = W f © SX = W ! © SZ © SV

so dim coker S = dim W* = codim SV - dim SZ = dim coker T - dim Z. other hand,

a bisec­

Y.It fol­

is one-to-one on

S is one-to-one on

and dim SZ = dim Z.

consider the

this map is

isomorphically onto a closed subspace

and codim S(V) X = ker

Y respectively such

ofV © W

the same is truefor

[VI, Theorem 6 ]

By

of

V

V

T.

On the

so

S - dim coker S = dim ker T - dim coker T = ind T. q.e.d.

COROLLARY.

If

T € F(X, Y)

ind(T + k) = ind(T) .

and

k € K(X, Y) then

VII s FREDHDLM OPERATORS PROOF. and since

123

By Corollary 1 of Theorem 2,

K(X, Y)

T + K(X, Y) C F(X, Y)

is connected (in fact a subspace of

L(X, Y)) the corol­

lary follows. LEMMA.

Let X

and

with

ind(T) > 0

dim

V = ind(T) .

component of

find

asub space

and extend Then tP

P

W

as

such that

L(X,

as S.

T1:

P

of

by letting

in

K(X, Y)

W

from

Theorem

zero on

T

there is a unitary map Let T 1 = SU.

U

to

W 1. hence

by Corollary 1 of

ind(S) = dim ker(S)

T1

we can

onto T(X)1

P be

and since U

of

X

Then ker T 1 = U ”1S“1(o)

is in the same component of

But by [VI, Theorem 15] there is a path

from Corollary 3 of

is

for* 0 < t < 1 ,

L(X, Y) as

by Theorem

X

X -► Y

dim T(X)1 = ind(T) > 0

Y)

U(V) = ker S.

of automorphisms of

to

T such that

is surjective so

so itwill suffice to prove that

F(X, Y)

with

T 1in the same

and an isomorphism

to an element of

S

X

ker(T.j) = V.

of ker T

ind(S) = ind(T) = dim V onto itself

exists

Since dim ker (T) -

Clearly,

T £ F(X, Y)

be a subspace of

Then there

is in the same component of

Theorem 2.

G

V

has finite rank and hence is

S = T+ P

= V

Hilbert spaces.,

and let

F(X, Y)

surjective and PROOF.

Y be

I-^. Since

2 that SU(t)

U(t)

in the group

G C_ F(X, X)

is a path in

it follows

F(X, Y) from

T1

S. q.e.d. THEOREM 5.

If

two elements

X

and

S and

Y

T

index, then they are in PROOF. *

and

S

have non negative index.

T

ind(T ) = -ind(T)

the same components of and S1

S1 and

group from

G A

F(X, Y)

are surjective and T 1 map

V1

of

have the same

F(X, Y)

F(X, Y) .

topologically onto

it suffices to consider the cage where

as

By the lemma choose T

and S

Y,

T1

respectively,

ker T 1 = ker S1 = V.

isomorphic ally onto

of automorphism of

F(X, Y)

the same component of

Since T -► T* maps *

F(Y , X ) and

are Hilbert spaces and if

and S1 such that

V1

lying in

T^

Then by [VI, Theorem 1 ]

hence

A = T-1S1

is in the

V 1. By [VI, Theorem 15] there is a path

to the identity map of

in

G C F(VL, V 1) .

A^

The orthogonal

121*-

PAIAIS

projection

P

of

of Theorem 2

,

T-jA^P

clearly

X

T 1AP = S1

onto

V1

is clearly in

is a path in

and

component of F(X, Y) ,

T.,P = T1,

F(X, Y) hence

hence so also are

F(X, V 1), from

S1 S

T1AP

and

T1

and

so by Corollary 3 to

^P.

But

are in the same

T. q.e.d.

Let

X

be a Hilbert space of infinite dimension.

Corollary of Theorem 11 ] A = L(X, X)/K(X, X)

K(X, X)

is a closed ideal in

is a Banach algebra and we let

the canonical homomorphism.

Let

the identity component

of

U.

corollary of Theorem ^

there is awell-defined map iffd:

ind

° it = ind,

-1

so

X

and let

ind(it(T)) = -1

T

ind

A

U

Z

is a group homomorphism.

ker(iffd) = UQ . Let be an isomorphism of

V X

and since -1 generates

hence

L(X, X) -► A

F(X, X) = jt_1(U).

ind „ o - UQ - U ----- ► Z -► o is exact.

L(X, X) ,

be the group of units of

By Theorem 2,

and by Theorem 3

Theorem 5 it follows that space of

U

it:

By [VI,

be

and

U0

By the such that From

be a one-dimensional sub­ onto V 1.

Then

Z the sequence

ind(T) =

CHAPTER VIII CHAINS OP HILBERTIAN SPACES Richard S. Palais

§1 .

Chains If

X

is a Banach space and we change the norm in

equivalent norm, then the norm on valence.

In other words, if

of a Banach space then

X

X

X

is the underlying topological vector space

is a well defined topological vector space. X

then

x -► < x,

>

H.

X).

If H

write

X

H

is the with

X

*

*

is a hilbert space

we shall always regard H

*

= H.

this

It follows

is an hilbertian space then there will be no confusion if we

X as the anti-dualrather

canonical isomorphism as an identification, so that that if

If

is an isomorphism of

(this is precisely the reason we chose to define than the dual of

We

hilbertian if it is the underly­

ing topological vector space of a hilbert space, H

to an

is also changed only to within equi­

shall call a topological vector space

inner product in

X

>

for4(x)

when

x € X

and

ofhilbertian spaces wecan identify =

if x e X

DEFINITION.

and

Let o f

real numbers and

H € X . Moreover, by the reX**

=

X

and then

ft e X*.

denote either the

o f +

with

{k e o/ |k > 0 } .

hilbertian spaces is a set

{H^}

integers or A chain of

of hilbertian

spaces indexed by

o f+

1)

If

then the underlying vector space

of

H

k > £ > 0

space of

is a linear H^

such that:

subspace ofthe underlying vector

and the inclusion map

-► H^

tinuous; 2)

H°° =

3)

H°

Hk

is dense in each

is a hilbert

space 125

(so

Hk ; (H0)* = *H°) .

is con­

126

PAIAIS If whenever

k >

> o

I

§1

the inclusion

Hk -*■

is not only

continuous but even completely continuous then we shall say that the chain {H^}

satisfies the Rellich Condition or is a Rellich chain. If

=Z

spaces, while if o/ = R

we call {Hi we call

(Hk)

a discrete chain of hilbertian a continuous chain of hilbertian

spaces.

H"k = (Hk)* ity,

If

{H }

if

k > 0 . If k = 0

while if If

define

is a chain of hilbertian spaces then we define

k >

> 0

H

3 (j^ k) : ^

-i

“*■H

H”k = (Hk)*

then by 2 )

Hk

_4 )

is dense in 30

J*(£,m) = J*(4, m) = ^(4,k)^(k,m)

the inclusion map being

of

H

H 00( 0 0 > k

ous inclusion

ft

j(_k _ji y

then

Hk

f€

and if

- 00)

k): H -*• H

the pairing of k >

>

k

H

If

< g, f > 0 defined if

B°

e

f € H

and

H“°° and

then

00

>

k

>

S.

*

j^_^ ^

k) • Denote, temporarily,

g

g €

c

k < ,

We topologize

H‘"k and

H“^ then

f >k

< g, f >k =

< g, f >Q

is then well

H°° as the inverse limit

H“°° as the inductive limit

with the finest topology such that each inclusion Then it follows that H~°° with

< , >Q

•

extends the pairing

3jjn Hk, i.e., with the least fine topology such that each inclusion is continuous and we topologize

=

we have a continu-

> -°o

and

g € H~ . We note that

f e H00.

8X1(1 i1:

onto a dense linear subspace k we can regard each H as a vec­

>a = < 3(.k, . a ) e, f >k =
0 > -JI

< , >k . Iff e Hk

It follows in particular that for each of

m > k

as a dense linear subspace of

. If it should so happen that also

< ^ i >1 =
0

is injective it follows

and

and H_k by

= ,

j^

H

d-eIInes “the inclusion of —°° k . Thus if we define H = Ik H

tor subspace of

j(_k

^(-m,-£) = ^ (-m, -k) ^ (-k, -S .)

k)

o )j(o

n , hence if we

H_k. Moreover, if

follows that we may consistently regard

—ft

is dense in

is injective, and since

by reflexiv-

by 3) of the definition.

to be the inclusion map then

-k

that the image of

H“k,

H"k = (Hk)*

we then have

we still have

k) : Hk

*

k < 0

H°° -► Hk

lim H”k, i.e.,

H -► H~°° is continuous.

is a continuous conjugate bilinear pairing of

(in fact it is easily seen that

and

H“°° are locally

§1

VIII: CHAINS OF HILBERTIAN SPACES

convex topological vector spaces and that

H -00

127

is the antidual of

H00,

i.e., all continuous conjugate linear functionals on H°°) .

DEFINITION.

If

(H^)

and

(Hg)

are chains of hilbertian

spaces a morphism ( - * • (Hg) is a collection {T^) k k that T^: H 1 -► H2 is for each k e o9 a continuous linear map, and

Tk = T^|H^

set of such morphisms by

k > I . We denote the

if

Hom(CH^), {Hg}).

We note that the operation Hom({H^), (H^))

a vector space.

such

+ PT^}

Also if

H({H^})

makes

and clearly with

this composition law the class of discrete (continuous) chains of hilbertian spaces forms an additive category (not abelian; there are kernels but not cokernels). If

{Tk) € Hom({H^}, (H^))

T^|H^

if

k > i it follows that

since

Toof

= L J£ f

of

H^

namely

into Tk

H^.

d

e for all

Since

H^

and

Tkf = k, 7

f e H^° then since

T^f= T^f

T oof € H^, d 7

is the unique continuous extension

DEFINITION.

If

(H^)

and

is well defined.

i.e.,7

is dense in each

H^, of

T00

we define

each map

k € Z( R)

Tk : H^ - K*~r

Remark. that and

Tk

T

T:

the corresponding

T^

H^°

k > ft,

it follows that

is clearly

THEOREM 1 .

If

CH^j

H^ -► H^

such that for

is dense in

T, and

is continuous if r = 0

to be the vector

.

We note that since

In particular, if

r e

extends to a continuous linear

is uniquely determined by

H2”rH^_r

is a linear map

(Hg} are twodiscrete

OPr ({H^}, {H^f})

space of all linear maps

Also

Tto determines each Tk , v to a map of H 1 into

(continuous) chains of hilbertian spaces and Z( R)

Tk =

T and

since

H^

it follows

H^ -*• H^ is continuous

it follows that Tk = T^|H^. (Tk) €Hom({H^), (H^)).

we have: (h |)

are two discrete

(continuous) chains of hilbertian spaces then the

Since

128

PALAIS map

(T^) — T^

of

defined above is a linear isomorphism

Hom( {H^}, {Hg))

DEFINITION.

§1

If

with

(H^J

0PQ ({H^), {Hg}) .

and

(H^l

are two discrete r e

(continuous) chains of hilbertian spaces and Z ( R)L(H^, H2 r)

is a Banach space (in the topology

of uniform convergence on bounded sets) and we take the weakest topology on maps T -►

of

(i.e. a net

[Ta] in

T

0Pp

0Pr

L(H^, H2”r)

the net

formly to Remark.

T^

is continuous

OPr ({H^}, (H^))

if and only if for each

B C

such that each of the

k

converges to

and each bounded set

{T^} in L(H^, H2”r) on

converges uni­

B) .

Note in particular that by Theorem 1 this topologizes

Hom({Hb, (Hg}) . THEOREM 2.

Let

{H^}

and

bertian spaces and let OPa((H^}, {Hg})

{Hg}

be chains of hil-

r, s € o f with

is a subspace of

tion

{H^} and

or

[H2)

Then

OPr ({H^), (H^))

and the inclusion map is continuous. either

r > s.

Moreover, if

satisfies the Rellich condi­

T e OPg({H^}, (H^))

then for each

k € of

the unique continuous extension of T to a linear map k k—r of H 1 into H2 is in fact completely continuous. PROOF. and

j 1: H2“s

continuous and

H2~s_t j

condition while

j'T^

proves

t = r - s = H2~r

j*

sion is immediate. B

(H^)

is completely continuous if T € O P s ({H^}, {h |})

let

T

j: j

-► and

j1

are

satisfies the Rellich

(H2)

satisfies the Rel­

Tk_t :

be continuous extensions of

give continuous extensions of

T € 0Pp .

let

be the inclusion map, so

- H2"t_s

= H|"r

T.

Then either T^-t*^ k k—r to linear maps H 1 -+ H2 which

Since the composition of a continuous and a completely

continuous map is completely continuous

and let

k e of

and given

is completely continuous if

lich condition. If k k—s and T^: H 1 -*• H2 or

Let

Finally, suppose

be a bounded set in

H^.

[VI, Theorem 11] the final conclu(ex')

T v ’ -► T Then

in

0Pg

-*■ T^

and let

k e of

uniformly on

B,

§1

129

VIII: CHAINS OF HILBERTIAN SPACES

hence,

j

_► j'Tk

uniformly on

B

w h i c h proves continuity.

q.e.d. THE O R E M 3.

Let

{Hk }, (Hk ), and {Hk }

be discrete

(continuous) chains of hilbertian spaces and let r, s € o f . map of

Then the map

PROOF.

Obvious. Let

{Hk }

bertian spaces and

T:

linear map

-►

if

into

(Hk }).

DEFINITION.

T

is a continuous

(Hk ) ) x OPr ({Hk ), (Hk ))

O P s ({Hg},

O P r + s ((H^},

(S, T) -► ST

S:

and

CHk }

be chains of hil-

a linear map.

A

is called a transpose of

< Tf, g >Q = < f, Sg

for all f t

>Q

,

g €

Remark. < f, (S-S')g >0 = ° dense in

H°

If

ST

is a second transpose for

for all

it follows that

f €

and

S = S ’,

it is unique and will be denoted by THEOREM it-.

Let

(Hk )

T^. and

(Tt)k :

- H^_r

of

k.

hence

(T_k+r)*:

Hk - H g k

for some

By assumption

T_k + p : then

f f

< Tf, g > Q= < T 0f, g > 0 - < f, so

f

has

of.

k €

for each

k € Q = < f,

on

-

(Tt )k =

k

f f

€ H^,

g e H^

then

hence

and

(Tt )_r g > 0

T. Since for each

h£

independent

k,

f e

r estriction of the continuous linear ma p

the final conclusion is immediate.

T

(H^)) and

for each

is i n fact the transpose of

n ition the

Then

the adjoint map of

H" k+r -*• H ‘k .

then

exists

belongs

T^ € OP

T_ i+r = T _ k + r |H~£+r

is a linear m ap of

T

Ttt: = T.

-►

(t ) k |H^1 is a well-defined function

If f e f

is

is

be two chains of

r € of .

k > £ so

(Hk )

to

uous linear m a p

(TtjjlHg

We note that

T:

PROOF.

If

hence since

hilbertian spaces and suppose OPr ({H^), (H^))

then

i.e., if a transpose for

a transpose and moreover

Hk_r.

g € H^j

T

= < f, f g f

(T_ k+r) * '■

>Q

is b y d efi­ H k -*• H k_r

130

PALAIS THEOREM 5.

Let

(H^)

and

§1 {H2)

be discrete (con­

tinuous) chains of hilbertian spaces and suppose T:

H^

-► H^

is a linear map with a transpose

r, s € o f

Let with T^:

admits a continuous linear extension

-► H2”r

and that for each

k > r - s,

T^

(Tt)k : h £

- H^‘r .

if

Tk

k < s,

PROOF.

If

we

and, since

have H“

with

Then T € 0Pr ((H*}, {H^})

is the adjoint of then

H^

-k + r > r - s

into

so

>hence

H2”r .

and

(Tt)_k+r*

(Tt)_k+r>:H2k+P

is a continuous linear map of g e H^

k e of

admits a continuous linear extension

k < s

tinuous linear extension

k e of

and suppose that for each

k > s, T

T^.

If

T"k

admits a con­

Tk = (Tt)*^+r

f €

then

forall

< T^f, g >Q = < f , (Tt)_k+pg >Q = < f, ^ g >0 = Q

is dense in

H2,

Tkf = Tf

so

Tk

extends

T. q.e.d.

COROLLARY 1 . If map

Tk :

-*■

T

extends to a continuous linear

H2

for all

to a continuous linear map k> 0 T

then

T e OP0 ({H^), {Hg}),

defines amorphism

PROOF.

k > 0 and T1" extends t k k (T )k : H2 -► Ej for all

of {H^1}into

hence (Theorem 1) {H^} .

Take r = s = 0.

COROLLARY 2.

If

{H^} and

of hilbertian spaces and if T

(H2)

are discrete chains

extends to a

continuous

linear map

Tk :

and if

extends to a continuous linear map

T^

(T^) k :

H^ -*• H2_1 for each integer

for ea°h integer

k > 1

k > 1

then

T € OP 1 ({H^}‘, {H|}) . PROOF. equivalent to

Take r = s = 1

k > 1since o¥

andnotethat

k > r - s = 0

is

= Z.

q.e.d.

§2

VIII:

§2.

CHAINS OF HTLBERTIM SPACES

Quadratic interpolation of pairs of hilbert

131

spaces.

Given a continuous chain of hilbertian spaces, by restricting the indexing set to the integers we get a discrete chain, and this corre­ spondence is clearly functorial from the category to the category

of continuous chains K In this and the next section we

of discrete chains.

C

* Cr

will construct a sectioning (i.e. right inverse) functor

called-

quadratic interpolation. Let (Hq , H 1) -► H 1

H^q ^

denote the category whose objects are pairs

of hilbert spaces such that

is a continuous linear map of

whose morphisms TQ : Hq -► note by

T:

HQ C_ H 1 HQ

and the inclusion map

onto a dense subspace of

H 1,

and

(HQ , H 1) -► (H^, H.J) are all continuous linear maps

which extend to continuous linear maps

H^q ^

HQ

:

H1

Hj . We de­

the analogous category formed with hilbertian rather than

hilbert spaces and note that there is an obvious

"weakening of structure"

or "forgetful" functor

Similarly, we define

F:

H^Q ^

— ► H^Q

to be the category whose objects are indexed sets indexed by inclusion of

[0,1],

such that if

Hg

map

Tq :

HQ -►

T^_: H^

then

is a continuous linear map of

H^, and whose morphisms

maps

0 < s < t < 1

{H^}

T:

{H^}

which for all

H|.

{H£)

of hilbert spaces, H g C_

onto a dense subspace

Hj-0 1 ]

forgetful functorF: If

i^: * *

i^:

tive.

H q -*■ Ht, *

-► H q

{H^}

is

to be the analogous category

object of

anti-dual of

an objectof

H*

which

{H^} .

If

*

i^

with its image under we also denote by TQ=

T:

T*: H.J* -► H*.

{H^} -► {H|.} TQ ,

Thus if we defineD(T) =T*

is a contravariant functor. we define

or

E(HQ , E^) =

Similarly, if for

, HQ)

inclusion

itsadjoint, the restriction map

has dense range because

-*■ H.J. is the continuous extension of to

Hj-Q 1 j then since the

]has denserange,

is injective, and

Hj-Q 1 j

have an obvi­

Hj-0 -j] — * H j-q 1j.

t e [0,1

If we identify

linear

extend to a continuous linear

formed with hilbertian rather than hilbert spaces and again we ous

and the

are allcontinuous

t e [o, 1]

Again we define

Hg

H^Q ^

i*

i^

then

is another

ECH^}

and

is a morphism and then T * : then

Hj *

call the T^:

H* extends

D:Hj-Q 1 j ^ — ► H^Q 1 j

(HQ ,H 1)

(where we identify

is injec­

H*

an object of H^Q with itsimage under

^

PALAIS

132 the adjoint of the inclusion -+ (H^, H.j)

we define

i:

HQ -+

D(T) = T*

then

§2 ) and for a morphism D: H^Q

^

of H^Q

^ let

T:

— ► H^Q ^

(HQ ,

)

is a contra-

variant functor, the anti-dual functor. object (HQ , E^)

Given an inner product of norm

C)

if

H^.

Since theinclusionHQ -► ^

x eHQ

then yH- (x, y) 1

is a unique elementAx y € Hq .

of

Clearly,A:

HQ

HQ -*■ HQ

< ||x]l1 ||y||1 < C2 ||x||0 ||y||0 Since

DEFINITION.

If

is continuous

is an element

such that

(x, y) 1

is linear and since

if

x ^ 0,

(HQ , H , )

A

(HQ , H 1)

If

(x, y)1 = (Ax, y)Q

0 < t < 1

then

| (Ax, y)Q | = |(x, y) 1 | and

product on

HQ

= ||At/2x||0

by

A is continuous.

H {0

is the unique HQ -+ HQ

A:

for all

x, y € HQ .

[VI, Theorem 18, h ) ] is

A^

strictly positive continuous operator on (x, y)^ = (A^x, y)Q

for all

is a strictly positive operator.

bounded, strictly positive operator such that

hence there

= (Ax, y)Q

is a n object of

the defining operator of

(say with

of HQ ,

HAH^ < C2

it follows that

(Ax, x)Q = (x, x)1 > 0

( ,denote the

HQ

hence

is a positive definite inner

and we denote the completion of HQ t 1/2 relative to the corresponding norm l|x||t = (A x, x)0'

the indexed set (H^*, E[)

Q t (H0 , H,)

and write

(HQ , H 1 )} .

is a morphism of

If

H^Q ^

T:

Q(Hq , H , )

for

(HQ , H.j) -*■

we define

Q(T) = T. Remark. functor

^

The fact that

Q

is, as the notation suggests, a

-- H^Q 1 j will be proved in Theorem 1 .

The proof of

functoriality was discovered independently by several persons^ including Stein whose previously unpublished proof is used here.

E.

Q, is called quad­

ratic interpolation. It is clear that identification)

If

< ||A(t_s) /2 \\^ ||x|| H

and that (with an obvious

Q 1 (HQ , H 1) = Ej . For simplicity we shall often put

Qt(H0 , H,) = Ht .

linear map

Q^Hq, H 1) = HQ

“♦ H^.

0< s < t < 1 then

||x||t = ||At/2x||0 = ||A(t_s)/2A s/2x|l0

so that the identity map of Since the composition

HQ

extends to a continuous

HQ -► H g -►

H1

is just the

§2

VIII: CHAINS OF HILBERTIAN SPACES

inclusion of

HQ

in H 1

we may regard each

Hg -► Ht

it follows that

it follows that a fortiori

Hg

with the above convention H q -*■ H1

is injective and that

E^(the image of

H^ as included in

become inclusion maps.

suppose

H s -+ H^

133

Since

H^ —►

so that

)

HQis by definition dense

is dense in

Q,(Hq , ^ ) =

H^

if

0 < s < t < 1,

{H^} is an object

is completely continuous

in H^,

and let

of Hj-q

Now

(xn ) be a bounded se­

quence in

Hq .

Then passing to a subsequence we can suppose

(xn )

Cauchy in

^ ,

i.e. that

hence

A 1/2 : Hq

-*• H q

Ar/2o < r < 1 . Hq

such that

so that

Hx^ - xn ||1 = ||A1/2 (xn - x ^ ||Q — 0,

is

is completelycontinuous and by [VI, Theorem 18, 3)1 so is Then if

0 < s
Q =

given by

Clearly

2

lows that

H^q ^

fn : X -*■ V

and the

2

k > 0,

2

so

f -*■ cck /2f

Given

we have

L^(X, V, cc) C l|(X, V, cc)

||f||| < m ^ ”k^ ||fH^. which proves that

integer

k > £

m = Inf{cc(x) |x € X) . Then if

L^(X,

V, cc)

with respect

i.e.,

^ ( L ^ X , V, a), l |(X, V, a)) - l£+t(i_k)(X, V, a) 2

which proves that

{L^(X, V, cc))

is a scale. q.e.d.

The following class of scales will be of particular interest in the sequel. DEFINITION. for each

If

V

k € R

is an hermitian vector space then

we define

bert space of functions l|f|lk

=

£

fL^( Zn , V)

f:

Zn

llf(v) ||2(1

+

to be the hil­

V such that ||v||2)k

< 00

v€ Z 11 THEOREM 3.

If

V

is an hermitian vector space then

p

(^k( X , ^ ^ k e R is' foP each non-negative integer

n,

a continuous Rellich chain and a scale. PROOF.

If we make

zn

into a measure space by declaring each

set measurable and each point to have mass one, and define then clearly

Zn , V) = L^( Zn , V, cc) ,

remains is to verify the Rellich condition. F

be the finite set of

v e zU

such that

cc(v) = (i + ||v|| )

so by Theorem 2 above all that Let

k > &.

(1 + ||v||2)

Given > e2/2.

e > o Then the

let

VIII: CHAINS OF HILBERTIAN SPACES subspace if

v

4

V)

such that 2 is finite dimensional, so the unit ball of ^ ( F , V)

F

of

and we can choose unit ball of

l|( Z n , V)

f 1, ..., fm ,

£k ( Z n , V)

consisting of

143

e/\/2 -dense in it.


||f(v)||2(U||v||2)k v€ Zn

< M k f^

If

f(v) = o

then

£ ||f(v) ||2(1 + IM|2)* veF

hence we can choose

f

< 1

so that

||f(v) - f± (v) ||2(l + ||v||2)4 < ^

v€F On the other hand since if

v

4

(1 + ||v||2)^_k
1/2

bk = Jj” (l+y2)”kdy.

bk
1/2

-► V

defined by

p(f)(X) =

/2( Zn_1, V) . More-

p: «2( Zn , V) - f2., /£( Zn_1 , V)

is a con-

tinuous linear map and has a continuous linear sec­ tion (i.e. right inverse)

X: J^2_1j ^ ( Z n-1 , V) -►

*£( Z n, V) . PROOF.

Taking

a^ = ||f(x, p) || and

c = (1 + I|i||2) in Lemma 2

gives

(Z

I l f U , ^ ) l l ) 2 (1 + I|x||2) k -1 / 2 < Bk

pez

Z

t1) II^ ( l +ll^ll2 +^2) k

Since the sum on the right is clearly no greater than the absolute convergence of gives

•

^Z

Z^ f(x, p)

to an element

ll^ll^-

this proves

p(f) (X.) e V

and

§!)•

VIII: CHAINS OP HIIBERTIAN SPACES

1^


in Lemma 1 l|x (g )(x ,

gives

= g(x)

r(A.) < 1 /b^.

H )||2 (1 + | | X ||2 +

^ 2) k

aH

A, e Zn 1 .

Taking

c = (1 + ||x||2)1

so

=

| | g ( x ) | | 2 ( i + | | x ||2)k - 1 r ( x ) 2 f

l + M

8

l k

1+ ||X||2+|i2J


H > 0,

be integers, k > s

f e Jt£( Zn , V)

and

the series

f(x, n)

f Z ,n-i converges absolutely, and the function

p(f) : z^

defined by

belongs to

*£_s(

p(f)(x) =

zf, V).

Moreover,

f(^, 1-0

p: i£( Zn , V) - Je^._s(

V

zf

V)

is a continuous linear map and has a continuous linear section X : *£_s( Z£, V) PROOF.

Z,n , V)

.

Immediate from Theorem k by induction on

COROLLARY.

If

(n-£).

n

is a positive integer and k > n/2 2 n there is a constant B such that f e , V) then

z

Zv€ j n f(v)

converges absolutely and

I

f(v) || < B||f||k

.

r7n vc€ L PROOF.

This is just the case

It = 0

of the theorem.

.CHAPTER IX THE DISCRETE SOBOLEV CHAIN OF A VECTOR BUNDLE Richard S. Palais In this section bly with boundary, and M.

M

will denote a compact

|, t], £

We shall also assume that

C°° manifold, possi­

will denote hermitian vector bdndles over

M

is equipped with a fixed strictly positive

smooth measure.

§1.

The spaces

Ck (g)

We let vector space of

Ck (|),

Ck

which it is complete.

in

M

for

and if

\|r:

f € C°°(|)

Thus C°(g)

a map

We give

C°(g) the compact-open is a norm for

| 10 ^ 6 x Y

R n —► ®|a |-< k V

by the map

I

k-jet extension map

C0 (Jk (g))

the requirement that Ck (|)

Jk?f) (y) = (Daf (y)) (a j < k .

jk :

C°°(g) -► C°°(Hk (0)

It follows

extends naturally to

Jk .

is clearly injective and we topologize Jk

shall be a homeomorphism into.

Rn

then

Dafn

from which we see that

jk

maps

Ck (|)

Then

(so

Ck (|)

Then

Ck (|)

by

fR -► f

|a| < k

converges uniformly to

D^f

and on

in K K,

isomorphically onto a closed sub­

is the underlying topological vector space

of a Banach space) and that the topology of "Ck topology."

then

so that the same coordinate representation still

is a compact subset of

C°(Jk d))

and if

f° cp-1 : R n -► V,

f =

means that, in terms of the above "coordinates,” if

space of

is a chart

0f

over

holds for this extended map, which we continue to denote by jk : Ck (|)

in

jk (f) I® is then represented by the map

defined by

Ck (|) -*■ C°(Jk d))

cp: 0 ^ R n

is a trivialization of f| 0

we represent

C°(|)

is the underlying topological vector space

We recall from Chapter IV that if

and

that the

g.

|||f|||0 = Sup{ ||f(x) |||x e M)

Jk (|) jk~f):

a non-negative integer, denote the complex

cross sections of

topology and note that

of a Banach space.

k

k7

Ck (|)

is just the usual

1^8

PAIAIS

§2. The hilbert space

§2

H°(g)

We recall that by Section b of Chapter IV /M (f(x), g(x))^ define

H°(|)

defines a prehilbert space inner product on

to be the completion of this prehilbert space.

can (and will) regard I

< f, g >^ =

such that

H°(|)

C°°(i).

We

Clearly we

as the set of measurable cross-sections

||f||g = /M (f(x), f(x))^
1,

D:

If

D € Diff^g, ti)

C°°(g) -► C°°( r|) By Theorem 5,

Diff^ri, g),

then by Theorem 6 for each integer

extends to a continuous linear map

D

has a transpose

D^

and in fact

so again by Theorem 6 for each integer

to a continuous linear map

Hk (T]) -*Hk_1(|).

k > 1,

D^: Hk (|) -»■ D 1" = D* e D^

extends

Then [VIII, Corollary 2

of Theorem 5 ] completes the proof. q.e.d. LEMMA.

If

k and

JI

are non-negative integers then

Diffk+Jf(|, T,) = D i f f 4(Jk U ), n) • Diffk(S, Jk (I)). PROOF. and

ff

thefirst means

phism

jk : C” (?) - C“ (Jk (|)), Jk+£: C“ (|) - C°°(Jk+*(!))

: C°°(Jk (|)) -*• C“ (J^(Jk (|))

Jk+£(f)p = 0

= 0

Let

if

only if j^Cj^Cf))

Da?(p)

|a| < It and k+ ( S of J (g)

^ J*k+£ ^p^ can find

= 0

if |a| < k+H

=0

|p| 1

and, inductively, that

k-1 . By the lemma we can write

D2 e D i f f ^ d ,

where

Diff 1 (Jk_1 (|), r\) . By the inductive hypothesis by Theorem 9 is in

D1 € O P ^ J ^ U ) ,

r\) ;

Jk-1(g))

and

D1 e

D 2 e 0Pk-1 (g, Jk_1(g))

hence by [VIII, Theorem 3]

and

D = D-jDg

0Pk (i, t]) . q.e.d. If M = [0, 1],

D

e Diff^l, |)

transpose

D

t

g

is the trivial line bundle over

is the usual derivative, we would have for all

Df = f ’, then if 00 f € C (g) that

D

M and had a

1 f(l) - f(0) = y

f 1(s)ds = < Df, 1 >t = < f, d S

>?

°1 - J 0

f(s)(Dti)(s)ds

or in other words the measure with mass one at

1

and mass minus one at

0

is absolutely continuous with respect to Lebesgue measure, a contradiction. Thus

does not have a transpose so D 4 OP^g,

D

0Pk (g, g) dM = 0

for any

k

by [VIII, Theorem k ] .

g)

and in fact

D 4

This shows that the hypothesis

is in fact essential in Theorems 5, 9 , and 10.

CHAPTER X THE CONTINUOUS SOBOLEV CHAIN OF A VECTOR BUNDLE Richard S. Palais §1.

Continuous Sobolev chains We recall that in VII, §3 we defined a functor (quadratic in­

terpolation) {H^)^e ^

C^

— ► Cj^ which to each discrete chain of hilbertian spaces

associated a continuous chain

on morphisms. measure and

If £

M

is a compact manifold with a strictly positive smooth

is an hermitian vector bundle over

functor to the discrete Sobolev chain (H^( | c a l l e d rem 1)]

if

{Hk (r|))keZ

anc^ which was the identity

{H^( t)}

M

we get a continuous chain

the continuous Sobolev chain of

r e Z (where

then 11

then applying this

|.

By [VIII, §3, Theo-

0Pr ({Hk (5)) kcR , (Hk (11)) k£R) = OPp ({Hk (l) )keZ ,

is a second hermitian bundle over

M) so we define

OPr (e, n) = OPr (CHk (|)}k£R , tHk(ii)}keR } for

r e R. In the following theorem we are identifying

(Hk (n))k € R )

with

OP0 (| , n)

[VIII, §1, Theorem 1 ].

is a bundle map then the induced map

Hom( (H^( |)}

f: 6 — i\

Thus if

f# : H “ (|)-*,H“ (ri)

is in Hom( (Hk (|)}keR,

{Hk (ii)}k e R ) [IX, §3, Theorem 71. THEOREM.

Let

M

be a compact manifoldwith strictly

positive smooth measure.

Then

a functor from the category bundles over

M

V(M)

V(M)

-►

then

sociated morphism of

If

f* : H°°(g) Cj^ . 155

is

of hermitian vector

to the category

chains of hilbertian spaces. phism of

|

of continuous f: | H°°(t^)

,

^ is a mor­ I s the as­

156 PROOF.

§2.

The chains

with

|or| < k

[IX, §3, Theorem 8] and [VII, §3, Theorem].

{Hk (T11, V)]

If

k

oc

is a non-negative integer and

then we define

nomial theorem if

y € Rn

Ck =

then

kl /(a! (k- |a|) !

cc

one for each n-multi-index

is a n-multi-index so that by the multi­

(1 + ||y||2)k = E|a |^ k C^y2^.

hermitian vector space we denote by V,

§2

PALAIS

®|q;| < k ^

with

If

V

is an

direct sum of copies of

|a| 0

f € C°°(Tn , C),

< Daf , Daey >Q = (-i)'a U

D^e^ >Q . for all

Z ^ < k Ckv2c* = (i + ||v||2) k

and recalling the identity

mentioned at the beginning of this section Since

C°°(Tn , C)

f e Hk (Tn , C) .

for all

{(1 + 1|v ||2)-k//2e }

„n

=

H ^ T 11, C)

that

1

and by Lemma 1,

< e € v*, e v >n 0 = 5(J.v

is the dual basis for

^

the same equality holds

< f, ev > k = 0

If

v € Zn

ey >k = ^|a| < k

^r.

{( 1+ ||v ID ~k//2ey)

a11 v

for

f = 0

is an orthonormal basis.

v v€ Z'

(1+ I "Iv "||2)k^2ev >n 0

< f,

It follows from Lemma

Hk (Tn , C) .

is orthonormal in < f, ev >Q = 0

is dense in

f, D 2aev >Q

then

e Zn

so < (1 + ||v ||2) _k^2e ,

Since

it follows that

{(1 + ||v ||2)k//2e v }v€ L 7n

H'k (Tn , C). q.e.d.

We recall that in [VIII, §^] we defined hilbert space of maps

g: Zn “*■V

llgll£ -

J Z . n , V)

to be the

such that

£

l|g(v)||2(l+ ||v||2) k


Q= f ( v ) , Then by Lemma 2,

if

V 1© V 2 .

Since

V

we can restrict attention to the f e H”00(Tn , C ) ,

< f,

ey e C°°(Tn , C) C H°°(Tn , C)

so we

and that i f

define

f(v)

f e H^c (Tri, C)

= < f, ey >Q

we have

f , ® f2 e

from which the validity of the

implies it for

C

Zv€ ^n f(v)ey

H ^ T 11, V) .

f2( Zn , V,® V 2) = «2( Z n , V,) © £2(Z n , V 2)

C°°(Tn ,

onto ^ f.

>0 .

If

is isometric case

V =

C.

is a well defined f e C00^ ,

for all

C)

f € H"°°(Tn ,

then

C)

•

§2

X: CONTINUOUS SOBOLEV CHAINS

l |f | l k

-

£

l< f ,

159

0 + llv||2 ) k / 2 e v > Q | 2

v e Zi =~y

and the series

EVt,7 L n

if(v) 12(i+ HvH2) 2' k

■ I lf |l k

< f,' (i+ ||v||2)k/2ev > u (i + ||v||2) k^2eV V= tZ / j™n f(v)e v

converges absolutely to

in H ^ T 11, C) .

f

q.e.d. COROLLARY.

The chains

are isomorphic. (H^(T11, V)}

( H ^ T 11, V))

In particular, [VIII, §^, Theorem 3]

is a Rellich chain and a scale.

In the following lemma, maps

f:

Tn -► V

of the form

ECT11, V)

f:

will denote the space of

f(x) = Zy€ ^n ev(x)g(v)where

is zeroexcept for finitely many values of Also given

{Jfc£(Z n , V)}

and

Tn

V

and

g:

Zn

V

v.

a e Tn we define

f : Tn -► V SL

by

fa (x) = f (a+x). LEMMA.

If

1) ECT11, V)

is dense in

llf a llk = llf llk

and

for all

V = C.

a e T11 then

a

then

for all

k eR.

fa e ECT11, V)

k eR.

is an n-multi-index

then Daf e ECT11, V)

< r

PROOF.

H ^ T 11, V)

for a11

f e ECT11, V),

3) if

lemma with

is any hermitian vector space

f e EfT11, V),

2) If

M

V

and

with

||Daf||k_r < ||f||k

k £ R. As in the proof of Theorem 1, it suffices to prove the

Now

ECT11, C)

is just the linear span of

C (1 + ||v||2)'k/2e }

Since by Lermia 2 of Theorem i

n is m

{e^}

n .

orthonormal basis

V v€ Z for

H^CT11,* C)

and since (e

)

v a

= e^v ’ae

v

while

D^e

v

= (i) ^a U ae

v

the

lemma is immediate.

q.e.d. THEOREM 2. (Sobolev) space,

If

V

is an hermitian vector

r a non-negative integer and

k

a real number

1 60

PAIAIS with

k > § + r

then

§2

H ^ T 11, V) C Cr (Tn , V)

and

the inclusion map is completely continuous. PROOF.

It will suffice to prove that the inclusion map is

continuous, for we can factor it as Hk(Tn > where

k > I > ^ + r,

(T31, V)

_ jj^rpn^ y) _ cr (Tn^

and by the corollary of Theorem i

Hk (T11, V) -*■

is completely continuous. f € EfT11, V ) ,

Let

(e.g. by Theorem 1 )

so

f = Z

ef(v), v€ Z

where

Z

f(v) = 0

„ f(v), v€ Zn

v e Zn .

except for finitely many

In particular

V

f(o) =

hence

I l f ( o ) II = II

£

f(v) II < B||f||k_r

Un v€, Z by

[VIII,§^,corollary

ofTheorem 5]

since

k -r >

||f(0)|| < B||f|lk_r If

a € T11 then replacing

f

by

f

^ . By Theorem

1 above

.

and using2) of

thelemma gives

||f(a) || < B||f||k_r since

fo £L(0) = f (a) .

Since this holds for each

a e Tn

and

||| f || U =

SupC ||f(a) |||a e T11}, |||fIII Q If

oc

is an

n-multi-index with

< B||f ||k_r

|cc| < r

ll|D“ f III .o Ck IIDaf II I IoT] < r

§2

X: CONTINUOUS SOBOLEV CHAINS

i.e., the identity map of on it by

H^CT11, V)

ECT11, V)

is continuous from the topology induced

to that induced by

Cr (Tn , V) .

Then 1) of the lemma

completes the proof. q.e.d. COROLLARY. then

If

Z

r

e f(v)

v€ Z

Cr (Tn , V) f

is a positive integer

k > ^ + r,

converges absolutely to

f

in

V

f e H^CT11, V) .

for each

€C°°(Tn ,V)

then

Z v€ Z

e f(v)

In particular if converges to

f

V

in the C°°-topology. PROOF.

Immediate from Theorem 1 and Theorem 2.

THEOREM 3. and let

Let

n

s = (n-£)/2.

larger than

s

it

and

be integers with

Then if

k

n > it > 0 ,

is any real number

the restriction map

Cco(Tn , V) -+ C°°(T^, V)

extends uniquely to a continuous linear map p: H^CT11, V) -*• H^~S(T^, V). tinuous linear sectionX : PROOF. (y, z) € T^ x T31-^

Let

vergenceis

has a con­ V).

f € C (T11, V) .By the above corollary,

if x =

then

=

thesummation

p

Hk-S(T^, V) -► H^CT11,

f(y, z) = Z

where

Moreover

^

is over all

uniformon

Tn .

e (x kl)(y, z)f(\,n)

ex(y)e^(z)f(x, p )

p)

Then since

f(y, °) = ^

(f|TfM *0

x Zn_^ = Zn

e [1( ° ) = (2j0 ~ S

ex,W(

\e

where the convergence is uniform on

in

we have

^ pe Zn_^

o

T .

It follows that

= (2rt)-s

Z

r, n-f t peC Z

and the con­

v)

162

PALAIS

§3

The theorem is now an immediate consequence of Theorem 1 above and [VIII, § 14-, Theorem 5 ]. q.e.d.

§3.

An extension theorem Let

V

Bn = (x e R n |||x|| < 1 }

is an hermetian vector

we define

C~(X, V)

and let

space then, letting

to be the space of

pact support, and for

k

Banach space that results

B^ = (x e Bn |xn > 0 }.

X

C°° maps of

denote either X into V

Bn

we define

from completing

with respect to the

Ck (X, V)

norm lllflll k = Sup{

and similarly we define completing

C“ (X,V)

with

,,f^ It is

Hk (X, V)

Ck ||D“f(x) II2 |x e xj

Y \a\
r’ which

.

C°°(£) -*■ C (||N)

by a straightforward weleave to the reader.

7

is a section for p is a formal conse(r i) 7 [m ] , 7[m _s ] and are sections for P[m ]>

quence of the fact that fr) P [m-s ] *p respectively.

q.e.d. In the sequel we will need only the following special case:

X: CONTINUOUS SOBOLEV CHAINS COROLLARY. C°°(|)

into

The restriction map C°°(||dM)

linear surjection

f -+ f |dM

of

extends to a continuous

CHAPTER XI THE SEELEY ALGEBRA Richard S. Palais In this chapter

M

will denote a compact

boundary and with a fixed choiceof strictly

C°° manifold without

positive smooth measure, and

i, t), (; will denote hermitian vector bundles over

M.

In his paper, "integro-differential operators on vector bundles," to appear in Trans.

Amer. Math.

a subspace

t])

Int^(|,

-*■ Smbl^.(|, ti)

of

Soc., Seeley constructs for each integer

0Pk (

t})

and a linear symbol map

k

a^.: Int^(^, ti)

with the following properties:

(51)

0P^._1(|, n) £ Int^.(|, t))

and is precisely the kernel of

(52)

Diffk (|, n) £ Intk (|, t])

if

Int^Xl, i)

k > 0

and the symbol map

extends the symbol map on

Diff^l,

on

defined in

Chapter IV > (53)

If

T e Intk (|, t\)

If

S e Int^Tj, 0

then

ST e Intk+J^(|, t])

^k+J^(ST) = a^ (S) crk (T) j

and (St)

and

T

Int^.( |, t\)

is in

and is in

then the transpose

T^

of

T

(which exists

by [VIII, Theorem k ]) is in Int^X^, |)

OP^(t], §)

and

= (-1)kak (Tl; (S5)

There is a map

X^: Smb 1^.(1, t}) -► Int^( |, r^)

is a right inverse for

a^.

Smbl^/l, n)

C°°-topology) to

(with the

duced topology from

Moreover

x^.

(n o t c a n o n i c a l )

which

is continuous from

Int^( |, r\)

(with the in­

0?k (I, *0 [VIII, Definition following Theorem 1]).

It should be remarked that is easily shown that its kernel,

is n o t

0P^._1(|, n) ,

175

continuous and indeed it

is not closed in

0P^( £, n) .

176

PALAIS In addition, there is a sixth property (S6) which will be in­

troduced in [XIV, %k] . Xk ,

The actual construction of

Intk (g,

, f e

Then by Theo­

(T' Tf -f) = (T'T - Ig)** € Hm+1(i). Tf e Hp (ti)

and

T'T e Hr+kd ) C H01*1 (|) . Hence

On the other

by hypothesis, Theorem 2

f = T'Tf - (T'Tf - f) e Hm+1( | ) . q.e.d.

COROLLARY 1.

If

Tf e H°°(ti)

In other words, if

then

f e

Tr : Hr (|) -► Hr_k(T))

continuous extension of

T

then

is the

Tr (Hr (|)) n B“ (t|)

= T(H“ (|)). COROLLARY 2.

If

Tf = o

other words, if

r

If

f e H“ (|).

In

Tr : Hr (|) -*• Hr’-k(ri) is the con-

tinuous extension of THEOREM 6.

then

T

then

T € E^( g, n)

ker Tr = ker T. then for each integer

its continuous 3inear extension

Tp : Hr (|) -►Hr_k(T})

is an F-operator. PROOF.

e OP^U,

I) and

By Theorem k choose T T f- I

of Theorem 3 ],

Tp-kTr “

into itself and

similarly

of

Hp""k'(tj)

€ OP^Cn, n) .

T' e E_ic(q, g)

so that

Then by [X,

Corollary 2

^g is a completely continuous map TrT^_-^ - 1^

into itself so that

Tp

T'T - 1^

of

Hp (g)

is a completely continuous map

is an F-operator by [VII, Theorem 2]. q.e.d.

COROLLARY. and ker T

T(H°°(g))

is a closed subspace of

H°°(tO

is a finite dimensional closed subspace of

rd). PROOF.

Immediate from Corollaries 1 and 2 of Theorem 5

the fact that the topology of duced from any

H°°(|) = lim H^(g)

is stronger than that in­

Hr (g).

THEOREM 7. Moreover, sub space of

If

T e E k (|, n)

ker T^ H°°(ti)

then

and

Tt e Ek (r], I).

is a finite dimensional closed complementary to

T(H°°(g));

XI s THE SEELEY ALGEBRA and in

fact T(H°°(|))

and

ker T^

complements of each other in inner product of PROOF. isomorphism of

If

gx

T(H°°( |)) so

and

T(H°°(g))

h

e

and

r\x ,

gx ,

so

T^

< g, h

= < Tf, h

ker T^ = ker(Tt)Q

h e T(H“ (g)).

h = g - Pg

s o

Now, by Theorem 6,

[VI, Theorem 10] (Tt)*(Hk (|))

extension of

If

g = Tf €

= < f, ^ h > g = 0

is the continuous extension of

that

in

E k (ii, |) .

are orthogonal with respect to

Pg € ker T^ _CH°°(t])

ker T^,

e

ker T^

then

is an

ak (Tt)(v, x) = (-1)kcrk (T) (v, x)*

hence there is an orthogonal projection

or,

ak (T)(v, x)

then

lary 2 of Theorem 5,

ker(Tt)0

relative to the

ker Tt

(Tt)Q : H°(ti) —► H~k (g)

so by

H°°(t])

then since

by (SM

r\x with

is an isomorphism of

are orthogonal

H°( ti).

(v, x) e T (M)

with

179

< , >^.

If

T^, then by Corol­

is a closed subspace of

H°(t])

P:

g € H°°(t})

e

H°(t]) -► ker T^. H°°( ti) .

If

It remains only to show

(T^) 0 (H°( tj))

is closed

in H “k (|)

is the orthogonal complement of

by Corollary 2 of Theorem 5, the orthogonal complement of H°(t}). T

But by [VIII, Theorem b]

to a map

Hk (|)

= (ker T^)1 . Now since h e Tk (Hk (|)) n H°°(ti)

H°(ti).

(T^)* = Tk ,

Thus we have finally,

h = g - Pg e (ker T^)1 or

h € T(lf°(|))

thecontinuous

h e Hf°(*n),

and

Tk (Hk (|)) we have

by Corollary 1 of Theorem 5. q.e.d.

THEOREM 8.

If

T e E k (g,

tj)

then all the following

integers are equal 1)

ind(Tp) where Tp : Hr (g) -► Hr_k(ti) continuous extension of

is the unique

T (an F-operator

by

Theorem 6); 2)

dim ker T

- dim ker T^;

3)

dim ker T

- dim coker T.

PROOF.

ind(Tp) = dim ker Tp - dim ker (Tp)*.

ker Tby Corollary 2 of

Theorem 5 and since (Tp)*

Theoremt] ker(Tp )* = ker by Theorem 7, so

ker T^

by Corollary

=(Tt)r _^

2 ofTheorem

is a complementary subspace of

Now ker Tp = by

5again.

[VIII, Finally,

T(lf°(g)) in

Hc°(ti)

dim coker T = dim ker T^.

q.e.d.

180

PALAIS DEFINITION.

If

T € E^( I, t})

then the integer de­

fined by any of the equivalent expressions of Theorem 8

is called the index of

THEOREM 9.

If

T

T

and denoted by

ind(T).

6E k(|, n)

then

ind(Tt) = -ind(T).

€ E^( |,|)

and

T^ = XT for some

PROOF. Trivial. COROLLARY.

x e C

then

PROOF. not zero,

If

T

|x|

=1

T = (Tt)t

\x\ = 1. Also,

ind(T)= ind(XT) = ind(Tt)

and

ind(T)= o.

=x ^ = |x 12T.

ker XT = ker T

Since

T, being elliptic, is

and coker XT

= coker T

so

= -ind(T). q.e.d.

THEOREM

10. If

T, T ’ e E k (|,

n) and if

afc(T) =

ak (T’) thenind(T) = ind(T’). PROOF.

By Theorem 3,

Corollary 1 of Theorem 3] ous.

T - T ’e

n)

Tr “ Tp : Hr (|) -► Hr_lc(Ti)

Then by [VII, Corollary of Theorem b]

hence by [X, §t,

is completely continu­

ind Tp = ind T^,

so

ind(T) =

ind(T f). q.e.d. THEOREM then

11. If

T € Ek U ,

ST e

^

PROOF. ST e if

Tr : Hr (|) - Hr_k(n),

Hr_k_^(?) Sr-kTr

so

ti)

and S € E^ (rj, £)

ind(ST) = ind S + ind T. 5)

is obviousfrom (S3).

Sr_k : Hr'k (Ti) - Hr_k"f (0,

are the continuous extensions of tVIII, Theorem 3l

T, S

and

and ST

Forany

r e Z

(ST)r : Hr ( ? ) ~ then

ind(ST)p = ind(Sp _k) + ind(Tr),

(ST)r = hence

ind(ST) = ind S + ind T. q.e.d. THEOREM and

ker T = 0.

Moreover, Also

12. Let

T e E^( Then

T:

T ”1 e E_k (ri, |)

Tp : Hr (|) -*• Hr-^(ii)

cal spaces for all

r e Z.

ti)and suppose H°°( t) -** H“ (ti) and

ind(T)

= 0

is bijective.

a_k-(T_1) =

is an isomorphism of topologi­

181

XI : THE SEELEY ALGEBRA PROOF. ker

= 0

Since

ind(T) = 0

so the surjectivity of

and

T

ker T = 0

it follows that

follows from Theorem 7.

Since

ker Tr = ker T = 0 (Corollary 2 of Theorem 5) and ind Tp = ind T = 0 follows that

Tp

is also bijective and hence [IV, Theorem 1] and isomor-

phsim of topological vector of

T-1

it

to a map of

spaces.

Hr_ic(Ti)

into

Thus

T~1 is a continuousextension

Hr (|)and it follows that

OP_k (t, £) • As in Theorem k, choose

T !€

|)

so that

T ’T - Ig € OP_.j (g, %) .

cjk(T)-1

it will suffice by Theorem3 to provethat No¥

To prove

with

= ak^T^"1 and a _k (T”1)

T_1 € E_k (r], l)

e n d slnce

S = (ST)T_1

T”1 €

=

S = T ! - T-1 €

ST = TtT "

6 0P_i(£> 5) while

T“1e OP_k (ri, g), [VIII, Theorem 3] completes the proof. q.e.d. Assume a € Smbl^g, |) is elliptic * and satisfies a = a. Then there exists A e

THEOREM 13.

E 1(|, I)

such that

1)

a 1 (A) = a;

2)

At

=

3)

A

is bijective;

A;

-

and M

=

0

^

a11

Ak e E k (g, g), ind(Ak)= 0 k e Z;

(Ak)r : Hr (|) -► HP-k(|)

is an isomorphism of

topological vector spaces, all 5)

T-► TAk

with for

is a (linear) bijection of

Intk (g, ti) all

k, r € Z.

and of

E Q (S, n) with

IntQ (g, ti) E k (g, i\)

T e E Q (g, tj) ,

k e Z . Moreover, if

ind(TAk) = ind(T) . PROOF. Let =

Aq = 1(S - St) . - (-“)) =

By Theorem 3

there exists

Then clearly

AQ = -a£

Since

is separable a q

H°(0

many eigen values, hence we can choose A

has zero kernel.

in fact so

ker A = 0 ,

By Theorem 3 (since

(A) = a1 (aq) = a.

A"k = -a.

Then ind(A) = 0 a

S e E ^ ? , g)

is bijective,

Since

X

x

and

with

= ^(0,(3)

cr.,(S) = a . -cr1(St))

has at most countably

pure imaginary so that

AQ - XI

xi^ e OPQ (g, I)), A e E ^ g , is pure imaginary

g) and

(Xl^)1" = -(XI^)

by the corollary of Theorem 9 and, since A-1 € E_1(g, g),

and

ty_1(A_1) = a_1

by

=

182

PALAIS

Theorem 12.

By Theorem 11, Ak € Ek (g, g)

and by (S3)

=

Since

(Ak)p : HP (g) - H P “'k (g) k, r e Z.

all

ker A = o,

n)

into

rem 11,

T

€ EQ (g, t])

ind(Ak) = 0 ker Ak = 0

for all

k e Z,

and by Theorem 12,

is an isomorphism of topological vector spaces for

Finally by (S3),

IntQ (g,

and

Intk (g, n)

T

TAk

is a (clearly linear) map of

and S S A _k

a two sided inverse.

if and onlyif TAk €

n)

and also

By Theo­ ind(TAk) =

ind(T) + ind(Ak) = ind(T). q.e.d. THEOREM 1t. S = S^. of

S € E^(g, g) ,

Let

k ^ 0

Then there is an orthonormal basis

H°(g)

consisting of eigenvectors of

particular,

fn e H°°(g))

eigenvalues

[Xn)are real and satisfy

if

k < 0

then

and suppose

and

X"1

S

(in

and the corresponding

° if

f € Hkr(g)if and

{fn 3

k >

\n -*■

0. If

0

f e H"°°(g)

only if

£ u n l2r •!< f. f „ > s!2 < n In fact if ker S = 0 then

•

(f, g)kr = I l " n l 2r < fn > 4 < ^ > 4 n is an admissible inner product for Hkp(g). PROOF.

We assume

first that

§ b , Corollary 2 of Theorem 3]

[X, map

S: H°(4)—

H°(4).

Since

S

f,

g € H°°( g)it follows

H“ (4)

that

of Theorem 163 there is a sequence basis

{fn )

H°(!)

such that

is dense in H°(g)

so

f

e H_mk(g)

Sfn = Sfn = \nfn . Now

for

then by

S isself adjoint. [\n 3 Sf

Sr € E kp(g, g)

By [VI, Corollary

of real numbers and an orthonormal

= ^nfn

m > 0

and

= < f, Sg > g = < f, Sg > g

ker S = o [Corollary 2 of Theorem 5 3 hence Then inductively

If k < 0

extends to a completely continuous

< Sf, g > g = < Sf, g > g = < f,Stg >i for

ker S = 0.

and

\ -► ° . Since

^ °

and

and since and

ker S = 0

fn = (^n )_1Sfn . k < 0, fp € H°°(g)

(S1*)^: Hkr(g)

-H°(g)

is

an isomorphism of topological vector spaces by Theorem 12 (and the Corollary

XI: THE SEELEY ALGEBRA of Theorem 9 ).

Thus

inner product for

183

(f, g ) ^ = < (Sr)krf, (Sr)kr>g

Hkr(|).

But since

{f )

is an admissible

is an orthonormal base for

H°( I) (f,

=

< (Sr)k r ^ fn > t
|

*

n Now< (Sr)jj-pf, fn >| = < f, Theorem i]. Also

(Sr)t = (St)r = Sr

^xn f r< f > fn >i < S, fn > 6*

E_k (|, I)

and

-k < 0 .

an eigenvector of from the case In any case

S

ker S

30

S)^ -

k > 0

ker S

onto

< f, em

Q. € Z

em

we have

H°°(g).

ker S, kerS

q = k - 1 . By Theorem 3,

it follows from Theorem 7 that

then

S1

P

qe Z

S1 = S + P e E^( £, £) ker S 1 = o,

Let

P

de­

em e H ^ d )

to a map and in particular

and since

S = S1 .

so we have the theorem for

S

from

eigenvectors belonging to

1

into eigenvectorsbelonging to zero,

S

0.

Pf =

On the other hand, we get

theorem follows for

is

restricted to

em e H_i(|) and

for all

x

is not necessarily

is a continuous extension of P € 0P^(|, |)

S"1e

follows trivially

is a finite dimensional subspace of H°(|)

(Sr)Qfn =

S - 1 belonging to

the case

{e.j, ..., e }' is an orthonormal basis for

f -► E^ =1

e H“ ( 0

by [VIII,

then by Theorem 1 2 ,

k < 0 . Finally, assume that

H^(S) -*■ H^-CJ-(|) and hence for

x- 1

belonging to

*ta=l < f > em > | em ’ Given so

k > 0

If

f

fn >| >

Since an eigenvector of

note the orthogonal projection of If

and since

30 < ^ k / ’ fn >| - xn
t = < f, ( ( S ^ b o f n > ?

by making a certain finite number of so the

also. q.e.d.

COROLLARY.

Let

S e Ekd,

has an orthonormal basis S^S.

n), {f }

k> 0.

{(1 +xn )

' fR )

H°(i)

of eigenvectors of

The corresponding eigenvalues

negative and

Then

{xn)

XR -► °°. Moreover, for each

are non­ r e Z

is a complete orthonormal basis

with respect to an admissible inner product for H ^ d ) . PROOF. StStt = StS, so the Also

< Sfn , Sfn > n

S^S € E 2 k ^ ^ existence of

^ (fn )

Theorems 7 and 1 1 is

and

(StS)t =

immediate from the theorem.

= < S^Sfn ,fn > 5 - xn g

from which

XR >

o.

PALAIS Since

k > 0,

I = I ^ e OPQ d ,

|) _C opk_i(£> 0 ,

(StS + I) € E 2k(i, £) • Moreover if < Sf, Sf > n + < f, f > s, and since (Tr)^

T = T^,

maps

ind T = 0.

H^d)

It follows that

hence

with itself.

linear combinations of the hence in

follows that respect to

f

so

H'^d)

f = 0, i.e., T1* € E 2kr^ > = H^d)*

ker T = 0 t)

for all

and r € Z.

defines a non-degenerate bi­

It follows from the theorem that

are dense in

H^d)

for arbitrarily large

H ^ d ) . Moreover, since

(TP)ia?fn =

= ^1+X,n^Pfn

(fR ,

^ 1+x,n^ "P ^2fn^

is or'tilonorinal with

( , )lQ?.

= (1+x,n )r

30

It follows from this that

nite and hence an admissible inner product for ^ 1+Xn )

T =

0 = < Tf, f >^ = < Sf,

Then by Theorem 12,

isomorphically onto

H^d)

then

< f, f > g = 0

(f, g ) ^ = < (Tr)krf, g >£

linear pairing of

H,

Tf = 0

so by Theorem 3,

( , ) ^r H^d)

is positive defi­ and that

;i's 8X1 OI*thonormal basis. q.e.d.

CHAPTER XII HOMOTOPY INVARIANCE OF THE INDEX Richard S. Palais In this section we assume

M

is a compact Riemannian manifold with­

out boundary and we use the Riemannian structure ofM with T(M) by

T*(M) in the canonical fashion. We denote S(M) . Asusual

and wedenote by

|

ti,

the

and

T(M)

unit spherebundle of

* |

over

M

T'(M)

to

{;.

a of Smbl^S, l)

defined by

* cr = a .

clearly elliptic and satisfies

identity

are hermitian vector bundles over

the restriction of the bundle

S(M), and similarly for The element

£

to

o(v, x)e = ||v||e

is

It follows from [XI, Theorem 13J

that THEOREM 1.

There exists

is bijective and

A e E.,(S,

1)

A

2)

ak (Ak)(v, x)e = ||v||ke

k € Z;

3)

If

TA_k e EQ (I, n)

T € Ek (|, n)

Ak e

|) such that

then

|) for all

k e Z;

and

ind(TA_k) = ind(T). THEOREM 2.

The map

a -► o' defined by

a bisection of Smbl^d, t)) with k € Z,

the inverse map

l|v||k ?(v/||v||, x) . if and only if PROOF.

Hom(T,

is

tf) * for each

being given by o(v, x) =

Moreover

a € S m b l ^ d , n)

is elliptic

o' € lso(|, n) •

Immediate from the definitions.

DEFINITION. hy

o' = a|S(M)

We define the map

a ^: Intk (|,

^ ( T) = aiC(T) IS (M) • We note that

linear, that if

T e Int^d, ti)

and

cJk

t j)

-»-Ham(S,Ti)

is clearly

S e Int^(rj, O

then

ak+J^(ST) = oA (S)ak (T) [XI, (S3)], and that by [XI, Theorem 3]

186

PALAIS and Theorem 1 above 0 ->■ O P ^ / i ,

is exact.

Intk (|, ti) — & — ► Hom(|,

11) —

0

Finally it also follows from Theorem 1 that

T e E k (|, ti)

if and only if

k ^ S.

LEMMA.

If

PROOF.

Suppose

Int^._ 1(i, t])

T)) —

ak(T) = 0

then

ok (T) e Iso (?, t^) .

E^U,

k > H.

If

n) n E ^ U ,

n) = 0.

T € E^(|, n)

E^(|, r\) C_

then since

T 4 Ek (£, r\) .

by [XI, Theorem 3] so

q.e.d. DEFINITION.

We define the set

operators from T € E(|, r\)

If

r\ by

I to

E(|, rj)

E(|, i\) = Uke Z E k ^ >

we define the order

the integer (unique by the

k

Remark. orders

k

and

hy

1

and has order

Z(T) =

LEMMA 1.

If

so

T e>*0

that

Z: E(£, t]) -►

kis the order of

T e E(£, tj)

and

T.

S € E(t], (;)

have

ST e E(£, £)

Z(ST) = ^ +J^(ST) = 0 we can regard

Hk 0 ifij as a linear subspace of

H^ 0 H ^ . For (4, m > o),

k > 0

we define

so in particular

inverse limit of inclusion

Hk C_ H ° . We topologize

( , )^m

on

( , )k

gives an admissible inner product

f t 1 > It

H^ 0 ifj ,

and

m* > m

H C H

H^ 0 l£J

JM +

regard

Hk H^

is and

H^

as described at the beginning

m ’= k,

(f, S)j^m ~ * 1 + m < k then we

so that

C

^ 0

Hk as theintersection of the

ft + m = k.

It is immediate from the k1

®

as the finite

It is easily seen that

defined by (f, g) k = k on H .Note that if

with

so we may equally well

H^ 0 H?J with

k

Hk

H° =

n£+m< k

In fact a choice of admissible inner products for

section and then

can find

as a vector space to be

is continuous.

gives an inner product of this

We define

i.e., with the weakest topology such that each

Hk -►

hilbertian.

Hk

H^ 0 H^.

definition that if

and the inclusion map is continuous.

Since

k H7

> k* > 0

then

is dense in each

§1

XIV: TENSOR PRODUCTS

h !?

it follows that

H^ 0 H^,

H^ 0

hence in each

is dense in each Hk .

Since

H = {H^}

H°°

so of course

hence in each

is clearly also the in­

H^ 0 iSJ

(4, m > 0),

is dense in each

H^1.

H^ 0 H^

It follows that

is a chain of hilbertian spaces which we define to be the tensor

product of the chains We put

lf°

H^,

lf° = lim Hk

verse limit of the doubly indexed family is dense in

199

H 1 and

H^ 0 H^ = H°° = Now suppose

H2

and denote by

H 1 0 H2

or {H^} 0

[H2) .

0 H2)°°.

f € H “^ 0 H ”m ^ (H^ 0 ifi?)*

since the inclusion of

into

ft + m < k. Then

where

is continuous,

f gives, by re­

H^. Now the H ”^ 0 H~m where H + m -k ^ -k H 1 0 H2 and we can take their linear

striction, a continuous functional on < k

can be considered sub spaces of

span

k

H2m •

ff|H^

gives a linear map of is dense in each

it follows that this restriction mapping is injective.

call a theorem of Let

Then

H2m t n t o (Hk)* = H_k, and since H“ 0 H^

£j£+m< k 0

®

X

G. W. Mackey concerning locally convex vector spaces.

be a complex vector space,

pseudonorm

N

We now re­

on

X

L(N) C X

X let

its algebraic dual, and given a denote

the linear

functionals on

which are N-bounded.

Then Mackey's theorem [TAMS, vol.

states that if

N 1, ..., N^

supremum, then

L(N)

are pseudonorms on X and if N is their k is the linear span Zi_1 L(N^) of the L(N^) in

If we apply this to the present situation, taking

X =

as the restrictions of admissible norms for the the above map of

^ +m 0

(H1 0 H2)k

with

into

S 0 T. q.e.d.

COROLLARY 1 . Let

r = 0

Then

T

(S, T) -► S 0

0P s (H2, H^) PROOF. 0Pq (H2, H2)

and

If

into I2

is

or

1

and let

s = 0

or 1 .

is a bilinear map of0Pp (H1,

H^) x

0Pr+s(H1 0 Hg, H3 0 H^) . the identity map of

I^ € 0PQ (H3, H 3)

H^,

anelement of

is the identity map of

H^, then clearly

§1

XIV: TENSOR PRODUCTS

S 0 T = (I^ 0 T)(S 0 I2) ,

so by [VIII, Theorem 3l it will suffice to show

that

S 0 I2 € 0Pp (H1 0 H2, H 3 0

201

Hg) (and hence, symmetrically, that

Ig 0 T e 0Pg(H3 0 Hg, H^ 0 H^)) . But this follows from the theorem and the two corollaries of [VIII, Theorem 5 ], COROLLARY 2 .

Let

be a

C°° compact manifold without

boundary and with a strictly positive smooth measure and (i = 1 , 2 , 3 , k ) .

an hermitian vector bundle over Then if

r

and

°Pr(^1 > 6 3 ^

s

811(1

are non-negative integers,

T € 0 Ps ^ 2 '

then

S

€

e

S 0 T

0 Pr+s(CHk(l1 )) ® (Hk (?2)}, {Hk} ® (Hk)) .

By Corollary 1 we can proceed inductively and assume the

PROOF.

theorem true for smaller values of

r

satisfying Theorem 1 of Chapter XII. Since

Sa ^

€ 0Pp - 1 (H.J, H^)

and

and Then

s.

€ E ^ l ^ , |^)

Choose

S 0 T = (SAj-1 0 > Ta“ 1 )(a1 ® Ag) .

Ta“ 1 e 0Ps_ 1 (Hg, H^)

the corollary

follows from the inductive hypothesis and [VIII, Theorem 3l. q.e.d. In view of [VIII, Theorem 1 ] if we take

r = s = 0

in Corollary 1

above, we get: THEOREM 2 .

If

C

denotes the additive category of

discrete chains of hilbertian spaces, then covariant functor from

C x C

into

C

0

is a

which is ad­

ditive in each variable separately. COROLLARY.

There are natural equivalences of functors

(E, © Hg) 0 H 3 2

0 H3) © (Hg 0

H3)and

E, 0 (Hg © H3) s* (E} 0 Hg) © (E, 8)H 3) . Remark. with

Hg 0 Ej

Clearly there are also natural equivalences of H 1 §>

and of

H 1 0 (Hg &> H 3)

in natural equivalence, 0

with

(Ej 0 Hg)

0

Hg

i.e., to with­

is commutative, associative, and distributive.

§ 2 . The Sobolev chain of a tensor product of bundles Let

M.j

and

Mg

be compact

with strictly positive smooth measures.

C°° manifolds without boundary and. Integration on

M 1 x Mg

will be

PALAIS

§2

with respect to the product measure which is likewise strictly positive and smooth.

If

^

is a vector bundle over

0 12

is an hermitian vector bundle over M1 x M2

with an hermitian structure = (el> e- P i / e2> e ? h 2 for

characterized by- ((e1 0 e2), (e« e1, ej e (i*|)p H°°(12)

and

e2,

e (l2)p.

with subspace of

f1 0 f2

f1 (p) 0 fg(q) .

whose value at

In the same way the prehilbert space

H0 ^ )

= 1

^

H00^ )

®

then

(p, q)

0 g2) . Moreover, if

< f, 0 f2, f. 0 f * >

then

e H00^ )

namely if

0 l2)

identified with a subspace of H°(|.)

We recall that we identity

® i2);

is the element of

then [VI, §8]

is

0 H°(£2)

is

f^, f| €

( f , « f2, f. 0 f«)

J

M 1x M 2

1

M 1x M 2

"(

y

^"^2, f2

M1

M2

= < f1, f.| X

so that the inclusion of metric.

Now

H°-dense.

B*^)

H0 ^ )

® H°°(l2) is

On the other hand

H°(i1) 0 H ° ( | 2) definition,

0 H°(l2)

C°° dense in

H00^

0 l2)

0 H°°(|2)

H0 ^ )

0 H°(l2)

into

< f2, f£ >

H0 ^

0 l2)

is dense in

hence a f o r t i o r i

H00(S1 0 l2)

is dense in H0 ^

is the completion of

is even iso­

H0 ! ^ 0 52)

0 i2) H0 ^ )

and, since by 0 H°(l2) we have

LEMMA 1 . H°( ^ ) 0 H°(i2) =• H0 ^

0 g2 ) ,

it, m > 0

^ ( J ^ C ^ ) ) 0 H°(Jm (l2))

we have more generally

LEMMA 2.

If

k

is a non-negative integer and

are non-negative integers with

e H o m ^ ^

is a unique such that if

hence if

0 Jm (i2)).

=

K

with

0 S2), J^d,) ® Jm (l2))

Moreover if £+m = k

e Hom(Jk (|

then

0 1 ),

©

J = o,

i+m=k

i ,m

then there

T (1>jn)Jk (f1 0 f2)(pjq) = J ^ V p

f± €

m > o

4+m = k

m )(j) = 0

® for all

hence

1

0 Jm (Sp))

so

i,

§2 defined by

^U)

=

identification of ®£+pi=k

-► H°°(

gives a canonical

with a sub-bundle of

By the theorem of [IV, §8] the map

^1)) lf°(Jm (l2))

® V

m )(j)*4+m=k

Jic(l1 ® l2)

® j D1(52) •

PROOF.

h

203

XIV: TENSOR PRODUCTS

j^ ® Jm : H00^ )

® H°°(i2)

extends uniquely to a map, still denoted by

® l2)

® Jm (l2))

which belongs to

Diff^S, ® i 2 ,

J£ (8,) ® Jm (l2)) and by [IV, §3, Corollary 1 of Theorem i] thereis a

e

unique

Hom(Jk (l1 ® l2), jtl-j) ® Jm (l2))

® jm )(f)(p,q) = H“ (l, ® l2)

T (£,m)jk(f)(p,q)-

and If

such

that

It remains to show that if

(jf ® jm )(f)(p>q) = 0 for all

i+m = k

then

fe J V ftp,q)

= 0.

Since this is a local question we can assume that M . = R 1 and n. n. np = R x V^ so that f e H°°(R x R , V 1 ® V 2)and we must show

that if = 0.

is an (n^ + n2)-multi-index with

?0

We can write

and

is

an

rQ

uniquely as

aQ

n p-multi-index and

where

and

cc0

then(Drf) (p, q) is an n 1-multi-index

|aQ | + |pQ | = |rQ |(the notation is

that of the proof of the theorem of [IV, §8]). < 4

|r0 I < k

|p | < m. Then the proof of

U f ® 3m) ( f ) ( p ,q ) = (DQ®P f (P, «lU|a|f,m > 8 , 0 6 3 = < fn ® V

it follows that

{cp^ ^)} n ,P

f t ® fm > 6 , ® ^ = < fn> f£ > 6 , < fp ’ fm >6

is an orthonormal basis for

note that

H k (£

^ .

kTJfc,m

ft

( X ^

v

+ X ( 2 ) )(p i k )

m

'

0 £ ).

< = ■

Next

^

.x

p

i+xix }

m

Recalling from [XI, Corollary of Theorem 1b ] that

y £,m

— °° ,

it follows

that x.(1>+ x" + V _

i+xi1t x £ ^ it m from which it follows that the range of

Ck

the

Also

T ^ ^ 2^

such that

+

^ °*

is the closed linear span of ker

is the linear span

0f

of the

such that

=0

x.^1^ = x / ^ =

0,

= (number of JI such

(number of

such that x/2^ = o) = (dim ker S^ S ^

m

(dim ker

hence dim ker

or, since

x (dim ker S2) . Also since

lows that

ker

£ H00^.,) 0 H°°(£2)

x

e if0^ )

and so

ker

su°h that

that

= °) x

(dim ker S2 S2) = 0 H°°(£2)

it fol­

= ker C. q.e.d.

§3.

The # operation The following notation will be maintained for the remainder of

this chapter.

For

i = 1, 2,

without boundary and

£^

will denote a compact Riemannian manifold

and

hermitian vector bundles over

will denote the Riemannian manifold bundles over

M

ti

€Hom(£^, £^)

and

£

and

ti

IVL . M

the hermitian

defined by I =

1^

M 1 x M2

and

1^

(£1 0 |2)

© (tj1 0 n2)

= (T]1 0 i2)

© (11 0 n2)

e Hom(r^, t]^) will

denote the obvious identity

maps. If

a e C°(L(ir*|^,

and

r e R

we shall say that

a

is

§3

XIV: TENSOR PRODUCTS

homogeneous of degree Smbl^-U^, t^) , which are such a

C°°

a

r

k € Z,

if

a(pv, x) = pra(v, x)

is precisely the set of

the zero map of

|

less

o(v, x)

T(M)

is possible.

into

tix ,

is independent of

v)

o € C°(L(jt*i1,

v 1 7^ 0

and to be the zero map of

then

to be

r = 0

(and also for

un­

a

r > 0

over

we

x.,) 8 aQ (I^ )(v2, x2) into

if

0 ^ 2 ^xg

^i^x

) € C°(L(n*(i1 8 |g), **(1^ 8 i2)) r.

will not necessarily be in

such that

More generally if and

t

[Note that even if

a 8 aQ (I^ )

a e Smblr (£ 1,

Smblr (l1 8 £2, ti1 8 tj2)

since it

C°°, at elements ((v^ v2) , (x1, x 2))

v 1 = 0 . As we saw in [IV, § 8 ] if

o 8 a0 (I^ ) € Smbl^U-j 8 £g,

r

a(o, x)

then

Then from what has just been said it follows that

cept in this case

degree

r > 0,

homogeneous of degree

will in general be only continuous, not T 1 (M)

r < 0

U-,)x ® ^ 2 ^x2

and in fact is homogeneous of degree

of

(so that

a e C°(L(jt*!^,

if we define

)) ((v 1,v 2) , ( x^ x 2)) =

a 8 aQ (I

o < 8> a Q( 1 ^ )

p > 0

no continuous extension of

))

( a 8 aQ( I

v1 = 0 .

T(M^)

while if

define

if

for

and homogeneous of degree k) . We note that if

extends to be continuous on

Given

20 7

(l1 8 £g,

8 £g).]

r, s > 0 and a € C°(L(jt*l1,)) is homogeneous of 0 * € C (L(n £g, it r\2) ) is homogeneous of degree s then

o 8 t = (cr0 (I) 8 t) (a 8 aQ (1 ^ ))

is in

C°(L(n*l1 8 £g), n*(ri1

8 Tjg)) and

is homogeneous of degree r+s. In case r = s we define an element o # t o * * of C (L(jt |, jt T) ) ) f which again is homogeneous of degree r; namely the matrix of components of

where

a # t

[XIII, § 3 ]

is

/ f f ® a 0 (IS2)

-«0 (Ini) ® ( - l ) V \

\ a 0 (I|i) ® T

(-I)V ® ^(1^)

(-l)r = e1Plt. Recall that the matrix of

reflecting the matrix of joint of each entry.

a # t

Since

(o # t)*

J

is obtained by

in the main diagonal and taking the ad­

(-1 )p = (-1)-P

this gives

),

208

PALAIS

§3

°* ® °0(Il2) -«0 (I

hence

(a #

t )*(cj

)

#

°0(I|1) ® T*

® (-1)'rT

t)

'a*a 0

(-l)"ra ® x 2 ^ :^i2 ^x2 “*‘ ^ 2 ^x2

t e C°(I(jr*t2, * % 2))

is non-sln® llar>

It follows that if (which implies

dim ^

=

dim | = dim t])then

# t)((Vi, v2), (x,, X 2 ) ) =

l(Xi;X2) - n(x1 ,x2)

is a monomorphism and hence an isomorphism for all € T ’(M).

at each

at least one of the maps

is non-singular at such a point.

a e C°(I(it*^1, *%.,)) dim

T'(M) where

v

((v^ v2), (x1, x 2))

This proves: THEOREM 1 . of

degree

If

ck e C°(I(jt*ii ,

r > 0 , then

a 1 # a2

and is homogeneous of degree Recall from Chapter XII that

is homogeneous is in

C°(I(jt*|, *%))

r. is defined as the restriction of

£-

jr

to the unit sphere bundle of

T(M^) . Recall also that for any

r e R

the restriction mapping is a bisection of the set of elements of C°(L(jt*|^, the inverse map

which are homogeneous of degree r ( r')' being of course defined by a -► av

with

C0 (L('Tj_, ?j_)),

a^rt v , x) = M | r cr(v / 1|v||, x) Finally, recall that

a

(|^, tj^)

is the set of arc components of

c°(i(?'1 , t i p ) . THEOREM 2 . -► 8 , # 8 g

There is a uniquely determined map Of

A d , , n,) X A d 2 ,n2) -*• A d , n)

(&,, 8 2) such that

209

XIV: TENSOR PRODUCTS if

ai e &i €

Tij_)

r > 0

and

then

(o|r)# a^r))|S(M) € 5, # S 2 PROOF.

If

then a e C°(I(k*I1 , «*’i1))

o± e C°(I(|'1 , n±))

ishomogeneous of degree

r

for any

r > o,

C°(I(«*4 , «%)) and hence

(°jr^ #

a^tlS^)

map

a 2) - (0 jr) # a^r))|S(M) of

(a,,

C°(I(T,

.

e

C0 (I(?1, n ,)) x C°(I(t2, n2))

+ (1 -t)r

A,(t) = ts

continuous arc in

s > 0 then as

If

varies from

r

to

t

and (0 ^ ^

s

into

a(I.,, 11 ) x

varies from zero to unity

(0 ^ #

C°(I(T, *0 ) from

hence the above map

e-

C°(I(T, tD ) • Moreover the

is clearly continuous and hence induces a map

a(£2, n2) -*• a( I, t\) .

and

1,

so by Theorem

# a2 ^ ) | S ( M )

a 2 r^)|S(M)

a(I1# t\^) x a(I2, ti2) -*• a(£, t\)

is a

1:0 ( a \3^ # a^s^)|S(M)

is independent of

r

and it is clearly the unique map having the property stated in the theorem. q.e.d. Recall that in Chapter XII we defined an index map ind: Ad^ , -►

Z.

ity,"

t^)

Our goal in the rest of this chapter is to prove its "multiplicativi.e., that ind (&1 # 52) = (ind(51 ))(ind(6 2))

The property (S6 ) of the Seeley Algebra If D 0 I^ e

k

is a non-negative integer,

Diff^( 1 1 0 i2, T|1 0 |2)

and

ak (D 0 1 ^ )

might hope naively that analogously if

Intk d.| 0 l2,

would be in

0 ^))

is in

0 l2)

must be disappointed.

If

k

unless

k

0 |g))

ak (T) ® a0 ^ | ^

ak (T) ® ao ^ | ^

no^

so this hope

31101 is homogeneous of

® ^2^

so we can clearly uniformly approximate it on any compact set by Smblk ( | 1 0 £2,

an element of

0 l2) .

by Seeley that we can approximate elements

T 0 1^

OPk d 1 0 l2,

T e Smbl^d-,, t^),

0ne

is positive, then at least we know that

ak (T) 0 aQ (I^ ) € C° (L( it*d -J ® S2) , degree

then

and its symbol would be

However, as pointed out in the preceding section, even in Smbl^d-, ® l2,

= 0 k (f>) ® ao^I|2^*

T € Int^f^, t^)

(which by Corollary 1 of the Theorem of §2

then [IV, § 8 ]

D € Diff^d.,, t^)

A

in

Intk (1 1 0 l2, ^

T 0 1^ ® l2)

Not so obvious is the fact shown in

0Pk (I 1 ® i2>

so that

0 k (A)

® ^

approximates

210

PAIAIS

a , (T) 0 a (I ) a-^(T) ao(I| K U |p '

in the compact open topology.

This is a basic fact on a

par with the properties (Si)-(S5) listed in Chapter XI.

We state it formal­

ly as (S6) and will prove it along with (Sl)-(S5) when we give the explicit construction of the Seeley algebra. (56)

Let

M1

and

M2

be compact C°° manifolds without boundary and with

strictly positive smooth measures. vector bundles over M 2,

and let

{An ) (An in in

in

and

k > 0.

l2, t]1 0 |2) Int^C^ 0 S2, i2)

0Pn_(l, ® 0 Z2’ 1^. 111 ti„ ® 0 ^

^

and

behermitian

l2l2 ananhermitian hermitianvector vectorbundle bundle over

T € Int^( 11, t^),

a^.(T) 0 cr0 (I^ )

§5.

M1

Let

Then there is a'‘sequence

such that

and such such that that

An

converges to T 0 1^

on (A._) aiJAn)

converses converges to to

in the compact-open topology.

Multiplicativity of the index Given

element of

= 1 , 2)

S^ € IntU^, ^ ( i

0P(|, n)

THEOREM 1.

k> 0

then

S^ e OP^^i* ^

S1 § s1 # S2 b2 € 0Pk(|, ti ]) ;.. ir If

k > o 0

then there is a sequence

CAn) CAn ) in

that t*

(

An \

converges to r» n m r c s v i r f Q e i

+• /^>

ak(An) converges to

and

S S1 1 # S2

.-r

S 1 # S2 to be the

of components [XIII, § 2 ]is

whose matrix

If

we define

/ Q ^

/-r

implies

S± € Int^^, t t^) ]±;

Int^d, ttjj)

such

in 0Pk( 0Pk (|, tt] j) and / C!

& ak ^ 2 \^

-? v~i

+- V i o

n r n n o r * +• comPact-

open topology. PROOF.

Since

1 ^ e 0PQ

1^,

1 ^ e OP^i^, n1)

and

e

_L

OPkC^i, thefirst firststatement statementisisa a consequence of [XIII, [XIII, Theorem Theorem 2]2]and OPjJ’l-p ljl_j)_)the consequence of Corollary 1 of of the the theorem theorem of of §2 §2 above. above.

€ € Int^d^, ii\/),

If If

then by (S6) (S6) of of the the preceding preceding section we we can can find find a sequence I nt^£ ^Int^( |0 ® i2, i2 ^ t)1 ^1 00 l2) ^ with

converging 1 0® 1^ converging to to SS1

a^(Bn) a^(Bn ) converging converging to to 0k 0k( (S. S|) .|) 00 aQ aQ(I^ (I )

Similarly, we can find sequences

in

k > 0,

(Bn) (Bn ) in

0Pk (S1 0 tj j1 ® 0 S2) 0Pk( i2, t

in the compact-open topology.

, and

[En) in

§5

X IV : TENSOR PRODUCTS

Intk(T)' 0 n2» ’11 ® 82)»

Intjc(11 ® S2, h ®

respectively, converging in the respective and

211 8X1(1 Intk^nl 0 n2 ' 6l 0 ^

0 Pk*s

1^

to

/V

t S2,

0

1^

^

0 S2

sf J I

and with the corresponding sequences of symbols converging I T)n k * in the compact-open topology respectively to 0 o^tj ^ ® °k^2^ > aQ (I^ ) 0 crk (S2),

(-1 )kak (S1)* 0

and

B

)•

n

-C n

Dn

En

Then by [XIII, §^, Theorem]

An =l

is in

Intk (|, rj)

while

a^(An )

and clearly,

converges to

0 Pk (|, n)

An converges in

CTk (S.,) #

to

S 1 # S2

in the compact-open topology. q.e.d.

THEOREM 2 . PROOF. S 1 # S2

(S, # S2 ) 1 = S^ # (-S2) (S1 # Sg) ^

The matrix of

.

is obtained from that of

by reflecting in the main diagonal and replacing each entry by its

transpose, i.e., S* 0 1 .

-I

n-,

I,

2

61

0 S0

® Sp

S J l

2

1

which is also clearly the matrix of

t2

# (-S2) . q.e.d.

COROLLARY 1 .

Q I

°)

n-|

where

C = S^S-, 0 1 ^ + 1 ^ 0 S2 S 2

0 S0S^ + S.S^ §> I 2 2

(S, # Sg)t(S1 # Sg) is

The matrix of

1 1

COROLLARY 2 . If

PROOF.

s j_ e E1 ( g±, t)±)

(b) (c)

then for all

ker T C ker T^

forany T e OP.j(£, ;

ker T^T = ker T ; (TtT)k - T^_lTk

k e Z

and has dimension

S-jMdim ker S2) + (dimker S^) (dimker

We first note that

(a)

D =

n2

ker(S1 # S2)k = ker(S1 # S2) (dim ker

and

.

S2) . 0

212

PALAIS

Applying(a) and

T = S1

(c) with

§5

# S 2 gives ker (S1 # S2) C ker(S1 # S2)k

C ker((S1 # S2 )t(S1 # S2))k . And applying (b) gives

ker(S1 # Sg) =

ker(S 1 # S 2 )t(S1 # S2) . Hence it will suffice to prove that ker((S1 # S2 )t(S1 # S2))k = ker(S, # S 2 )t(S1 # S2) Now by Corollary 1 it follows that

and has the required dimension.

kerUS, # S 2 )t(S1 # S2))k = ker Ck © ker D k - By Corollary 2 of the theorem of §2 , ker C^ = ker C

and has dimension (dim ker S-jMdim ker S2) .

changing the roles of

^

ker

= ker D

and

r\^

and of

and has dimension

S^

and

S^,

Inter­

it follows that

(dim ker S^)(dim ker S2) .

It follows that

ker( (S1 # S 2 )t(S1 # S2))k = ker C © ker D = ker(S, # S 2 )t(S, # S2)

and has

the right dimension. q.e.d. 1 . Let

LEMMA

X

and

Y

be hilbert spaces and

T: X -*■ Y a continuouslinear map. closed range, then so does PROOF. jectively vn hence

Let

(vn )

T

maps

isomorphic to

has

T. Then

T*T

so by [VI, Theorem 1 ]

T*T

is a sequence in

V.

V isomorphically onto V ,

T*T

V = (ker T ) 1 = (ker T^T)1 .

with the range of

0if

If

Now

TvR -*• 0

T*Tvn -► 0

T(V) = T(X) . Hence

V

bi-

implies

T*Tvn -♦ 0 ,

implies

is a hilbert space, hence closed in

maps

T(X),

being

Y. q.e.d.

LEMMA 2 . Let spaces,

Hj

be chains of hilbertian and suppose that

has closed range.

Then

A^: H^ -*•

has closed range.

PROOF. to an isomorphism

JA*: H° - H*

H2

A e 0Pk (H1, H2)

(A^A)k : H^ -*• H “k H^

and

A choice of admissible inner product for j: H~k s H^.

Now

(A^A)^ = A^A^

is the adjoint of

Ak ,

so

and, by Lemma 1 , so does

lr H 1 gives rise

and by [VIII, Theorem k]

j(AtA)k = A^Ak

has closed range

A^.

q.e.d.

213

XIV: TENSOR PRODUCTS

§5 THEOREM 3.

Let

S^ €

r^)

so that (by Theorem 1 )

(S, # S2), : H 1 (I) - H°(n)

S, # s2 € OP, (I, n) •

Then

is an F-operator and

ind(S1 # S2) 1 = (ind S1)(ind S2) .

PROOF.

Writing

v = dim ker

v((S1 # S2) 1) =

we have

v (S 1 )v (S2) + v(S^)v(S2) by Corollary 2 of Theorem 2 . Since [VIII, Theorem k] •X* +■ (S1 # S2)1 = (S1 # S2)Q , from Theorem 2 and its second corollary we have v((S, # Sg)*) = v (S^ # -S2) = v(S^)v(S2) + v(S 1 )v(Sg)

.

Then

v ((S, # Sg),) - v((S, # Sg)*) = (v(S,) - v(S^))v(S2) - v (Sg)) = (ind( S-j)) (ind( S2)) Hence (cf. [VI, Corollary 2 of Theorem (S1

# S 2 ) 1 has

closed range, and by Lemma 2 above it will suffice to prove

((S- # S 0 )^'(S1 # SQ)) -

that

above ((S1 # S 2 ) (S, # Sg)), -

/C 1

Corollary 2 of the theorem of § 2 .

of

S^

((S1 #

that of

and

S^

D1

0

\

D/

C1

where

has closed range by

Interchanging the roles of D1

it follows that

S 2 )t(S1 # S 2 ) ) 1

By Corollary 1 of Theorem 2

has closed range.

t

and of

1 0 ] it will suffice to prove that

and

has closed range, hence the range

which isthe direct sum of the range of

C1

and

is also closed. q.e.d. THEOREM k.

(Multiplicativity of the index)

Si e A(li , i\±)

If

then

ind(5 1 # 52) = (ind 5 ^ (ind s2) . PROOF. 6 ± = 6 (Si)

By [XII, Theorem k]

= [a1 (Si)]

so

as in Theorem 1 then

(An)

we can choose

S^ € Ei (t^, n^)

6 1 # 6 2 = [(of1 (S1) # o 1 (S2)) |S(M)].

o'1 (An) = a ^ A )|S(M)

with

If we choose

converges uniformly to

(a 1 (S1) # a 1 (S2 )) |S(M). Recalling (see proof of the lemma of [XII, Theorem t]) 5 1 # S2

that

It follows that for and

§

# 52 = s(An )

converges to

( a ^ S ^ # c?2 (S2 ))|S(M)

is a neighborhood of

S1 # S2

n

large, so that in

(Ar ) is in

# &2,

ind(s1 # 5 2) = ind A^.

OP^I, r\)

so that

(An ) 1

in

hence

C°(I(I, ti) An € E ^ l , n)

On the other hand, An converges to (S1 # S2)1

21 k in

PALAIS L(H1(£),

H°(t])) . Hence for

(ind(S1 ))(ind(S2))

n

large,

ind An = ind(An ) 1

by Theorem 3 above and [VII, Theorem U].

CHAPTER X V D E F I NITION OF i

A N D i t ON K(M)

Robert M. Solovay

In §1, we show that

ia (D)

depends only on

7 (D)

e K ( B ( M ) , S(M)).

Paragraphs 2 and 3 are devoted to a study of the multiplicative properties of

i^.

In

we associate to each vector bundle

sional manifold

M,

an operator

i(M, for

1 = 1+. or

1 ,

U

cL

DM 0 1 .

marizes the p rincipal properties of

over the even-dimen­

Putting

T,) = i(DM ® 1^)

we m ay consider

t\

,

i to be defined on

ia

and

i^

on

K(M).

K(M) .

§5 sum-

For the most

part these properties are proved in § § 5 — 1, though one property w il l be e s ­ tablished (after m u c h labor)

§1.

in XVII.

D e f i n i t i o n of the analytical index on Let

(resp. S(M)) Let

M

shall denote

be the projection map. If

C°°(t])

(;

and

^

Let

(resp. unit sphere bundle)

the restriction of

N ow let D: C°°(E)

be a closed oriented R iemannian manifold.

be the unit ball-bundle

it: B(M) -*■ M

K ( B ( M ) , S(M))

(;

B(M) of

M.

is a bundle over

M,

bundles over

and

X

to S ( M ) . be complex vector

an elliptic operator.

The symbol of

D

M,

gives an isom o r ­

p hism o(D) :

•

The difference construction yields an element

7 (D) = d ( A ,

A ,

7 (D) e K ( B ( M ) , S(M));

°(D))

216

§1

SOLOVAY THEOREM 1 .

There is a homomorphism

(1 )

ia : K(B(M), S(M) - Z such that

(2 )

ia (D) = ia (7 (D)) for e ach elliptic operator index of

D

depends only on

K(B(M), S (M ) ), operator PROOF.

7 (D).) If

for some elliptic

W e prove the last assertion first.

* £,

where

£

D: C°°( £) —► C°°( t])

tD •* with

over B(M)

-► M is

iso-

£, r], and

aQ ,

«*n, o0 )

B y [XII, Theorem a(D)

B(M)

is a smooth vector bundle over

[II, Theorem 2 ] shows that for suitable

aQ e Iso(f,

Since

every complex vector bundle

r0 = d(**5, where

rQ e

D.

morphic to one of the form Thus

(Thus the analytical

r0 = 7 (B),

then

is a homotopy equivalence,

M.

D.

there is an elliptic operator

homotopic to

aQ . Then

7 (D)

= 70 ,

and the

last sentence of the theorem is proved. Acco r i i n g to [XII, class of

a(D)

operator

D'

in

Theorem k]

Iso(Z, ?) •

of order zero.

Moreover

ir £ s ir

tj

over all of

B y [II, Lemma 1 ] it: B(M) -*■ M t: £ = n. operator since

a(D)

7 (D) = 0 .

/

of order zero, and

a(D')

foh some elliptic

of the proof, we shall con­

extends to an automorphism of

B(M) .

is a homotopy equivalence, O n the other hand,

=

of order zero.

The automorphism

*

a(D)

depends only on the homotopy

For the remainder

sider only elliptic operators, D, Case 1 .

ia (D)

We show that tj(D)

ia (D) = 0 .

is homotopic in

• C (£) -*■ C (11)

Since

it \|r where

is an elliptic differential

a(\|r*) = ir%|S(M) .

Thus

ia (D) = i (>|r*) = 0 ,

is a n isomorphism. Case 2 .

D

is an elliptic operator of order zero, wit h

7 (D) = 0 .

*

W e allow the dimension of the fibres of

nectedness components of

M.

£

to differ on different c on­

§1

XV. DEFINITION OF i We show

over

M

ia (B) =

AND it ON K(M)

Bytll, Lemma 1(v) ] there is a bundle

cd

such that a(D) + 1

f

® ^ if ® m

extends to an isomorphism over all of (cf. XIII, §U) . By Case 1,

V Thus

217

B(M) . Now

ia (D ® 1CD) =

D ® ’J

■ V

°*

o(D) 0 1~ = o(D ©

0n

other hand,

D> + W

■ V

D>

•

ia (D) = 0. Case 3.

D^, D 2

We show

ia (D.j) = ia (D2) .

with 7 (D3) =

- 7 CD, ) -

are elliptic operators with LetD^

By [XIII, §k]

be elliptic

a(D1 0 D 3) =

rCD^

of order zero

a(D1) 0 a(D3) .

additivity of the difference construction [II, Lemma 1 (vi)] 7 (D1) + r(D3) = 0 .

Case 2 , Thus

Similarly,

r(D2 + D^) = o.

o = ia (D1 © D 3) =

;

+

= r(D2) .

By the

r(D1 0 D 3) =

By [XIII, Theorem 1 ] and

similarly,

0 = 1& (D2) + ia (D3) .

ia (D,) - ia (D2). By Case 3,

ia

K(B(M), S(M)) by(2).

is well-defined on

In

view of [XIII, Theorem 1] and the additivity of the difference construction, i

is a

homomorphism. Remark.

This completes the proof of Theorem 1 .

The definition of the topological index makes it clear,

a priori, that

depends

only on

r(D).

Thus

i^. defines

ahomomor-

phi sm it : K(B(M) , S(M)) - Q it(B) = 1^.(7 (D)).

such that

operators on

M,

To prove the index theorem for all elliptic

it suffices to show that

ia

and

i^

agree on

K(B(M), S(M)).

§2.

Multiplicative properties of 2.1.

ChapterXIV. and

i^

We review the results

For

and

hermitian bundles

i = 1, 2,

isa

on tensor products that we need from closed oriented Riemannian manifold

are hermitian bundles over I

and

r\

over

M

M^.

Let

by

I = ( 1 1 ® i 2 ) © ( t^

®

ti2)

ri = (ti! l2) 0

0 r\2 )

M = M1 x M2;

define

218

§2

SOLOVAY Let

S^:

1, 2 . Define

-► C°°(ri^)

S, # S 2 e OPr ( , r\) S, ®

(3)

for

i =

by the matrix

I.

1

r > 0

be elliptic or order

-I

i 2

® S* m

2

I II1 ® s2 S1

If

S2

and

S1 ® t,2

are differential operators, then

elliptic differential operator of order will not be elliptic of order

M^,

is an

S 1 # S2

However, in general,

r.

S^, a(S^) is, a p r i o r i

The symbol of tangent vectors of

r.

S 1 # S2

T'(M^).

Since

a(S^)

defined only on the non-zero is homogeneous of degree

r > 0 , it has a continuous extension to all of

T(M^) :

let

a(o, x)

be the

zero map. a(S1 # S2) € C°(L(jt*|, jt%)

We define

by giving its matrix

components 0(3,) ® °0 (l|2)

" V 1^

® (-1F « ( s 2)*)>

o0(I5 ) ® o(S2)

(-l)ro(S,)* ® o0 (I^)

W 51 If

S1

bol.

S2

and

are differential operators, a(S1 # S2)

In general,

(I.e., if

w

(v, x) e T(M),

a(S1 # S2)

is the usual sym­

will be homogeneous of degree

v ^ 0,

then

''2

r

and elliptic.

a ( S 1 # S2) (v, x) e Iso(lx , *lx) •)

Finally, we remark that the analytical index enjoys the following multiplicative property: Suppose that

a( S)

(it: T»(M) -► M) . (5)

let

is homotopic to

is an 2 .2 .

C°°(t})

be elliptic of order

a(S1 # S2)

r.

in* C° (Iso( it*£, *%))

Then ia (S)

(S1 # S2

S: C°°(|)

F-operator,

= ia (Sl # S2) = V S , ) so

ia (S1 # S2)

• ia (S2) .

is defined.)

We recall the definition of the topological index of an

elliptic operator S: C°°( |) - C°°( r|) (Here

|

and

.

r\ are complex vector bundles over

M.

is an arbitrary closed oriented Riemannian manifold.)

For the moment,

M

X V: D E F IN IT IO N OF

§2 Let

m

be the dimension of

be the Thom class of H*(B(M) , S(M) ; Q)

AND ± t ON K(M)

U e Hf^BfM), S(M); Q) * determined by its orientation. Let cp*: H (M, Q) s

M

M,

S

and let

be the Thom isomorphism: cp*(a) = / a u U

Since

219

is elliptic, the element

(*: B(M) - M) r(S) e K(B(M), S(M))

is defined (cf. § 1 ).

We put (6 )

Ch(S) =

where (7)

e(m) = £l|±l1

.

Next recall the definition of ^r(M) e H*(M, Q) . Let | vector bundle over

M;

we view the rational Pontrjagin classes of 2

elementary symmetric functions in + pm (£) = Hi=i (1 +yi)-

bea real

y 1, ..., y

2

(n large):

1 + p^l)

I as + ...

Then n

(8)

Tr

^(e) > n

_Tr

,_e-yi

i =1

1 -e

Finally, ST ( M) =^(T(M)). The following formula results from (8 ): (9)

^(l-, © l2) =^(1-,) We recall the argument.

• ^ ( l 2)

Write

r p(5,)

(where the

y^'s

=

s

n ( 1+y f ) i=l

and

P ( l 2) = II i=r+1

are indeterminates and

formula for Pontrjagin classes says that

[ji = i ( 1 + y ? ) .

r, s-r

( 1+yf)

are large).

The product

p ( ^ © t2) = p(l-,) * p(l2) =

Now (9 ) follows easily from (8 ): s

~£fd,® i2) = n (-~fz'~~y~)= n (•••)■• n (•••) =3^) • 1=1

1 -e

Finally, if

1

a € H (M, Q) ,

top-dimensional component of orientation on

M.

1 -e 1

a

i =1

a[M]

I=r+1

is the result of evaluating the

on the fundamental cycle determined by the

The topological index of

S

is given by

S0L0VAY

220

(10)

i t (S) = Ch(S) 2.3.

§2

£ T { M)

[M]

We r e t u r n to the n o t a t i o n o f 2 . 1 .

d efin e

y(

S.,# S2) € K(B(M), S(M))

D efin e

i ^ S . j # S2) e Q THEOREM 2.

y(S^#

= dim M^.

S2) = d (*

We

* n, * ( 3 ,# S2) | S (M ) ).

by (6) and ( 1 0 ) . The t o p o l o g i c a l i n d e x i s m u l t i p l i c a t i v e : V

PROOF.

by

Let

s!

S2> ' W

#

■

W

*

The b a s i c p o i n t i s t h a t a l l the " i n g r e d i e n t s " o f

have m u l t i p l i c a t i v e p r o p e r t i e s .

i^

The s i g n i n (6) w i l l s e r v e t o compensate

f o r s i g n s a r i s i n g from the skew -com m u tativity o f the c u p -p ro d u c t. We s h a l l f i r s t show t h a t (11)

^(M) =

x ^T(M2)

In f a c t , we have a d i r e c t sum d ecom po sition ( 9 ) , ^ (M )

= K*

£T 1}

lying between

S(M)

. and

We have a homotopy commutative

diagram of homotopy equivalences; (B(M1) x B(M2), ^(B(M1) x B(M2)) ( 2 2 )

r

(B(M), S(M)) ----- !--- Here

i,

and

i2

are inclusions and

The map

a (S., # S2) |E(M)

of non-zero vectors.

Then by (2 2 ), (2 0 ),

= d( A

,

A

r !i^(7 *) = i ’(7 *).

i 2 (7 *) = 7 (S1 # S2) .

is as in Lemma 1 (cf. Figure 1 ).

r

is an isomorphism since E(M)

7 * e K(B(M1) xB(M ), E(M))

Define 7*

(B(M1) x B(M2), E(M))

,

"(S ,

Now

#

consists

by

S2) |E (M ))

i.j(7 *) = r(S1) x 7 (S2)

by (2 1 ); by

Thus

r *7 (S 1 # S2) = 7 (8 ,) x 7 (S2)-

,

and Lemma 1 is proved.

§t.

Definition of

i^

and

ia

on

K(M)

t . 1 . We shall need the following material from [V, §3]. M

beaclosed oriented even-dimensional manifold;

dim M =

2m.

Let

Let | =

A*(T*(M)) C be the bundle of complex-valued exterior differential forms. Then there is a canonical direct sum decomposition:

|

The

22k

SOLOVAY

elliptic differential operator d + 5 : C°°(|) - C°°( |) restricts to an elliptic differential operator Dm : C°°( |+ ) - C°°(D The character of §6 ].

may be computed using the results

To state the result, we view the Pontrjagin classes of 2

elementary symmetric functions of y 1, m p 1 + p^M) + ...+ Pm (M) = J|i = 1 (1 +yj_)- Then

The element

y^

is of

y ,

(

M

[III,

as the

in the usual way:

~J ±

J 1

m Ch(DM ) = _n

(23)

2

of

f f

)

2.

formal degree

•

Since

- y ix .

y± (e - e

)/y^ = 2 + positive-dimensional terms,

(2 3 ) implies

( 2 k)

Ch(D^)

k, 2 . -*■ M

Let

tj

= 2 m + positive-dimensional terms.

be a complex vector bundle over

be the projection map. a(DM ) 0

M.

Let jt:T*(M)

Then it (|" 0 r\)

1 * : **(| + 0 ti) Jt T|

lies in Symb^( |+ 0 t),

0 n) .

It is elliptic of

order one since

>

|f(x) |

|f(y) I < ||f|| « • Also if L each x € R n

i.e.,

g e L** ( R n)

(f*g) (y) = ^ dx y f(x-z)g(z)e"lx y dz = y dx y f(x)g(z)e_i^x+z^ d z = f(y)g(y) THEOREM

2 . If

DEFINITION. f: R n

C

S( R n)

N

(f*gf =

denotes the space of

fg.

C°° maps

is bounded on R n . We topologize

by means of the family of pseudonorms p (f) = Sup{ | x V f ( x ) || x e R n}

We note that if differential operator

then

such that for each pair of n-multi-indices

a, p Ix^D^fCx)| S( R n)

f, g £ L 1 ( Rn)

on Rn

P: R n -► C

is a polynomial map

and

D

a

with constant coefficients then

Np ,D(f} = Sup{|P(x)Df(x)|| x e R n } is likewise a continuous pseudonorm on

S( R n) .

In particular

S( R n)

is

§1 .

XVI: CONSTRUCTION OF Intk

237

stable under differential operators with constant coefficients and multipli­ cation by polynomials. there exists

M^ > 0

If

f € S( R n)

so that

|(1

and

k

is a positive integer then

+ ||x||2 )kf(x) | < Mk

or

|f(x)| -1,

dp

(1 +P2)k

o

.2 N k J (1 + p c ) J

o

which is finite if

n-1 ,

x

k > n/ 2 .

i.e., if

p

2 k-n-i

It follows that

S( R n) S( R n)

and in particular the Fourier transform of an element of

is defined. j = 1 , ..., n

e. = (5.., ..., 5 .) e R n “ - t j ' •••’ °nj standard base for R n . Let

e-ix-(y+te.) =

=

|g( tx) | < 2 .

If

e_.x .y e -itx.

e lx *y (1

i ex . g(tx) = 2 e J

where by Taylor's Theorem

f e L 1( R n )

denote the

_ itXj - t2 x^g(tx)) with

0 < 6 < t

and hence

then

l(f(y+tej) - f(y)) = § e~±x J

(-ixj)f(x)dx

+ t y e-ix y g(tx)xjf(x)dx Now if < 2

f € S( R n)

we have

then

x?f e S( R n) C_ L 1 ( R n),

|/ e_ix*yg( tx)x?f (x) dx |


Mk

then

(1 +x2 + ... +

y


Q

of

is an isometry

with itself. f, g € S(R n)

then

by Schwarz's inequality

C |e 1Z y f (z) g(y) Idydz < ||f|| 0

“

and Fubini's Theorem gives < f,

But by Theorem k g € S( R n) . S( R n)

of

Thus

g

>0= y

f -► (2tt)

“n /^

f

t

'

||g|| T L

< 00

?(y)g(y)dy

=

y d y y e"l z ‘y f(z)g(y) dz

=

y f(z)(y elz’y g(y)dy) dz

=

y f(z)g(-z)dz

g(-z) = (2 jr)ng(z),

L2( R n)

L2( R n)

extends uniquely to a continuous

linear automorphism for

is dense in

L2(R n) .

dense in

THEOREM 5 . (Plancherel Theorem) form on

C“ ( R n)

so Parseval^s formula holds for

f,

is an isometry of the dense subspace

onto itself and extends uniquely to an isometry of L 2( R n) . q.e.d.

21+2

§1

PALAIS AMD SEELEY DEFINITION.

We denote by

chain of hilbert spaces the measure on R n

{Hk ( R n )}

{L^( R n , C, (i + ||x||2))) (2 ^)_n

is taken to be

Lebesque measure.

the continuous

Thus the norm

|| ||g. on

where

times Hk ( R n ) is

given by llfllk = (2 n)‘n j> lf (x ) |2(i + l|x||2)k dx Remark. .The definition of Chapter VIII, §t. a scale.Note that

S(R n)

L^( R n , C , (i + ||x||2))

is given in

By Theorem 2 of that section it follows that H°( R n)

Also f (x)(i+ ||x||2)_k/^zf (x) since

.

= L2( R n)

hence

is an isometry

S(

R n )is dense in

of

H°(R n ) onto Hk ( R n) and

is stable under multiplication by

S( R n ) isdense in each

Hk ( R n ),

DEFINITION.

hence in

(i + ||x||2)k

Hk ( R n ) to be the hilbert space obtained

If

it followsthat

k

we define

by completing

the norm

l|fHk = I CallDO!fllo M 0 and such that

for

for all positive integers

We topologize

the linear

and

n-multi-indices a, p

5) I | x e R n , 5 e Sn "1} < »

Smblp ( R n)

.

by this family of semi-

norms. DEFINITION 2 . 0:

A patch function on R n is a C°° map

R n —► [0, 1]

such that 0

is identically zero

near the origin and identically one near infinity.

Rn)

0

DEFINITION 3.If

a e Smblp (

function on R n

we define a linear map

S( R n) -

S( Rn)

and

is a patch

A( 0 , a ) :

by

[ A ( e , a ) f ) ( x) = ( 2 « ) ' n J e l x ' ? e ( 5 ) a ( x , 5 ) f ( S ) d5

•

We have the following results which will be proved below. THEOREM 1 . For each patch function number

s, and integer

0 (0 , s, r)

r

0

on

such that n

2 p

[1

7^7 1=1

an integer £ € Sn_1 . element of

f e S( R n)

and

01

sequence

(cpm )

xQ e R n S( R n)

p

is

and

A (0 , a) as an

a*-* A( 0 , a)

Smblp ( R n)

is also a patch

in

regard

and the map

A( 0 , a) - A( 0 ', a) e O P ^ C R n) THEOREM 2 . Given

Here

and the Sup is over all

In particular we can 0Pr ( R n)

a ( x , 0 |dx

1

a e Smblr ( R n) .

>(|s|/2 ) + (n/2 )

a continuous linear map of Finally if

real

there is a constant

l|A(e, a)f||g < ||f|lr+ s c ( e , s , r ) Sup

for each

Rn,

into

is then 0Pp ( R n) .

function on R n then for all

a e Smblp ( R n) .

(; e Sn_1

such that:

there is a

2 kk

§2

PALAIS AND SEELEY (i) (ii) (iii)

||g = m -*■ oo

D^ak (x,

O

a =

D^a^.. . Let D“ak+|a | / o

only if

is an

only if

|cc| < r-k,

and it

k € Z

\oTT D^D? 7 | | x ^ ak+|a| \aj> 0 is a finite sum of elements of Smbl^( R n) denotes complex conjugate).

Similarly

hence unless

|ar| < r + s-k

I

ieZ

and

finition.

and

and

I

(Ds ak-f+ M

is an

0 < |a| < r- k+4,

k - r < 1 < s. Hence for each

Smbl^.( R n )

follows that we have well-defined elements ( R n)

1 < s

(the

k e

Z

)(( r Dl “ V

\a |> 0

is a finite sum of elements of

ing to

k > r

(D^ a^-_j^+ |a |) ((^ Dx )^ ^ 4 )

Smbl^( R n)which is zero unless

element of

and is zero if

£p+s ( R n )

and is zero for a*

respectively,

and

acr’

of

k > r + s. Z( R n )

It

belong­

given by the following de­

§2

X V I: CONSTRUCTION OF I n t k DEFINITION 5.

in

^

E( R n)

elements of Eg ( R n)

If

ak

and

2k5

o' =

Rn)

(belonging to

are

( R n)

(belonging to

respectively) then we define

Z( R n)

t>k

a*

and

and

and

oo'

£p+g( R n)

respectively) by

I IOfI> 0

■ keZ I

I

J

aa' -

,

|a| > 0

^

(D“a)(( ± D x )a o')

IOfI > 0

1

=

1

keZ THEOREM 3.

leZ

1

^

ak-£+ |a|)(( r Dx)“ V

\a\ >o

With respect to the operations introduced

in Definition 5 ,

Z( Rn)

is a filtered *-algebra.

DEFINITION 6 . A linear map

A:

S( R n) — S( Rn)

be called a Calderon-Zymund operator of order there exists

o =

each integer

m > 0

Rn

ak

in

^r ^

r

will if

such that for

there is a patch function

G on

such that r A-

A(6, ak) e 0Pr _m_ 1 ( R n)

J

.

k=r-m The collection of such A and we define

will be denoted by

CZp ( K n )

CZ( R n) = Ur e 2 CZp ( R n) • ^e define

o ( A) = o . Remark. from

the corollary of Theorem If

0Pp

A e

2 that

CZp ( R n ) and

a(A) is well-defined.

a (A) = Zk ak

then

A - A(0, ap) e

R n) and since A( 0 , ap) e 0Pp ( R n), it follows that

Moreover, if hence

It follows from the last statement of Theorem 1 and

ap ^ 0

A 4 0Pp _ 1 ( R n) .

then by the corollary of Theorem 2 Thus

A e 0Pp ( R n).

A( 0 , r) 4 0Pp _ 1 ( R n)

•

2 k6

PAIAIS AND SEELEY CZr ( R n) = {A e CZ( R n) |o(A) €

( R n)) = CZ( R n) n 0Pp ( R n) 0P_oo( R n) C_ CZp ( R n)

It is immediate from the definition that o(A)

= 0 for

and that

A £ OP( R n ) . By the preceding remark it follows that in

factO P ^ R 11) =

CZp ( R n) = (A e CZ( R n)|o(A) = 0 } .

flr

Since that if

§2

CZp ( R n) C 0Pp ( R n )

A e CZp ( R n )

then

A

it follows from [VIII, Theorem it]

has a transpose

A t e 0Pp ( R n) .

The following theorem states the basic algebraic facts about

o.

the collection of Calderon-Zygmund operators and the mapping THEOREM 14-.

The collection

Zygmund operators for

CZ( R n)

Rn

is closed under sums,

products and

A *-►

The CZp ( R n)

are subspaces of

CZ( R n)

the

of Calderon-

and hence forms a *-algebra. CZ( R n)

and give

structure of a filtered *-algebra.

Moreover the mapping *-homomorphism and

o:

CZ( R n)

a (A) € Z^ (Rn)

Z( R n)

is a

if and only if

A € CZp ( R n) . Finally the sequence 0 - OP _ J R n ) - CZ( R n ) - Z( R n) - 0

is exact. DEFINITION 7.

If

ak e Smblk ( R n ),

A e CZp ( R n ) then

and

we define

0r (A) = (-i)r ap

COROLLARY OF THEOREM k. CZjJ R n)

into

a (A) = Zk ak,

ap

Smblr ( R n)

.

is a linear map of and if

A € CZp ( R n)

then

o ^ )

= (-1 )r ^ J A )

Moreover the sequence 0 - CZr _ 1 ( R n) - CZr ( R n) - Smblr ( R n) - 0

is exact. If

A e CZr ( R n)

for all

(x, £) e

and B e CZs( R n) R n x ( R n - Co})

thenAB e

CZr+s( R n ) and

§2

XVI: CONSTRUCTION OP Intk

2k 7

o r + s ( AB ) ( x , 0

= [op (A)(x, 5) ][ 0,

There is a sequence

CZp ( R n x Rm ) that

A e CZp ( R n)

Let

and

in

and let

{Bv}

in

0PQ ( R n x R m ) such

0 Pr ( Rn x R m )

to

®r (Bv)((x,, x2)(£,, ?2))

verges uniformly on R n+m x Sn+m_1

con­

to

®r (A) (x,, £.,)y(x2) . Finally we consider coordinate transformations. setin R n

open

then we shall consider

extending an element y:

U -► R n

maps

of

C“ (U)

and we define

cp is an element of

S( Rn)

and

A

maps

S( R n)

U is an

Rn) by

of S(

to be identically zero outsideU.

y*: C“ (U)

C“ (U) into C“ (U) C S( R n) .

C“ u (U)

as a subspace

is a C“ diffeomorphism then we define

y*(f) = f ° y‘ 1 that if

f

C“ (U)

If

y*:

C” (U) -*■ C“ (y(U))

by

y*(f) = f • y.

Note

S( R n) — C°°(U)

by

then

S( R n)

M^

maps

It follows that if

into itself then

cp1

into

and

C” (U)

cp2

Let

U

Mcp1 o y * o A o y # » M cp2

be an open set in

cp1 , cp2 e C“ (U) .

a C°° diffeomorphism and

0 Zr( R n)

so is

or (B)(x, 5) where at

J(x)

is

B = M

Rn,

0

cp1

o A o ijr o M

i|r: U -+ R n Then if

cp2

A€

and

= cp, (x)or (A) (y(x),J _1 ( x ) O t 2 (x) the adjoint of thedifferential of \|r

x.

We now proceed to the proof of the above theorems.

and

are in

into itself. THEOREM 7.

If

maps

248

§2

PALAIS AND SEELEY LEMMA 1 . For each real number ( 1

+ || £| | 2 ) s ( 1

2 IS I(1 + |U-1 ||2) I S I


• R < const.

y ||e + m£o r l p ' *| R/m

In this last expression, the integral for J |q/7^ (S) |d|.

The integral for

< const.

y

||| + m50 II

> 1 is dominated by

R/m < ||I + m50 1| < 1

is

h | r |p| •|cp^r^ (ti - m50) |dn

R/m < ||nil m-1

and evaluating the

2

XVI: CONSTRUCTION OF Intk

25

R/m T

< const, const. p (the first factor is

"R

l°g(^)

a bound independent of

m.

I^ ^

- (— )

P

(1 + (m-1) )

|p| = n). Now we have

Choosing -*■ t

— lr

k > i(|p| - n)

pointwise and

gives

l^m (x ) I
IK II

( 1 + I U - U I ) m + 1 _ q (i + l | x - 8 l l) " p de

,

§2

XVI: CONSTRUCTION OF Intk

since

(1 + IUI|2)1^2 ~ (1 + Il£ll)-

other

kg.

In

k2

take

||1 1|< ||£ || >

Call the expression

m + 1 - q = -p

separately over the two half spaces get

253

where

p > n.

||5-t|| < ||t-t;||

k2(x, 5) < Const. (1+ ||^ ||)s+r_m_1 (l+i||T-^ ||)

and ’tiie

Then by integrating

| U - t | | > Wt-^W

and

we

since in each half space

one of the factors in

5*

(u||s-?ll)m+1“q (n||T-i||rpa&

is


r.

Zj a^ €

( R n),

so

a^ €

Choose patch function

0j

so that for

j < 0 |(i+ ||x||S)“J D“aj(x, ?)0j(?)| < 2J’ for

|ck| < |j|.

This is possible since

—

zero uniformly as

||£|| -►

because

(i + ||x||2)-^ D“a.(x, (;)

X J

a^(x, £)

converges to

is homogeneous of degree

j < 0. Let ly on R n x Rn

a(x, O

= Zj aj(x, O ej(£)»

The series converges uniform-

and has the properties: (i)

(l + ||x|| )

R n x Rn (ii)

Dx a(x, 0(1 + IUII) for each

k, ot.

(l+ ||x||2)k D“ [a(x, O is bounded on

Rn x

is bounded on

sm +1

- Z^=_m aj(x, C)e j(£) ](1 + |U I Rn

for each

k, a.

25b

§2

PALAIS AND SEELEY From (i) it follows that if for

(Af) (x) = (2it)-n / into itself;

a(x, £)f(£)d£

f € S( R n)

then

A

maps

we define

S( R n)

linearly

from (ii) and Lemma 2 it follows that r A-

J

A(®j, a,) e 0P_m _,( R n)

j= -m

Thus

A e CZ(

Rn)

and

r a(A) -

J

a>5

.

j=-o°

q.e.d.

For the proof of Theorems 5 and 6 we will need LEMMA 3.

If

a € Smblr ( R n );

e

mates of Theorem 1 apply when constant 1.

e

Moreover if

PROOF.

and

A( 1, a) e

because if

r > 0

then

2

( 3 x 7 ) )P a^x ’ ^ j=1

M

and hence

We need only invoke Lemma 2;

IK1

6 - 1

is replaced by the

or (A(i, a)) = (-i)ra .

n

and because

then the esti­

is any patch function

A(0, a) - A( 1, a) € OP.*/ R n ) C Z ^ R n)

r > 0,

< c°nst-(i+n^n2)r//2

>

J

vanishes outside a compact set 2\ 3 ( a r ) 2)P a(x> e>[0(x) = (2*)~n y y ei(x_y » O a(x > Oe(5)q>2(7)f(y)d.yd?

y = T|f~1(y),

a(\|r(x),O

x = t|r”1(x).

= a ’(x, J(x)0,

(2)

Replace

J(x)(; = - A’

=

I,and

0 (0

by

0(J(x)O

v(x) = |det J(x)|.

and set Then

S e 0 ? _ J R n)

where (3)

(A'f)(x) = (2n)-n cp1(x)v(x)-1 j* fj ei(i(x) -i(y) ,J(x)

l)a i(x> |)e(s) («p2fv) (y)dyd|.

256

PALAIS AND SEELEY We use the expansion 1

k ez

=

J

ztj! + zk+1 J* d - t ) k ezt dt/k!

J=0

,

o

and III \ff(y) - V(x) = d^x (y-x) + ^ |a|=2 or

(x) (y-x)°7aj + Rm (x, y)

m dt"1(t(x)-*(y)) = x-y + Y

Qa (x)(x-y)“ + R^(x, y)

l«|-2 where

d^x

is the differential of

of unity we can suppose that where (*0

\|r at

By multiplying by a partition

cp1(x)cp2(y) / 0

we have

Hd^x (y~x) + tR 1(x , y) || > C||y—x||

where

Set

0 < t < i

R ^ x , y) = 0( ||x-y||2) . Note that gi( t(x) -i|r(y),J(x) _11)

of

x.

z = i(R*(x, y), l) f

to

m

=

ei(x-y,S)

in the above expansion of

terms and of

ez

to

k

ez . Using the expansion

terms, and neglecting all remainders

we get (5)

(Am>kf)(x) = k

m

.j

J (i \ Qa (x)(y-X)a , | ) 7 J ! j=0 |a|=2

(2 , ) A l(x)v(x)-1 | J

a'(x, I) e(t) (^(y)f(y)dydt

|p| > i.

We shall show the follow­

ing two facts. LEMMA k.

Given

s > 0,

y

k

and

m

can be chosen

arbitrarily large so that (6) and (7) have the form (8)

K(x, y)f(y)dy

where the support of

R n x Rn order

and

K

K

,

is a compact subset of

has continuous derivatives of

< ts.

LEMMA 5.

Let

T:

S( Rn) —

S( R n)

be a linear

map and suppose there exists an even integer and a

K

in

C^s( Rn x

j

2s > 0

Rn), with compact support

so that (Tf)(x) = y K(x, y)f(y)dy

§2

PALAIS AND SEELEY

258 Then if

j > -2 s

HJ ( Rn)

into

and

t < 2s,

T

extends to a continuous linear map of

H t R n) . A*-A^ J^ _ 1

Granted this, Theorem 7 is concluded by showing that has order

< r -£,

and hence that the expansion for A^ J^ _ 1 .

obtained from that of that for each f

in

j

||A'f - A^

R n ) . Let

s

To check the order of f*ll^ _r+J£ < cj||f||j,

be an integer with

HJ*

continuously into

H^_r+^.

2&

teger

Take

k

is

we show

||f||k = the norm of

2s > max(|j|,

||A1 f

r-£+i

A'- A^ J^_1,

|j-r+4|).

so that

Since Am k - A^ J^ _ 1

we obtain by the triangle inequality that PROOF of Lemma k.

to order

where

m > H and k > £ - 1

by Lemmas ^ and 5, we may choose maps

A’

Then

A 1- A^ k

has order

< r-A,

- A^ J^ _ 1 f||j_r+Jp
2 4 > r

+ k+

i + i 4-s + n 2

Since (6)

a

. e ^ z'^

= - ||z||2 e ^ z,T1^

(where

At = Z? , —

)

we can write

as a sum of terms of the form

-2s and

is continuous from

HJ*(Rn )

to

H ^ R 11)

t ± (p)q>j(p)a(v, p) • Smblr (M)

into

o e Smblp (M)

into

0Pp (M) ,

Xp (a)

such that support qK p

with image in

Definitions 6 and 7 of §2.

and

ffr(Aij(°)) (v > It follows that

into

= Z^. A^.(a), support cp^ ^ 0 ,

Then by

Ps a continuous lin­ € CZp (M)

and

= cPi (p)cpj(p)cy(v, p) where the sum is over all pairs i, j satisfies the requirements ofthe

theorem. q.e.d. COROLLARY.

The sequence

0 - CZr _1 (M) - CZr (M) is exact.

Smblr (M) - 0

PALAIS AND SEELEY

266

THEOREM 6.

Let

M 1 at

M2

§3

be compact Riemannian

manifolds without boundary and let There is a sequence

(By] in

K € 0PQ (M1 x M 2)

such that

in

and

0Pp (M1 x Mg)

PROOF. is a chart for

If

M1

A

r > 0.

and a

converges to A

o (A) (v1, x 1)

0 I- K

( x 1, x 2) )

on the unit

M 1 x Mg.

is of the form T € CZr ( R n)

and

By

or (By)((v1, v 2),

converges uniformly to sphere bundle of

A e CZr (M1)

CZr (M1 x M 2)

A(T, tjT1)

where \|r!

n

=

.

and

X

Let

e C°°(ti*)

g

T = and

with respect to the Her­

a trivial computation shows that

h € C°°(M) = C°°( C^)

then

y (f(x), h(x)g(x))T)dn(x)

I

(x*)^ =

C°°(|*).

X e

|. Then

f e C°°( t})

namely if

and similarly

§5

(f(x), g(x))^h(x)d^(x)

=

J

g(f(x) )h(x)d|i(x)

=

< g*(f) , h > r

>

M

Then

Tt =

= M xStg#

and the theorem is now an easy consequence of Theorem 3 above and Theo­ rem t of § 3 . q.e.d. THEOREM 7.

Let

M1

and

M2

manifolds without boundary, over

M 1,

CZr (£,

£a vector bundle

and

where

£

i)

and

over

vector bundles

M2

and let

r]), r > 0.Then there is a sequence

CZr (| 0 5, t] 0 O that

be compact Riemannian

B^

K e 0PQ (| 0 5, rj 0 £)

T 0 I - K

converges to

ar (^v)

in

in such

0Pp (| 0 5, n ® 5)

converges uniformly on the unit sphere

bundle

of

PROOF.

ByTheorem

A e CZr (M),

and a

{B^}

Te

M1x M2

g € C ° ° ( ti)

to

crp (A) 0 aQ (I^). M °A ° X* § X e C°°(| ), and the theorem then fol­

3 above we can assume and

T=

lows easily from Theorem 6 of §3. q.e.d.

§5- Definition and properties of Again

M

Int (£, ti) .

is a compact manifold without boundary and

are vector bundles over

M.

rj, 5

§5

271

XVI: CONSTRUCTION OF Intk THEOREM 1.

For each integer

czr.i(5’ n) = czr u , PROOF.

n) n OPr_,(s, n)

•

As noted in the remark following Definition 6 of §2,

CZr ( R n) =CZC R n) n 0Pp ( R n) CZr ( R n) n0Pr _1 ( R n) . CZp (M)

r

and Theorem

CZr>_1 ( R n) =

from which it follows that

It is then immediate from the definition of

1 of §3 that

CZr _1 (M) = CZp (M) n 0Pp _1 (M) ,

theorem follows directly from this and the definition of DEFINITION 1.

= CZp (|, t f

T e Int (|, ti)

A e CZp (|, r\)

and

0Pr _l(£> n)

If

then A

T = A + S

by

T = A ’ + S'

- A ’ = S' - S.

where

and define

ap (T) = ap (A) . where A ’e CZp (£, q) Now

and

A - A f e CZp (£, t f

S’ e

and

S' - S

^ > hence

€ 0Pr_i

A - A ’ € czr ( i , By

write

+ 0Pr_1(|, t])

S e OPp _1(l, ti)

cjp (T) e Smblr (|, t f Remark.

CZp (|, ti) .

We define

Intr (|, t f Given

and the

Theorem b

of §t

tf

n OPr _1(i, t f

= CJZ^d,

o (A) = a (Af), proving that

ap

tf

is well defined.

It is clearly linear. Since

CZp (£, ti) C_OPp (£, t])

0Pr _l(£> n) £ opr ( ^ perties of

n)

it is clear that

(Theorem b of §t) and Intp (£, ti) £ 0 P p (|, t f . Pro­

(S1), (S2) and (S3) of Chapter XI are also immediate consequences

Theorem b

Chapter XI

of §t and Definition 1 above. Properties (St) and (S5) of

follow from Theorems 6 and 5 of §t respectively.

Finally,

Property (S6) of Chapter XIV, §t follows directly from Theorem 7 of §t. This completes the construction and verification of the properties of the Seeley Algebra.

2 72

§6

PALAIS AND SEELEY

§6.

An element of IntQ (S1)

with analytical index -1

Thissection considers the situation in which analytical indices for

singularintegral operators were first computed, namely on the mani­

fold

M = S1 ,

plane. ing

realized as the unit circle

Here we have standard coordinate systems

eix

x,

with

in'the complex

X : z -► -i log z

and thus identify the cotangent bundle

infinite cylinder vdx^,

{|z| = h)

S1 x R 1

by letting

T*(S1)

send­

with the

(z, v) e S1 x R 1 correspond to

a standard coordinate system.

X

Let

oo”= (z11}.a,00

the usual orthogonal basis of

L2(S1). For M

f e C^CS1)

and

o < r < 1,

set

P^ffz) = -— y- / &£ p 1 5-rz

= 1.

THEOREM 1. Ppf

Let

f € C°°(S1).

P e C Z ^ S 1)

when

=0

v > 0,

n > o,

PROOF.

when

Pcpn = o

if

^^ we

and for

joint supports. K(z,

O

^

when

= cpn

n 1 -

Choose

( |x| > Jt/2),

r -► 1-.

P

\|r in

cp = 0

^

on

i|r with dis­

= / K(z,Of(£)d£

-oo . in standard coordi­ C°°(R1)

with

{ |x| > it] .

dy

00

J-»

For

= 0.

cp and

f(Odt

M^PM^ has

pPf (z) =

z) = rn zn -► zn =

M^PM^fCz) = ^U- /

€ C°°(S1 x S 1),

30

converges uniformly as

Finally, consider the representation of

on

Pcpn

lr z l
0

and

v < 0. Further

For the first,

Pcpn with

r — 1-,

converges uniformly to a limit, denoted

The operator

if

Thenas

+(x)g(y)^(y)

Tfcg). i-re '

dy

,

\|r = 1

Then

§6

XVI: CONSTRUCTION OF Intfc

since

cp(x-y)

= 1

be defined b y

when

Tg(|)

+(x)\|r(y) £ 0.

Let

273

K^x)

= g7

^^"X |x >

811(1

T

We can conclude the proof of

= i/2( — I— + i)g(g).

III the theorem b y showing that the map

g *-►

lim

K_*g

differs from

r- > 1-

g *-► Tg

b y an operator of order

v < >

- w

-oo .

I -n< :1I1X

I

—00 00

I

■ h

Now

dX

0 v d - n T r 11

0 converges boundedly to 00 K(g)

= U.

by h(cz,

C) and we c) =

(z,c) .

then we have e(cz, c) = e(cz, ^

) = (z, cz)

and id^) (z)h(cz, c) = i d ^ ( z ) ( z , c) = (z, cz) i.e.,

we have commutativity in the diagram (H|D0)|S1

h S

hi!

► C D lo1 0

1

identity

C D Is1 -------- . C IS1 o id^j o

It follows (formally, by [II, §3, (v) and (ix) of Lemma 1]) that d(H|Dn , C n , elS1) = d( C n , u o

o

, idM x) 1 '

811(1 hence

d( C D^, C D , id(l)) = -g = 5 (-M-)

de Now

280

PALAIS AM) SEELEY

Recalling our identifications

U(1) = S1

§7

and

= DQ

we get the follow­

ing result by comparing with Definition 1. THEOREM 3.

Identify

U( 1)

with

\i

in the standard fashion and let of

H 1 (S t Q)

S1 = (z e C|

M

= 1)

be the generator

corresponding to its standard orienta­

tion (i.e., as the boundary of

DQ = {z e C|

|z| < 1},

the latter having its orientation as a complex mani­ fold) . Then °h( i) = -[I Remark. vay.

The idea of the above proof was suggested by R. Solo-

Note that we have nowhere used the explicit definition of the dif­

ference construction, just its functorial properties given in [II, §3, Lemma 1 ]. THEOREM k.

Let

A € IntQ (S1)

Theorem 2 of §6. of

A

be an operator as in

Then the topological index,

it(A) ,

is minus one.

PROOF.

Since

S1

is parallelizable

3(B(S1)) = 1

so by

Theorems 2 and 3 above it(A) =- Q = < f, L*g >Q = < f+ , L*g|X+ >Q + < f_, L*g|X_ >Q

/M (G(f+ |M), g|M)^ - /M (G(f_|M), g|M)^ = 0,

since

Lf+ = 0

and

= f+ |M =

f_|M.

q.e.d.

283

PALAIS AM) SEELEY Now let

X

by

p

be the metric on

t(x) = p(x, M)

if

x € X ,

X, and define a function

t(x) = -p(x, M)

is a coordinate in a tubular neighborhood suppose that

eQ = 1,

the unit interval Rg •

and that

(-1, 1).

£|{|t| < 1)

Then for

C°°(U( |t| < 1}) - C°°(UM)

of

M.

on

Then

t

We may

t, |M

is the product of

-1 < e < 1

given by

x e X_.

if

{|t| < eQ}

t

and

there is a map

(R£f)(m) = f(m, e)

for

m

in

M.

By [X, H , Theorem 7], R£ is continuous from Hs(f;) to H s _1/2((;|M) / * for s > 1/2. For 6 = 0 we need also the adjoint RQ defined by

< f, RQ*g >Q = JM (f |M, g)^ a continuous map of Define

\

Since

(L)

R£L

HS(^|M)

and

L

R q LQg

P+

then as

LQ

differential operator

P+g,

with the symbol de­

-P_.

If

then also

converges uniformly as

is bijective.

e - o+

The same holds as

isreplaced by

extends to

(LQf)(x) = \ ( v , x)f(x).

by

converges uniformly to a limit

scribed in Theorem 2.

R Q*

s < 0.

is elliptic, it follows that

P+ e CZq ((;|M, £ |M)

when

By duality,

H s-1^2(^) for

If g e C°°(£ |M) ,

*

where

into

g.

LQ : C°°(UM) -►C00(£|M)

LEMMA B. -1

for smooth

e

o+

D

e -► o-

is any

R£DL

-1

*

RQ LQg

or e -*• o-.

The proof of Lemma B is postponed for

a

few paragraphs. It is

the essential part of Theorem 2. LEMMA C.

If

P

and

P_

constructed in Lemma B, PROOF. any

q>

e C°°(£),

Let

are theoperators

then

f € C°°(?|M),

P+ + P_ = identity.

and set

/M (LQf, 0

< Lu, cp >Q = < u, L*cp >Q = lim

/|t | ^ £

u| ( 111 > e)

Lu = R0*LQf

in

is

{ 111 > e} .

u=

C00,

and since

L'1Ro*L0f.

= < R0*L0f, ^ >o =

(u, L*cp) . From Lemma B, has support in

Thus from Theorem 1

y (L0f, 0+

M -e

Then

(L_g (u|M_e ), cp |M_e )

M,

Lu = 0

for

289

XVII: COBORDISM INVARIANCE OP THE ANALYTICAL INDEX where

M£ = {t = e}

and

Lg

is the appropriate analog of

everything

in sight is

to obtain

/M (LQf, q>|M) = /M (LQ (P+f + P_f) ,

LQ .

Since

C°° inthe right places, we may pass to thelimit cp|M) ,

since

P+ =

-1 * 00 I I + lim R L Rn . But every C section of £ |M has the form cp|M e- > 0+ 6 u cp in t;, so LQf = LQ (P+f + P_f) . Finally,LQ is bijective, so

for a f =

P +f + P f. q.e.d. PROOF of Theorem 2.

as in Lemma B. In view of + Lemmas A and C, it suffices to show that the range of P+ lies in H+, for then ment of

f = P+f + P_f H+

Define

is the unique decomposition of

plus an element of

C“ (5|M),

then

supported

in

Similarly

P_f € H_.

P

H_.

But

f

Lemma B shows that if f €

u|X+ = (L_1R0*L0f)|X+ e C” (C|X+). Since

M,

into an ele­

Lu = 0 in the interior of

X +, and

Lu= R0*LQf

is

P+f = u|M e H+ .

q.e.d. The proof of Lemma B uses LEMMA D.

L~1 e CZ_1(5, £). T e CZ_1(5,

PROOF. Choose Then

oQ (TL) =

s°

TL= I - K

(I + K + ... + K ^ T L = I - K®4"1

5) with

with

a_1(T) = X-1 =o ^ L ) -1.

Ke C

on the right by

Z L-1

£)•

Multiplying

yields

L“1 = T + KT + ... + K®T + K ^ ’l -1 Now

K^T e CZ_J_1(^,

Thus

L -1

CZ_1 ,

O,

and

since L'1 e

0P_, .

can be approximated within arbitrarily low order by members of

and the lemma follows easily. PROOF of LemmaB.

when

kP+1L _1 e 0P_m _2U , O

cp1

and

cp2

It

suffices to consider

M

L~1M R *L„g cp1 cp2 0 Oto have disjoint support, or are supported in an appro-

continuous from

for all s < 0, and a fortiori for -1 * L R Q LQg converge uniformly to a

all limit

s

H S(M)

whatsoever, Hg, and

H

is of order cp1

M

R^

is

q>2 0

H°°(X)

Thus

Suppose now with coordinates

to

T-,

L

*

In the first case,

9

M

-1

priate coordinate neighborhood.

and

-oo. cp2

have support in a neighborhood

U -► Rn+1 , and such that

£ |U

is trivial.

If

U C X U

290

PALAIS AND SEELEY

does

*

not intersect M,

RQ LQg = 0 and there is nothing

\|r= (Y-,,

we suppose that the support of

..., Yn ,^n+-j)with

|cp1 | + |cp2 |,

support

tn+1 =

(cp) C U.

to prove; so

t. Let

cp = 1 on

Then according to Lemma

D , the operator

e CZ -I.

B = y»McpL"1M cpf It is crucial to know Let

ff(B)(y, t; r\,

= a1 ti

+ aQ ,

certain details about

t )

=

where a^

Z J^


|r(support( |cp1 | + |cp2 |))

the

7

b^

(y, t ;

^ = (t^,

is "dual to"

t j,t )

...,

t. Because

as matrices of functions.

fwith support (f)C support ( |cp1 |

M L-1M M LM g = g, it cp cp cp cp

\|r*)

+ |cp2 l)

we have

7 Theoreirf 3l that on

satisfy

b _ 1a 1 = identity matrix (1)

b _ 1aQ + b _ 2a,

- ±(Sb_.,/St ) (Sa1/St)

- i E (Sb_1/Sijj) (Sa 1/Syj)

6

Let

A( 0 , b^)

< k,

^[Mp

as in [XVI, §2, Definition 3l.

L”1

Xj£ ^ £

-

JfcQ < -1 - k - (n+l)/2 tinuity of

B^l has

etc.

^n

R D[M L e 9-|

—

1 M

0

92

order < i Q + k,

/ a = 1. weakly in

Set

D

has order

so that for

B c ]Rn*Ln

remarked above show that

is continuous from

-K* - E* N B B 0 ]Rn L g 2* o

a € C°°(R1) ,

a(s)

am (s) = ma(ms) .

HS(X) for each

I (g(y), q>(y, 0))dy for

cp

H “1^2(^|M)

converges uniformly as

and it remains to prove a like result for Let

Then if

Soboleff's theorem [X, H , Theorem ij-] and the con­

R0*: H_1/2(UM) — H"1( O

L"1M Cp^ Y2

Hence

0

be a patch function, and set

with

D[M

=

s < in H“s .

** = ^M>2g and

C°(0-

s -* o,

* R£DB^R0 LQg.

vanish except for

Then as

to

-1 < s < -i,

m -► + 0 }.

finite path in

equations ( 1 ) for the

(ti, t )

for

We obtain an integral in which

lim may be taken m- > 00 with an integral over a

oo / dT

under the integral sign by replacing

of

0 = 0

To see that this is possible, consider the

b^.

There

a ^ y , tj n, x)

11 12 + |x |2 > 0,

which is non-singular for

because of the ellipticity of

L.

is a linear function ^

Thus there is a compact set

support(\|r( |cp1 | + | 0 } Extend Since

0

to

Rn x C / 1^

a(x) =

{Im(x) > 0 }

,

and

so that e"itT e

0(t],

x

a(t)dt

)

is bounded for

J , >. dx.

still for

h | 2 + |x|2 >_ 1 .

decays exponentially as

integrand in (2 ) is holomorphic between may be replaced by

= 1

Then

t > 0 r(^)

lim m- > 00

and

and

|x|-* «> in

Im(x) > 0,

(Im(x) = 0 },

and the /“^ dx

may pass inside the integrals,

yielding for (2 ) the expression (3)

(2n)_n'1tp1#

y

y

Rn since

a(x/m)

e ^ £eitTdT ei < y ’ n> dr)

converges boundedly to

immediate that the expression (3) tives in

y

and

t

a(o) = f a = 1 as

isC°° for

extend continuously to

the uniform convergence of

,

r( ti)

RgDB^R0*g.

t > 0, {t

Further

m

00.

and all its

It is deriva­

> 0 }. This establishes

292

PALAIS AND SEELEY lim 6-

for if

|n | > 1

and

>

€ CZ,

H B R 0+

U

R |n I > 1

(UM)

;

+'

then the function

b

T)bj^(y, 0; n, t) dx r(n)

is homogeneous of degree

£+1

r\ .

in

(Note that the homogeneity of

is preserved in its holomorphic extension, and for the given = 1

and

is the boundary of

r(r\)

r\

b^

6(n, t)

{|t| < R |t]|} n (Im(T) > o}.)

It remains only to compute

o (lim

*

R B.R

u 6 — > 0+ 6

a (P ) = a (lim R B -R *Ln) . We have for u + u e - > o+ u u

),

and hence

u

x in M

(y, 0) = y ( x )

and

that

( k)

ReB -iRo*s)(x)

o+

t)-n J

( 'l'*.(v, x)- 1 \(v, x)

generalized eigenspaces of

,

with positive imaginary

part along the direct sum of the remaining generalized eigenspaces. Since

k(v, x ) = -l(v, x), by what was remarked above

a (P )(v, x) = i(I + ik(v, x)X(v, x)). B±(x)

=

i(I + ix(v, x)).

We also have

Keeping in mind that -1

anticommute and have square

wehave + aQ (B”)(v, x) =

X(v, x)

and

x(v, x)

a straightforward computation gives

aQ (B+P+)(v, x) = i(I -

ix.(v, x) - iX(v, x)\(v, x)

+ \ ( v , x))

crQ (B~P_) (v, x) = v(I +

il(v, x) + ix(v, x)X(v, x)

+ X(v, x)) .

So if we define

C e CZQ (5|M, U M )

by

cr0 (C)(v, x) = 2

Sincex(v, x)

= -I

and

x(v, x)

C = B+P+ + B ”P_ then

i ( I +\ ( v , x)) *

= -X(v, x)

aQ (C)*(v, x) = i d

- ^(v, x))

and a0 (C)“1 (v, x) = (I - \ ( v , x))

so in particular skew-adjoint so by

C

is elliptic.

Since index(C ) = 2 index (C) the other hand ker(C) = 0 . B+ and B ”

1

2

a0 (C )(v, x) = 2 l(v, x)

XI, Corollary of Theorem 9 ,

2

Since

Now

it follows that

Indeed suppose

C C

0 = Cf

2

has index zero. has index zero. =B+P+f

+ B -P_f.

are orthogonal projections itfollows that B+P+f = B “P_f

=

0.

is

On

296

PALAIS AND SEELEY

Let

P+f = u|M,

u e

Then by the G-reen-Stokes formula (1) with

u =

we have (since Lu = o)

=

so

B P +f shows that

= 0,

(B P+f, P+f)

=

| ||B"P+f||2

hence

P_f = 0

P f+ = B P f+ + B P f+ = 0. A similar argument hence f = P f + P_f = 0. Since C has kernel

and index zero it is bijective and, by an argument similar to that of Lemma D,

C“1 e CZ (£|M, {;|M),

and of

course

a (C"1)(v, x) = a (C)"1(v, x) = (I - X(v, x)) Now define

T e CZ (|+ , g )

by T = B P C"1i

where

i+ e Diff0 (|+, 5 |M)

is the inclusion of

C°°( |+)

into

C°°(t;|M).

Then aQ (B_P+)(v,

x) = i(I

+ i d v,x))(I - ix(v, x)x(v,x))

= i(I

+ idv, x) - idv , x)dv, x)

= i(I

+ i d v ,x))(I - d v , x))

- d v , x))

so a0 (B"P+C_1) (v, x) = i(I + idv, x))(I - X(v, x))2 = -i(I + i d v , x))d v , x) = -idv, x) (I - i d v , x)) Now on

g*,

d v , x)

is multiplication by

i,

hence

I - idv, x)

is

multiplication by two, so a0 (T)(v, x) = a0 (B“P+C"1i+) (v, x) Thus

a

(T) agrees

with

-a

T.

-dv, x) |g*

analytical index of

M, so

T

is

a is equal

to

Thus to complete the proof that

alytical index zero it will suffice to show that bijective.

= -a(v, x)

on the unit sphere bundle of

elliptic and by [VII, Theorem 3 ] the the analytical index of

=

T:

a

has an­

C°°(|+) -► C°°( g~) is

297

XVII: COBORDISM INVARIANCE OF THE ANALYTICAL INDEX Define a linear map B+g

into B ”g.

B+g

= 0 then

we take

To seethat B “g = 0.

u = v

0

in

=

U

U

as follows:

with

J (B+g - B _g, g)

is injective.

H°

=

norms.

=

B “g

&

T

B+ ;

from [XI, Theorem 73 that

B"P+ h

h € C°°(i“).

= 0

;

(ii)


Q

= 0

;

(iii)


Q

The first and second becauseB+h = 0

P_h

=

is surjective.

B"P+h

Then

Then


Q = < P+h, B+h >Q is in the range of T,

Then

=

< (B+ + B')(P+ + P")h, h >Q

=

< B “P_h - B+P_h, h >Q

€ H_, say P_h = v|M

with

v € H__.

=

< B"P_h, h >Q

Since

Lv = 0 if

we take

in the Green-Stokes Formula (1) we get 0 = < B"P_h - B+P_h, h >Q

=

||h||2

so

h = 0

which completes the proof that the analytical index of

cr is zero.

298

PALAIS AM) SEELEY To prove the final statement of the Theorem we note that by

[X, H , Theorem 5 ]

Z = £ §> n

Let

t)

is the restriction to

and define

M

A e Smbl^CT, T)

of a bundle by

^

A = A 0 1^.

over

X.

Clearly

X is skew-adjoint and satisfies x ( v , x)2 = -1 if ||v|| = 1. Moreover, ^jb ^ ^ ^jb if we put Tx = (e £ r x l^( v, x)e = +ie) for x e M then clearly T” = 0

£(v, x) IT*

Moreover it isclear that

a

that the analytical index of analytical index of

= (v,x) . The

proof

is zero now implies the same for the

o’.

q.e.d.

We now show how Theorem 3 leads to the main goal of this chap­ ter; namely the proof of part c) of Theorem k of [XV, § 5 ] for the analy­ tical index. mension

We assume that

2m+1,

and we let

M

X

has dimension

be the double of

n = 2m, Y.

so

Y

has di­

As usual we put

2m+1 A*(T*(X) ®

C)

=y

Ap (T*(X) ® C)

,

p^O m v e n (T * (X) ® C)

A 2 p (T*(X) ® C ) ,

= y P=°

and

m A°dd

(T*(X) ® C)

A 2 p + 1 (T*(X) ® C )

= ^

P=° We recall certain linear maps. terior multiplication nap

e -► v A e

(recall the identification of into

aP+1

First for each

T (X)x

and hence interchanges

is the interior multiplication by

V for

odd..

that

Aeven

(vi> •••> vP -i}

Since iv

and

=

y

e

maps

hP into

A°dd. If we define uy :

uve = v A e A

with

v,

- iye, then

a ]_ s 0

ct1(s )(v

uv

iv ,

T*(X)x ), and

into itself

which carries

Aodd.

Secondly, there

identified by

A^3"1

iv

is the adjoint of

it likewise interchanges

A*(T(X) 0 C)x — A*(T(X) 0 C)x

is skew adjoint and interchanges

and so

Ap

e(v> v 1' **•' VP-1}

recall from [IV, §6c] that

, x)e = -ive

we have the ex­

A*(T*(X) 0 C)x

0 e A^(T*(X) 0 C)x . We recall [IV, §6c] that

e »-* v A e. Aeven

of

v e T(X)

a1(d)(v, x)e

a1(d+5)(v, x) = u^..

=

Aeven

v A e,

Also

by and

hence

o 2(a )(v

, x)

299

XVII: COBORDISM INVARIANCE OF THE ANALYTICAL INDEX = cr2((d+e>) 2) (v, x) = u2 Since

v -► uy

which we computed to be multiplication by

— 1|v||2 .

is linear the argument at the beginning of the proof of

Theorem 3 shows that uv, % and in particular if

v-

+ UV2 uv,

vp

and

■

- 2 ( V 1 ’ V2>

are orthogonal then u

and u

V1

v2

antieommute .

We also have a bundle automorphism mapping

AP

into

A2m+1~p

basis element (the volume element) for ®x ,

and defined by

cdx = v., a

any oriented orthonormal basis the uniqueelement of f e AP ,

where

that in

T*(X)x ®

A2m+1'"p

(f, e) C

for

T(X)x

then

and

e

*x .

where

for

Then if

T (X)x .

and

Ae v e n

There is a canonical

... a v 2m+1,

denoted by

v 1 ,..., v2m+1 e€ Ap ,

is

*^-e

is

? o r aH

f A *-^e = (f,

hermitian structure for Ap

induced by

is the complex conjugate

of

v 1, ..., V2m+1

*x (v1 a ... a v_^) =

since the elements of

*

A (T (X) ® C)

A2m+1 (T (X) ® C)x ,

such that

is the

cular it follows that if

*

of

and hence again interchanging

reca3_^ the explicit definition of

Aodd ^

*x

e.

Inparti­

as an orien1:ed- orthonormal base ... a v 2m+1 .

a

In particular

of the form v. A ... A v with * * Jp j-1 < ... < jp form an orthonormal basis for A (T (X) ® (D x it follows •0 2 that is a unitary map. It also follows that on A-^ we have *x = (_i )P(2m+1 “P) _ i

A*(T*(X) C d

i.e., that

*x

is the identity.

cause interior and exterior multiplication by have

forany

f

v

If

e € Ap

then be­

are adjoint maps we

e AP “1

f a va *^e =

(f A v,e)cox

=(-1) p _ 1 (v

A f, e)^x

= (-i)p_1(f, ive)»x = (-1 )p_1f A *x d ve) and hence

v A *^e = (-i)p_1 *x (ive)Define a bundle automorphism

^(p-lj-m+l ^

on

Ap^

Then clearly

and is a unitary map satisfying i.e.,

a^. i s

v a *xe

oP

A*(T*(X) ® C)

interchanges

A0^

2

a^. = -1 . It follows that

s k e w - a d j o i n t . Also if

e e Ap

= (-T)p-1 *x (iye) = -(i)"2p *x U ve)

by and

*

= Aeven _*| = -c^,

then multiplying the equation by

iP(P“1)“m+1

we get

300

PALAIS AND SEELEY

v A a^e

=

a^i^e.

Replacing

e

by

a^e

a^.(v a e) = L^a^e.

sides of the latter equation gives

aXv Ae ” ^ v 6 = ive = ^v^X6 ” v A aXe = ^ v ^ X 6 * and

oc^

and applying

to both

Hence

axuve =

other> words

anti-oommute .

We now define the bundle Recalling that

both

and uy

(;

interchange

and in fact

considered as an element of

\

X

by

Aeven

f; = Aev0n(T*(X) 0 C ) . anaa °

\ e Smbl^((;, £)

lows that we can define an element x(v, x)e = U-Va x e ,

over

j_t

fol­

by the formula (d+5 )Q^

is the symbol of the operator

Diff1(5, 5)•

We note that

*(v, x)2 =v w x Also since

u^

and

are skew adjoint we have

\ ( v , x)* = a* u* Thus the hypotheses of Theorem

i+

and

symbol

over

M

by

a

ax uv

=

by

a(v, x) = \(v, x)|

A*(T*(X) 0 C)|M.

T*(M)

by

I = A*(T*(M) 0 C ) .

as asub-bundle

of

T*(X)|M

e

We note that if v A e

on

£,

and their difference

the corresponding maps on ment of

£

v e T(M)

then the exterior multipli­

its adjoint the interior multiplication u^,

are just the restrictions to

A*(T*(X) 0

C)x

when we regard

v

£x

of

as an ele­

T(X) .

fined by

f(e) = e

if

e € Aeven(T*(M) 0

e € Aodd(T*(M) ® C).

fCu^e)

(via the

is a sub-bundle of

There is a canonical isomorphism

if

and the analytical

over M

|

Riemannian structure) it follows that

iv ,

and an elliptic

is zero.

Since we are regarding

map

= -X(v, x)

3 are satisfied and we can define bundles

We now define a bundle

cation map

= -uy

|+ = Ce e £x l^(v, x)e = + ie),

a e Smbl^(£+, |“)

index of

= _uv “x = -||v|!2 •

=

u^u^f ( e)

If

v e T(M)

)P

£ *=* £; |M = £+ ©

and

de­

f(e) = uve = -e A v

then it is easily seen that

by considering the two cases

The bundle automorphism *2 . (_1)P(2m-P) =

C)

f:

of £

and lf we define ^

e

€

Ae v e n

analogous to . iP(P-D-m ^

and

e e Aod d .

*x

satisfies on

301

XVII: COBORDISM INVARIANCE OF THE ANALYTICAL INDEX aP(T*(M) 0 C)

then

satisfying

= 1. x € M

Given basis for

is a unitary and self adjoint automorphism of

T(M)X ,

let

v2, ..., v2m+i

so that if we put

an oriented orthonormal basis for form

e = v. A ... A v. J1 Jp formula f(*Me) = (-l)puv the latter by

with

^e an oriented orthonormal

v 1 = -v

T(X) .

£

then

v 1, ..., v 2m+1

is

Checking on elements of the

1 < j1 < jp < ... < j p e 6 aP(t*(m ) ® C)

we verify the and multiplying

iP^P-1)"111 we get

f(«Me) = I uvaXf ^e' )

=

T ^ v'

or finally faMf 1 In particular since x(v, x)e = + ie,

|

=

T X,(v’ x)

is the set of

e € £x

such that

we have

(1)

r

=

{e € 5 |M|fQ^f_1e

=

+ e}

Now if T(M) .

1 j

If

u axfco,

co e | we have seen that fu co = u u fco for any ve v v v c co = f”1e, where e e £+ , then cc^co = co, so fco = fCcc^cp) = hence since

2

u^ = -1

we get

f^co = -iuva^f(co) = -ia(v, x)f(co),

Thus (2)

fuvf“1e = a(v, x)e Now by Theorem 3,

if

restriction of a bundle over Y, Smbld (£ + 0 n,

r\ is any bundle over then

a 0 I

M

which is the

is an elliptic element of

f: | » £+ © |” + is an isomorphism it follows from (1) and (2) that if we redefine £~ by + £~ = (e e £ |c^e = + e) and redefine a by a(v, x) = uv , then the same result holds.

® ti)

e € |+

with analytical index zero.

Since

To recapitulate we have the following theorem whose proof

was the main goal of this chapter. THEOREM k.

Let

and define

a

i.e.,

e

§* = {e e A*(T*(M) 0 C) |o^e = + e] Smbld (£+ ,

a(v, x) = uy .

| ”)

by

a = a1(d+&)|| + ,

302

PALAIS AND SEELEY If

bundle over

t] is any bundle over

Y, then

M

which is the restriction of a

a ® 1^ e Smbl^(| + ® ti,

symbol with analytical index zero.

®> ti)

is an elliptic

In other words c) of Theorem b of

[XV, §5 ] holds for the analytical index.

CHAPTER XVIII BORDISM GROUPS OF BUNDLES E. E. Floyd

§1 .

Introductory remarks The purpose of this chapter is to prove the uniqueness theorem

for index functions stated in Chapter I.

The proof can he based directly

upon the results of Conner and myself in our Ergebnisse tract [2],

In view

of the fact that torsion does not enter into the proof, it is also possible to give the following complete proof based directly upon Thom's work [^]. We first collect some definitions and elementary remarks, be­ ginning with Thom's L-equivalence classes. X

without boundary and an integer

k

entiable k-dimensional submanifolds L- e q u i v a l e n t X x I

such

with

M , M1

Fix a differentiable n-manifold 0 < k < n.

Two closed differ­

with oriented normal bundle are

if there exists a compact differentiable submanifold that

W intersects

X x 0, X x 1 transversally in

of

MQ x 0,

M 1 x 1 respectively

and such that

normal bundle of

W

can be oriented so as to extend the orientation of the

normal bundle of

MQ x 0, M 1 x 1

Lk (X)

W

W

in

has no other boundary, and if the

X x 0, X x 1

respectively.

the set of L-equivalence classes of closed k-manifolds in

oriented normal bundle. (1)

If

X

X

with

There are the following remarks. is oriented, there is a one-to-onecorrespondence

between orientations for the normal bundle of the tangent bundle of

Denote by

M.

MC X

andorientations

In this case, in the definition of

for

L-equivalence

we suppose the tangent bundle oriented instead of the normal bundle. (2) isotopy of

Mq

orientation and

Given an with

MQ

with oriented normal bundle and a smooth

M 1 C X, then the normal bundle of

MQ and

M1

are L-equivalent. 303

M1

receives an

§1

FLOYD (3)

If

smoothly isotopic to

M1

with

k < dim X/ 2 ,

MQ

and

we can define an abelian operation in Since

L^-(X)

Mj

given

MQ

disjoint.

L^(x )

and

M1

there is an

Hence in this case

^y disjoint union.

is defined only for manifolds, does not always

have a natural operation, etc., it is natural to consider also more function­ al constructs (see Atiyah [1], Conner-Floyd [2]). Let pairs

(M, f)

f: M - * X

X

be a space, and

where

is a map.

M

n

a non-negative integer.

is a closed oriented differentiable n-manifold and

Two pairs

(MQ, ^q) > ^1 >

are b o r d a n t if there

exists a compact oriented differentiable manifold joint union

MQ U -M1

and a map

F: W -► X

There results an equivalence relation. sented by

(M, f),

and by

nn (X)

the set of all For

¥

with boundary the dis­

F|Mq = f , EM, f]

the class repre­

[M, f ].

X = point,

F|M1 = f1 .

Then

ftn (X)

fln (X)

is

is the Thom

consisting of all bordism classes of closed oriented differentiable

n-manifolds.

There are the following straightforward remarks. (t)

defined by

A map

cp: X -► Y

cp*[M, f] = [M, cpf].

cpQ , cp1 : X -► Y

gives a homomorphism Moreover,

are homotopic then

homotopy equivalence, then (5)

If

boundary and if ^^(X)

with

Denote by

an abelian group under disjoint union. group

Consider

X

cp*: ^(X) -► ^^(Y)

)* = T2*cp1*;

cpQ* = cp.,*.

Hence if

if

cp: X -*■ Y

is a

cp*: nn (X) « ^n (Y) .

is an oriented differentiable manifold without

2k + 2 < dim X,

there is the natural homomorphism

L-^(X) -♦

which takes the L-equivalence class represented by the closed oriented

submanifold

M

[M, i] C n^(X),

into

where

i

is the inclusion map.

It

follows from the Whitney embedding theorem that this is an isomorphism Lk (X) ~ nk (X) • (6) Sn . Denote by boundary. plies that

Let N

X

be a finite complex embedded as a subcomplex of

the closed regular neighborhood of

Then ( b )

implies

n^(X) « ft^N-dN),

X

and by

for

dN

its

2k+2 < n, (5) im­

flk (X) « L^(N-^N) . (7)

Given

f: M -*■ X, where

M

is a closed oriented differen­

tiable k-manifold, there are the Pontrjagin numbers of given a cup product p. ... p. e H-^(M) 1r

f

[2],

Namely,

of Pontrjagin classes of the tangent

Mj

§1

XVIII: BORDISM GROUPS OF BUNDLES

bundle of

M

x e H^-(X)

and

with

305

r(i1 + ...+ i ) + q = k,

there is the

integer P1

••• Pn- f*(x) [M] r

1 the value of

,

H^(M) .

... p. f*(x) e H^M) on the orientation class of M 1 r These numbers are functions only of the bordism class [M, f ] e

ft^(X),

and are the Pontrjagin numbers of

x €

p.

Q),

[M, f ].

,

where

G

is a compact Lie group.

can reinterpret this group in the following way. M

We may also let

in which case the Pontrjagin number is a rational number.

Consider now

where

in

is a

Consider pairs

over

M.

Two pairs

(MQ , a ) and

(M1 , a )

there exists a compact oriented differentiable manifold disjoint union

MQ U -M1,

stricts to

on

M^,

and a principal G-bundle

i = 0, 1.

p

(M, a)

a is a prin­

closed oriented differentiable k-manifold and

cipal G-bundle

We

are bordant if

W

with

over

dW

W

the

which re­

This is an equivalence relation; we see

that the set of equivalence classes is in one-to-one correspondence with (M, a)

ft^BG) .

In fact, given

for

Letting the bordism class represented by

of.

there is a classifying map

ft^BG),

we get the one-to-one correspondence.

for

any classifying space for

BG

consider

BG

G

(M, a)

f: M

-*• BG

map into [M, f] €

It is seen that we may use

in dimensions

< k + 1.

Hence we may

a finite complex. In a similar fashion, we can consider bordism of pairs

where

M

is a closed oriented differentiable manifold, and

pairs

(Mq , aQ)

and

(M1, a 1)

differentiable

manifold

b € K(W)

the restriction

i = 0, 1.

with

W

(M, a)

a € K(M). Two

are bordant if there exists a compact oriented

with boundary the K(W)

disjoint unionMQ

K(M^) sending

b

into

a^

U -M1 ,and for

It can be seen that this is an equivalence relation, using, in

proving transitivity, exactness of K(W1 U W 2)

K(W.,) + K(W2) - K(W1 n W 2)

We are thus lead to an abelian group of bordism classes. one-to-one correspondence with homotopy classes of maps points not considered). correspondence with

But

K(M)

is in

M -+ Z x BU

(base

Hence, as above, the bordism group is in one-to-one x BU).

306 §2.

§2

FLOYD The computation of

fl^(X) ® Q

We first recall somewell-known facts abouthomotopy theory. Given

spaces X, Y

classes of maps X

with base point, denote

X

Y

by

[X, Y]

which preserve base point.

is a finite CW-complex and that

Y

the set ofhomotopy

We suppose always that

is a CW-complex whose skeletons are

finite. If

Y

is

stable range), then [3].

(n-1)-connected and

[X, Y]

(i.e., in the

is an abelian group, also finitely generated

In the stable range there is a homomorphism [X, Y]

where

dim X < 2n - 2

Hom[H*(X), H*(Y) ]

H*(X) -►H*(Y),

- Hom[H*(X), H*(Y) ]

,

is the group of degree preserving homomorphisms

which sends the homotopy class of

f

into

f*:

This is an isomorphism modulo the class of torsion groups [3 ].

H*(X) -*■ H*(Y) . It is equally

true that [X, Y] is an

Hom[H*(Y) , H*(X) ]

isomorphism modulo torsion.

Here homology and cohomology are

taken to

be reduced. Let

A

-► BSO(n)

closed n-ball, and let Then

MSO(n) = A/dA

dA C A

be a universal

S0(n) -bundle with fiber the

denote the union of the boundary spheres.

is the universal Thom space; it is (n-1)-connected.

The cohomology is given by the Thom isomorphism i J k;

Also

consisting of all

flk (1 x BU) = «k (BU) ~

choosing a classifying space

BU(n)

in dimensions

be a finite complex, we may apply (9 ) to compute rank the number

of pairs

(j.,, j2, ...) For

( cd,

cd

’ )

fi*(Z x BU)

of partitions

cd

=

(i

[M,

ftk (BU(n)) < N

flk (1 x BU) . i2,

. . . ) ,

a]

cd

1

to It is

=

with t(i1 + i2 + ...) + 2 (j 1 + j2 + ...) = k. a £ K(X),

dimensional component.

let

ch a = Z ch^ a

Writing

symmetric functions we obtain,

Z

where

ch^ a

is the 2q-

as a polynomial in elementary

308

FLOYD

chqa

Denote by bundle. by

ch^ It

chq,

If ch(a 0 b) =

(-I)11'1 _ ! a _

Q)

€ H2c^(BU;

is seen that

q > 0.

integers with

=

H*(BTJ;

is

where

a

V1

is the universal

the polynomialalgebragenerated

t = (v1, v2, ...) i,

.

of non-negative

let

vp

= (ctLj)

• (ch2)

and

b e K(Y)

are virtual dimension one, then

ch a 0 ch b

implies that ^ chpa ® chqb p+q=n

ch0 = 1 . Consider now

generator sion

cq_^)

chT

c h ^ a b) =

where

Q)

for almost all

a e K(X)

+

the class °hqa

Given a sequence

v^ = 0

§3

g

S

n\r ,

of H2k(S2k;

one. InK((S2k)v),

Z)

C H 2k(S2k;Q),

let

ch(Vk)v =

(Vk)v = V k ®

(chk)v (Vk)v = g ® ... ® g

by

with

Vk

withch^V^.

a

ofvirtual dimen­

... V k . Then

(1 + g) ® ... ® (1 + g)

ohk(Vk)v = g ®

Denote

OI/-

e K(S )

and the element

1

v!

® ... ® 1 + . . . + l ® l ® . . . ® l ® g , g ® ... ® g .

gk v - Note also that ch^(Vk)v = 0

if

i < k.

Consider again the sequences t as above. For each t , let o Voh- v-i S = (S ) x ... x (S )where v. = 0 for i > k, and let V € Vlr V1 * K(S_) (V1) 0 ... 0 (V.) . Write elements of H (Sd T be given by V =T I K T * Pi i as linear combinations of elements x 1 0 ... 0 where x^ € H ((S ) ). Then

V1

(ch )

v p

(ch2)

VT = v 1 ! g.,

0 1 0 ... 0 1

V T = v 21 1 0 g2 v

8) 1 0 ... 1 + Z x^ 0 y^ 0 1 0 ... 0 1,

deg x 1 > 0 ,

§3

309

XVIII: BORDISM GROUPS OP BUNDLES Order the partitions

...+ 2kvk .

If

there exists

t

k

x

’ = (v.J, v£, ...) with

v£ = v.^ for

as follows. has

deg t

Let

deg x* = deg x,

i > k

while

x' < t

let

< v^.

+

= 2v1 + if

Computation as

above shows (11)

chT V TT = 0

if

Consider now elements

deg x 1 = deg x [M, a] e

and

fl^(BU),

x’ < x

and consider the x e H (BU; Q) .

characteristic numbers of such elements as in (7) except that

given finite sequences co = (s^ s2, ...), t = (v1 , v 2, ...), let P^ = S1 S2 ^ (P-|) (P2) ... be the appropriate cup product of Pontrjagin classes of M and let POJ°hT [M' “ ] = PtB ' ChT (aH M ] These rational numbers are functions only of the bordism class

[M, oc ].

They are zero unless (is1 + 8s 2 + 12s, + ...) + (2v1 + W 2 + 6v^ + . . . ) = k (3.1)

THEOREM. [M, oc]

classes

Consider

fl*(BU)

where

is a closed oriented differ­

entiable manifold and one.

PROOF. (v1# v 2,...) x ...,

Pontrjagin classes of Pa>' • cV Hence, p^, deg x and

where

H2j'(S2f

V. e K(S2j)

Consider finite sequences Let

ch.V.

J J

a gen-

cd

= (s1 , s2, ...), t =

P^ =

(P2(C )) 1x (P^( C)) 2

[P^ x S , 1 V ] € n^(BU). Note that since the ST

are all zero, we have

[Pm x ST , 1 ® V T ] - P ^ . E P J

• ch^jP^ x ST , 1 V ] = 0 unless

• (chT ,VT)[ST ]< . degcd! = deg oj,

deg x f

x 1< x . Supposethe characteristicnumbers [M, a] = Z n ^ t P ^ x

are allzero.

has

[S2j*, Vj ],

Q) .

Z) C H2;>(S21

of non-negative integers.

and consider

is of virtual dimension

[P2i(C), 1], i = 1, 2 , ..., and

j = 1, 2, . . . 9 erator of

a € K(M)

n*(BU) Q is the polynomial algebra gen­

Then

erated by

M

as the ring of bordism

We show inductively that

of alinear

combination

ST ,1 ® T = 0 f°p aH

a>,T.

Suppose

=

310

FLOYD

ncD t = 9

fop a11

03' T

§3

deS T =

0 “ Pm' • C\

[M> a] =

T0

^

=

Y n ^ p ,[P ] ■ ch T, [S, ] L »,T ) = < -v, u >.

316

§2

SOLOVAY

By (3),

T(T*(X))

inherits a complex structure (in which the real part

is "vertical" and the imaginary part is "horizontal"); in this way, T*(X)

becomes an almost-complex manifold.

By construction, we have an

isomorphism of complex vector bundles: (10

T(T*(X)) s n*(T(X) ® C) Remark.

*

The almost-complex structure on

T (X)

depends on

the choice of a Riemannian metric; however, it is well-defined up to a homotopy since any two Riemannian structures on Let

B(X)

be the unit ball bundle of

some fixed Riemannian structure on structure from

-ft

T (X) ; we give

almost-complex structure. H 2 n (B(X), S(X); Q) tion of

X

Let

X.

B(X)

B(X)

are homotopic.

T*(X)

inherits analmost-complex

the orientatidn determined by its

n = dim X,

and let

[B(X), S(X) ]

e

be the fundamental class determined by the orienta­

B(X) . Finally, let ^(B(X)) = ^(T(B(X)))

the almost-complex manifold THEOREM

determined by

2 . Let

be the Todd class of

B(X). X

be a compact differentiable mani­

fold, not necessarily oriented.', let S:

C°°(t) - C°°( r\)

be an elliptic operator on X. (5)

Then

ia (S) = ch(7 (S)) • ^(B(X))[B(X), S(X)] (Note the absence of signs!) PROOF.

[X] e H ^X; Q) tion of

X.

We consider first the case when

X

is oriented.

Let

be the fundamental class determined by some fixed orienta­ LetU e Hn (B(X), S(X); Q)

be the Thom class associated to

this orientation.(From now on, rational coefficients are Let cp*:

H:‘+n(B(X) , S(X)) - HJ'(X)

be the Thom isomorphism: cp* - 1 (a) = 7t*a U U.

understood.)

§2

XIX: LEMMA 2.2.

THE INDEX THEOREM: APPLICATIONS Let

x € H*(B(X), S(X)).

317

Then

x[B(X), S(X)] = (-l)n(n+1)/2 l) • Let

ir*S:

C°°(**i)

be the map which makes the diagram

Then

ir^S e E k («*|,

7 (ir*S) = ttj (7 (S)) ,

C ° ° ( ti)

tj) ;

moreover

32 2

SOLOVAY

§3

and (22)

i-a^*3 ) = V PROOF,

The isomorphism (1 1 ) induces isomorphisms

i.

H^( 6 )

by

Opj(|, ti) = Opj( ,

jt*S € Op^.

it follows that

cp(x) = cp(jt(x)).

If

cp in

C°°(X),

cp, \|r € C°°(X),

Then if

(23)

also.

Thus

cp and

M~ SM~

0 p^._i(7t*|, jc*h) .

Thus it *S

2 . Now let

over

U.

We put

1L

Let

cp

Let

U

cp

and

H

do

lies in

be a small neighborhood of

j , j2 : U Let

= J 1 (U ) .

= 1 . We write

.

so by (2 3 ),

,

C°°(X)

satisfies (1) of Lemma 5.6.

p € X.

diffeomorphic to R n .

t\)

Op^Ci,

cp in

we have

\|r have disjoint supports, then €

;

«*n)

we define

«.(M~ SM~) = Mq? n#SM^ If

cp(p)

s)

= cp1 + cp2 ,

X

be the two sections of

C°°(X)

cp e

where

cp^

have support in has support in

U,

U^.

p X and By

(1 *0 , we have canonical isomorphisms ( 2i0

* * 1 |U = j-| I ® j2 5 >

**^ 1 ^ = j 1 t] © j2 |

which we treat as identifications. It will be convenient in the following to identify U2

with

U.

In this way, we get identifications j

If follows that

In

termsof (2 U)

M

*

2.)

Let

it: Spin(n+2)

S0(n+2)

Put Sc(n) = it~1 (SO (2) x SO(n))

If

P

is a principal

is a principal Spin(k)-bundle, then

P > A ) = ^

as a su^r^n 8

E*(Bgc (k)>Q)

We shall also use

(where

X

identified with to denote the

332

§6

SOLOVAY

characteristic class for 6.7.

Sc( k) -bundles determined by

PROPOSITION.

Theorem 6.2.

Let

z e H2(X, Z)

Then X admits an

principal bundle, (Here

X,

and

the

2

be as in

Sc(m)-structure (with

P,say) such that

V is R m ,

X € H

X(P)

action of

= z.

Sc(m)

on

V is

given by the composition Sc(m) - iL— ► S0(2) x SO(m)

PROOF.

We give

X

■

SO(m) .)

a Riemannian structure with principal

Q2 . Recall that the integral Euler class sets up a 1 - 1 2 correspondence between principal S0(2)-bundles over X and H (X, Z) .

SO(m)-bundle

Let

Q

be an S0(2) -bundle over

w2(Q1) +

X

with

w 2(Q2)= p2(X(Q1)) + w 2(X) =

p

SO(2) xS0(m)-bundle corresponding to

It follows that

that

S0(m) s Q2 .

PxSc^

is the desired 6.8.

P

covers the "inclusion" Sc(k+i) fixing

Let Q ’ be (Q0,2) .

such that

Since

Sc(k) -► Sc(k+1)

of X,

V s T(X) .

this

Thus

X.

be the "inclusion” map that

map S0(2) x S0(k) -► S0(2) x S0(k+i).

acts transitively on

Ac­

cohomology) and

Q2is the frame bundle \|r: P x ^ ^

the

P Xsc(m ) (S0(2) x

x(P) = X(Q1) = z (rational

Sc(m)-structure on

l:

Let

Now by hypothesis,

the ordered pair

isomorphism gives rise to an isomorphism (P, y)

= z.

+ w 2(X) = 0.

2 (z )

cording to §6.5, there is an Sc(m)-bundle S0(m)) s Q 1.

X(Q^

S^ CR^ +1,

and

Sc(k)

Then

is the subgroup

ek+1. The following Propositionsummarizes the

representations

of

Sc(m).

PROPOSITION. and

N1

(1 )

M 1 and

(2)

in

remarks of [3l on

(In [3 ], Sc(m)is referred to as

There are complex

Sc(2n)-modules

Gm *) M*

such that N ’ are isomorphic as

H*(BSc(2n), Q),

ch(M') - ch(N') = e

x / 12

Sc(2n- 1 )-modules.

we have

tt

j]

1=1

(e

yV2 1

-yn-/2

- e

1 )

the

§6

XIX: THE INDEX THEOREM: APPLICATIONS 6 .9 .

333

The following paragraph summarizes the results we wil

need on quaternionic spin-representations (cf. [1 , k ]). H i s the field of quaternions; A^ the Clifford algebra on R m (cf. IV, § 1 0 ).

a unique irreducible tible

A^-module.)

S; © *£,

Then there is, up to isomorphism,

A^ ®H-module,

As

S .

A* ®H-module,

into irreducible modules. Since

Spin(m) £ A^, + S”

Consider C C H).

m = k(Q) ,

We suppose

Let

S~

(S

Sm

splits uniquely,

Moreover,

S*

£

S* .

Spin(m)-module (via the inclusion

be the vector bundle on

Then, with appropriate labeling of

Bgpin^m ^

associated to

S^ .

S^,

y./s

ml 2

=

A^ S^

Sm =

is a quaternionic Spin(m)-module.

as a complex

ch(S^) - ch(Sm )

is, in fact, an irreduc-

-j./2

(e 1

[]

- e

1

)

i=i H*(BSpin(m ) ; Q)

(We are identifying

induced from the canonical map

H*(BS0(m ) ; Q)

with

p:

-* BSq ^

via the map

.

PROOF of Theorem 6 .2 . We may assume, by Proposition 6.7, that X

has an

Give

W

Sc(m)-structure, with associated principal Sc(m)-bundle a hermitian structure.

U(n)-bundle

P2 .

U(n)-bundle

P,

The principal bundle of

The ordered pair and

X

(P1, P2)

W

P1.

is then a

corresponds to an Sc(m) x

may thus be viewed as a manifold with an

Sc(m) x U(n)-structure. For brevity, let projection

G

be

G -► Sc(m) , on the

the unit sphere of R m G that leaves Let N = N1 ® C n.

em M’

and

Sc(m) -module R m

since

fixed

Sc(m)

G

tion 6 .8 , it follows that

M. M

Sc(m)

N M, N

on

H,

of

Sc(m-1 ) x U(n).

M 1 and of

on

Similarly,

and

can apply the results of §5.3 to

acts, via the

and G is transitive

N r be as in Proposition 6 .8 .

on

G

is.Moreover the subgroup,

is precisely

From the actions of

we get an action of

Sc(m) x U(n).Then

G

acts on

are isomorphic as and

G.

Put

M = M 1 ® Cn,

U(n)

on C n ,

N. From Proposi­ H-modules.

The result is that

We

33*4-

§6

SOLOVAY

y± / 2 -y± /2 v /„ (e - e ) • y. oh(W)e ' J] i [X]

■i 1 - e 1) (1 - e is integral.

1)

(As usual, the Pontrjagin classes of

elementary symmetric functions in

y^,

y^

X

where

are viewed as the r = m/2.)

The result now follows from the identity y± /2 -y± /2 e 1 - e 1

1

PROOF of Theorem 6.3.

-1

The proof is, for the most part, quite

similar to that of Theorem 6.2; we omit many details. 0(m).

Then, using the hypothesis

w 2(X) = 0 ,

we give

Let X

G = Spin(n) x a suitable

G-structure (cf. first paragraph of the proof of Theorem 6.2).

Let M = Sm % Rn N

arequaternionic

by means of

N = Sm

G-modules.

Let V = R

m

Rn ( c f ' §6-9) - Then M .

the natural inner product of a 1*.)

Proposition 1^.

We are going to

with

V

*

apply

Thus we need a map :

Since

(We identify V

S(V*)

Rm C A^,

- S".

If

and v

— HomH (M, N) . A^ S* C_ S^,

€ S(V*)> s e S*,

there is a bilinearpairing w € R n ,we put

ff1 (v) (s ® w) = \|r(v ® s) ® w Then

a1

satisfies the hypotheses of Proposition 5.U. The remainder of the proof follows the last twoparagraphs

the proof of Theorem 6.2. struct the details.

of

The interested reader will be able to recon­

§6

XIX: THE INDEX THEOREM: APPLICATIONS

335

REFERENCES

[1]

M. F. Atiyah, R. Bott and Arnold Shapiro, Clifford modules , Topology, vol. 3, supp. 1 (196*0 pp. 3-38.

[2]

F. Hirzebruch, Reue Topologisahe Methoden in der Algebraisohen Geometrie > Springer-Verlag, 1956.

[3]

F. Hirzebruch, A Riemann-Rooh theorem for differentiable manifolds , Seminaire Bourbaki, Fevrier, 1959 -

[*4-]

John Milnor,

The Representation Rings of Some Classical Groups ,

mimeographed notes, Princeton, 1 9 6 3 .

APPENDIX I THE INDEX THEOREM FOR MANIFOLDS WITH BOUNDARY M. F. Atiyah Introduction In this appendix we shall indicate briefly how the index theo­ rem may be extended to manifolds with boundary. manifold with boundary X

Y

and

d

If

X

is a smooth compact

is an elliptic differential operator on

then there is a definition of an "elliptic boundary condition” (or bound­

ary operator) b.

This definition, due originally to Lopatinski, imposes an

algebraic restriction on the symbol of

b

relative to the symbol of

d.

In §1 we shall give this definition, taking care to state everything intrin­ sically.

Such a pair

(d, b)

then has an index and the problem is to ex­

press this index in terms of the topological data provided by the symbols ff(d)

and

o(b) . In the case when the boundary is empty then

difference element ch d e H*(X; Q)

[a(d)] € K(B(X), S(X))

defines a

and hence a cohomology class

by the formula ch d = ( _ D n (n+D/2 ^

cp*

a(d)

ch[cJ(d)]

being the Thom isomorphism. In the general case we shall show in §2 that the pair

a(b)

defines a difference element [o(d, b)] e K(B(X), B(X)|Y u S(X))

and hence a relative cohomology class ch(d, b) € H*(X, Y; Q) by the formula

337

,

a(d),

338

ATIYAH

§1

ch(d, b) = (-i)n (n+1>/2 cp*

oh[o(d, b)]

being now the relative Thom isomorphism H*(B(X), B(X) |Y U S(X); Q ) — H*(X, Y; Q) The topological index

it(d, b)

.

is then defined, in analogy

with the closed case, by it(d, b) = (ch(d, b) 0,

x^,) ^ 0

f = °

n

' p j(ax7 , 1

343

=

0

(J=1> •••'

r)

which is of the form

= exp(ix11, +...+ lxn_1ln _1)h(xn )

(and real) and

h ^ 0.

First let us observe that (ii) and (iii) may be reformulated as (ii) 1

for each

..., in _1) 7^0

(g-,

and real, m. gj e C J ,

and for every set of vectors the boundary problem (*)

•••7

(**)

d i ^ dxR ) ' h

5n _1,

P(^>

gn-1' -1 d x ^ h ^xn=° = Sj-

has a unique solution for

0

h

^* = 1'

which is bounded

xn > 0.

In fact, writing

v = (l