The Atiyah-Singer index theorem

The Atiyah-Singer Index Theorem

Simon Kitson

A thesis submitted in partial fulfilment of a Degree of

Bachelor of Science with Honours

Department of Mathematics

Australian National University

2

Acknowledgements

First I would like to thank my friends, my family and my homeland for their love,

support and inspiration. I would like to thank Adam Rennie for being such a diligent

supervisor throughout the year. Also, RRHS, ANU and the MSI as well as all of my

teachers and lecturers over the years - I am privileged to have had access to such a good

education.

It has been five years since I moved to Canberra to study. During this time there have

been many moments which, in one way or another, have seemed as long and significant

as its entirety. Most of all I would like to thank the people with whom I shared these

moments.

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4

Contents

1 Introduction 9

2 Vector Bundles and Topological K-Theory 13

2.1 Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.2 Operations on Vector Bundles . . . . . . . . . . . . . . . . . . . . . 16

2.1.3 Principal G-bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 K-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Compact Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.2 Relative K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.3 Compactly supported K-theory . . . . . . . . . . . . . . . . . . . . 24

2.2.4 Complexes and the the multiplication map . . . . . . . . . . . . . . 25

2.2.5 Homogeneous complexes . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3 Equivariant K-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4 The Thom isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 ΨDO and the Principal Symbol 35

3.1 Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Pseudo-differential operators on Rn . . . . . . . . . . . . . . . . . . . . . . 38

3.2.1 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.2 Pseudo-differential operators on Rn . . . . . . . . . . . . . . . . . . 42

3.2.3 The algebraic properties of symbols . . . . . . . . . . . . . . . . . . 44

3.2.4 ΨDO defined by kernels . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Pseudo-differential operators on manifolds . . . . . . . . . . . . . . . . . . 54

3.3.1 ΨDO on vector bundles . . . . . . . . . . . . . . . . . . . . . . . . 57

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6 CONTENTS

3.4 The Symbol Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.5 Classical ΨDO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4 Analytic Properties of Elliptic ΨDO 61

4.1 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2 Elliptic operators on Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3 Elliptic operators on manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 68

5 Fredholm operators and the Analytic Index 77

5.1 Fredholm Operators and the Index . . . . . . . . . . . . . . . . . . . . . . 78

5.2 The Analytic Index Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6 The Topological Index and its Axioms 85

6.1 The Topological Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.2.1 Refining the axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.2.2 The normalization axiom . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2.3 A sufficient condition for (B2’) . . . . . . . . . . . . . . . . . . . . . 96

6.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7 The Atiyah-Singer Index Theorem 103

7.1 Verification of the axioms for the Analytic Index . . . . . . . . . . . . . . . 103

7.1.1 Excision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.1.2 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.1.3 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

8 An Example: The Dirac Operator on S2 111

8.1 An atlas for S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

8.2 The bundles on S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

8.2.1 Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

8.2.2 The spinor bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8.2.3 The spinor bundle connection . . . . . . . . . . . . . . . . . . . . . 113

8.3 The Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

8.3.1 Extension to Hilbert space . . . . . . . . . . . . . . . . . . . . . . . 115

8.3.2 Spinor Harmonics and the spectrum of D . . . . . . . . . . . . . . . 115

CONTENTS 7

8.4 Twisted Spinor Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

8.4.1 Twisted Dirac operators . . . . . . . . . . . . . . . . . . . . . . . . 118

8.4.2 The index of the twisted Dirac operator . . . . . . . . . . . . . . . 118

9 Summary 121

8 CONTENTS

Chapter 1

Introduction

The index theorem provides a fundemental connection between topology, differential

geometry and analysis. As we will see the theorem ties together many ideas which illu-

minate relationships between these fields. The theorem states that a map, known as the

topological index, is in fact equal to the map taking an elliptic operator (regarded as a

Fredholm operator on a suitable Hilbert space) to its index. On the one hand this allows

the index of an elliptic operator to be computed topologically. On the other, it allows

the topology of a space to be investigated by computing the index of canonical elliptic

operators. This makes it important in a range of applications, in addition to its theoretical

value.

There are several proofs of the index theorem. The original proof made use of cobordism

theory and was announced by Atiyah and Singer in 1963 [AS]. In 1968 they published a new

proof using topological K-theory [AS1]. At the time topological K-theory was relatively

new itself and this was its first major application (see the preface of [A]). This enabled

Atiyah and Singer to avoid cohomology entirely and also make generalisations, such as

the equivariant index theorem and the index theorem for families [AS2] (by Atiyah and

Graeme Segal), [AS3]. Later proofs based on heat kernel methods were investigated by,

among others, Getzler, Gilkey and Patodi [BGV], [Gi], [ABP].

We will present the proof of the index theorem published in 1968 in [AS1]. We have

chosen to present this proof, not only due to its intrinsic interest, but also because it

opened up new areas of research, such as K-homology and index theory, which are still

active today. Due to the relatively accessible nature of the mathematics in the proof it

provides a good introduction to many of ideas which underlie these fields of research. We

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10 CHAPTER 1. INTRODUCTION

make our presentation with the aim of introducing these ideas. Certainly, it is the author’s

experience that writing this thesis has made the modern index theory literature much more

accessible.

A presentation of this proof poses several problems for both the reader and expositor.

The theorem and the proof which we present are primarily concerned with the strong

interrelations between topology, differential geometry and analysis. As such, a working

knowledge of each of these fields is required. For this reason we have opted to give an

account which is essentially self-contained. We have tried, as far as is possible, to give

a clearly delineated development of the analysis and topology involved. Hopefully, this

will aid the reader who is not so familiar with analysis or topology and also emphasise

the way in which the proof relates them. There are many topics which we have had

to omit. In particular, the theory of Dirac operators is fundamentally related to index

theory. Although, we have not included this, we give an example in Chapter 8. This

example demonstrates the main ideas in the proof and is the best way to appreciate their

interaction.

We make no claim to originality of the mathematics within this thesis. Our main source

for the proof is the original paper [AS1] by Atiyah and Singer. This paper gives the proof of

the equivariant index theorem. The proof we present has been extracted from this and we

should note that it can easily be extended again to the equivariant case. Our development

of K-theory and equivariant K-theory is based on Atiyah’s book “K-theory” [A] and

Segal’s paper [Se] respectively. The majority of the development for pseudo-differential

operators and the analysis of elliptic operators was sourced from Gilkey’s book [Gi] on the

heat equation proof of the index theorem. The example which we present is computed in

[Va]. Finally, other accounts of the K-theoretic proof of index theorem given in [La] and

[LM] provided reference.

Broadly speaking, this thesis may be split into two parts. The first develops the rela-

tionship between K-theory and elliptic operators by showing how the symbol of an elliptic

pseudo-differential operator can be considered to represent a class in K-theory. This is an

important relationship in itself and provides the setting for the proof of the index theorem.

Along the way we will develop material, such as equivariant K-theory, which is used later

in the proof. In the second part of this thesis we begin working on the proof of the index

theorem. With the symbol class of a pseudo-differential operator defined we will be aiming

to prove results at the symbolic level. We will define the analytic and topological indices

11

and then verify that they are in fact equal, proving the index theorem.

In Chapter 2 we develop the basics of K-theory for vector bundles. The main points

of this chapter are the characterizations of K(X) using complexes, the definition of a

multiplication map which turns K(V ) into a K(X)-module for a vector bundle V → X,

and the Thom isomorphism. There are also sections discussing equivariant K-theory and

principal bundles, both of which will be needed in the proof of the index theorem in later

chapters.

Chapter 3 develops the theory of pseudo-differential operators and their symbols. We

finish by showing the relationship between elliptic pseudo-differential operators (ΨDO) and

K-theory via the principal symbol. Here we deal only with pseudo-differential operators

between smooth sections and leave their analysis on Hilbert spaces for the next chapter.

Having seen in the previous chapters that K(X) can be represented by both vector

bundles and the symbols of ΨDO on these bundles, the motivating question is “what

analytic information is present at the symbolic level?”. In the fourth chapter we begin

this investigation, extending elliptic ΨDO to Sobolev spaces. The main results include

the existence of approximate inverses for these extensions and the fundamental elliptic

estimate.

In Chapter 5 we define Fredholm operators. The existence of an approximate inverse

shows that the extended elliptic operators from Chapter 4 are Fredholm operators, and thus

have a well defined index. We then show that the index of any extension is the same, thus

the index may be consider an invariant of elliptic ΨDO (independent of their extensions to

Hilbert spaces). Finally, we investigate the structure of the space of Fredholm operators

to show that the index is a homotopy invariant. This in turn shows that the index of an

elliptic ΨDO depends only on the homotopy class of its principal symbol. We then define

the analytic index map on K-theory, and our previous results show that it is well-defined.

By Chapter 6 we have seen that index of an elliptic operator may be considered as an

invariant of its symbol class in K-theory. The Atiyah-Singer index theorem computes the

index topologicaly from this class. The topological index map is presented as a candidate

for computing the index and we can then state the index theorem, that the analytic and

topological indices coincide. Next, axioms are presented which characterize the topological

index. These are then refined to give sufficient conditions for the proof of the Atiyah-Singer

index theorem.

In Chapter 7 we verify that the analytic index satisfies each of the axioms for the

12 CHAPTER 1. INTRODUCTION

topological index, proving the index theorem.

In Chapter 8 we give an example of the Dirac operator on the 2-sphere which demon-

strates the dependence of the index on the topology of the underlying bundles. First,

representative bundles for each class in the reduced K-theory are given. We define the

Dirac operator on the spinor bundle. Then we compute the index of the Dirac operator

and the index of the Dirac operator after being tensored with the representative of the

m-th K-theory class. Our result for the index of the twisted Dirac operator is −m.

Chapter 9 gives a brief summary and points out the relationship of the material we

have presented to some of the modern theory.

Chapter 2

Vector Bundles and Topological

K-Theory

In this chapter we will develop the K-theory we need in the context of vector bundles.

In this context, K-theory is an algebraic abstraction which “counts” the number of vector

bundles over a space. In this way K-theory provides information about the global topol-

ogy of the underlying space. The theory we develop here will be used later to show the

relationship between K-theory and elliptic operators. It will also allow us to define the

topological index, and give some tools for the proof of the index theorem.

2.1 Vector Bundles

First we will outline the basic theory of vector bundles. This will provide the basis of

our K-theory development. We are interested in the global properties of vector bundles

and the effect which various operations have on these properties.

2.1.1 Basics

The idea of a vector bundle is that locally it is the product of a vector space and an

open set. We will primarily be interested in the case when the base space is a manifold

and the vector space is complex.

Definition 2.1.1. Let X be a locally compact Hausdorff space. A pair (E, π), where E

is another locally compact Hausdorff space and π : E → X a continuous map, is called a

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14 CHAPTER 2. VECTOR BUNDLES AND TOPOLOGICAL K-THEORY

complex vector bundle over X under the following condition: About every point x ∈ Xthere exists an open subset U such that π−1(U) is homeomorphic to U × V , where V is a

finite dimensional complex vector space. The vector space Ex = π−1(x) is called the fibre

over x. The set U together with the implicit homeomorphism π−1(U) ∼= U × V is called a

local trivialisation.

Vector bundles with real fibres are defined similarly and share the basic properties of

complex vector bundles. The first of the following examples is a real vector bundle over

the circle.

Example 2.1.1. Consider S1 as the interval [0, 1] with the points 0 and 1 identified. The

Mobius bundle over S1 is then given by π : F → S1 where the total space is F = [0, 1]×Rmodulo the identification of (0, x) and (1,−x). The map π is projection onto the first

coordinate.

Example 2.1.2. We can also consider a complex vector bundle over a point π : V → pt.

The total space V is then a complex vector space of some dimension k and π maps the

entire space to the point.

Definition 2.1.1 makes no reference to a metric. However, a vector bundle on a para-

compact Hausdorff space can always be given an inner product using a partition of unity

(a Hermitian inner product on a complex bundle, or a positive definite inner product on a

real bundle).

Given a vector bundle π : E → X we note that the dimension of the fibre, dimEx, is

locally constant. Thus we may make the following definition.

Definition 2.1.2. If π : E → X is a vector bundle over a connected base space then we

define the rank of E to be the dimension of its fibres.

Our next definition is that of a section. A section is a function which continuously

picks a vector from each vector space of a bundle. It generalises the idea of a vector-valued

function. As we will see this is a useful concept for topological purposes (i.e. defining

explicit isomorphisms between bundles). Also, the pseudo-differential operators which we

will define later, map between spaces of sections.

Definition 2.1.3. Let π : E → X be a vector bundle. A section of E is a continuous

map s : X → E such that π s = IdX . The space of sections of E is denoted Γ(E).

2.1. VECTOR BUNDLES 15

Later we will characterise K-theory in terms of sequences of vector bundles. The maps

between each bundle will be bundle homomorphisms.

Definition 2.1.4. Let π1 : E → X, π2 : F → X be vector bundles. Let f : E → F be a

continuous map such that π2f = π1. If fx : Ex → Ff(x) is linear for all x ∈ X then we call

f a homomorphism of vector bundles.

The isomorphism classes defined below formalise what we mean when we speak of the

“global properties” of vector bundles.

Definition 2.1.5. Two vector bundles π1 : E → X, π2 : F → X are said to be isomorphic

if there exists a continuous map f : E → F such that π2f = π1 and fx : Ex → Fx is an

isomorphism of vector spaces for all x ∈ X.

Definition 2.1.6. We denote the set of isomorphism classes of vector bundles over X by

Vect(X).

From here on in we will mostly be interested in the isomorphism classes of vector

bundles over a space. In fact, the K-theory classes in the next section will be constructed

from these isomorphism classes. We should note that if X is a manifold we may require

all of the maps in the above definitions to be smooth. This is because the isomorphism

classes of smooth vector bundles coincide with those of continuous vector bundles. When

working on manifolds this can be show from the description of vector bundles in terms of

transition functions, see [D]. Also, Blackadar’s book [Bl], Sections 4.5 and 4.6, contains

results which imply that these classes coincide; these results are given in the context of

K-theory for operator algebras.

Example 2.1.3. Considering S1 as in Example 2.1.1 we may form a bundle with total

space E = [0, 1] × R where (0, x) and (1, x) are identified. Although this bundle has the

same base space and fibres as the Mobius bundle F , the two are not isomorphic. This can

be seen as follows. Consider a potential isomorphism f : E → F . Each map ft : Et → Ft

is required to be an isomorphism. As such, f must map the constant section (t, 1) ∈ Γ(E)

continuously to F without crossing the zero section (t, 0) of F . Even if it does so for

0 < t < 1 continuity fails at t = 1 ∼ 0 as (0, x) is identified with (1,−x) in the Mobius

bundle.


Example 2.1.3 displays the motivation behind the definition of vector bundle isomor-

phism. It captures the global properties of vector bundles. By considering a map between

sections we see that the Mobius bundle is “twisted” while S1×R is not, even though locally

they are identical. Bundles which can be written as a product π : V ×X → X where V is

the fibre and π(x, ξ) = x have the simplest global properties and are therefore referred to

as trivial bundles. We denote a rank N trivial bundle by N , so the trivial bundle of rank

1 is just denoted 1.

2.1.2 Operations on Vector Bundles

The topological operations we define here will be the basis for the algebraic operations

in K-theory. Our main aim is to show that the pair (⊕,Vect(X)) forms a commutative

monoid (semi-group with identity), where ⊕ is the direct sum of bundles which we define

below. Note that all of the following operations easily descend to isomorphism classes.

Definition 2.1.7. Let E,F be vector bundles over a locally compact Hausdorff space X.

Define the direct sum E ⊕ F of E and F to be the vector bundle π : E ⊕ F → X such

that π−1(x) = Ex ⊕ Fx.

Example 2.1.4. The direct sum of the Mobius band F with itself is a vector bundle which

is isomorphic to the the trivial bundle S1 × R2. To see this consider F ⊕ F to be given

by [0, 1]×R2 modulo the identification (0, x, y) ∼ (1,−x,−y) and S1 ×R2 to be similarly

paramaterized. In this case the following matrix gives a linear isomorphism between the

fibres at t (cos(πt) sin(πt)

sin(πt) cos(πt)

): (S1 × R2)t → Ft

The above example demonstrates the effect of the direct sum on isomorphism classes.

In particular, note that the although both bundles were twisted the direct sum was not,

this suggests that the algebraic structure of (⊕,Vect(X)) for general X may be quite

complicated. It is true in general that for every bundle there is at least one bundle which

“untwists” it. This is shown in the next two results.

Lemma 2.1.1. Let π : E → X be a vector bundle over a compact Hausdorff space X.

Then for some N there exists a continuous map f : E → CN which is injective and linear

on each fibre.


Proof. Take a covering of X by local trivializations. By compactness there exists a finite

subcover Uimi which yields trivializations αi : π−1(Ui)→ Ui×Cki . Set ai = pi αi where

pi : Ui × Cki → Cki is projection. Let ψimi be a partition of unity subordinate to the

Uimi . Then f = (ψ1a1, . . . , ψmam) : E → Ck1 ⊕ . . . ⊕ Ckm gives the map we are looking

for.

Corollary 2.1.1. For every bundle E over X there exists a bundle E⊥ such that E ⊕E⊥

is a trivial bundle. Such a bundle is known as a perpendicular bundle of E.

Proof. Let f : E → CN be the map from Lemma 2.1.1. Then we can define E⊥ =

∪x∈Xf(Ex)⊥, where f(Ex)

⊥ ⊕ f(Ex) = CN . Note that if M = X × Cn is a trivial bundle

then E ⊕ E⊥ ⊕M is also trivial, so perpendicular bundles are not unique.

Example 2.1.5. As seen in Example 2.1.4 the direct sum of the Mobius bundle F with

itself is isomorphic to the trivial bundle S1 × R2. So it is its own perpendicular bundle.

Next we have the tensor product. This will give us a ring structure on K-theory.

Definition 2.1.8. Let E,F be vector bundles. Define the tensor product E ⊗ F to be

the vector bundle π : E ⊗ F → X such that (E ⊗ F )x = Ex ⊗ Fx.

The following pull back construction is fundamental in relating the K-theory of two

spaces. It provides the contravariance of the K functor.

Definition 2.1.9. Let X and Y be compact manifolds, and π : F → Y a vector bundle

over Y . Then any continuous map f : X → Y induces a pull back bundle f ∗(F )

characterized by f ∗(F )x = Ff(x).

The pull back construction is basic to K-theory. It also allows us to define the following

product.

Definition 2.1.10. Given vector bundles p : E → X and q : F → Y over locally compact

Hausdorff spaces X, Y , their exterior tensor product is a vector bundle over X × Y

defined by E F = π∗XE ⊗ π∗Y F , where πX and πY are the coordinate projections from

X × Y to X and Y respectively.

The exterior tensor product can be used to construct the product E⊗F = 4∗(EF )

where E,F → X are vector bundles and 4 : X → X × X is the diagonal map. This

product will yield an important multiplication map when lifted to K-theory. We collect a

few facts about the operations we have defined.


Lemma 2.1.2. Let X and Y be compact Hausdorff spaces and f : X → Y a continuous

map.

(a) The direct sum, tensor product and exterior product are well-defined on isomorphism

classes.

(b) The pull back map, considered on isomorphism classes f ∗ : Vect(Y ) → Vect(X) is

well defined.

(c) The pull back operation is compatible with the direct sum and tensor product on vector

bundles.

Proof. These facts are all straightfoward consequences of the fibrewise definitions

of isomorphism, pull back, direct sum and tensor product. We will just prove the

last two items: Let E,F be vector bundles over Y which are isomorphic via the

(continuous) map g : E → F . Also, let f : X → Y be continuous. Then

f ∗(E)x = Ef(x)

gf(x)∼= Ff(x) = f ∗(F )x. (2.1)

The map defined fibrewise by gf(x) is continuous and thus gives an isomorphism

between f ∗(E) and f ∗(F ), proving (b).

For the direct sum, (c) is shown by

f ∗(E)x ⊕ f ∗(F )x = Ef(x) ⊕ Ff(x) = (E ⊕ F )f(x) = (f ∗(E ⊕ F ))x.

The proof for the tensor product is similar.

The above proposition implies the following.

Proposition 2.1.1. The assignment defined for compact Hausdorff spaces by X 7→ (⊕,Vect(X))

is a cofunctor with respect to continuous maps. Furthermore, maps induced by this functor

respect the multiplicative structure on Vect(X) given by the tensor product.

The next lemma shows that homotopic pull back maps give isomorphic vector bundles.

Lemma 2.1.3. Let X and Y be compact Hausdorff spaces. If f : [0, 1] × X → Y is

continuous and F → Y is a vector bundle f ∗0F∼= f ∗1F .


Proof. Let I := [0, 1] and p : X × I → X be projection. We have the isomorphism

s : f ∗F |X×t → p∗f ∗t F |X×t. Using an extension theorem for sections of vector bundles

(which is a consequence of the Tietze extension theorem, see [A]) we can extend this

isomorphism to an open set U containing X × t. We see that f ∗F and p∗f ∗t F are

isomorphic in some strip X × (a, b) where t ∈ (a, b). Thus the isomorphism class of f ∗t E is

locally constant in t. As I is connected this implies that the isomorphism class is constant

and f ∗0F∼= f ∗1F .

An immediate corollary is that a homotopy equivalence between two spaces induces an

isomorphism between their monoids of vector bundles.

Corollary 2.1.2. Suppose f : X → Y is a homotopy equivalence. Then f ∗ : Vect(X) →Vect(Y ) is an isomorphism of monoids.

2.1.3 Principal G-bundles

We will now introduce principal G-bundles. In general principal bundles are not vector

bundles, they are a more general type of object called a fibre bundle. Fibre bundles are

similar to vector bundles in the sense that they satisfy a local triviality property. Principal

G-bundles have fibres which are homeomorphic to a locally compact group G, rather than a

vector space. From a principal G-bundle P and a finite dimensional complex representation

of G, π : G → MN(C), one can form a vector bundle P ×π CN . For our purposes we will

only need to consider the case where G is GL(n), or one of its subgroups. Example 2.1.6

shows how the tangent bundle and its exterior powers may be expressed as vector bundles

constructed from the same principal bundle. Although this point of view is not needed

in our development of K-theory it is related and we will use it when proving then index

theorem.

Definition 2.1.11. Let E,X and F be three topological spaces. A mapping π : E → X

is called a fibre bundle with fibre F under the following condition: about every point

x ∈ X there exists an open subset U such that p−1(U) is homeomorphic to U × F . Then

E is called the total space of the fibre bundle, X its base and Ex = π−1(x) the fibre over

x.

Note that if F is a vector space then above definition agrees with that of a vector

bundle. Before defining principal bundles we define a G-space.


Definition 2.1.12. Let G be a topological group. A G-space is a topological space X

together with a continuous left action G × X → X, written (g, x) 7→ g.x, satisfying the

conditions g.(g′.x) = (gg′).x and e.x = x where e is the identity of G.

Definition 2.1.13. A principal G-bundle denoted (P, π,X;G) is a fibre bundle with

fibre homeomorphic to G where

(a) P is a topological space and G a topological group acting freely from the right on P ;

(b) π : P → X is such that for every x ∈ X there exist an open set U, x ∈ U ⊂ X and a

homeomorphism ΦU : π−1(U)→ U ×G such that

ΦU(p) = (π(p), φU(p))

where φU : π−1(U)→ G has the property that φU(p · g) = φU(p) · g.

Note that this is a fibre bundle with the fibres homeomorphic to G. Now that we have

defined the principal bundle we can describe the associated bundle construction.

Proposition 2.1.2. Let (P, π,X;G) be a G-principal bundle and F a G-space (G acting

on the left), then (E, p,X;F ) with E = P ×G F is a fibre bundle. Here P ×G F is the

quotient of P × F by the G-action g(p, f) = (pg−1, gf). The fibre bundle (E, π,X;F ) is

called the bundle with fibre F associated to the principal bundle P .

Example 2.1.6. Let X be an oriented manifold. We associate to the tangent bundle TX

the bundle of oriented bases P . This is a principal GL+n -bundle, where + denotes the

connected component of the identity. Taking the identity representation i of GL+n on Rn,

we have TX ∼= P ×i Rn. If we let ik : GL+n → GL+

n (Λk(Rn)) be the k-th exterior power

then

Λk(TX) = P ×i Λk(Rn)

If a metric is choosen then we can take orthonormal bases, allowing us to replace GL+n

with SOn.

2.2 K-Theory

We now construct theK-theory, K(X), first for compactX and then for locally compact

X. This is done using the operations defined in the last section. We will then investigate

2.2. K-THEORY 21

some alternative characterizations of K(X) in terms of complexes. These will allow us to

relate K-theory to elliptic operators. Extra structure will then be defined on K-theory by

lifting operations on (isomorphism classes of) vector bundles. In particular we will define

a multiplication map K(V )⊗K(X)→ K(V ) where V → X is a vector bundle. This will

enable us to define the Thom isomorphism which in turn is used to define the topological

index. The multiplication map and Thom isomorphism will be vital for our proof of the

Atiyah-Singer index theorem.

2.2.1 Compact Case

If X is compact then, as we saw in Section 2.1, the set Vect(X) with binary operation

⊕ forms a commutative monoid with identity given by the zero vector bundle. The group

K(X) is then constructed from (⊕,Vect(X)) by the following process.

Proposition 2.2.1. Let S be a commutatuve monoid and 4 : S → S × S the diagonal

map. Then A(S) = S×S/4(S) is a group. The group A(S) is known as the Grothendieck

group of S.

Proof. We form the product monoid S×S with addition given by (a, b)+(c, d) = (a+c, b+d)

and identity (0, 0) where 0 is the identity of S. The quotient with the submonoid 4(S) is

again a monoid. It is easily seen that inverses will be given by interchanging the factors as

(a, b) + (b, a) = (a+ b, a+ b) ∈ 4(S). This makes A(S) into a group.

Example 2.2.1. The natural numbers form a commutative monoid under addition. The

group A(N) is formed by cosets (a, b) + (g, g) for g ∈ N. By taking the homomorphism

(a, b) 7→ a − b to Z we see that (a, b) + (g, g) = (a + g, b + g) 7→ a − b + g − g = a − b∀g so A(N) ∼= Z. The inverses in A(N) are given by elements (b, a) = (a, b)−1, under our

isomorphism into Z this is just negation.

The above example shows the idea behind the construction, we have formally added

inverses. We now form K(X) by applying the construction to (⊕,Vect(X)) and give some

characterisations of classes in K(X).

Definition 2.2.1. LetX be a compact Hausdorff space. The groupK(X) is the Grothendieck

group K(X) = A(Vect(X)).


Proposition 2.2.2. Let X be a compact Hausdorff space. The group K(X) consists of

pairs of isomorphism classes of vector bundles [E]− [F ] under the equivalence relation

[E1]− [F1] = [E2]− [F2]⇔ E1 ⊕ F2 ⊕G ∼= E2 ⊕ F1 ⊕G

for some G ∈ Vect(X)

Viewed in this way the elements of K(X) are often refered to as virtual bundles, as not

every element of K(X) can be realized as the isomorphism class of a vector bundle.

Example 2.2.2. Consider K(pt). Isomorphism classes of vector bundles over a point

are in one-to-one correspondence with isomorphism classes of finite dimensional vector

spaces, i.e. they are classified by dimension and dim(V ) + dim(W ) = dim(V ⊕W ). Thus

K(pt) = A(Vect(pt)) ∼= A(N) ∼= Z by Example 2.2.1. Note that there is no vector bundle

representing a class −n for non-zero n ∈ N.

Using the existence of a perpendicular bundle, shown in Corollary 2.1.1 we can prove

the following two corollaries.

Corollary 2.2.1. Let X be a compact Hausdorff space. Every equivalence class in K(X)

can be represented by an element of the form [V ]−N where N = [X ×CN ] is the class of

the trivial bundle of rank N .

Proof. For any vector bundle E we have that [E] = N − [E⊥] for some N . So an arbitrary

element of K(X) can be written as [F ]− [E] = [F ]−N + [E⊥] = [F ⊕ E⊥]−N

Corollary 2.2.2. If [E] = [F ] there exists a trivial bundle N such that E ⊕N ∼= F ⊕N .

Proof. Suppose [E] = [F ]. By definition there exists a G ∈ Vect(X) such that E ⊕ G =

F ⊕G. This implies that for any bundle G⊥ such that G⊥⊕G = N we have E⊕G⊕G⊥ ∼=F ⊕G⊕G⊥. Thus E ⊕N ∼= F ⊕N . The converse is immediate.

We now look at the effect of the tensor and pullback operations on K-theory.

Proposition 2.2.3. Let X be a compact Hausdorff space. The tensor product extends

to K(X) making it a commutative ring with identity. The identity for multiplication is

represented by the trivial bundle of rank 1.

2.2. K-THEORY 23

The tensor product is a special case of the more general exterior product on K-theory,

defined in Definition 2.2.7 and extended in Section 2.2.4.

Next we show that the pullback operation on isomorphism classses of bundles turns K

into a contravariant functor.

Proposition 2.2.4. The map K(·) is a contravariant functor from the category of compact

Hausdorff spaces and continuous maps to the category of commutative rings with unit.

Proof. We know from Proposition 2.1.1 that X 7→ (⊕,Vect(X)) is a cofunctor with respect

to continuous maps. Noting that K(X) = Vect(X)×Vect(X)/4(Vect(X)) we see that K

is a cofunctor with respect to continuous maps.

In a similar fashion we see that the homotopy invariance of isomorphism classes of

vector bundles is passed on to K-theory.

Corollary 2.2.3. If f : X → Y is a homotopy equivalence, then f ∗ : K(X) → K(Y ) is

an isomorphism of rings.

Proof. This is immediate from the definition of K(X) and Corollary 2.1.2.

The functoriality shown in Proposition 2.2.4 allow us to investigate the K-theory of one

space via the K-theory of another. An example which we will be particularly interested in

later is the pullback of K(X) to K(V ) where V → X is a vector bundle over X, however

we will need K-theory with compact supports in order to do this. The following example

of a pullback on K-theory will be used to define the relative K-groups in the next section.

Example 2.2.3. The inclusion i : pt → X induces the homomorphism i∗ : K(X) →K(pt). Let E and F be vector bundles over X with fibres Ei(pt) and Fi(pt) of dimensions

m and n respectively. Then

i∗ : [E]− [F ] 7→ [Ei(pt)]− [Fi(pt)]

where Ei(pt) and Fi(pt) are considered as bundles over the point. As we saw in Example

2.2.2, K(pt) ∼= Z and [Ei(pt)]− [Fi(pt)] represents the equivalence class of m− n.


2.2.2 Relative K-theory

We now consider spaces X with basepoint ptX . In order to define the relative K-groups

we examine the map i∗ : K(X)→ K(ptX) ∼= Z as in Example 2.2.3.

Definition 2.2.2. Let X be a compact Hausdorff space with basepoint ptX . The reduced

K-group is defined by∼K(X) = Ker(i∗). The group

∼K(X) is an ideal of K(X).

As K(X) ∼=∼K(X)⊕K(pt) this can be thought of as K(X) modulo the trivial bundles.

The next example has particular importance in K-theory.

Example 2.2.4. We may view S2 as the one point compactification of C1, (C1)+, which

in turn may be viewed as CP 1. The space CP 1 is C2 modulo the equivalence relation

[v] ∼ [λv] for all non-zero λ ∈ C1 and v ∈ C2. The Hopf line bundle H over CP 1 ∼= S2 is

given by associating to the equivalence class [v] the complex line λv : λ ∈ C containing all

of its elements; for this reason it is sometimes refered to as the tautological line bundle. We

will see in Section 2.4, that the reduced K-group for S2 is given by∼K(S2) = ([1]− [H])Z.

Definition 2.2.3. Given a non-empty closed subset Y of X we define the relative K-

group to be

K(X, Y ) =∼K(X/Y )

here Y is considered to be the base point of X/Y . If Y is empty then we define X/Y =

X+ = X ∪ pt where pt is a disjoint point, considered to be the base point.

2.2.3 Compactly supported K-theory

Our motivation for defining K-theory for locally compact spaces is that we will need to

consider K(V ) where V → X is a vector bundle over a compact space X. Such a bundle

V is not compact (unless it is the zero bundle) as the fibres are non-compact. Using the

relative K-groups defined in the previous section we can define K-theory with compact

supports.

Definition 2.2.4. We define the K-group of a locally compact Hausdorff space X by

K(X) :=∼K(X+) where X+ = X ∪ pt is the one point compactification of X taking pt as

basepoint. In the case that X is compact we let X+ = X ∪ pt where pt is a disjoint point.

2.2. K-THEORY 25

When X is compact the above definition coincides with Definition 2.2.1. The group

K(·) as defined above is a cofunctor on the category of locally compact spaces and proper

maps (those for which the inverse image of a compact set is compact). We will only be

using locally compact K-theory to define the K-groups over vector bundles on compact

spaces. In this case the lack of compactness is only due to the fibres, so our use of the

extended definition is mild.

In the next definition we define an extension map on K-theory, induced by the inclusion

of an open manifold.

Proposition 2.2.5. Let h : U → X be the inclusion of an open manifold U into a man-

ifold X. The map which trivially extends a vector bundle over U to one over X − h(U)

lifts to a homomorphism on K-theory. This homomorphism is known as the extension

homomorphism and denoted h∗ : K(U)→ K(X).

An extension homomorphism will form one factor of the shriek map which we define in

Chapter 6.

2.2.4 Complexes and the the multiplication map

Next we work on obtaining alternative characterizations for compactly supported K-

theory in terms of complexes of vector bundles. There are two reasons for this. Firstly,

they facilitate a nice description of the exterior product and multiplication map. Secondly,

these characterizations will be used in relating K-theory to elliptic operators. We begin

with some definitions.

Definition 2.2.5. Let X be a locally compact Hausdorff space. A complex of vector

bundles E• over X is a sequence of vector bundles Ei and homomorphisms αi

0→ E0 α0→ E1 α1→ · · · αn−1→ En αn→ 0,

where αiαi−1 = 0 except on some compact set K ⊆ X. The support of E• is defined to

be the set of points x ∈ X such that the restriction of E• to x gives a sequence which is

not exact. Two complexes E• and F • over X are homotopic if there exists a complex G•

over X × [0, 1] such that E• = G•|X×0 and F • = G•|X×1.


Proposition 2.2.6. Let X be a locally compact Hausdorff space and C(X) be the set of

equivalence classes of compactly supported complexes over X, with two complexes being

equivalent if there is a compactly supported homotopy between them. Then

K(X) ∼= C(X)/C∅(X)

where C∅(X) is the set of complexes with empty support and K(X) is the K-theory with

compact supports.

For a proof of this theorem which uses a slightly different development see [A] Section

2.6. We will see an example of such a complex in Section 2.4.

Proposition 2.2.7. Let X and Y be locally compact Hausdorff spaces. The exterior tensor

product extends to a product

: K(X)⊗K(Y )→ K(X × Y )

(here the mapping is from the algebraic tensor product). For complexes E•, F • on X, Y :

(E• F •)i =⊕

j Ej ⊗ F i−j and

αE•F •

i =⊕j

(αE•j 1F i−j) + (−1)j(1Ej αF•i−j))

We now define a multiplication map K(V ) ⊗ K(X) → K(V ) for the K-theory of a

bundle V → X (real or complex) over a compact space. This is of fundamental importance

to the structure of K-theory. Among other things, the Thom isomorphism is defined using

this map.

Let V and W be vector bundles over X and i : X → X×X be diagonal inclusion. This

inclusion induces a map j from vector bundles over V ×W → X × X to vector bundles

over V ⊕W → X. The map j is given by restriction to the diagonal in X. In K-theory

this gives a homomorphism

j∗ : K(V ×W )→ K(V ⊕W ).

We then compose our exterior product with this homomorphism to get a product K(V )⊗K(W )→ K(V ⊕W ). Finally, we may consider X as a vector bundle with zero dimensional

fibres, in which case we get a product K(V ) ⊗K(X) → K(V ). In other words, K(V ) is

2.2. K-THEORY 27

a K(X)-module. Stating this more explicitly, if E• is a complex with homomorphisms α•

which represents a class in K(V ) where π : V → X and φ : F → X are bundles then

[(E•, α•)]⊗ [F ] 7→ [(E• ⊗ φ∗F, α• ⊗ Id)] ∈ K(V ).

When we come to proving the Atiyah-Singer index theorem we will split it up into a set of

axioms. The most significant axiom is focused on the relationship between this map and

the index of an elliptic operator.

As we have seen K(X) can be defined in terms of complexes of arbitrary length. How-

ever, it is sufficient to consider complexes of length one only.

Proposition 2.2.8. Classes in K(X) may be represented by equivalence classes of com-

pactly supported complexes of length 1

0→ Eα→ F → 0

we will denote such complexes by [E,F ;α].

Proof. Let i : C1(X) → C(X) be the inclusion map. A left inverse p : C(X) → C1(X)

is constructed as follows. Given a complex E• of length n choose a Hermitian metric on

each of the vector bundles Ei comprising E•. Adjoint maps α∗ : Ei+1 → Ei may then be

defined by 〈αi(v), w〉i+1 = 〈v, α∗i (w)〉i. We then set p(E•) to be the complex

0→⊕i

E2i α→⊕i

E2i+1 → 0

where α =⊕

i α2i +⊕

i α∗2i+1. Note that p i = Id. However, we are not guaranteed

that i p = Id unless p is injective. If p(E•) ∼= p(F •) then there exists a length one

homotopy G• between p(E•) and p(F •); denote the map of this complex γ. Let πj denote

projection onto Ej ⊆ p(E•) ⊆ p(E0) the jth bundle in E•. Then the maps πj+1γπj

give a homotopy between the original complexes E• and F •. So p is injective and thus

C(X)/C(X)∅ ∼= C1(X)/C(X)∅.

Example 2.2.5. One example of such a complex over S2 is

0→ 1α→ H → 0

where H is the Hopf bundle from Example 2.2.4. If we consider S2 as complex projective

space paramaterized by (z1, z2) then the homomorphism is given by α(z1,z2) : ξ 7→ z1/z2ξ.

This complex corresponds to the element [1]− [H] in K(S2). See Example 2.4.1 for more

details.


We will need to compute the exterior product of two length one complexes during the

proof in Chapter 7, so we give the formula here explicitly.

Proposition 2.2.9. Given two complexes of length one

0→ E0 α→ E1 → 0 and 0→ F 0 β→ F 1 → 0,

over X and Y respectively, their exterior tensor product is the complex

0→ E0 F 0 α1+1β→ E1 F 0 ⊕ E0 F 1 −1β+α1→ E1 F 0 → 0,

over X × Y . The corresponding complex of length one is

0→ E0 F 0 ⊕ E1 F 1 θ→ E0 F 1 ⊕ E1 F 0 → 0,

where

θ =

(α 1 −1 β∗

1 β α∗ 1

).

2.2.5 Homogeneous complexes

Homogeneous complexes are complexes over vector bundles. Homogeneous complexes

over the cotangent bundle will provide the final characterisation of K-theory which we will

need in order to show the relationship to elliptic operators. We carry out the development

with respect to homogeneous complexes over real vector bundles, although it applies to

complex bundles by ignoring the complex structure; we regard a rank n complex bundle

as a rank 2n real bundle.

Given a real vector bundle π : V → X the projection π induces a pullback map π∗

between bundles on the base and bundles on the total space. If E → X is a bundle then

π∗E is a bundle over V .

π∗E

E

π∗oo

V π// X

If E• is a complex over X then the pullbacks of its bundles Ei and homomorphisms αi

under π∗ form a complex

0→ π∗E1 α1→ π∗E2 α1→ . . .αn−1→ π∗En → 0

2.2. K-THEORY 29

over V . Note that we have simply extended the αi : Eix → Ei+1

x along the fibres of V so

that

αi,(x,v) : π∗Ei(x,v) = Ei

π(x,v) → Ei+1π(x,v) = π∗Ei+1

(x,v),

where (x, v) denotes the vector v ∈ Vx. We may also pull back the homomorphisms αi

so that they vary along the fibres of V in a specific way; this is the purpose of the next

definition.

Definition 2.2.6. Let π : V → X be a real vector bundle over a compact Hausdorff space

X. A homomorphism α : π∗E → π∗F , where E and F are vector bundles over X, is said

to be (positively) homogeneous of degree m if for all real λ > 0 we have:

α(x,λv) = λmα(x,v) : Eπ(x,v) → Fπ(x,v)

where (x, v) denotes the vector v ∈ Vx. Note that π(x, v) = x. If E• is a complex over

V and all of its homomorphisms αi are homogeneous of degree m then we say that E• is

homogeneous of degree m.

We now show that the appropriate homogeneous complexes can be considered as ele-

ments of K(B).

Proposition 2.2.10. Let X be a locally compact Hausdorff space, π : V → X a real vector

bundle over X with a positive definite inner product and E• a complex over X. If π∗E•

is exact on the sphere bundle S(V ) and is homogeneous of degree m then it represents an

element of K(V ).

Proof. Our inner product determines a sphere bundle S(V ) = (x, v) : 〈v, v〉x = 1. As

the homomorphisms αi of E• are homogeneous they are determined by their restriction to

S(V ). The complex E• is exact on S(V ), so homogeneity implies that the sequence

0→ E1vα1→ E2

vα1→ . . .

αn−1→ Env → 0

is exact except when v is in the zero section of V , which is the image of X. Thus E• has

compact support, and so represents an element of K(V ).

Finally, we are able to characterise K(V ) in terms of homogeneous complexes.


Proposition 2.2.11. Let V be a real vector bundle, with a positive definite inner product,

over a locally compact Hausdorff space X. Let m ∈ R, Cm(V ) denote the set of homotopy

classes of homogeneous complexes of degree m over V , and let C∅,m(V ) be the subset of

Cm(V ) consisting of complexes where the homomorphisms αi are constant on each fibre of

the sphere bundle S(V ). Then

Cm(V )/C∅,m(V ) ∼= C(V )/C∅(V ) ∼= K(V )

Proof. Since we have the inclusion Cm(V ) ⊂ C(V ), we would like to construct an inverse

map C(V )→ Cm(V ). Given a complex E• ∈ C(V ) with compact support L

0→ E0 α0→ E1 α0→ . . .α0→ En → 0,

we choose a metric on V , and let D(V ) be a disc bundle containing L (i.e. D(V ) is a bundle

of closed balls over X such that L ⊂ D(V )). Take F i to be the restriction F i = Ei|X to

the zero section. The disc bundle D(V ) is homotopic to X so by Lemma 2.1.2 pulling back

the F i gives an isomorphism Eiγi∼= π∗F i on D(V ). We define βi := γi+1αiγ

−1i so that βi

agrees with αi on the sphere bundle S(V ) (the boundary of D(V )). We then extend βi to

V as a homogeneous map of degree m, obtaining the complex G• on V given by

0→ π∗F 0 β0→ π∗F 1 β1→ . . .βn−1→ π∗F n → 0.

This complex is homogeneous of degree m and is homotopic to E•. As E• and G• are

homotopic they both correspond to the same class in K(V ). So the map C(V )→ Cm(V )

is an isomorphism.

Let E• ∈ C∅(V ) be a complex with empty support, i.e. it is exact at all points of

V . We follow the above construction. As E• has empty support we can take the D(V )

from the above argument to be X. Take F i to be the restriction F i = Ei|X to the zero

section. Then the trivial homotopy X = X gives an isomorphism Eiγi∼= π∗F i on X. Now

we define βi := γi+1αiγ−1i on X. We then extend it as a degree 0 homogeneous map, giving

a complex G•. Note that E• is homotopic to G• which is homogeneous of degree zero and

βi is constant along each fibre of S(V ).

On the other hand, if G• ∈ C∅,m(V ) is a complex where βi is constant on each fibre of

S(V ), then we have the homotopy αi(v, t) = ‖v‖tmβi(v/‖v‖), t ∈ [0, 1], to an exact complex

E• ∈ C∅(V ) at t = 0. So, we have Cm(V )/C∅,m(V ) ∼= C(V )/C∅(V ).

2.3. EQUIVARIANT K-THEORY 31

2.3 Equivariant K-Theory

Equivariant K-theory enters the proof of the index theorem after an application of the

associated bundle construction. The development for equivariant K-theory is essentially

the same as in the non-equivariant case. We will only sketch it here along with the results

which we need. For a more comprehensive introduction, see Segal’s paper [Se].

A G-bundle is a vector bundle over a G-space (see Definition 2.1.12) such that the

projection map commutes with the action of G.

Definition 2.3.1. Let X be a G-space. A vector bundle π : E → X is a G-bundle if E is

a G-space and the following two conditions are satisfied:

(a) g.π(v) = π(g.v) for all v ∈ E, g ∈ G,

(b) The map Ex → Eg.x given by v 7→ g.v is a linear map for all x ∈ X

Lemma 2.3.1. Given a continuous group homomorphism φ : H → G a G-space can be

consider to be an H-space by taking the action h.x = φ(h).x. This construction gives a

map φ∗ : KG(X)→ KH(X).

Definition 2.3.2. A G-bundle isomorphism, of G-vector bundles E and F over a

locally compact G-space X, is vector bundle isomorphism f : E → F which commutes

with the action of G. The collection of isomorphism classes of G-vector bundles is denoted

VectG(X).

Definition 2.3.3. Let X and Y be compact G-manifolds, and π : F → Y a vector bundle

over Y . Then any continuous G-equivariant map f : X → Y induces a pullback bundle

f ∗(F ) characterized by f ∗(F )x = Ff(x).

Note that the direct sum ⊕ of G-vector bundles will again be a G-vector bundle. As

in the non-equivariant case we have that (⊕,VectG(X)) is an abelian semi-group with

identity.

Definition 2.3.4. LetX be a compact Hausdorff space. The groupKG(X) is the Grothendieck

group K(X) = A(VectG(X)).

Example 2.3.1. The G-equivariant K-theory of a point is equal to the representation ring

of G i.e KG(pt) ∼= R(G). More generally, KG(X) is an R(G)-module.


2.4 The Thom isomorphism

The Thom isomorphism theorem is described in [AS1] as the fundemental result of

K-theory. It gives an isomorphism between the K-theory of a vector bundle over a space

and the K-theory of the space itself. Our interest in the Thom isomorphism lies in the

fact that it is used to define the topological index.

Let π : V → X be a complex vector bundle over a compact space X. We may then

form a complex of vector bundles Λ•(V ) over V by pulling back bundles of exterior powers

with π:

0→ π∗Λ0(V )α1→ π∗Λ1(V )

α2→ . . .αn−1→ π∗Λn(V )→ 0,

here the homomorphisms are αi+1 : (v, w) 7→ (v, v ∧ w) for all v ∈ V and w ∈ π∗Λi(V )v =

Λi(V )π(v). The sequence terminates at n = rank(V ) due to the the anti-symmetry of the

wedge product. Note that the maps αi satisfy αiαi+1 = 0 and are homogeneous of degree

1. Furthermore, the complex is exact outside the zero section of V and so defines a class

λV ∈ K(V ). Recall that K(V ) =∼K(V +) where we may consider V + = D(V )/S(V ). Here

the sphere bundle represents the union of the points at infinity for each fibre of V .

Theorem 2.4.1. (The Thom isomorphism theorem) Let π : V → X be a complex vector

bundle over a compact space X. Then multiplication by λV gives an isomorphism ϕ :

K(X)→ K(V ), where E 7→ π∗E ⊗ λV . This map is know as the Thom isomorphism.

In the above theorem we are using the multiplication K(X)⊗K(V )→ K(V ) discussed

in Section 2.2.4. We will not prove the Thom isomorphism theorem here, for details of the

proof see [A] Section 2.7. In order to define the topological index in Chapter 6 we will only

need a special case of the Thom isomorphism which we give now as an example.

Example 2.4.1. Taking the bundle π : Cn → pt and applying the Thom isomorphism we

obtain

ϕ : K(pt)∼=→ K(Cn) ∼= K(S2n).

So we see that K(S2n) ∼= Z.

We can find a representative of the generating class of K(C1) ∼= K(S2) ∼= K(CP 1) by

looking more closely at the definition of the Thom isomorphism. The generator of K(pt)

is the one dimensional trivial bundle 1. Under the Thom isomorphism this will map to

1 · λC1 which will be the generator of K(C1). Thus, all we need to do is determine the

2.4. THE THOM ISOMORPHISM 33

element λC1 . We will consider this complex over CP 1 parameterized by (z1, z2), so that

(z1, z2) corresponds to the point z1/z2 ∈ C1. From the definition we see that λC1 is

0→ π∗Λ0(C1)α→ π∗Λ1(C1)→ 0.

which is just the complex

0→ C1 α→ C1 → 0.

over C1 where α(z1,z2)(ξ) = z1/z2 ∧ ξ. So we can fix the first bundle of the complex to be

trivial giving

0→ 1α→ L→ 0.

for some line bundle L. The fibre of L over the point (z1, z2) is given by α(z1,z2)(1(z1,z2)).

Thus, to each point (z1, z2) ∈ CP 1 the complex line z1/z2C is associated. This is the Hopf

bundle H described in Example 2.2.4, and so our generator is ([1]− [H]).

We now note that the Thom isomorphism is transitive. If V,W are vector spaces then

we have

Λ∗(V ⊕W ) ∼= Λ∗(V )⊗ Λ∗(W ).

It follows that for vector bundles V,W over a compact space X, we have the multiplicative

formula

λV · λW = λV⊕W .

Using the multiplication described in Section 2.2.4, this implies that the Thom isomorphism

K(X) → K(V ⊕W ) is transitive in the sense that it agrees with the composition of the

Thom isomorphisms

K(X)→ K(V ) and

K(V )→ K(V ⊕W )

where we regard V ⊕W as a bundle over V , with projection onto the V factor.


Chapter 3

ΨDO and the Principal Symbol

This chapter focuses on developing the basic theory of pseudo-differential operators

(ΨDOs). The theory is initially developed on Rn with operators acting on sections of

trivial line bundles i.e. complex-valued functions. It is then lifted in a straightfoward

manner to manifolds and vector bundles. An object known as the principal symbol will be

used to show the relationship of pseudo-differential operators to K-theory. We begin with

a discussion of the specific case where our operator is a differential operator. The principal

symbol will then be defined in the more general context of pseudo-differential operators.

Once this is done we finish the chapter by showing the relationship to K-theory and briefly

discuss some of the implications. The majority of the material in this chapter and the next

comes from Gilkey’s book “Invariance Theory, the Heat Equation, and the Atiyah-Singer

Index Theorem”, [Gi], which gives a watertight account of the basic theory of ΨDOs.

3.1 Differential Operators

Our aim is to relate elliptic operators to K-theory. Recall that K-theory contains

information about the global topology of a space. In order to make the connection to

K-theory we will define the principal symbol of an operator and show that it is globally

well-defined. We will then explain how, for elliptic operators, it can be considered to

represent a class in K-theory.

Notation 3.1.1. If α is an n-tuple of non-negative integers α = (α1, . . . , αn) we define

|α| =∑k

αk and α! = α1!α2! . . . αn!.

35

36 CHAPTER 3. ΨDO AND THE PRINCIPAL SYMBOL

Given a vector ξ ∈ Rn

ξα = ξα1 . . . ξαn

and given standard co-ordinates we define the differential operators

∂αx = ∂|α|/∂xα = ∂|α|/∂xα11 . . . ∂xαnn

Dαx = −i|α|∂αx .

Definition 3.1.1. Let E and F be smooth vector bundles over a compact manifold X. A

differential operator of order m on X is a linear map P : Γ(E)→ Γ(F ) which satisfies

the following: Given a local coordinate chart (U, x) = (U, (x1, . . . , xn)), local trivializations

E|U ∼= U ×Cp, F |U ∼= U ×Cq and ξ ∈ Γ(E) a smooth compactly supported section we can

write P :

(Pξ)(x) =∑|α|≤m

Aα(x)(Dαxξ)(x)

Here Aα(x) is a q × p matrix of smooth complex valued functions, with Aα 6= 0 for some

α with |α| = m.

Straight from this definition we can calculate the effect of a change of coordinates or

trivialization. We are particularly interested in the effect of these on the highest order

term, that is the terms with coefficient Aα(x) where |α| = m. These top order terms will

define the principal symbol.

Observation 3.1.1. Given a representation of P over U , a change in trivialization given by

smooth maps gE : U → GLp(C), gF : U → GLq(C) gives P the new form:

P =gF (∑|α|≤m

Aα(x)Dαx )g−1

E

=∑|α|≤m

Aα(x)Dαx .

The coefficients Aα will be determined by the Leibniz rule and in general the effect on lower

order coefficients will be complicated. However, the leading coefficients where |α| = m are

given by Aα = gFAαg−1

E (x).

Observation 3.1.2. The effect of a change in local co-ordinates for U given by x = x(x) is

computed by applying the chain rule

∂

∂xj=

n∑k=1

∂xk∂xj

∂

∂xk

3.1. DIFFERENTIAL OPERATORS 37

This gives

P =∑|α|≤m

Aα(x)∂|α|

∂xα

For |α| = m we have

Aα =∑|β|=m

Aβ[∂x

∂x]αβ

where [∂x∂x

]αβ represents the symmeterization of the mth tensor power of the Jacobian matrix

(∂xk/∂xj).

These two calculations show that the coefficients Aα(x) where |α| = m are global invari-

ants of the operator P . They may be considered as a section of the bundle (⊙m T ∗X) ⊗

Hom(E,F ), where⊙

is the symmetric tensor product. This section is know as the prin-

cipal symbol of P and we will denoted it σ(P ). For each cotangent vector ξ ∈ T ∗xX the

principal symbol gives a homomorphism

σξ(P ) : Ex → Fx.

If we fix coordinates and trivializations we can represent the principal symbol locally

by

σξ(P ) =∑|α|=m

Aα(x)ξα.

We can now define ellipticity for a differential operator.

Definition 3.1.2. A differential operator P is elliptic if for each non-zero ξ ∈ T ∗X the

map σξ(P ) : Ex → Fx is invertible.

This condition allows us to consider the symbol as an element of K(T ∗X) as follows:

Suppose that we have an operator P : Γ(E1) → Γ(E2) and that π : T ∗X → X is the

projection for the cotangent bundle. Using this projection map we can pull back E1 and

E2

π∗Ei

Ei

π∗oo

T ∗X π// X

Recall that the fibre of π∗Ei over the point (x, ξ) ∈ T ∗xX is given by (π∗Ei)(x,ξ) = Eiπ(x,ξ) =

Eix. The principal symbol can then be considered as a map σξ(P ) : π∗E1

x → π∗E2x. This

gives a complex [π∗E1, π∗E2;σ(P )]. If σ(P ) is invertible for non-zero ξ ∈ T ∗X (i.e. P is


elliptic) then this complex represents an element of K(T ∗X) via the characterization given

in Proposition 2.2.11.

Although every symbol of an elliptic differential operator represents a class in K(X),

the converse is not true. To represent every class in K(X) we need to consider pseudo-

differential operators, a larger class of operators for which the principal symbol and ellip-

ticity can be appropriately defined.

3.2 Pseudo-differential operators on Rn

3.2.1 The Fourier Transform

First we introduce some basic tools which we shall need when defining and working with

pseudo-differential operators. The Fourier transform will allow us to write differentiation

as the transform of multiplication. We will use this point of view to generalize differential

operators to pseudo-differential operators in the next section. The Fourier transform is

defined on the space of Schwartz functions.

Definition 3.2.1. The Schwartz space S is the vector space of smooth complex valued

functions f on Rn satisfying the following condition: for all multi-indicies α and β there

exist constants Cα,β such that |xαDβxf | ≤ Cα,β

Definition 3.2.2. The Fourier transform is a map F : Sn → Sn given by

(Ff)(ξ) = (2π)−n/2∫

Rne−ix·ξf(x)dx

The Fourier transform of a function will be denoted f(ξ) = (Ff)(ξ).

The convolution product will be used in some of our proofs later on. Its relationship

to the Fourier transform is given in Lemma 3.2.1.

Definition 3.2.3. The convolution product on S is defined by:

(f ∗ g)(x) :=

∫Rnf(x− y)g(y)dy =

∫Rnf(y)g(x− y)dy.

We now prove some properties of the Fourier transform. Part (b) shows how the Fourier

transform can be used to express differentiation in terms of multiplication. Part (d) is

known as the Plancherel formula, it shows that the Fourier transform is an L2 isometry.

3.2. PSEUDO-DIFFERENTIAL OPERATORS ON RN 39

Lemma 3.2.1. The Fourier transform has the following properties:

(a) F has an inverse given by: f(x) = (2π)−n/2∫

Rn eix·ξf(ξ)dξ

(b) ξαf(ξ) = F(Dαxf)(ξ), Dα

ξ f(ξ) = (−1)αF(xαf)(ξ)

Dαxf(x) = F−1(ξαf(ξ)), xαf(x) = (−1)αF−1(Dα

ξ f(ξ))

(c) f · g = F(f ∗ g) and f ∗ g = F(f · g)

(d) 〈f, g〉L2 = 〈f , g〉L2

Proof. We integrate by parts and use the Lebesgue dominated convergence theorem to

establish the first two parts of (b). Consequently f ∈ S implies f ∈ S and the Fourier

transform defines a continuous map from S to itself. Let

T (f) := FF(f)(−x) = (2π)−n/2∫eix·ξf(ξ)dξ : S → S

We must show that T is the identity on S to prove (a). We do this in four steps. Let

f0(x) = e−|x|2/2 be the Gaussian distribution;

∫f0(x)dx = (2π)n/2.

(1) We will show f(0) = 0 implies Tf(0) = 0.

(2) We will show f0 = f0 and T (f0) = f0.

(3) We will use (1) and (2) to show Tf(0) = f(0) ∀f ∈ S

(4) We will use the linear structure on Rn and (3) to complete the proof.

Step 1: If f(0) = 0, define gj(x) :=∫ 1

0∂jf(tx)dt. Then:

f(x) = f(0) +

∫ 1

0

∂t[f(tx)]dt =∑j

xjgj(x).

Let φ ∈ C∞c (Rm) be identically 1 near x = 0. Let

hj := φgj + |x|−2xj(1− φ)f(x);

we may decompose

f(x) = φf(x) + (1− φ)f(x) =∑j

xjhj. (3.1)


Since φgj ∈ C∞c (Rm), φgj ∈ S. Since φ = 1 near x = 0, we see that

|x|−2xj(1− φ)f ∈ S.

This shows hj ∈ S. We take the Fourier transform of (3.1) to see:

f =∑j

F(xjhj) =∑j

i∂ξj hj.

We integrate by parts to see Tf(0) = i∑

j

∫∂ξj hj(ξ)dξ = 0.

Step 2: We compute that

f0(ξ) = (2π)−n/2∫e−ix·ξe−|x|

2/2dx

= (2π)−n/2e−|ξ|2/2

∫e−(x+iξ)·(x+iξ)/2dx.

We make a change of variables to replace x + iξ by x and then perform the standard

integration of the Gaussian. This shows that

f0(ξ) = e−|ξ|2/2

so f0 = f0. Since f0 is an even function, T (f0) = f0.

Step 3: Let f ∈ S be arbitrary. Decompose f = f(0)f0 + g where

g := f − f(0)f0.

Clearly g(0) = 0. We apply the first two steps to compute that

T (f)(0) = [f(0)T (f0) + T (g)](0)

= f(0)f0(0) + 0 = f(0)

Step 4: Let g(x) = f(x+ x0). We compute that

f(x0) = g(0) = T (g)(0) = (2π)−n/2∫e−ix·ξf(x+ x0)dxdξ

= (2π)−n/2∫e−ix·ξeix0·ξf(x)dxdξ = T (f)(x0).

Thus T (f) = f and F is a continuous bijective map from S → S. This completes the proof

of (a); The final two assertions of (b) now follow from (a) and the first two assertions.


We prove the first assertion of (c). We compute that

f · g = (2π)−n∫e−ix·ξf(x)e−iy·ξg(y)dxdy

= (2π)−n∫e−i(x−y)·ξf(x− y)e−iy·ξg(y)dxdy

= (2π)−n∫e−ix·ξf(x− y)g(y)dxdy.

These integrals are absolutely convergent so we can interchange the order of integration to

see

f · g = F(f ∗ g).

We replace f by f and g by g and use (a) to see:

(f · g)(−x) = F(f ∗ g).

We take the Fourier transform and use (a) to complete the proof of (c):

F(f · g)(−ξ) = (f ∗ g)(−ξ).

We begin the proof of (d) by computing

〈f , g〉L2 = (2π)−n/2∫f(x)e−ix·ξg(ξ)dxdξ = (2π)−n/2

∫f(x)e−ix·ξg(ξ)dξdx

= 〈f, g(−x)〉L2 .

We replace g by g and use (a) to complete the proof:

〈f , g〉L2 = 〈f,FFg(−x)〉L2 = 〈f, g〉L2 .

We shall also want the following simple but useful lemma.

Lemma 3.2.2. (Peetre’s Inequality) Let s ∈ R and let x, y ∈ Rn. Then

(1 + |x+ y|)s ≤ (1 + |y|)s(1 + |x|)|s|.

Proof. For the case s > 0 taking the sth power of the inequality

1 + |x+ y| ≤ 1 + |x|+ |y| ≤ (1 + |x|)(1 + |y|)

gives the result. For s < 0 we let u = x+ y and v = −y. Applying the our previous result

to u and v proves the lemma in this case.


3.2.2 Pseudo-differential operators on Rn

Let P be a differential operator on Rn given by P =∑|α|≤mA

αDα. If f is a compactly

supported function we may use the inverse Fourier transform to write

f(x) = (2π)−n/2∫

Rneix·ξf(ξ)dξ.

Using the Fourier transform to express differentiation as the transform of multiplication

(see part (b) of Lemma 3.2.1) our differential operator may then be written

Pf(x) = (2π)−n/2∫

Rneix·ξp(x, ξ)f(ξ)dξ

where

p(x, ξ) =∑|α|≤m

Aαξα.

The function p is called the (total) symbol of P . Notice that this is just like the princi-

pal symbol, except that it includes all the lower order coefficients. A pseudo-differential

operator is defined by replacing the polynomial symbol p with a more general function.

Definition 3.2.4. A smooth complex valued function p(x, ξ) on Rn×Rn, with compact x

support is called a symbol of order (at most) d if for each pair of multi-indices α, β there

exists a constant Cα,β such that

|DαxD

βξ p(x, ξ)| ≤ Cα,β(1 + |ξ|)d−|β|.

The space of such symbols is denoted Symd(Rn). Also, we denote the intersection of the

Symd(Rn) by Sym−∞(Rn) :=⋂d Symd(Rn).

The symbols define all the pseudo-differential (and differential) operators which are

constructed from them as follows.

Definition 3.2.5. Given p ∈ Symd(Rn) the associated operator P : C∞c (Rn) → S is

defined to be

Pf(x) :=

∫Rneix·ξp(x, ξ)f(ξ)dξ

=

∫Rnei(x−y)·ξp(x, ξ)f(y)dydξ.


Definition 3.2.6. The operators constructed in Definition 3.2.5 are called pseudo-differential

operators of order d on Rn. The space of all such operators is denoted ΨDOd(Rn).

Notation 3.2.1. We denote the map taking a pseudo-differential operator to its symbol by

σ : ΨDO(Rn)→ Sym(Rn). From here on in we will use the convention that p = σ(P ) and

q = σ(Q).

We will also define symbols and their associated operators on an open set U which has

compact closure. This will be necessary in the next section when we look at the algebraic

properties of symbols.

Definition 3.2.7. Let U be an open set with compact closure. The spaces Symd(U) ⊆Symd(Rn) and ΨDOd(U) ⊆ ΨDOd(Rn) are the subspace of symbols with compact x support

in U , and their associated operators respectively. We denote

Sym(U) =⋃d Symd, Sym−∞(U) =

⋂d Symd(U)

ΨDO(U) =⋃d ΨDOd(U), ΨDO−∞(U) =

⋂d ΨDOd(U).

Before investigating the algebraic properties of symbols we will need the following

lemma about symbols and their associated operators.

Lemma 3.2.3. Let U be an open set with compact closure in Rn. If p ∈ Symd(U) is the

symbol of P ∈ ΨDOd(U) then

(a) If Pf = 0 for all f ∈ C∞c (Rn), then p = 0.

(b) Let q(ζ, ξ) =∫

Rn e−ix·ζp(x, ξ)dx. Then

F(Pf)(ζ) =

∫Rnq(ζ − ξ, ξ)f(ξ)dξ, and

|DαζD

βξ q(ζ, ξ)| ≤Cα,β,k(1 + |ζ|)−k(1 + |ξ|)d−|β| ∀k = 0, 1, 2, . . .

Proof. The first assertion follows from the fact that the Fourier transform of C∞c (Rn) is

dense in L2(Rn). To prove (b) compute that

F(Pf)(ζ) =

∫Rneix·(ξ−ζ)p(x, ξ)f(ξ)dξdx

This integral is absolutely convergent as p has compact x support and f(ξ) decays to

infinite order. Thus we may interchange the order of integration and the first part of (b)

follows immediately.


By assumption p has compact x support and we are given the uniform estimates

|Dγxx

αDβξ p(x, ξ)| ≤ Cα,β,γ(1 + |ξ|)d−|β|.

Now using Lemma 3.2.1 we complete the proof of (b) by estimating

|ζγDαζD

βξ q(ζ, ξ)| =|

∫U

e−ix·ζDγxx

αDβξ p(x, ξ)dx|

≤Cα,β,γvol(U)(1 + |ξ|)d−|β|.

3.2.3 The algebraic properties of symbols

We have seen that pseudo-differential operators are defined by their total symbol. How-

ever, we have yet to define an analogue of the principal symbol for a general ΨDO. Thinking

back to our discussion on the principal symbol of a differential operator we would expect

that the principal symbol of a ΨDO would in some sense be the “highest order term” of its

total symbol. In order to formalise this notion we look at the algebraic structure of Sym.

In this section we first give an equivalence relation on Sym and ΨDO. Then we will define

a wider (but more or less equivalent) space of symbols and use these to prove results about

the algebraic properties of Sym.

When we come to discussing adjoints and compositions of pseudo-differential operators

we will need to specify support of the symbols p and restrict the domain of the associated

operators P . Throughout this section we let let U and O be open sets with compact closure

in Rn such that U ⊂ O.

Pointwise multiplication makes Sym(U) into a ∗-algebra, that is, an algebra with an

antiautomorphism ∗ : Sym(U)→ Sym(U) such that

(x+ y)∗ = x∗ + y∗, (xy)∗ = y∗x∗, 1∗ = 1 and (x∗)∗ = x.

The space Sym(U) has Sym−∞ as a two sided ideal. Also, if d1 ≤ d2, then

Symd1(U) ⊆ Symd2(O) and ΨDOd1(U) ⊆ ΨDOd2(O). (3.2)

So we can make the following definition.


Definition 3.2.8. (Equivalence classes of symbols) The equivalence relation ∼ on Sym is

defined by the following property:

p ∼ q if and only if p− q ∈ Sym−∞.

We also extend this relation to the associated operators

P ∼ Q if and only if P −Q ∈ ΨDO−∞.

The following definitions and lemma are essentially technical. Among other things, they

will allow us to prove Proposition 3.2.5 which deals with the compositions and adjoints of

symbols.

Definition 3.2.9. We donote by Sym′d a wider class of symbols r(x, ξ, y) satisfying:

(a) r ∈ C∞(Rm × Rm × Rm) has compact x and y support.

(b) For all 3-tuples of multi-indices (α, β, γ), there exist constants Cα,β,γ such that

|DαxD

βξD

γyr(x, ξ, y)| ≤ Cα,β,γ(1 + |ξ|)d−|β|.

Definition 3.2.10. If r ∈ Sym′d, define the associated operator R : S → C∞c (Rm) by:

R(f)(x) :=

∫Rnei(x−y)·ξr(x, ξ, y)f(y)dydξ.

The class of such operators will be denoted ΨDO′d.

The next lemma shows the relationship between the extended symbols defined above

and Sym. The two spaces are essentially the same, aside from some technicalities regarding

domains.

Lemma 3.2.4. (Properties of extended symbols) Let U be an open subset of Rn with com-

pact closure, P ∈ ΨDOd an operator and p its symbol.

(a) If P ∈ ΨDOd(U) there exists R ∈ ΨDO′d(U) such that Rf = Pf for all f ∈ C∞c (U).

(b) If R ∈ ΨDO′d(U) then R ∈ ΨDOd(U) and

σ(R)(x, ξ) ∼ [∑α

∂αξDαy r(x, ξ, y)/α!]|x=y.


(c) If P ∈ ΨDOd(U) and if Pf = 0 for all f ∈ C∞c (U) then p ∼ 0.

Proof. Let P ∈ ΨDOd(U). Choose a function φ ∈ C∞c (Rn) such that φ|U = 1 and let

r(x, ξ, y) = p(x, ξ)φ(y) ∈ S ′d(U)

Since φf = f for f ∈ C∞c , Rf = Pf . This proves (a).

Now let R ∈ ΨDO′d(U). Taking the Fourier transform of r in the variable y define

q(x, ξ, ζ) =

∫Rne−iy·ζr(x, ξ, y)dy.

Using Lemma 3.2.1 we see that F(r · f) = F(r) ∗ F(f). Thus∫Rne−iy·ξr(x, ξ, y)f(y)dy =

∫Rnq(x, ξ, ξ − ζ)f(ζ)dζ.

Noting that r has compact y support we apply Part (b) of Lemma 3.2.3 to get the uniform

estimate

|q(x, ξ, ζ)| ≤ Ck(1 + |ξ|)d(1 + |ζ|)−k (3.3)

for all k ∈ N. As f ∈ S|f(ζ)| ≤ Ck(1 + |ζ|)−k.

So we may use Peetre’s inequality (Lemma 3.2.2) to estimate

|q(x, ξ, ξ − ζ)f(ζ)| ≤Ck(1 + |ξ|)d(1 + |ξ − ζ|)−k(1 + |ζ|)−k

≤C ′k(1 + |ξ|)|d|−k(1 + |ζ|)|d|−k.

This shows that q(x, ξ, ξ − ζ)f(ζ) is absolutely integrable. Define

p(x, ζ) =

∫Rneix·(ξ−ζ)q(x, ξ, ξ − ζ)dξ

We now interchange the order of integration to see that

Rf(x) =

∫Rnei(x−y)·ξr(x, ξ, y)f(y)dydξ

=

∫Rneix·ξq(x, ξ, ξ − ζ)f(ζ)dζdξ

=

∫Rneix·ξq(x, ξ, ξ − ζ)f(ζ)dξdζ

=

∫Rneix·ζp(x, ζ)f(ζ)dζ


So in order to show that R ∈ Ψd(U) we must show that p ∈ Symd(U).

It is immediate that p has compact x support. Changing variables we see that

p(x, ζ) =

∫Rneix·ξq(x, ξ + ζ, ξ)dξ.

Using (3.3) and Peetre’s inequality gives

|q(x, ξ + ζ, ξ)| ≤Ck(1 + |ξ + ζ|)d(1 + |ξ|)−k

≤Ck(1 + |ζ|)d(1 + |ξ|)|d|−k.

This is integrable in ξ for large k so |p(x, ζ)| ≤ C(1+|ζ|)d. Using the bounds for derivatives

of r and estimating similarly to above gives bounds for the derivatives

|DαxD

βζ q(x, ξ + ζ, ξ)|

which show that p ∈ Symd. Thus R ∈ ΨDO(U).

Now using Talyor expansion on the middle variable of q(x, ξ + ζ, ξ) we get

q(x, ξ + ζ, ξ) =∑|α|≤k

∂αζ q(x, ζ, ξ)ξα/α! + qk(x, ζ, ξ). (3.4)

The remainder qk decays to arbitrarily high order in (ξ, ζ) and after integration gives rise

to a symbol of arbitrarily high negative order which may therefore be ignored. To complete

the proof of (b) we integrate (3.4) and use Lemma 3.2.1 to check that

p(x, ζ) =∑|α|≤k

∫Rneix·ξ∂αζ q(x, ζ, ξ)ξ

αdξ/α! + remainder

=[∑|α|≤k

∂αζDαy r(x, ζ, y)/α!]|x=y + remainder

Now let p ∈ Symd(U), define P ∈ ΨDOd(U) and suppose that Pf = 0 for all f ∈C∞c (U). Choose a function φ ∈ C∞c (U) such that φ|supp(p) = 1 and let Pφ(f) = P (φf) have

symbol p(x, ξ)φ(y). This defines a pseudo-differential operator R with symbol r so that by

(b):

r ∼∑α

∂αξ pDαyφ ∼ p

since φ = 1 on the support of p. As R = 0, r = 0 by Lemma 3.2.3.


With our technical lemma completed we are able to prove the following lemma. Note

that in what follows we must restrict the domains in order to apply Lemma 3.2.4. However,

once we pass to compact manifolds such restrictions will be unneccesary.

Lemma 3.2.5. (Adjoints and compositions) Let P ∈ ΨDOd(U), Q ∈ ΨDOe(U), where U

and O are open subset of Rn such that O has compact closure and U ⊂ O. Then

(a) There exists P ′ ∈ ΨDOd(O) such that 〈Pf, g〉L2 = 〈f, P ′g〉L2 for f ∈ C∞c (U) and

g ∈ C∞c (Rn). Furthermore,

σ(P ′) ∼∑α

∂αξDαxp/α!,

where p is the pointwise complex conjugation of p.

(b) There exists R ∈ ΨDOd+e(U) such that Rf = PQf for all f ∈ C∞c (U). Furthermore,

σ(R) ∼∑α

∂αξ pDαxq/α!.

Proof. Let f · g = fg be the Hermitian inner product on sections of the trivial line bundle

C → Rn. Choose a function φ ∈ C∞c (Rn) such that φ|U = 1. We then have f(y) = φf(y)

for f ∈ C∞c (U) so

〈Pf, g〉L2 =

∫ei(x−y)·ξp(x, ξ)φ(y)f(y) · g(x)dydξdx

=

∫f(y) · ei(y−x)·ξp(x, ξ)φ(y)g(x)dydξdx

This integral is not absolutely convergent. However, by approximating p(x, ξ) by func-

tions with compact ξ support, we can justify the use of Fubini’s theorem to replace dydξdx

by dxdξdy hence

〈Pf, g〉L2 =

∫Rnf(y) · ei(y−x)·ξp(x, ξ)φ(y)g(x)dxdξdy

=〈f, P ′g〉L2

where

P ′g(y) :=

∫Rnei(y−x)·ξp(x, ξ)φ(y)g(x)dxdξ

Note that if we reverse the roles of x and y we have that P ′ ∈ ΨDO′d as in Lemma 3.2.3.

Thus P ′ ∈ ΨdDO(O) and since φ|supp(p) = 1, Lemma 3.2.4 applies to prove the second part

of (a).


Now using Lemma 3.2.3 to expand

F(Qf)(ζ) =

∫Rne−iy·ξr(ζ − ξ, ξ)f(y)dydζ.

for r(η, ξ) =∫

Rn e−iz·ηq(z, ξ)dz. So we may write

PQf(x) =

∫Rneix·ζp(x, ζ)e−iy·ξr(ζ − ξ, ξ)f(y)dydξdζ. (3.5)

We use Lemma 3.2.3 and Peetre’s inequality to estimate

|r(ζ − ξ, ξ)| ≤Ck(1 + |ζ|)k(1 + |ξ|)e−k, and (3.6)

|DαηD

βζ r(η, ζ − η)| ≤Ck,α,β(1 + |η|)|e|+|β|−k(1 + |ζ|)e−|β|. (3.7)

Thus the dydξ integral from (3.5) is absolutely convergent and we may apply Fubini’s

theorem. Doing this along with a change of variables shows that

PQf(x) =

∫Rnei(x−y)·ζp(x, ζ)eiy·(ζ−ξ)r(ζ − ξ, ξ)f(y)dξdydζ

=

∫Rnei(x−y)·ζp(x, ζ)eiy·ηr(η, ζ − η)f(y)dηdydζ (3.8)

Using (3.6) and (3.7) we have

p(x, ζ)t(ζ, y)φ(y) ∈ Sym′d+e, where

t(ζ, y) :=

∫Rneiy·ηr(η, ζ − η)dη ∈ C∞(Rn × Rn)

Now using (3.8) gives

PQf(x) =

∫Rnei(x−y)·ζp(x, ζ)t(ζ, y)φ(y)f(y)dydζ

Applying Lemma 3.2.4 completes the proof of the first assertion in part (b).

Using Taylor expansion on the middle variable of r we get

r(η, ζ − η) =∑|α|≤k

(−1)|α|∂αζ r(η, ζ)ηα/α! + rk(η, ζ). (3.9)

the remainder term rk decays to arbitrarily high order in (η, ζ) and after integration gives

a symbol of arbitrarily high negative order which may therefore be ignored. Integrating

(3.9) gives

t(ζ, y)φ(y) ∼∑α

(−1)|α|∫

Rneiy·η∂αζ r(η, ζ)ηαdη · φ(y)/α!


Let q1 be the symbol of Q1 = Q′ and let q2 be the symbol of Q′1. Since φ|supp(p) = 1 we

have

t(ζ, y)φ(y) ∼∑α

(−1)|α|Dαy ∂

αζ q(y, ζ)φ(y) ∼ q1(y)φ(y)

p(x, ζ)t(ζ, y)φ(y) ∼ p(x, ζ)q1(y, ζ)φ(y)

σ(PQ) ∼∑α

[∂αζDαy p(x, ζ)q1(y, ζ)/α!]|x=y

by the Fourier inversion formula and (a). We now use Leibniz’s formula and apply (a)

again to see

σ(PQ) ∼∑β

∂βζ p(x, ζ)Dβx [∑γ

Dγx∂

γξ q1/γ!]/β!

∼∑β

∂βζ p(x, ζ)Dβxq2(x, ζ)/β! (3.10)

For f, g ∈ C∞c (U), we have

〈Qf, g〉L2 = 〈f,Q1g〉L2 = 〈Q2f, g〉L2

Using Lemma 3.2.4 shows that q ∼ q2 on U and so, noting that p has compact support in

U , we may replace q2 by q in equation (3.10).

Before stating the main theorem of this section we will prove one more result which

will be used in the next chapter.

Lemma 3.2.6. Let pj ∈ Symdj for dj −∞. Let U ⊂ O. Then there exists p ∈ Symd1(O)

such that p ∼∑

j pj .

Proof. Let φ be a smooth function such that

0 ≤ φ ≤ 1, φ(ξ) = 0 for |ξ| ≤ 1, and φ(ξ) = 1 for |ξ| ≥ 2.

We will use φ to remove support near 0. Let tj 0 and define

p(x, ξ) =∑j

φ(tjξ)pj(x, ξ).

For a fixed ξ, φ(tjξ) = 0 for all but a finite number of j values; so the above sum is well

defined and smooth in (x, ξ). For j > 1,

|pj(x, ξ)| ≤ Cj(1 + |ξ|)dj = Cj(1 + |ξ|)d1(1 + |ξ|)dj−d1 .


By increasing the size of |ξ| we are able to reduce (1 + |ξ|)dj−d1 ; making it as small as we

like. Thus by passing to a subsequence of tj we may assume that for j > 1,

|φ(tjξ)pj(x, ξ)| ≤ 2−j(1 + |ξ|)d1 .

This shows that |p(x, ξ)| ≤ (C1 + 1)(1 + |ξ|)d1 . Using a similar argument we may bound

derivatives of p. A diagonalisation argument is then used on the resulting subsequences,

showing that p ∈ Symd1(O).

Now we apply the same argument to construct a subsequence so that∑j≥2

φ(tjξ)pj(x, ξ) ∈ Symd2 .

Continuing in this fashion and using a diagonalization argument on the resulting subse-

quences we construct a final sequence (which we again denote tj) such that∑j≥k

φ(tjξ)pj(x, ξ) ∈ Symdk .

As pj(x, ξj)− φ(tjξ)pj(x, ξj) ∈ C∞c (O), it belongs to Sym−∞(O). Therefore for all k

p−∑

1≤j≤k

pj ∈ Symdk(O).

Lemmas 3.2.5 and 3.2.6 prove our main result about the algebraic structure of ΨDO

and Sym.

Proposition 3.2.1. We may regard ΨDO and Sym as filtered ∗-algebras i.e.

ΨDOd ΨDOe ⊂ ΨDOd+e and (ΨDOd)∗ = ΨDOd

Symd Syme ⊂ Symd+e and (Symd)∗ = Symd

Furthermore, Sym is complete in the sense of Lemma 3.2.6.

Using Proposition 3.2.1 we are able to define the principal symbol of a pseudo-differential

operator.

Definition 3.2.11. Let π : Symd → Symd/Symd−1 be the natural projection. We define

the principal symbol of a pseudo-differential operator of order d by

σ = σ π : ΨDOd → Symd/Symd−1.


In Section 3.3 we will lift pseudo-differential operators to manifolds. We will then show

that the principal symbol of a general pseudo-differential operator is globally we defined,

as in our earlier discussion of differential operators.

3.2.4 ΨDO defined by kernels

We take a slight digression to introduce ΨDO defined by kernels. We will not need to

consider them explicitly but the following three lemmas will be used as tools, first in our

proof that the principal symbol of a ΨDO is a global invariant and then in Chapter 4.

The ΨDO defined by kernels are constructed by integrating against a kernel function.

Definition 3.2.12. Let K(x, y) ∈ C(Rm × Rm) satisfy ‖K‖∞ <∞. Let

P (K)(f)(x) :=

∫K(x, y)f(y)dy for f ∈ S;

K is called a kernel function and P (K) is the operator defined by this kernel.

In our first lemma we show that an operator defined by a smooth compactly supported

kernel lies in ΨDO−∞.

Lemma 3.2.7. If K ∈ C∞c (Rn × Rn) then P (K) ∈ ΨDO−∞.

Proof. Choose φ ∈ C∞c (Rn) with∫

Rn φ(ξ)dξ = 1. Let

r(x, ξ, y) = ei(y−x)·φ(ξ)K(x, y) ∈ Sym′−∞

by Lemma 3.2.4, R ∈ ΨDO−∞. It is immediate that

P (K)(f)(x) =

∫Rnei(x−y)·ξr(x, ξ, y)f(y)dydξ = R(f)(x).

The next lemma provides a converse (if we take d to be −∞).

Lemma 3.2.8. Let U ∈ Rn, r ∈ Sym′d(U) for d < −n− k and let

K(x, y) =

∫Rnei(x−y)·ξr(x, ξ, y)dξ.

Then K ∈ Ckc (Rn × Rn) and Rf(x) =

∫Rn K(x, y)f(y)dy for all f ∈ C∞c (U).


Proof. Using Leibniz’s rule we compute that

DαxD

βyK(x, y) =

∑α=α1+α2,β=β1+β2

(α1

α2

)(β1

β2

)

· i|α1|−|β1|∫

Rnei(x−y) · ξα1+β1Dα2

x Dβ2y r(x, ξ, y)dξ.

Estimating gives

|ξα1+β1Dα2x D

β2y r(x, ξ, y)| ≤ Cα1,α2,β1,β2(1 + |ξ|)d+|α1|+|β1|

so this will be integrable for |α| + |β| ≤ k. Thus K is well defined and is Ck. Applying

Fubini’s theorem to the integral defining K we interchange the order of integration, showing

that P (K) = R.

Finally, we give a lemma which will be used in the proof that the principal symbol is

invariant under coordinate change.

Lemma 3.2.9. Let r ∈ Sym′d(U). Suppose there exists ε > 0 such that r(x, ξ, y) = 0 for

|x− y| < ε. Then R ∈ ΨDO−∞ and there exists K ∈ C∞c so that R = P (K).

Proof. We would like to define

K(x, y) =

∫Rnei(x−y)·ξr(x, ξ, y)dξ.

However, this integral may not converge. Let

∆ξ := −∑ν

(∂νξ )2

be the ξ Laplacian. Since ∆ξei(x−y)·ξ = |x− y|2ei(x−y)·ξ, we can integrate by parts k times

to give

Rf(x) =

∫Rnei(x−y)·ξ|x− y|−2k∆k

ξr(x, ξ, y)f(y)dydξ

this is well defined since we may restrict the integral to |x − y| ≥ ε. This formal process

can be justified by approximating r by a sequence of functions with compact ξ support.

Define

Kk(x, y) :=

∫Rnei(x−y)·ξ|x− y|−2k∆k

ξr(x, ξ, y)dξ.

If k is large, ∆kξr decays to arbitrarily high order in ξ so the integral converges. Furthermore

Kk is independent of k. The same argument as that given in the proof of Lemma 3.2.8

now shows that K is Ck for any k.


3.3 Pseudo-differential operators on manifolds

Having looked at pseudo-differential operators on Rn we now define them on manifolds.

Throughout this section take X to be a compact manifold. We begin by lifting the notions

of previous chapters. First we check the effect of coordinate changes on ΨDO.

Let

h : U → U ′

be a diffeomorphism. We can pullback objects from U ′ to U via h∗ and push them fowards

using h∗. For example, if f ∈ C∞c (U), f ′ ∈ Cc(U ′), and if P is a linear operator in C∞c (U),

then

h∗(f ′)(x) := f ′(h(x)),

h∗(f) := (h∗)−1f, and

(h∗P ) := h∗ P h∗.

Let dh be the Jacobian matrix. We regard (x, ξ) as giving coordinates to the cotangent

bundle T ∗U and extend h to a diffeomorphism which will continue to denote h from T ∗U

to T ∗U ′:

h(x, ξ) := (x′, ξ′) = (h(x), (dh(x)−1)tξ).

If p ∈ Symd(U), we transform the symbol under this correspondence:

h∗(p)(x′, ξ′) := p(x, ξ).

Definition 3.3.1. Let P : C∞(X) → C∞(X) be a linear operator. If (O, h), h : O →Rn is a coordinate chart and if φ, ψ ∈ C∞c (O), the operator φPψ on C∞c (O) is called

a localization of P . We push-forward to define h∗(φPψ) on C∞c (h(O)). We say P is a

pseudo-differential operator of order d on X if

h∗(φPψ) ∈ ΨDOd(h(O)) for all φ, ψ,O, h.

We denote the set of such operators by ΨDOd(X).

Note that this definition assumes that an operator is already globally defined on X,

and also that the localisation might be quite different from chart to chart. Next we define

the space of symbols.

3.3. PSEUDO-DIFFERENTIAL OPERATORS ON MANIFOLDS 55

Definition 3.3.2. Let φ ∈ C∞c (O), we define

Symd(X) := p(x, ξ) ∈ C∞(T ∗X) : h∗(φp) ∈ Symd(hO) ∀O, h, φ.

This definition also assumes a symbol is already globally defined. However, if we do

have a symbol p ∈ Symd(X) as defined above we can always construct the associated

pseudo-differential operator using a partition of unity.

We would now like to define the principal symbol of an operator on a manifold. To

do this on a chart (O, h) we localise the operator and push it forward with h∗, take the

principal symbol of the localised operator and then pull this back via h∗. However, we

need to check that this is globally well defined. We do this before giving the definition of

the principal symbol.

Lemma 3.3.1. (Invariance of σ) Let h be a diffeomorphism from U to U ′. Let U1 ⊂U1 ⊂ U . If p ∈ Symd(U1), and P ∈ ΨDOd(U1), then there exists P ′ ∈ ΨDOd(U ′1) with

σ(P ′) = h∗(p) so that P ′ = h∗(P ) on C∞(U ′1).

Proof. First we localise the problem. Let φν be a partition of unity and let Pνµ = φνPφµ

so that

P =∑ν,µ

Pνµ.

If the supports of φν and φµ are disjoint, then Pνµ ∈ ΨDO−∞ and has a smooth kernel

Kνµ(x, y) by Lemma 3.2.9. Therefore h∗Pνµ is also given by a smooth kernel and is a

pseudo-differential operator by Lemma 3.2.7. Thus we may restrict our attention to pairs

(ν, µ) such that the supports of φν and φµ intersect. This allows us to assume P is defined

by a symbol p(x, ξ, y) where p has small support in x and y. We first suppose that h is

linear to motivate the construction in the general case. Let ht be the transpose matrix.

Let

x′ = hx, y′ = hy, and htξ′ = ξ.

Then:

(x′ − y′) · ξ′ =(x′ − y′) · (h−1)tξ = h−1(x′ − y′) · ξ = (x− y) · ξ

dy′dξ′ =| det(h)|dy| det(h)−1|dξ = dydξ.


Let f ′ ∈ C∞c (U ′) and let p′(x′, ξ′, y′) = p(x, ξ, y). We compute that

(h∗P )(f ′)(x′) =

∫Rnei(x−y)·ξp(x, ξ, y)f ′(hy)dydξ

=

∫Rnei(x

′−y′)·ξ′p(x′, ξ′, y′)f ′(h′y)dy′dξ′.

This proves P ′ = h∗P is a pseudo-differential operator on U ′ with symbol p′; there is

no localization necessary in this special case. If h is not linear, the situation is more

complicated. Let

T (x′, y′) =

∫ 1

0

d(h−1)(tx′ + (1− t)y′)dt;

T (x′, y′) is a smooth square matrix valued function of x′ and y′. We decompose

x− y =h−1(x′)− h−1(y′) =

∫ 1

0

∂th−1[tx′ + (1− t)y′]dt

=[

∫ 1

0

d(h−1)(tx′ + (1− t)y′)dt](x′ − y′) = T (x′, y′)(x′ − y′).

If x′ = y′, then T (x′, y′) = d(h−1)(x′) is invertible since h is a diffeomorphism. We localize

using a partition of unity and suppose henceforth that the supports are small enough so

that T (x′, y′) is invertible at all points of interest, using the inverse function theorem. We

make a varying, linear change of fibre coordinates to define

ζ := T (x′, y′)tξ;

(x− y) · ξ = T (x′, y′)(x′ − y′) · ξ = (x′ − y′) · ζ.

Let J = | det(dh−1)| = | det(T (x′, x′))|. Then

dydξ = J | det(T (x′, y′))|−1dy′dζ.

Let p1(x′, ζ, y′) = p(x, ξ, y). If p has compact ξ support, then p1 has compact ζ support

and the integrals in question are absolutely convergent. We use Fubini’s theorem twice to

change the order of integration and then use the change of variable formula to compute

3.3. PSEUDO-DIFFERENTIAL OPERATORS ON MANIFOLDS 57

that

P (f ′)(x′) =

∫Rnei(x−y)·ξp(x, ξ, y)f ′(hy)dydξ (3.11)

=

∫Rnei(x−y)·ξp(x, ξ, y)f ′(hy)dξdy

=

∫Rnei(x

′−y′)·ζp1(x′, ζ, y′)J | det(T (x′, y′))|−1f ′(y′)dζdy′

=

∫Rnei(x

′−y′)·ζp1(x′, ζ, y′)J | det(T (x′, y′))|−1f ′(y′)dy′dζ. (3.12)

Both (3.11) and (3.12) are defined for general P ; we approximate p by functions of compact

ξ support and use a limiting argument to see (3.11) and (3.12) agree in general; even

though the second and third integrals in the above computation may not be defined. We

apply Lemma 3.2.4 to see P ′ is a pseudo-differential operator. We complete the proof by

computing:

σ(P ′) =[p1(x′, ζ, y′)| det(dh−1(x′) det(T (x′, y′))−1|]|x′=y′

=p(h−1(x′), d(h)tζ).

We can now globally define the principal symbol of a pseudo-differential operator.

Definition 3.3.3. Let P ∈ ΨDOd(X). If x ∈ X, choose a coordinate chart (O, h) so that

x ∈ O. Choose a function φ ∈ C∞c (O) which is identically 1 near x and define the principal

symbol

σ(P )(x) := h∗σ(h∗(φPφ)) ∈ Symd(X)/Symd−1(X).

By Lemma 3.3.1 we see that the principal symbol of an operator is invariantly defined

on the cotangent bundle, just as in our discussion of differential operators in Section 3.1.

3.3.1 ΨDO on vector bundles

Thus far we have been working only with pseudo-differential operators on line bundles.

We wish to define ellipticity by analogy with differential operators. However, in order to

do this we must consider operators which act on sections of vector bundles. The transition


to vector bundles is quite straight foward. Let X be a manifold and V,W → X be smooth

complex vector bundles of some rank k. We say that

P : Γ(V )→ Γ(W ) (3.13)

is a pseudo-differential operator of order d if for all φ, ψ ∈ C∞(U) the localisation φPψ

is given by a matrix of d-th order pseudo-differential operators of order d. In this case

the operator is associated to a matrix of symbols. The results about pseudo-differential

operators which we have proved generalize immediately to this situation.

Finally we may define ellipticity for a pseudo-differential operator.

Definition 3.3.4. Let X be a manifold, we say that P ∈ ΨDO(X) is elliptic if for every

coordinate chart (O, h) and every φ, ψ ∈ C∞c (O), h∗(φPψ) is invertible outside a compact

set containing ξ = 0 on the set where h∗(φψ) ≥ 1.

Observation 3.3.1. Let X be a manifold, P ∈ ΨDO(X) and p = σ(P ). Then P is elliptic

if and only if there exists q ∈ Sym−d(X) such that pq − Id ∈ Symd−1(X). By taking

p(x, ξ) = (1 + |ξ|2)d/2, we can construct elliptic operators of all orders.

3.4 The Symbol Class

Recalling our characterisation of K(X) in terms of complexes we have the following

result, just as in the case of a differential operator.

Lemma 3.4.1. Let X be a compact manifold and π : T ∗X → X the cotangent bundle. If

P ∈ ΨDOd(X) is elliptic, then [π∗E, π∗F ;σ(P )] represents an element of K(T ∗X).

We now obtain the converse result which was our motivation for developing the theory

of pseudo-differential operators. The symbols of elliptic pseudo-differential operators over

a manifold exhaust the K-theory of the cotangent bundle.

Lemma 3.4.2. Let X be a compact manifold. Given an element u ∈ K(T ∗X) and a real

number d > 0, there exists an operator P ∈ ΨDOd(X) such that [σ(P )] = u.

Proof. From section 2.2 we know that any u ∈ K(TX) may be represented by a complex

of the form [π∗E, π∗F ;α] where α is homogeneous of degree d. Choosing local coordinates

which trivialize E,F and TX, let p(x, ξ) = ψ(ξ)αm(x, ξ) where ψ(ξ) a smooth function

3.5. CLASSICAL ΨDO 59

which is zero in a neigbourhood of zero. The introduction of ψ eliminates any discontinuity

that may occur at the origin if d < 0. If we then let P be the pseudo-differential operator

associated to p, we see that P is an elliptic operator with [σ(P )] = u.

3.5 Classical ΨDO

For our proof of the Atiyah-Singer index theorem we will need to consider the following

class of ΨDO.

Definition 3.5.1. Let E,F be vector bundles over a manifold X. An operator P ∈ΨDOm(E,F ) is called classical if, over every compact subset K ⊂ X, there is a constant

c > 0 such that σtξ(P ) = tmσξ(P ) for all ξ ∈ T ∗X, ‖ξ‖ ≥ c and t ≥ 1. We will denote the

set of such operators by ΨCOm(E,F ).

Definition 3.5.2. For operators P ∈ ΨCOm(E,F ) we can define the asymptotic prin-

cipal symbol

σξ(P ) = limt→∞

σtξ(P )

tm

for all ξ ∈ SX := ξ ∈ T ∗X : ‖ξ‖ = 1.

Proposition 3.5.1. Let E,F be vector bundles over a compact manifold X. The map

σ : ΨCO(E,F )→ Γ(Hom(π∗E, π∗F )) is surjective. Here π∗E, π∗F are the pull backs of E

and F to the bundle SX.

Proof. Let [π∗E, π∗F ; s] ∈ K(T ∗X). Extend s smoothly to all of T ∗X to give a symbol

s which is homogeneous of degree m in ξ for ‖ξ‖ ≥ 1. Given local trivializations of

E and F over a coordinate chart U on X, one can then construct an operator P in U

with principal symbal s. Let Uj be a finite covering of X by such coordinate charts

and let χ2j be a partition of unity subordinate to this covering. Then the operator

P =∑χjPχj ∈ ΨCO(E,F ) has principal symbol σ(P ) = s and so σ(P ) = s.

The above proposition shows that we can always find a classical pseudo-differential

operator to represent a given symbol class in K(T ∗X)


Chapter 4

Analytic Properties of Elliptic ΨDO

We have shown that the principal symbol of an elliptic operator determines a class in

K-theory. We would now like to know about properties of these operators which are well-

defined at the symbolic level. The purpose of this chapter is to perform the analysis which

we will require later when we show that the index of an elliptic operator is determined by

its principal symbol. We extend our operators to Sobolev spaces and then prove two main

results. The first result is the existence of an approximate inverse for any elliptic ΨDO

between Sobolev spaces, later this will be used to show that an extension of an elliptic

operator has a well-defined index. The second result, which follows from the first, is that

elliptic operators are hypo-elliptic. This will allow us to show that an elliptic operator has

a well defined index, that is an index which is independent of any extension.

4.1 Sobolev Spaces

Sobolev spaces are Hilbert spaces containing functions with a certain degree of smooth-

ness. They will provide the appropriate setting for our analysis of elliptic ΨDO.

Definition 4.1.1. For s ∈ R and f ∈ S(Rn) the Sobolev s-norm is defined by

‖f‖2s = (2π)−

n2

∫Rn

(1 + |ξ|2)s|f(ξ)|2dξ

The Sobolev space Ls(Rn) is defined as the completion of S(Rn) with respect to the

Sobolev s-norm.

61

62 CHAPTER 4. ANALYTIC PROPERTIES OF ELLIPTIC ΨDO

Sobolev spaces are a natural setting in which to look for solutions to differential equa-

tions. However, these solutions need not be smooth. In order to obtain smooth solutions

from Sobolev spaces we will define the sup norm, another norm which measures derivatives.

This will then be related to the Sobolev norm.

Definition 4.1.2. Let Ck(Rm) be the space of functions with continuous partial derivatives

up to order k. If k ∈ N and if f ∈ S then we define

‖f‖∞,k := supx∈Rm

∑|α|≤k

|Dαxf |.

The next lemma shows the relationship between the Sobolev s-norm and the sup norm.

Lemma 4.1.1. (Sobolev Lemma) Let k ∈ N and let s > k + 12m. If f ∈ Ls(Rn), then f is

Ck and ‖f‖∞,k ≤ C‖f‖s.

Proof. Suppose first that k = 0 and that f ∈ S. We compute that

f(x) = (2π)−n2

∫eix·ξf(ξ)dξ

= (2π)−n2

∫eix·ξf(ξ)(1 + |ξ|2)s/2(1 + |ξ|2)−s/2dξ.

We use the Cauchy-Schwarz inequality to estimate:

|f(x)|2 ≤ (2π)−n2 ‖f‖2

s

∫(1 + |ξ|2)−

12dξ.

Since 2s > m, (1 + |ξ|2)−s is integrable. We take the supremum over x ∈ Rn to see that

‖f‖2∞ ≤ C‖f‖2

s (4.1)

for f ∈ S. Elements of Ls are limits in the ‖ · ‖s norm of elements of S. The uniform limit

of continuous functions is continuous so the elements of Ls are continuous and the norm

estimate (4.1) extends to Ls. If k > 0, we use the estimate

‖Dαxf‖∞,0 ≤ C‖Dα

xf‖s−|α| ≤ C‖f‖s

for |α| ≤ k and s − k > 12m to conclude that ‖f‖∞,k ≤ C‖f‖s for f ∈ S. We then argue

that the elements of Ls are Ck and (4.1.1) extends to Ls.

4.1. SOBOLEV SPACES 63

The following lemma shows that the identity mapping on C∞(K), for some compact

K, extends to a compact map from Ls(Rn)→ Ls(Rn) for t < s. This lemma will be used

to prove part its counterpart on manifolds, part (a) of Lemma 4.3.2. In the next chapter

Lemma 4.3.2 will be used to show that elliptic operators are Fredholm operators.

Lemma 4.1.2. (Rellich Lemma) Let K be a compact subset of Rm and let fn belong to

C∞c (K). Suppose that ‖fn‖s ≤ C for some constant C. Then there exists a subsequence

fn which converges in Lt for any t < s.

Proof. Choose g ∈ C∞c (Rm) which is identically 1 of some neighbourhoodof K. Then

gfn = fn so by Lemma 3.2.1, g is an identity for the convolution product on the Fourier

transform of C∞c (K):

fn = g ∗ fn.

Since ∂ξj (g ∗ fn) = ∂ξj (g) ∗ fn,

|∂ξj (fn(ξ)| ≤∫|∂ξj g(ξ − η)| · |fn(η)|dη. (4.2)

Define a continuous function hj by:

hj(ξ) := [

∫|∂ξj g(ξ − η)|2(1 + |η|2)−sdη]1/2.

We apply the Cauchy-Schwarz inequality to (4.2) to get

|∂j fn(ξ)| ≤ ‖fn‖s · hj(ξ).

We can estimate |(f)n(ξ)| similarly. This implies that the sequence fn is a uniformly

bounded equi-continuous family of functions on compact ξ subsets. We apply the Arzela-

Ascoli theorem to extract a subsequence which converges uniformly on compact subsets, we

again label this sequence by fn. Our proof is completed by checking that this subsequence

converges in Lt for any t < s.First we compute that

‖fj − fk‖2t =

∫|fj − fk|2(1 + |ξ|2)tdξ.

We decompose this integral into two pieces. Choose r so that

4C2(1 + r2)t−s ≤ 1

2ε.


We can then estimate∫|ξ|>r

∫|fj − fk|2(1 + |ξ|2)tdξ ≤ (1 + r2)t−s

∫|ξ|≥r|fj − fk|2(1 + |ξ|2)sdξ

≤ (1 + r2)t−s|fj − fk|2s ≤ 4C2(1 + r2)t−s ≤ 1

2ε

Lemma 4.1.3. (L2 pairing)

(a) |〈f, g〉L2| ≤ ‖f‖s‖g‖−s for f, g ∈ S(Rn)

(b) Given f ∈ S(Rn), there exists g ∈ S(Rn) so that 〈f, g〉L2 = ‖f‖s‖g‖−s

Proof. The first result follows by estimation with Holders inequality. To prove the second,

choose g such that

g = (1 + |ξ|2)sf ∈ S(Rn)

and note that 〈f, g〉 = 〈f , g〉 = ‖f‖2s and ‖g‖−s = ‖f‖s.

Observation 4.1.1. Lemma 4.1.3 implies that an equivalent norm for Ls is given by

supg∈L−s,g 6=0

|〈f, g〉L2 |‖g‖−s.

We also have the interpolation estmate

Lemma 4.1.4. Let s > t > u and let ε > 0 be given and f ∈ S. Then

‖f‖t ≤ ε‖f‖s + C(ε)‖f‖u.

Proof. This follows from the estimate

(1 + |ξ|2)t ≤ ε(1 + |ξ|2)s + C(ε)(1 + |ξ|2)u.

4.2 Elliptic operators on Rn

We now extend pseudo-differential operators on Rn to Sobolev spaces.

4.2. ELLIPTIC OPERATORS ON RN 65

Lemma 4.2.1. Let P ∈ ΨDOd(Rn). For any s, there exists C = C(s, P ) such that

‖Pf‖s−d ≤ C‖f‖s for all f ∈ S(Rn) and P extends to a continous linear map from Ls to

Ls−d.

Proof. Let p ∈ Symd be the symbol of P , and q(ζ, ξ) =∫e−ix·ζp(x, ξ)dx. Using Lemma

4.1.3 we see that

‖Pf‖s−d = supg∈S,g 6=0

|〈Pf, g〉|‖g‖−1d−s.

It is now enough to show that

|〈Pf, g〉| ≤ C‖f‖s‖g‖d−s∀f, g ∈ S(Rn). (4.3)

Let g1 = g. Using Plancherel’s formula (3.2.1) gives

〈Pf, g〉 = 〈FPf, g1〉 =

∫Rn×Rn

q(ζ − ξ, ξ)f(ξ)g1(ζ)dξdζ.

Now define

K(ζ, ξ) = q(ζ − ξ, ξ)(1 + |ξ|)−s(1 + |ζ|)s−d.

We may then write

〈Pf, g〉 =

∫Rn×Rn

K(ζ, ξ)f(ξ)(1 + |ξ|)sg1(ζ)(1 + |ζ|)d−sdξdζ.

Applying the Cauchy-Schwartz inequality gives

|〈Pf, g〉| ≤ [

∫Rn×Rn

|K(ζ, ξ)f(ξ)2(1 + |ξ|)2s|dξdζ]1/2

· [∫

Rn×Rn|K(ζ, ξ)g1(ζ)2(1 + |ζ|)2d−2s|dξdζ]1/2.

This shows that (4.3) will follow from the estimates∫Rn|K(ζ, ξ)|dζ ≤ C and

∫Rn|K(ζ, ξ)|dξ ≤ C (4.4)

Using (b) we estimate

|K(ζ, ξ)| ≤Ck(1 + |ξ|)−s(1 + |ζ|)s−d(1 + |ξ|)d(1 + |ζ − ξ|)−k

≤Ck(1 + |ξ|)d−s(1 + |ζ|)s−d(1 + |ξ − ζ|)−k.


Using Peetre’s inequality with x+ y = ξ and y = ζ we see that

(1 + |ξ|)d−s ≤ (1 + |ζ|)d−s(1 + |ξ − ζ|)|d−s|

Which implies that

|K(ζ, ξ)| ≤ Ck(1 + |ξ − ζ|)|s−d|−k

This is integrable for large k so (4.4) follows.

A differential operator P has the property that if f = 0 on a set U then Pf = 0 on U .

Because pseudo-differential operators are defined in terms of the Fourier transform they

do not have this property in general. This is easily seen in the case of a pseudo-differential

operator defined by a kernel. However, there is a weaker property which holds . This is

known as pseudo-locality.

Definition 4.2.1. Let U ⊂ Rn be open and f ∈ Ls(U). We say that f is smooth on an

open subset V ⊆ U if φf ∈ C∞c (V ) for every φ ∈ C∞c (V ). Let P ∈ ΨDOd(Rn), we say

that P is pseudo-local if f smooth on V implies Pf is smooth on V for every V ⊆ U .

Lemma 4.2.2. Pseudo-differential operators are pseudo-local.

Proof. Let P ∈ ΨDOd and let f ∈ Ls(U) be smooth on V . Let ψ ∈ C∞c (V ); we wish to

show ψPf ∈ C∞(V ). Choose φ ∈ C∞c (V ) such that φ|supp(ψ) = 1 and θ ∈ C∞c (Rn) such

that θ|supp(f) = 1. Since θf = f ,

ψPf = ψPφf + ψP (1− φ)θf.

By hypothesis, φf is smooth and hence ψPφf is smooth. The operator

P0 = ψP (1− φ)θ

has symbol

p0(x, ξ, y) = ψ(x)p(x, ξ)(1− φ(y))θ(y) ∈ Sym′d(U).

Since the x and y supports of p0 are disjoint, P0 is infinitely smoothing by Lemma 3.2.9 so

P0 is smooth.

The next lemma shows that an elliptic pseudo-differential operator has an approximate

inverse, known as a parametrix. This in turn shows that elliptic operators are hypo-elliptic.

The corresponding results on manifolds will later be used to show that elliptic operators

have a well-defined index which is independent of the extension taken. In order to state

the theorem we will first make a definition.

4.2. ELLIPTIC OPERATORS ON RN 67

Definition 4.2.2. We say that P ∈ ΨDOd(U1) is hypo-elliptic if for all f ∈ Ls(U ;N),

Pf is smooth on U1 implies that f is smooth on U1. Here N is the trivial bundle of rank

N .

Lemma 4.2.3. (Parametrix) Let U,U1 be open subsets of Rn such that U1 ⊂ U . If P is

elliptic on U1 then

(a) There exists Q ∈ ΨDO−d(U) and a function φ ∈ C∞c (U) which is identically 1 on a

neigbourhood of U1 such that φ(PQ−I) ∈ ΨDO−∞(U) and φ(QP−I) ∈ ΨDO−∞(U)

(b) P is hypo-elliptic on U1.

(c) (Garding’s inequality) ‖f‖d ≤ C(‖f‖0 + ‖Pf‖0) for f ∈ C∞c (U1).

(d) If d ≥ 0, ‖f‖0 + ‖Pf‖0 is an equivalent norm for Ld(U1).

Proof. We define a parametrix, or approximate inverse, for the operator P on U1 to prove

(a). Let qj ∈ Sym−d−j. We use Lemma 3.2.6 to find q ∈ Sym−d such that q ∼∑

j qj. Let

Q have symbol q. By Lemma 3.2.5,

σ(PQ) ∼∑α,j

∂αξ P ·Dαxqj/α!. (4.5)

We wish to choose the qj recursively so that PQ ∼ I on U1. Decomposing (4.5) into

elements of various Sym−k spaces leads to the recursive scheme over U1∑|α|+j=k

∂αξ p ·Dαxqj/α! ∼

I if k = 0

0 if k > 0

Now defining recursively

q0 =q, for k > 0

qk =− q0 ·∑

|α|+j=k,j<k

∂αξ p ·Dαxqj/α!.

Then PQ ∼ I on U1. Similarly we find Q1 so Q1P ∼ I on U1. We prove part (a) by

computing on U1:

Q−Q1 =Q−Q1PQ+Q1PQ−Q1

=(I −Q1P )Q−Q1(I − PQ) ∼ 0.


Next we use (a) to prove (b). Let f ∈ Ls(U) and assume Pf is smooth on U1. Choose

φ ∈ C∞c (U1). Then

φf = φ(I −QP )f + φQPf.

Since φ(I −QP ) ∼ 0, φ(I −QP )f ∈ C∞c (U1). Since Pf is smooth on U1, QPf is smooth

on U1 as Q is pseudo-local. Thus φQPf is smooth on U1.

We use (a) to prove (c). Chooose a function φ as in (a). Then

‖f‖d = ‖φf‖d ≤ ‖φf‖d ≤ ‖φ(I −QP )f‖d + ‖φQPf‖d.

We estimate the first norm by C‖Pf‖0 since (I −QP ) is infinitely smoothing. The second

norm is estimated by C‖Pf‖0 since φQ is an operator of order −d. This proves (c). Part

(d) follows as ‖f‖0 ≤ ‖f‖d and ‖Pf‖0 ≤ C‖f‖d for d ≥ 0.

4.3 Elliptic operators on manifolds

We would like to consider operators on manifolds between Sobolev spaces. We will

define Sobolev spaces on manifolds using a partition of unity. First however, we show that

Sobolev spaces are invariant under coordinate change.

Lemma 4.3.1. (Invariance of Sobolev spaces) Let U1, U2 and U ′ be open subsets of Rn such

that U1 ⊂ U1 ⊂ U2. If h is a diffeomorphism from U2 to U ′, then there exists a constant C

such that ‖h∗f‖d ≤ C‖f‖d ∀f ∈ C∞c (U1), d ∈ R.

Proof. Since |dh| is uniformly bounded on U1, the L2 norms on U1 and U ′1 are comparable

and h∗ defines an invertible map between these two spaces. Let P be elliptic of order d > 0

on U1. Then P ′ = h∗P is elliptic of order d > 0 on U ′1. Let f = h∗(f ′) for f ′ ∈ C∞c (U ′1).

We use Lemma 4.2.3 and Lemma 3.3.1 to see that:

‖f ′‖d ≤C(‖f ′‖L2(U ′1) + ‖P ′f ′‖L2(U ′1))

≤C(‖f‖L2(U1) + ‖Pf‖L2(U1)) ≤ C‖f‖d

and similarly ‖f‖d ≤ C‖f ′‖d. We use duality and Lemma 4.1.3 to establish the same

inequalities if d < 0. This proves the lemma.

4.3. ELLIPTIC OPERATORS ON MANIFOLDS 69

We define the Sobolev space Ls(X) on a compact manifold X using a partition of unity.

Let U = Ui, hi, φiηi=1 be an atlas for X. If f ∈ C∞ we define

‖f‖2U :=

∑i

‖hi∗(φif)‖s.

We note that if ψ ∈ C∞(m) then

‖hi∗(ψφif)‖s ≤ C‖hi∗(φif)‖s

as multiplication by hi∗(ψ) defines an order 0 pseudo-differential operator on C∞c (hi∗Ui).

Suppose that V = Vj, gj, ψjµj=1 is another atlas. As φiψjf ∈ C∞c (Ui ∩ Vj) we can use

Lemma 4.3.1 to estimate:

‖gj∗(ψjφif)‖s ≤ C‖hi∗(ψjφif)‖s.

We will now estimate the the norm defined by V in terms of the norm of U

‖f‖V,s =∑j

‖gj∗(ψj∑i

φif)‖s ≤∑i,j

‖gj∗(ψjφif)‖s

≤ C∑i,j

‖hi∗(ψjφif)‖s ≤ C∑i

‖φif‖s = C‖f‖U ,s

This shows that changing the atlas in our definition gives an equivalent norm. We

will denote any one of these equivalent norms by ‖ · ‖s. We define the sup norm similarly

‖f‖∞,k :=∑

i ‖hi∗(φif)‖∞,k.The next lemma verifies that our previous results involving Sobolev spaces on Rn hold

on manifolds. Part (a) is the Rellich lemma, Part (b) is the interpolation estimate 4.1.4,

Part (c) is the Sobolev lemma, Part (d) is our extension lemma 4.2.1 and Part (e) is the

L2-pairing lemma 4.1.3.

Lemma 4.3.2. Let X be a smooth compact manifold of dimension m and let s > t > u.

(a) The identity map on C∞(X) extends to a continuous, compact, injective, norm non-

increasing map from Ls(X) to Lt(X).

(b) If ε > 0 is given, then there exists a constant C(ε) such that ‖f‖t ≤ ε‖f‖s+C(ε)‖f‖u.

(c) If s > k +m/2, then Ls(X) ⊂ Ck(X) and ‖f‖∞,k ≤ C‖f‖s.


(d) If P ∈ ΨDOd(X), then P : Ls+d(X)→ Ls(X) is continuous for all s > 0.

(e) The L2 pairing Ls(X)× L−s(X)→ C is a perfect pairing.

Proof. Let U = Ui, hi, φiνi=1 be an atlas and define

ΦU(f) :=ν⊕i

hi∗(φif) ∈ν⊕i

C∞c (Rn);

ΦU embeds C∞(X) in⊕ν

i C∞c (Rn) and

‖f‖U ,s =ν∑i

‖hi∗(φif)‖s

Thus ΦU extends to a norm preserving embedding of Ls(X) in⊕ν

i Ls(Rn) (with norm

given by the sum of norms on its components). Thus previous results apply directly and

(a), (b), (c) are immediate.

By assumption, there exist coordinate charts (Oij, gij) (with open sets Oij ⊂ X and

chart maps gij) so that

U i ∪ U j ∈ Oij.

Let Pij = φiPφj. Then each Pij is a pseudo-differential operator on Oij and

P =∑i,j

Pij.

Using Lemma 4.2.1 we prove (d) by estimating:

‖gij∗(φiPφjf)‖s ≤ C‖gij∗(φjf)‖s−d ≤ ‖f‖s−d.

Choose non-negative functions ψi ∈ C∞c (Ui) such that∑

i ψ2i = 1. Then

|〈f, g〉L2(X)| =|∑i

〈ψif, ψig〉L2(X)|

≤∑i

|〈ψif, ψig〉L2(X)|.

We compare the volume form on X with the volume form on Rn to find smooth positive

functions αi such that

〈ψif, ψig〉L2(X) = 〈hi∗(αiψif), hi∗(ψig)〉L2(Rn).


Then using Lemma 4.1.3 we estimate

〈hi∗(αiψif), hi∗(ψig)〉L2(Rn) ≤C‖hi∗(αiψif)‖s‖hi∗(ψig)‖−s≤‖f‖s‖g‖−s.

This shows that the pairing 〈f, g〉L2(X) extends continuously to a map from

Ls(X)× L−s(X)→ C.

We use the Lemma 4.3.3 (which is proved next and independent of this lemma) to

complete the proof of Lemma 4.3.2. Let Q be elliptic of order d/2 and let P = Q∗Q + 1.

Then if g = P ∗Pf ,

‖Pf‖20 = ‖Q∗Qf‖2

0 + 2‖Qf‖20 + ‖f‖2

0 ≥ ‖f‖20

‖f‖2d is equivalent to 〈Pf, Pf〉 = 〈f, P ∗Pf〉 = 〈f, g〉.

The last result is obtained by applying part (c) of Lemma 4.3.2 to ‖f‖2, simplifying and

then using part (c) of Proposition 3.2.1. Since ‖P ∗Pf‖−d ≤ C‖f‖d, we conclude using

Cauchy-Schwarz

‖f‖d = 〈Pf, Pf〉/‖f‖d ≤C〈f, g〉‖g‖−d≤ C sup

h|〈f, h〉|/‖h‖−d.

Since the pairing of the Sobolev space Ld with the Sobolev space L−d is continuous, (d)

follows.

We will now extend the result of Lemma 4.2.3 to manifolds. This will show that an

elliptic ΨDO between Sobolev spaces on manifolds (as defined at the beginning of this

section) has an approximate inverse. This fact is key to showing that elliptic operators

have a well-defined index.

Lemma 4.3.3. (parametrix on a manifold) Let X be a compact manifold and P ∈ ΨDOd(X)

be elliptic.

(a) There exists Q ∈ ΨDO−d(X) such that PQ ∼ I and QP ∼ I.

(b) P is hypo-elliptic.


(c) If d > 0, an equivalent norm for Ld(X) is given by ‖f‖0 + ‖Pf‖0.

Proof. Let (Ui, hi, φi) be an atlas with an associated partition of unity φ, and let ψi ∈C∞c (Ui) be identically 1 on the support of the φi. In each coordinate chart, let Pi = ψiPψi.

Then

Pi − Pj ∈ ΨDO−∞(X)

on the support of φiφj. Construct Qi as the formal inverse to Pi on the support of φi,

then Qi − Qj ∈ ΨDO−∞(X) on the support of φiφj since the formal inverse constructed

in Lemma 4.2.3 is unique up to equivalence (see Definition 3.2.8). We compute that

P =∑i

φiP ∼∑i

φiPi.

Define the right inverse Qr =∑

j Qjφj. Then PQr ∼ I. We define Ql similarly and then

note that Qr ∼ Ql. This proves (a). Let p = σ(P ). Choose q belonging to Sym−d(X) so

qp− I ∈ Sym−1(X). Let σ(Q1) = q. Then:

Ql ∼Q1

∑k

(I − PQ)k

Qr ∼∑k

(I − PQ1)kQ1.

The proof of (b) now follows from (a) exactly as in the case of pseudo-differential operators

on Rn. Finally, let d > 0. We estimate:

‖f‖d ≤‖(QP − I)f‖d + ‖QPf‖d≤C‖f‖0 + C‖Pf‖0 ≤ C‖f‖d.

Our next result concerns the classical ΨDO defined in Section 3.5. In the proof of the

index theorem it will be important that we deal only with classical ΨDO. However, at one

point we must consider the exterior tensor product of two classical ΨDO - in general this

is not classical. In the discusssion and the two lemmas which follow we will show that this

product operator is in the closure of the classical ΨDO in a certain sense. This will be

sufficient for our purposes.

We defineOpms to be the set of continous linear maps T : Ls(X;E)→ Ls−m(X;F ) where

E → X and F → X are bundles. We give this space the operator topology. The extension


lemma in Part (d) of Lemma 4.3.2 then defines a map ΨCOm(E,F ) → Opms (E,F ). We

denote the image of this map by ΨCOms .

Now, to obtain smooth results from operators on Sobolev spaces we wish to consider

all of the spaces Opms simultaneously. We define Opm to be the set of linear operators on

smooth sections T : Γ∞(E) → Γ∞(F ) which extend by continuity to Opms for all s. The

Sobolev Lemma (see Lemma 4.3.2) implies that

Γ∞(E) =⋂s

Ls(X;E) and Γ∞(F ) =⋂s

Ls(X;F ), (4.6)

so an operator in Opm maps Γ∞(E) → Γ∞(F ) continuously. For each s there is an

embedding Opm → Opms , and thus of Opm →∏Opms . We take the induced topology on

Opm. This makes Opm into a Frechet space with a topology defined by the sequence of

norms qn(T ) := ‖T‖n,n−m (the operator norms from Ls to Ln−m). The space Opm is a local

space of operators in the sense that P ∈ Opm if and only if φPψ ∈ Opm for all compactly

supported functions φ, ψ ∈ C∞(X).

Let E, F be smooth vector bundles over X and G a smooth vector bundle over Y . If

P : Γ∞(E)→ Γ∞(F ) is a continuous linear operator then we can define the lifted operator

P ′ : Γ∞(X × Y ;E G)→ Γ∞(X × Y ;F G)

by

P ′(u v) = Pu v,

where u ∈ Γ∞(X × Y ;E G) and v ∈ Γ∞(X × Y ;F G). We have the following lemma,

for a proof in the case when m = 1 see [AS1] p.513.

Lemma 4.3.4. If m ≥ 0 then the lifting operation P 7→ P ′ defines a continuous map

Opm(X;E,F )→ Opm(X × Y ;E G,F G)

By extending the operators ΨCOm(E,F ) to ΨCOms (E,F ) we see that ΨCOm(E,F ) ⊂

Opm(E,F ). Thus we may define the closure ΨCOm

(E,F ) using the sequence of norms qn

introduced above.

Lemma 4.3.5. Let X, Y be compact manifolds, E,F vector bundles over X and G a vector

bundle over Y . Let m > 0 and P ∈ ΨCOm

(X;E,F ), then the lifted operator defined by

P ′(u v) = Pu v (4.7)


is in the closure

P ′ ∈ ΨCO(X × Y ;E G,F G).

Moreover, σ(P ′) = σ′(P ) where σ′ is the lift of σ defined by

σ′(ξ,η)(e⊗ g) = σ′ξ(e)⊗ g for ξ ∈ TX, η ∈ TY, e ∈ E, g ∈ G.

Proof. As ΨCOm

is a local class of operators , it will be sufficient to deal with the case when

X = U ⊂ Rn and Y = V ⊂ Rq are domains in Euclidian space, and all bundles are trivial

of dimension one. Moreover, by the continuity of of the lifting operation Opm(X;E,F )→Opm(X × Y ;E G,F G) it will be sufficient to prove that

P ∈ ΨCOm ⇒ P ′ ∈ ΨCO(U × V ).

Also, since ΨCOm

is a local class, it is sufficient to show that, for all φ, ψ ∈ C∞c (U) and

φ1, ψ1 ∈ C∞c (V ), Q = φφ1P′ψψ1 ∈ ΨCO(U × V ). To do this we shall construct a family

of operators Rt ∈ ΨCO0(U × V ) which are defined for t > 0, and have the properties

(a) Q Rt ∈ ΨCOm(U × V )

(b) Q Rt → Q in Opm(U × V ) as t→ 0.

First of all we choose a family of functions σt(ξ, η), defined for t > 0, taking values in the

unit interval such that

• σt is homogeneous of degree zero and C∞ outside the origin,

• σt =

1 for |ξ| < t|η|

0 for |ξ| > 2t|η|.

Also, let φ(λ) be a C∞ function of λ ∈ R such that

φ(λ) = 0 for |λ| ≤ 1

= 1 for |λ| ≥ 2.

We then take

ρt(ξ, η) = 1− φ(t(|ξ|2 + |η|2)1/2)σt(ξ, η).

Finally, we define Rt to be convolution by the inverse Fourier transform of ρt so that

Rtu(x, y) = (2π)−n−q∫ρt(ξ, η)u(ξ, η)ei〈x,ξ〉+i〈y,η〉dξdη.


Then the operator Qt = Q Rt is then given by the integral formula

(Qtu)(x, y) = (2π)−n−q∫φ1(y)φ(x)pψ(x, ξ)ρt(ξ, η)(ψ1u)(ξ, η)dξdη.

The properties of the function ρt show that pψρt ∈ Symm

0 (U × V × Rn+q) so that Qt ∈ΨCO(U×V ). The fact that, for m ≥ 0, Dβ

xpψ(x, ξ)/(1+ |ξ|+ |η|)m is bounded then implies

that Q ∈ Opm. Similarly for m > 0, the inequalities

|Dβx [pψ(x, ξ)(ρt(ξ, η)− 1)]

(1 + |ξ|+ |η|)m| < Cβt

m,

(this is a straight foward estimate using the homogeneity of σt which is part of the definition

of ρt) show that, as t→ 0, Qt −Q→ 0 in Opm. For the symbols we have

σ(Qt) = φ1φσ(pψ)(1− σt)ψ1 = φ1φσ′(P )(1− σt)ψψ1,

as t → 0 this converges to φ1φσ′(P )ψψ1. Since the symbol is also a local object, this

implies that σ(P ′) = σ′(P ) as required.


Chapter 5

Fredholm operators and the Analytic

Index

The aim of this chapter is to define the analytic index map a−ind : K(T ∗X) → Z.

This involves an investigation of Fredholm operators. It follows from their definition that

Fredholm operators have a well-defined index. The crux of our investigation will be to

show that the index is locally constant on the space of Fredholms.

After we have show that the index is locally constant we will relate this result to elliptic

operators. This is relatively straight foward as we have already done most of the work in

Chapter 4. We use the existence of a parametrix (proved in the last chapter) to show

that any extension of an elliptic operator is a Fredholm operator - this shows that any

extension of an elliptic operator has a well-defined index. Then we will show that all of

the extensions of a given elliptic operator have the same index - this shows that an elliptic

operator P ∈ ΨDOm has a well-defined index. We then use the local constancy of the

index to show that the index of an elliptic operator depends only on the homotopy class

of its principal symbol.

Once this is done the index of a symbol class u ∈ K(T ∗X) is well-defined. Recalling that

any class in K(T ∗X) may be represented by the principal symbol of an elliptic operator,

we define the analytic index map to be the map taking a class in K(T ∗X) to the index of

any representative operator.

77

78 CHAPTER 5. FREDHOLM OPERATORS AND THE ANALYTIC INDEX

5.1 Fredholm Operators and the Index

We now define Fredholm operators as linear operators which are invertible modulo

compact operators. Note the similarities of this definition to that of an elliptic operator’s

parametrix. We should note that a common definition of Fredholm operators is in terms of

their range, kernel and cokernel. However, this definition is actually equivalent to ours, as

Proposition 5.1.2 shows. In the following B(H1, H2) denotes the set of bounded operators

between the Hilbert spaces H1 and H2, and Com(H1, H2) the compact operators.

Definition 5.1.1. LetH1, H2 be Hilbert spaces. The space of Fredholm operators Fred(H1, H2)

consists of all bounded linear operators T : H1 → H2 which are invertible modulo com-

pact opertors. Let T ∈ B(H1, H2). So T ∈ Fred(H1, H2) if and only if there exists

S1, S2 ∈ B(H2, H1) such that

S1T − IdH1 ∈ Com(H1) and TS2 − IdH2 ∈ Com(H1) (5.1)

This definition has the following easy consequences.

Proposition 5.1.1. The adjoints and compositions of Fredholm operators are again Fred-

holm.

(a) T ∈ Fred(H1, H2)⇒ T ∗ ∈ Fred(H2, H1)

(b) T1 ∈ Fred(H1, H2) and T2 ∈ Fred(H2, H3)⇒ T2T1 ∈ Fred(H1, H3)

Proof. Taking adjoints of the operators in (5.1) shows that T ∈ Fred(H1, H2) ⇒ T ∗ ∈Fred(H2, H1) proving (a). For part (b) we again look at the definition. If S1T1 − Id ∈ComH1 and S2T2 − Id ∈ ComH2, then

S1S2T2T1 − IdH1 = S1(S2T2 − IdH2)T1 + (S1T1 − IdH1) ∈ Com(H1).

Similarly, T2T1S1S2 − IdH3 ∈ Com(H3).

We will now prove Atkinson’s theorem. This shows that Fredholm operators are equiv-

alently characterised as bounded linear operators with closed range and finite dimensional

kernel and cokernel. So, Fredholm operators have a well-defined index.

Definition 5.1.2. Given a bounded linear operator T : H1 → H2 between Hilbert spaces.

We define its kernel to be the subspace KerT = x ∈ H1 : Tx = 0, its image to be the

subspace RanT = Tx : x ∈ H1 and its cokernel CokerT = H2/imT .

5.1. FREDHOLM OPERATORS AND THE INDEX 79

Proposition 5.1.2. (Atkinson’s Theorem) The following conditions are equivalent for a

bounded operator T from H1 to H2.

(a) T ∈ Fred(H1, H2)

(b) dim(Ker(T )) <∞, dim(Ker(T ∗)) <∞,Ran(T ) and Ran(T ∗) are closed

(c) dim(Ker(T )) <∞, dim(Ker(T ∗)) <∞,Ran(T ) is closed

Proof. Suppose first T ∈ Fred(H1, H2). Let xn ∈ Ker(T ) with ‖xn‖ ≤ 1. Then

xn = (Id− S1T )xn = Cxn

Since C is compact, there exists a convergent subsequence xnj. This shows the unit

sphere of Ker(T ) is compact so Ker(T ) is finite dimensional; since T ∗ ∈ Fred(H2, H1),

Ker(T ∗) is finite dimensional as well. Next take a sequence xn ⊆ Ker(T )⊥ and set

yn = Txn. We must show y ∈ Ran(T ). We suppose without loss of generality that

xn ∈ Ker(T )⊥. Suppose there is a constant C with ‖xn‖ ≤ C. Then

xn = S1yn + (Id− S1T )xn.

Since (Id − S1T ) is compact, by passing to a subsequence if neccesary, we may assume

(Id−S1T )xn is a Cauchy sequence. Since yn → y, S1yn is a Cauchy sequence. This implies

xn is a Cauchy sequence so xn → x and Tx = y. We therefore suppose ‖xn‖ → ∞. Let∼xn = xn/‖xn‖. Then T (

∼xn) = yn/‖xn‖ → 0. We argue as above and pass to a subsequence

if neccesary to assume∼xn →

∼x with T (

∼x) = 0. But since Ker(T )⊥ is a closed subspace,

∼x ∈ Ker(T )⊥. This is impossible as ‖∼x‖ = 1. This shows Ran(T ) is closed and similarly

Ran(T ∗) is closed. Thus (a) ⇒ (b). The implication (b) ⇒ (c) is immediate.

Finally, suppose Ker(T ) and Ker(T ∗) are finite dimensional and Ran(T ) is closed. Then

Ran(T ) = Ker(T ∗)⊥. Decompose

H1 = Ker(T )⊕Ker(T )⊥ and H2 = Ker(T ∗)⊕ Ran(T ).

Then T is a continuous bijective map from Ker(T )⊥ to Ran(T ). Since these are Banach

spaces, we may use the bounded inverse theorem to find a bounded linear operator on S

so ST is the identity on Ran(T )⊥ and TS is the identity on Ran(T ). Extend S to be zero

on Ker(T ∗). Then

ST − IdH1 = πKer(T ) and TS − IdH2 = πKer(T ∗)


where π is orthogonal projection on the indicated subspaces. Since these two projections

have finite dimensional ranges, they are compact.

Atkinson’s theorem allows us to define the index of a Fredholm operator.

Definition 5.1.3. The index of a Fredholm operator T is given by

Index(T ) = dim Ker(T )− dim Coker(T ).

We have now reached our fundemental lemma on the structure of Fred(H1, H2) and

the index. The main point of this lemma is to show that the index is locally constant. In

fact, the index defines a bijection between the connected components of Fred(H1, H2) and

Z (for a proof of this fact see [LM] p. 201).

Lemma 5.1.1. (Properties of the Fredholm index) Let T ∈ Fred(H1, H2) and S ∈ Fred(H2, H3).

(a) Index(T ∗) = −Index(T ).

(b) Index(ST ) = Index(S) + Index(T ).

(c) Fred(H1, H2) is an open subset of B(H1, H2).

(d) Index : Fred(H1, H2)→ Z is continuous and locally constant.

Proof. Part (a) follows from the definition, noting that Ker(T ∗) = Ran(T )⊥

and Ran(T ∗) =

Ker(T )⊥. We compute that

Ker(ST ) =Ker(T )⊕ T−1(Ran(T ) ∩Ker(S))

Ker(T ∗S∗) =Ker(S∗)⊕ (S∗)−1(Ran(S∗) ∩Ker(T ∗))

=Ker(S∗)⊕ (S∗)−1(Ker(S)⊥ ∩ Ran(T )⊥).

We prove (b) by computing:

Index(ST ) = dim(Ker(T )) + dim(Ran(T ) ∩Ker(S))

− dim(Ker(S∗))− dim(Ker(S)⊥ ∩ Ran(T )⊥)

= dim(Ker(T )) + dim(Ran(T ) ∩Ker(S))

+ dim(Ran(T )⊥ ∩Ker(S))− dim(Ran(T )⊥ ∩Ker(S))

− dim(Ker(S∗))− dim(Ker(S)⊥ ∩ Ran(T )⊥)

= dim(Ker(T )) + dim(Ker(S))− dim(Ker(S∗)− dim(Ran(T )⊥)

=Index(T ) + Index(S).

5.1. FREDHOLM OPERATORS AND THE INDEX 81

Parts (c) and (d) are proved as follows. Fix T ∈ Fred(H1, H2) and decompose:

H1 = Ker(T )⊕Ker(T )⊥ and H2 = Ker(T ∗)⊕ Ran(T ).

The T is a bijective mapping from Ker(T )⊥ to Ran(T ). Give the direct sum Hilbert space

structures to

H ′1 = Ker(T ∗)⊕H1 and H ′2 = Ker(T )⊕H2.

If S ∈ B(H1, H2), define S ′ ∈ B(H ′1, H′2) by

S ′(f0 ⊕ e) := πKer(T )(e)⊕ (f0 + S(e)).

For S1, S2 ∈ B(H1, H2) it is immediate that ‖S ′−S ′1‖ = ‖S−S1‖ so that the map S → S ′ is

a norm preserving affine transformation from B(H1, H2) to B(H ′1, H′2). Let i1 : H1 → H ′1

be the natural inclusion and π2 : H ′2 → H2 the natural projection. Since Ker(T ) and

Ker(T ∗) are finite dimensional, i1 and π2 are Fredholm maps. We have

S = π2S′i1. (5.2)

If e ∈ H1 and f ∈ H2, let e = e0 ⊕ e1 and f = f0 ⊕ f1 for e0 ∈ Ker(T ) and f0 ∈ Ker(T ∗).

Then

T ′(f0 ⊕ e0 ⊕ e1) = e0 ⊕ f0 ⊕ Te1.

This shows that T ′ ∈ Iso(H ′1, H′2). Since Iso(H ′1, H

′2) is an open subset of B(H ′1, H

′2), there

exists ε > 0 so that ‖T − S‖ < ε implies S ′ ∈ Iso(H ′1, H′2). This implies S ′ is Fredholm.

We use (5.2) to conclude that S is Fredholm, which proves (c). Since S ′ is invertible,

Index(S ′) = 0. We show the index is locally constant by computing:

Index(S) =Index(i1) + Index(S ′) + Index(π2)

=Index(i1) + Index(π2)

= dim Ker(T )− dim Ker(T ∗) = Index(T ).

We now finish this section by showing that the index of an elliptic operator depends

only on the homotopy class of its principal symbol. In the proof of the following lemma

we first show that extensions of elliptic operators are Fredholm using the existence of an

elliptic parametrix, then show that all the extensions of an elliptic operator have the same

index, and finally use the local constancy of the index to prove that the index depends

only on the symbol.


Lemma 5.1.2. (Main result) Let P : Γ∞(E) → Γ∞(F ) be an elliptic pseudo-differential

operator of order d.

(a) P : Ls(E)→ Ls−d(F ) is Fredholm.

(b) Ker(P ) ⊂ Γ∞(E) is independent of s, and so Index(P ) is independent of s.

(c) Index(P ) depends only on the homotopy type of the principal symbol σ(P ) within the

class of elliptic pseudo-differential operators of order d.

Proof. Use Lemma 4.3.3 to find S : Γ∞(F )→ Γ∞(E) of order −d so that

SP − I ∈ ΨDO−∞(E) and PS − I ∈ ΨDO−∞(F ).

By Lemma 4.3.2, P : Ls(E)→ Ls−d(F ) and S : Ls−d(F )→ Ls(E) are continuous. We use

Lemmas 4.3.2 and 4.3.3 to see:

(SP − I) : Ls(E)→ Ls+1(E)→ Ls(E) and

(PS − I) : Ls−d(F )→ Ls−d+1(F )→ Ls−d(F )

are compact operators. This proves (a). If f ∈ Ker(P ), then Pf = 0 and hypo-ellipticity

implies that f ∈ C∞(E) so Ker(P ) and Ker(P ∗) are independent of the choice of s and

Index(P ) is defined independent of s. This proves part (b).

Let P (ε) be a smooth 1-parameter family of such operators. By Lemma 5.1.1, the index

is locally constant and hence Index(P (ε)) is independent of ε. If P0 and P1 have the same

principal symbol, then

P (ε) := εP1 + (1− ε)P0

has principal symbol p for any ε so it defines a smooth 1-parameter family of elliptic

operators. Thus

Index(P0) = Index(P1).

This shows that the principal symbol determines the index. If p(ε) is a smooth 1-parameter

family of elliptic symbols, we can construct a smooth 1-parameter family of operators P (ε)

with principal symbol p(ε) using a partition of unity to see the index is a homotopy invariant

of the principal symbol.

5.2. THE ANALYTIC INDEX MAP 83

5.2 The Analytic Index Map

We use the result of the last section, that the index of an elliptic operator depends only

on the homotopy class of its principal symbol, to prove that the index is well-defined on

K(T ∗X). The analytic index map will then be the map taking a class in K(T ∗X) to the

index of any representative operator.

Lemma 5.2.1. Let X be a compact manifold. If P ∈ ΨDOd(E,F ) and P ′ ∈ ΨDOd(E ′, F ′)

are elliptic and σ(P ) = σ(P ′) ∈ K(T ∗X) then Index(P ) = Index(P ′).

Proof. Let σ(P ) and σ(P ′) be represented by the K-theory classes [π∗E, π∗F ;σ(P )m], and

[π∗E ′, π∗F ′;σ(P ′)n] respectively, where σ(P )m and σ(P ′)n are homogeneous of degree m

and n. We first consider the case where m = n. As σ(P ) = σ(P ′) the complexes must

either be homotopic, or differ by a complex with empty support. If they are homotopic

then σm(P ) and σn(P ′) differ by a degree m homogeneous complex of the form [G,G; β]

where β is an isomorphism. By the homogeneity of β we have dim(G) = 0 unless m = 0.

When m = 0, β is constant in ξ so P = P ′Q where Qu = β(x, 0)u. As Index(Q) = 0 we

have Index(P ) = Index(P ′Q) = Index(P ′) by Lemma 5.1.1.

Now consider the case where m 6= n. By the above argument we may assume that E =

E ′, F = F ′ and σm = σn on the sphere bundle S(T ∗X). Construct the map ρ = σ′−1n σm.

We see that ρ is homogeneous of degree n−m and that it is the identity map on S(T ∗X).

Thus ρ is self-adjoint and it follows that P = P ′R where R is the operator associated to

ρ. The operator R is self-adjoint so IndexR = 0 and Index(P ) = Index(P ′).

Using Lemma 5.2.1 we can now make the following definition.

Definition 5.2.1. Let X be a manifold. The analytic index is the map

a−ind : K(T ∗X)→ Z

defined by a−ind σ(P ) = Index(P ).

We have now constructed “half” the statment of the index theorem.


Chapter 6

The Topological Index and its

Axioms

In the last chapter we saw that the index is an invariant of the symbol and we defined

the analytic index map a−ind : K(T ∗X)→ Z. The task now is to compute the index from

K(T ∗X) topologically. The Atiyah-Singer index theorem solves this problem by defining a

topological index map t−ind : K(T ∗X)→ Z, which coincides with the analytic index. In

this chapter we define the topological index and give a set of axioms which characterise it.

We will then refine these axioms to give a characterisation which is more easily verified for

the analytic index. This will give sufficient conditions for the proof of the index theorem

which we complete in the next chapter.

6.1 The Topological Index

The topological index must map K(T ∗X) → Z. One such map is induced by the

inclusion of a point i : pt→ (T ∗X)+ (recall from Section 2.2.2 that K(pt) ∼= Z). However,

by definition

K(T ∗X) ∼= K(T ∗X+) = Ker(i∗),

and i∗ : K(T ∗X) → pt is identically zero. So we must consider a more complicated type

of map, known as a shriek map. From here on in we will assume that we have a metric, so

that the tangent and cotangent bundles can be identified.

Given the inclusion i : X → Y of a compact submanifold X of Y we may construct

85

86 CHAPTER 6. THE TOPOLOGICAL INDEX AND ITS AXIOMS

a map i! : K(TX) → K(TY ) which we call “i-shriek”. Note that i! maps in the reverse

direction to the pullback map i∗. We construct this map in the discussion which follows.

Take a metric on Y and let N be an open tubular neigbourhood of X in Y which is

diffeomorphic to the normal bundle of X in Y . We may extend this construction to tangent

bundles. Here, TX is a closed submanifold of TY and TN is a tubular neigbourhood of

TX in TY which is diffeomorphic to the normal bundle of TX in TY .

We now aim to apply the Thom isomorphism (see Section 2.4). However, to do this we

must introduce a complex structure on TN .

Lemma 6.1.1. Given a submanifold X of a manifold Y . Let π : TN → N be the tangent

space to the normal bundle N of X in Y . Then

TN ∼= π∗N ⊕ π∗N

and thus may be given the structure of a complex vector bundle

TN ∼= π∗N ⊕ iπ∗N ∼= π∗(N ⊗R C)

Proof. Let π : E → X be a smooth real vector bundle. For an open set U on X we

have that π−1(U) is diffeomorphic to U × Rm for some m. So the tangent space to E at

(x, ξ) is given by TE(x,ξ) = TUx ⊕ TRmξ∼= TXx ⊕ Ex and the tangent bundle TE may be

decomposed TE = π∗TX ⊕ π∗E. Thus taking E = TX and E = TY |X we obtain the

following bundles over TX. Via the projection π : TX → X we have

T (TX) ∼= π∗TX ⊕ π∗TX.

Also, if we take the projection π : TY → Y and then restrict TTY to TX ⊂ TY we get

T (TY )|TX ∼= π∗(TY |X)⊕ π∗(TY |X).

Looking at N and TN , the normal bundles of X and TX in Y and TY respectively, we

have N ⊕ TX ∼= TY |X and TN ⊕ T (TX) ∼= T (TY )|TX . So using the previous results we

have that

TN ⊕ T (TX) ∼= π∗(TY |X)⊕ π∗(TY |X)

∼= π∗(N ⊕ TX)⊕ π∗(N ⊕ TX)

∼= π∗N ⊕ π∗N ⊕ π∗(TX)⊕ π∗(TX)

∼= π∗N ⊕ π∗N ⊕ T (TX)

6.1. THE TOPOLOGICAL INDEX 87

Therefore

TN ∼= π∗N ⊕ π∗N.

The complex structure is then introduced by

TN ∼= π∗N ⊕ iπ∗N ∼= (π∗N)⊗R C ∼= π∗(N ⊗R C)

In the discusion preceeding Lemma 6.1.1 we saw that the tangent space TN of the

tubular neigbourhood N is diffeomorphic to the normal bundle of TX. Lemma 6.1.1 shows

that TN may be given a complex structure; so the Thom isomorphism can be applied. We

now define the shriek map.

Definition 6.1.1. If i : X → Y is the inclusion of a compact submanifold we may define

an induced map

i! : K(TX)ϕ→ K(TN)

h∗→ K(TY )

where TN is considered to be a complex vector bundle as in Lemma 6.1.1, ϕ is the Thom

isomorphism and h∗ is the extension homomorphism of Proposition 2.2.5, induced by ex-

tending bundles trivially to TY . Such a map i! is known as the shriek map of i.

Referring to the proof of Lemma 6.1.1 we see that N is chosen to be diffeomorphic to

the normal bundle of X. Note that any such N will result in the same shriek map. Also,

because the Thom isomorphism is transitive, as shown in Section 2.4 the shriek maps of

inclusions Xi→ Y

j→ Z have the property (i j)! = i! j!.

We are now in a position to define the topological index.

Definition 6.1.2. Let X be a compact manifold, i : X → Rm an embedding for some m

and j : pt → Rm the inclusion of the origin. The topological index is defined to be the

map

t−ind : K(TX)i!→ K(TRm)

(j!)−1

→ K(pt) ∼= Z

The map t−ind is independent of the choice of embedding. To see this consider two

embeddings i : X → Rn and i′ : X → Rm. If we take the embedding i⊕ i′ : X → Rn+m we

see that the embeddings i⊕0 and 0⊕i′ are isotopic via the map Ht = t(i⊕0)+(1−t)(0⊕i′),that is, Ht is a smooth family of embeddings. Next, note that the linear inclusion f : Rl →Rk+l has shriek map f! : K(TRl)→ K(TRk+l). This is just the Thom isomorphism for the


bundle Ck+l → Ck. Thus i and i⊕0 determine the same shriek map, as do i′ and 0⊕ i′. So

the isotopy Ht and the homotopy invariance of K-theory imply that the topological index

is independent of the embedding used.

6.2 Axioms

We will now give two axioms which characterize the topological index. These axioms

will then be refined several times. The aim here is two-fold: on the one hand these axioms

give some insight into the properties of the topological index. On the other, the refined

axioms give a schematic for the proof of the Atiyah-Singer index theorem. We should note

here that, in what follows, it is clear t−ind satisfies our first pair of axioms. However, the

refined axioms are not so obvious.

Our axioms will be stated for a general index function.

Definition 6.2.1. A collection of homomorphisms indX : K(TX) → Z for all compact

manifolds X is called an index function if it satisfies the following property: Given

a diffeomorphism f : X → Y we have indXf ∗y = indY y for all y ∈ K(TY ), where

f ∗ : K(TY ) → K(TX) is the map induced by the extension of f to tangent bundles. In

other words, ind is functorial with respect to diffeomorphisms.

Our first set of axioms characterises the topological index in terms of its behaviour on

K(pt) and its commutation with the shriek map of an inclusion.

Proposition 6.2.1. If ind is an index function satisfing the following axioms then ind =

t−ind:

(A1) If X is a point, then indpt : K(pt) ∼= Z→ Z is the identity map.

(A2) The index map ind commutes with the homomorphisms i!. i.e. the following diagram

commutes for all inclusions i : X → Y .

K(TX)

indX

i! // K(TY )

indYxxrrrrrrrrrrr

Z

6.2. AXIOMS 89

Proof. Take an embedding i : X → Rm for some compact manifold X. Let i+ : X → Rm+

be the embedding given by i. Also, let j : pt → Rm be the inclusion of the origin and

j+ : pt→ Rm+ the embedding given by j. Consider the following diagram

K(TRm)

K(TX)

indX&&NNNNNNNNNNNN

i+! //

i!88qqqqqqqqqqq

K(TRm+)

indRm+

K(Tpt)j+!oo

j!ffMMMMMMMMMMM

indpt

xxpppppppppppp

Z

So that t−ind = (j!)−1 i! is the topological index. The top two triangles commute by

the definition of shriek maps i! and j!. Both the bottom left and bottom right triangles

commute by axiom (A2). Also, from the definition of the shriek map we see that j! is the

Thom isomorphism. Thus indX = indRm i! = indpt (j!)−1 i! = (j!)

−1 i!, using axiom

(A1).

6.2.1 Refining the axioms

When we come to proving the analytic index satisfies the above axioms we will see that

(A1) is quite straight foward to verify. However, (A2) is much more involved. For this

reason we will now split it into three axioms which are easier to verify, and show that these

imply (A2). The axioms are excision, normalisation and multiplication. They combine to

show that the analytic index commutes with the map i! : K(TX)ϕ→ K(TN)

h∗→ K(TY ).

Broadly speaking, the excision axiom shows that ind commutes with h∗, the multiplication

axiom gives a factoring of the map ϕ which reduces the whole problem to computing the

index of an operator on TRn and the normalisation axiom computes this index.

(B1) The Excision Axiom:

Let U be an open manifold and

j : U → X,

j′ : U → X ′


be two open embeddings into compact manifolds X and X ′. Then the following diagram

commutes

KG(TU)j∗ //

j′∗

KG(TX)

indXG

KG(TX ′)indX

′G

// Z

where j∗ and j′∗ are extension homomorphism as defined in Proposition 2.2.5. Note that

this axiom allows us to unambiguously define indXG for non-compact X via an embedding of

X into a compact manifold. If j : X → Y is any embedding of X into a compact manifold

Y then we define indXG := indYG j∗.

(B2) The Normalization Axiom:

If j : pt→ Rm is the inclusion of the origin, so that we have the induced homomorphism

j! : KO(m)(pt) ∼= R(O(m))→ KO(m)(TRm), then

indRmO(m)j!(1) = 1.

Here we will assume that (B1) holds so that indRmO(m) is unambiguously defined onKO(m)(TRm).

For example, we may define it using an embedding e : Rm → Sm into the m-sphere. This

won’t cause any problems in the following development as we will never assume (B2)

without (B1).

(B3) The Multiplicative Axiom:

This axiom is the most significant. It does most of the work to show that ind commutes

with the Thom isomorphism. It also requires some setting up. Recall the product defined

in Section 2.2.4, this gives a map

K(V )⊗K(W )→ K(V ×W )

Using this map, we could define a multiplicative axiom inda · b = inda indb, for a ∈ K(V )

and b ∈ K(W ). Then, if we had a class on V ×W which came from the product of classes

on V and W we could easily express its index in terms of these.

We will define a multiplication map

K(TX)⊗KH(TF )→ K(TV ) (6.1)

6.2. AXIOMS 91

for the twisted product V = P ×H F and give a multiplicative axiom for it. It will then

be shown that, for a real vector bundle V = P ×O(n) TRn the homomorphism

K(TX)⊗ λn → K(TV ),

using the new multiplication agrees with the Thom isomorphism

K(TX)⊗ λTV → K(TV ),

using the usual multiplication. We now constuct the multiplication map (6.1).

Let P → X be a compact differentiable principal H-bundle, where H is a compact Lie

group. Then H acts freely on the right on P and X = P/H. Given a compact differentiable

H-manifold F (here H acts on the left), as in Section 2.1.3 we can construct the associated

fibre bundle given by

V = P ×H F,

here V is the quotient of P ×F by the H action h(p, f) = (ph−1, hf), and π : V → X. The

action of H on F extends to TF . This means that we can form the P ×H TF which is a

vector bundle over V . Such a vector bundle is called the tangent over the fibres of V and

denoted T (V/X). The bundle T (V/X) is a sub-bundle of TV and introducing a metric on

TV we can make the decomposition

TV = T (X/V )⊕ π∗TX.

We see that π∗TX gives the components, of tangent vectors in TV , which run over V

(which sits over X), and T (X/V ) gives the components which run over the fibres. Now,

applying the first component of the multiplication map K(A)⊗K(B)→ K(A⊕B) for all

vector bundles A,B over V , we obtain the map

K(π∗TX)⊗K(T (V/X))→ K(TV ). (6.2)

The homomorphism π∗ is used to extend this to a map from K(TX) ⊗ K(T (V/X)) to

K(V ).

Now all that remains is to define a homomorphism KH(TF ) → K(T (V/X)). We use

the following maps

KH(TF )i∗→ KH(P × TF )

γ∼= K(P ×H TF ) (6.3)

= K(T (Y/X)). (6.4)


The first map is induce by the projection p : P × TF → TF , that is, it trivially extends

a complex u ∈ KH(TF ) over the P factor of P × TF . The second map is the canonical

identification given by applying the associated vector bundle construction. An H-complex

u ∈ KH(P × TF ) is collapsed to γu by letting (γu)[(p,f)] = u(p,f). Composing (6.3) and

(6.2) gives the desired multiplication map (6.1).

We can introduce a second topological group G throughout the above discussion. If we

assume that G acts on the left of P and F , and commutes with the action of H on these

spaces. Such a group has an induced action on X = P/H and V = P ×H F . Thus our

multiplication map becomes

KG(TX)⊗KG(T (V/X))→ KG(TV )

and our homomorphism becomes

KG×H(TF )→ KG×H(P × TF ) ∼= KG(P ×H TF )

= KG(T (V/X)),

composing these gives a map

KG(TX)⊗KG×H(TF )→ KG(TV ). (6.5)

Now, if V is a complex G×H-module, P ×H V is a G-vector bundle over X. This extends

to an R(G)-homomorphism

µP : R(G×H)→ KG(X). (6.6)

Recall, that KG(TX) is a KG(X)-module. Our multiplicative axiom will be a special case

of the following:

(B3 - preliminary): Let G,H, P and F be as above then

indVG(ab) = indXG (a · µP (indFG×H(b))

where a ∈ KG(TX), b ∈ KG×H(TF ), the product is taken as in (6.5) and µP is the

homomorphism of (6.6).

We will take our axiom to be the special case of this preliminary axiom where indFG×H(b)

lies in the sub-ringR(G) ofR(G×H). Then, as µP and indXG are bothR(G)-homomorphisms

our preliminary axiom simplifies to

indVGab = indXGa · indFG×Hb ∈ R(G).

6.2. AXIOMS 93

We now state our axiom.

(B3): If a ∈ K(TX) and b ∈ KH(TF ) then

indY ab = indXa · indFHb.

provided that a−indFHb is a multiple of the trivial representation 1 ∈ R(H). Here the

product ab is take as in (6.5).

In order to prove that the axioms (B1), (B2) and (B3) imply (A2) we will need

the following lemma which gives a factoring of the Thom isomorphims in terms of the

multiplication map (6.5).

Proposition 6.2.2. Let V = P×O(n)Rn be a vector bundle over X, i : X → V be inclusion

of the zero section, j : pt→ Rn be the inclusion of the origin and let · be the multiplication

map of (6.5). The homomorphism

K(TX)→ K(π∗TX)⊗KO(n)(TRn)→ K(T ∗V )

given by a 7→ a · j!(1) coincides with the homomorphism

i! : K(TX)→ K(TN)→ K(TV ).

Proof. We first break the multiplication map 6.5 into its parts. The first part which we

examine is the homomorphism

KO(n)(TRn)p∗→ KO(n)(P × TRn)

γ∼= K(P ×O(n) TRn)

= K(T (V/X)).

The map p∗ is induced by the projection p : P × TRn → TRn, it trivially extends a

complex to P . Mapping a complex u ∈ KO(n)(TRn) gives p∗u(p,f) = uf . The canonical

map γ then twists this complex via the associated vector bundle construction. The class

p∗u is pushed down to the quotient by O(n). Note that the class does not depend on the P

factor and retains its equivariance since p : P × TRn → TRn is an equivariant map. Thus

(γp∗u)[(p,f)] = uf . This means that if we take a local trivialisation P ×O(n) TRn|C for some

compact set C then in K-theory we have

K(P ×O(n) TRn|C) ∼= K(TRn)


and under this isomorphism γp∗u|C corresponds to u. An argument involving a Mayer-

Vietoris sequence can then be used to show that γp∗u is in fact the same class as λP×O(n)TRn =

λT (V/X). An example of this argument can be found in [LM] Appendix C.

Now we will compare the maps

K(TX)→ K(π∗TX)⊗ λT (V/X) → K(TX ⊕ T (V/X)) ∼= K(TV ),

and

i! : K(TX)→ K(π∗TX)⊗ λTN → K(TV ).

both of which is given by the usual multiplication map from Section 2.2.4. We see from

the decomposition TV = π∗TX ⊕ T (V/X) that the bundle T (V/X) is isomorphic to the

tangent bundle of the normal bundle of TX in TV , that is TN ∼= T (V/X). Thus λT (V/X)

is isomorphic to the element λTN . This completes our proof.

We can now show that our refined axioms imply (A2).

Proposition 6.2.3. Axioms (B1), (B2) and (B3) imply axiom (A2).

Proof. Let i : X → Y be the inclusion of a compact submanifold X into a compact

manifold Y . Recalling the definition of the shriek map we have the following diagram

K(TX)

indX&&LLLLLLLLLLL

ϕ // K(TN)

indN

h∗ // K(TY )

indYxxrrrrrrrrrrr

Z

where ϕ is the Thom isomorphism, h : N → Y induces the extension map h∗ and TN is

considered as a complex bundle (see Definition 6.1.1).

The map indN is defined via any embedding of N into a compact manifold, using the

excision axiom (B1). We can take this embedding to be precisely h : N → Y , which

immediately implies that the second triangle commutes. Thus it will suffice to prove that

indX = indV ϕ,

where π : V → X is a real vector bundle overX (we use excision to define indV := indD(V )ewhere e : V → D(V ) is an embedding of V into its disk bundle). This will be achieved by

the application of our multiplication axiom (B3).

6.2. AXIOMS 95

Let V = P ×O(n) Rm and j : pt → Rn be the inclusion of the origin. Then using the

multiplication map (6.5) we have the homomorphism

KG(T ∗X)→ KG(T ∗V )

given by a 7→ aj!(1). As proved in Proposition 6.2.2 this is just the Thom isomorphism

component of the map i!, induced by the inclusion of the zero section i : X → V . We

note that by excision indRm := indSm

k! where k is the embedding of Rm into its one point

compactification Sm. Thus we may apply (B3) giving

indY ϕ(u) = indY uj!(1) = indXu · indRmO(n)j!(1).

Applying the normalization axiom (B2) completes the proof.

6.2.2 The normalization axiom

We now work on obtaining a condition which can replace (B2), and which can be

shown more easily in the proof to come. As noted in Section 2.3, the G-equivariant K-

theory of a point is KG(pt) ∼= R(G). In the normalisation axiom we are interested in

the index of the class j!(1) ∈ KO(n)(pt) ∼= R(O(n)). However, products of the subgroups

O(1), SO(2) ∈ O(n) ⊆ O(n) contain all of the cyclic subgroups of O(n). We will show that

to determine the index of a class j!(1) ∈ KO(n)(pt) it suffices to determine it on KO(1)(pt)

and KSO(2)(pt). This is the idea behind axiom the following axiom.

(B2’): If j! is either

KO(1)(pt) ∼= R(O(1))→ KO(1)(TR1)

or

KSO(2)(pt) ∼= R(SO(2))→ KSO(2)(TR2).

Then indj!(1) = 1.

Before giving the proof that (B2’) can replace (B2) we make some observations about

the multiplication axiom. In the multiplication axiom we can take the group H to equal

1. This gives the multiplication property

indX×FG (ab) = indXG · indFGb (6.7)


for a ∈ KG(TX) and b ∈ KG(TF ). Now, if we let Xi be a Gi manifold and ai ∈ KGi(TXi),

we can form the product a1a2 ∈ KG(TX) where X = X1 × X2 and G = G1 × G2 then

applying the multiplicative property 6.7 with X1 = X,X2 = F get

indXGa1a2 = indX1G1a1 · indX2

G2a2.

Note that the this version of the multiplication axiom allows us to use different groups.

We will need this in the proof of the following lemma.

Proposition 6.2.4. Axioms (B1), (B2’), (B3) imply (B2)

Proof. First we extend axiom (B3) to open sets in compact manifolds via excision (as in

the proof of Proposition 6.2.3). So we can now apply (B3):

indYGab = indXGa · indFG×Hb,

with H = 1 and Y = X × F to ai ∈ KGi(TUi) where Ui is open in Xi (i = 1, . . . , k). This

gives

ind∏

ai =∏

indai ∈ R(∏

Gi).

The case we are interested in is when Ui = Rni , Gi ⊂ O(ni) and ai = ji! (1) where ji : pt→Rni . Now assume that all the ni = 1 or 2 so that Gi = O(1) or SO(2). In this case (B2’)

implies that ai = 1. So by the above result ind∏ai =

∏indai = 1 ∈ R(

∏Gi). As j!

is multiplicative∏ai is the restriction of j!(1) ∈ KO(n)(T

∗Rn). Thus indj!(1) ∈ R(O(n))

is 1 when restricted to any subgroup∏Gi (where Gi = O(1) or SO(2)). However these

subgroups are sufficient to determine the character of O(n) as they contain all of its cyclic

subgroups (see [FH] pp.65 and Chapter 23). So indj!(1) = 1 ∈ R(O(n)) and (B2) is

established.

6.2.3 A sufficient condition for (B2’)

The axiom (B2’) can be broken down even further. We will do this by computing the

index of a complex known as the de Rham symbol ρSn ∈ KO(n)(TSn). The index of ρSn

will be related to the index of j!(1) on KO(1)(pt) and KSO(2)(pt) and via this relationship

the next axiom will imply (B2’). First we define the de Rham symbol.

Let X be a compact manifold and consider the exterior algebra of the tangent bundle

Λ∗(TX). If we pull this back over TX we obtain a complex of real vector bundles which

6.2. AXIOMS 97

is exact outside of the zero section. We may then complexify to obtain an element ρX ∈K(TX). If X is a G-manifold then Λ∗(TX) is acted on naturally by G and we have

ρX ∈ KG(TX). Note that this element is distinct from the element λTX from Section

2.4, although it is related. The relation between these two complexes is as follows. Let

TXc = TX ⊗R C we have λTXc ∈ KG(TXc). If i : TX → TXc is the inclusion, then

ρX = i∗λTXc . (6.8)

If we take X = Sn = Rn ∪∞ and G = O(n), then we get an element ρSn ∈ KO(n)(TSn).

We will now give a sufficient condition for (B2’) which involves the index of this element.

(B2”): The map ind satisfies the following

(a) indρS2 = 2 ∈ R(SO(2)), i.e. 2 times the trivial representation in R(SO(2))

(b) indρS1 = 1− ξ ∈ R(O(1)) where ξ : O(1)→ U(1) is the standard representation.

(c) indj!(1) = 1 ∈ Z, where j : P → S1 is the inclusion of the origin.

All of the elements here belong to KO(n) for n = 1, 2. However, for items (a) and (c) we

restrict to the groups SO(n).

We would like to show that these conditions are sufficient to replace (B2’). The first

step in doing this is to relate the elements ρSn and j!(1).

Lemma 6.2.1. Let j0 : pt0 → Sn, j∞ : pt∞ → Sn be the inclusions of the origin and the

point at infinity, and let θ : TSn → TSn be multiplication of tangent vectors by −1. Then

we have

ρSn = j0! (1) + θ∗j∞! (1) ∈ KO(n)(TS

n).

Proof. The n-sphere Sn may be identified with the union Bn0 ∪ Bn

∞ of two copies of the

unit ball Bn ⊂ Rn, this identification is compatible with the action of O(n). We then get

an O(n)-isomorphism

TSn ∼= (Bn0 × Rn) ∪ (Bn

∞ × Rn),

where we identify points over the equator Sn−1 = ∂Bn0 = ∂Bn

∞ by

(x, v) 7→ (x, hxv) for x ∈ Sn−1, v ∈ Rn


here hx denotes reflection in the hyperplane of Rn orthogonal to x. If we then pass to

exterior algebras, we have an O(n)-isomorphism of complexes

π∗Λ∗T c ∼= (Bn0 × Rn × Λ∗(Cn)) ∪ (Bn

∞ × Rn × Λ∗(Cn))

where T c = TSn ⊗R C, π : TSn → Sn is the bundle projection and, on the right, we make

the identification

(x, v, w) 7→ (x, hxv, hx(w)), (6.9)

here we take hx(·) to be the action on Λ∗(Cn) induced by the reflection hx. Suppose now

that 0 ≤ s ≤ 1. We define a new complex As of vector bundles over TSn by changing the

bundle homomorphisms as follows. We define

Bn0 × Rn × Λi(Cn)→ Bn

0 × Rn × Λi+1(Cn)

by

(x, v, w) 7→ (x, v, (v − isx) ∧ w) (6.10)

and

Bn∞ × Rn × Λi(Cn)→ Bn

∞ × Rn × Λi+1(Cn)

by

(x, v, w) 7→ (x, v, (v + isx) ∧ w). (6.11)

Since hx(x) = −x, (6.9) implies that (6.10) and (6.11) agree over Sn−1, and thus define

a complex of vector bundles over TSn as asserted. Next, we observe the following facts

about the complex As

(a) for all s, it is exact outisde the zero section of TSn

(b) for s = 0, it is the original complex π∗Λ∗T c

(c) for s = 1, it is exact outside pt0 and pt∞.

The properties (a) and (b) imply that the element ρSn ∈ KO(n)(TSn) may be defined by As,

for any value of s. Property (c) implies that the complex A1 defines an element a = a0 +a∞

in

KO(n)(T (Sn − Sn−1)) = KO(n)(T0)⊕KO(n)(T

∞)

6.2. AXIOMS 99

where T 0 = T (Bn0 − Sn−1), T∞ = T (Bn

∞ − Sn−1). From their definitions we see that

a0 = k0! (1), a∞ = k∞! (1),

where k0 : pt0 → B0 − Sn−1, k∞ : pt∞ → B∞ − Sn−1 are inclusions. Applying the natural

homomorphism

KO(n)(T (Sn − Sn−1))→ KO(n)(TSn)

a becomes ρSn , k0! (1) and k∞! (1) become j0

! (1) and j∞! (1) respectively, so we obtain

ρSn = j0! (1) + θ∗j∞! (1)

as required.

We now prove another lemma which we will need.

Lemma 6.2.2. Let ψ : V → V be complex conjugation, where V = W ⊗C is the complex-

ification of a real G-module W . Then if ψ∗ is the induced homomorphism of KG(V ) we

have

(a) If W = R1, G = O(1), then ψ∗a = −a[V ],

(b) If W = R2, G = SO(2), then ψ∗a = a.

Proof. Case (b) follows from the fact that ψ is G-homotopic to the identity. We define

ψt(u+ iv) = u+ igt(v) for u, v ∈ W, 0 ≤ t ≤ 1

where gt ∈ SO(2) is rotation through πt. In the case of (a) we observe that the element

ψ∗λV + λV [V ] ∈ KO(1)(V ) is represented by the complex

C⊕ V αz→ V ⊕ C

where

αz =

(z 0

0 z

)Here C = Λ0(V ) is a trivial O(1)-module, and we have used the natural isomorphism

V ∼= V , V ⊗ V ∼= C. Let gt ∈ GL(2,C) be a path connecting the identity to the matrix(0 1

1 0

). Then (

z 0

0 1

)gt

(1 0

0 z

)


gives a homotopy from αz to (0 zz

1 0

).

Thus, on the unit circle S(V ), αz is homotopic to a constant and so

ψ∗λV + λV [V ] = 0.

The Thom isomorphism theorem for ϕ : KG(pt)→ KG(V ) then completes the proof.

Finally, we can prove that (B2”) is sufficient.

Lemma 6.2.3. Any index function satisfying (B2”) also satisfies (B2’)

Proof. Using Lemma 6.2.1 we have

ρSn = j0! (1) + θ∗j∞! (1) ∈ KO(n)(TS

n)

where j0 : pt0 → Sn, j∞ : pt∞ → Sn are the inclusions of the origin and the point at infinity

(considering S2 as R2+) and θ is multiplication by −1 on tangent vectors. Let f : Sn → Sn

be reflection over the equator so that f interchanges pt0 and pt∞ and commutes with θ.

We then have

f ∗(θ∗j∞! (1)) = θ∗j0! (1).

As ind is functorial with respect to diffeomorphisms we have

indSn

θ∗j∞! (1) = indSn

θ∗j0! (1) ∈ R(O(n)).

By the additivity of the index we see

indρSn = ind(1 + θ∗)j0! (1).

Now j0! factors through the group KO(n)(TRn) by definition and, on TRn = Cn, θ

corresponds to complex conjugation. So applying Lemma 6.2.2 we deduce that

indρS2 = ind2j0! (1) = 2indj0

! (1) ∈ R(SO(2)) and

indρS1 = ind(1− ξ)j0! (1) = (1− ξ)indj0

! (1) ∈ R(O(1)).

Parts (a) and (b) of (B2”) then give

2indj0! (1) = 2

(1− ξ)indj0! (1) = 1− ξ.

6.2. AXIOMS 101

Now the annihilator of 1− ξ in

R(O(1)) = Z[ξ]/(1− ξ2)

consists of integral multiples of (1 + ξ). Hence we deduce that

indj0! (1) = 1 ∈ R(SO(2))

indj0! (1) = 1 + a(1 + ξ) ∈ R(O(1))

for some integer a. Restricting the second equation to the identity of O(1), and applying

(c) of (B2”) gives

1 = 1 + 2a i.e. a = 0.

Thus indj0! (1) = 1 for both SO(2) and for O(1). This is precisely axiom (B2’) bearing in

mind that ind is defined on K(TRn) via the compactification Rn → Sn.

6.2.4 Summary

We now give a brief summary of the results of this chapter and how they can be used

to prove the Atiyah-Singer index theorem. The results of this chapter collect to give the

following proposition.

Proposition 6.2.5. Any index function satisfying axioms (A1),(B1), (B2”) and (B3)

is equal to the topological index.

The index theorem asserts that the analytic index of an operator equals its topological

index. For any elliptic pseudo-differential operator P over a compact manifold the analytic

index is determined by IndexP = a−indσ(P ). Thus the index theorem can be displayed

in the following diagram

K(T ∗X)t−indX // Z

E

σ

OO

σ // K(T ∗X)

a−indX

OO

where E ⊂ ΨDO is the set of elliptic operators. We verify the conditions of Proposition

6.2.5 for a−ind by representing classes in K(T ∗X) by elliptic operators and computing

their indices.


Chapter 7

The Atiyah-Singer Index Theorem

With both the topological and analytic indices defined we can now state the Atiyah-

Singer index theorem. We will then prove the index theorem using the refined set of axioms

given in the previous chapter.

Theorem 7.0.1. (The Atiyah-Singer Index Theorem) The topological and analytic indices

coincide as homomorphisms K(T ∗X)→ Z,

t−ind σ(P ) = a−ind σ(P ).

7.1 Verification of the axioms for the Analytic Index

In order to prove the index theorem we must show that the analytic index, a−ind

satisfies the axioms (A1),(B1), (B2”) and (B3) given in Chapter 6. We then finish by

invoking Proposition 6.2.5.

Proposition 7.1.1. (A1) If X is a point, then a−indpt : Z→ Z is the identity map.

Proof. First note that if X is a point then an elliptic operator on X is a linear transfor-

mation P : V → W between complex vector spaces V and W , and

[σ(P )] = dimE − dimF ∈ K(TX) ∼= Z.

Furthermore, we have

dim KerP + dim RanP = dimE, and dimF − dim RanP = dim KerP ∗.

103

104 CHAPTER 7. THE ATIYAH-SINGER INDEX THEOREM

Thus

IndexP = dim KerP − dim KerP ∗ = [σ(P )].

Axiom 1 is immediate.

7.1.1 Excision

Proposition 7.1.2. (B1): Let U be an open manifold and

j : U → X,

j′ : U → X ′

be two open embeddings into compact manifolds X and X ′. Then the following diagram

commutes

KG(TU)j∗ //

j′∗

KG(TX)

a−indXG

KG(TX ′)a−indX

′G

// Z

Proof. By Lemma 2.2.11 we know that any element u ∈ K(TU) can be represented as

[π∗E, π∗F ;σ] where

(a) E and F are trivial outside a compact set

(b) σ is homogeneous of degree 0 outside a compact set

(c) σ is exact outside a compact set

This means there is a compact set L ⊂ U such that we can take trivializations

a : E|U−L →(U − L)× Cm

b : F |U−L →(U − L)× Cm

which satisfy σ(x,ξ) = σx = b−1x ax ∀x ∈ U − L (using homogeneity and exactness).

So we see that when restricted to T (U − L), σ(x,ξ) is just a map σ0 : E → F over the

base space. Also, note that σ0 is the identity map on the above trivializations and is a

differential operator of order 0 (i.e. a smoothly varying linear map).

Next we choose a representative operator P ∈ ΨCO0(E,F ) such that σ(P ) = σ on TU

and is the operator σ0 on U − L.

7.1. VERIFICATION OF THE AXIOMS FOR THE ANALYTIC INDEX 105

Now we look at our embedding j. We may extend E and F trivially over X − j(U)

using the trivializations a and b. We also extend our operator P to a new operator j!P

by taking it equal to the identity (induced by b−1a) on X − j(U). By construction this

operator has the property

[σ(j!P )] = j![σ(P )] = j!u,

this is because j! is induced by the map j which trivially extends a representative of a

class in KG(TU) on the set X − j(U). Now if s ∈ Ker(j!P ) then s must have support in

L (as j!P is the identity elsewhere). This implies that s ∈ Ker(P ) (under the embedding

Ker(P ) ⊆ Kerj!(P )) thus dim Kerj!P = dim KerP .

The same argument applies for the adjoint operator P ∗. Thus we have

Indexj!P = dim Kerj!P − dim Ker(j!P )∗

= dim KerP − dim KerP ∗

= IndexP

Thus a−indj!P is independent of j! which proves that it satisfies (B1).

7.1.2 Normalization

Proposition 7.1.3. (B2”): The map a−ind satisfies the following

(a) indρS2 = 2 ∈ R(SO(2))

(b) indρS1 = 1− ξ ∈ R(O(1)) where ξ : O(1)→ U(1) is the standard representation.

(c) indj!(1) = 1 ∈ Z, where j : P → S1 is the inclusion of the origin.

Proof. For the circle, the de Rham complex is just f 7→ df = (df/dx) · dx where x mod 1

is a parameter for the circle. Hence Kerd consists of the constant functions, and Cokerd is

generated by dx. The generator of O(1) is the map x 7→ −x which induces dx 7→ −dx on

Cokerd and the identity on Kerd. Thus

a−indρS1 = 1− ξ.

For S2, the de Rham complex has three terms

0→ Ω0 d0→ Ω1 d1→ Ω2 → 0.


Again Kerd0 consists of constants. Taking adjoints, we see that Cokerd1 = Kerd1∗ is

generated by the volume form in Ω2. Since SO(2) acts trivially on both these spaces, it

remains to show that there is no contribution from Ω1 i.e., that

dw = 0⇒ w = df for w ∈ Ω1, f ∈ Ω0.

Now outside the north pole, the Poincare lemma (see Section 4 of [BT]) shows that if

dw = 0 then w = df0 and similarly w = df∞ outside the south pole. Then d(f0 − f∞) = 0,

and so f0 − f∞ is constant away from the poles. Thus f0 is actually defined everywhere,

i.e., f0 ∈ Ω0.

Now we wish to show part (c). We must compute the index of the symbol class j!(1).

This is not a differential operator. Using the multiplicative axiom (B3) we can compute

(a−indj!(1))2 on S1 × S1 or S2. Using part (a) which we have just verified, we find that

(a−indj!(1))2 = 1,

so a−indj!(1) = ±1, and the problem is reduced to showing that with our choice of sign

conventions we get the plus sign.

Consider first the operator P operating on functions on the circle S1 = R/2πZ defined

by

Peinx =

einx for n ≥ 0

0 for n < 0.

We shall show that P ∈ ΨCO0. Let f ∈ C∞c (R) have support in an interval of length < 2π

(which we identify with its image in S1). The Fourier coefficients of f(x)eixξ are f(n− ξ).Hence, in suppf ,

pf (x, ξ) = e−ixξP (f(x)eixξ) =∞∑n=0

f(n− ξ)eiz(n−ξ)

= f(x)−−∞∑n=−1

f(n− ξ)eix(n−ξ).

As ξ → −∞, the sum∑∞

0 converges to 0 faster than any power of |ξ|, and so do all of

its derivatives with respect to x and ξ. When ξ → +∞ , the sum∑−∞−1 has the same

properties. Thus P is pseudo-differential of order zero. Since

pf (x, ξ)→

f(x) as ξ → +∞

0 as ξ → −∞


if follows that P ∈ ΨCO0, and that its symbol σP is then given by

σP (x, ξ) =

1 for n ≥ 0

0 for n < 0,

Now define A = eixP + (1− P ). This is then an operator in ΨCO0, with symbol given by

σA(x, ξ) =

eix for n ≥ 0

1 for n < 0,(7.1)

Hence A is therefore elliptic. On the other hand by definition

Aeinx =

ei(n+1)x for n ≥ 0

eix for n < 0,.

and so KerA = 0 and CokerA is generated by the constant functions. Thus IndexA = −1.

To establish axiom (B2”) for the analytical index, it remains to show that the natural

extension homomorphism

K(TR1)→ K(TS1),

takes the element −j!(1) into the class of σA in K(TS1). To do this, we define a continuous

symbol σ on R1 by equation (7.1) for 0 ≤ x ≤ 2π, and by σ = 1 for x < 0 or x > 2π. It

is then clear that [σ] → [σA] in the above homomorphism. Thus we finally have to check

that

[σ] = −j!(1) ∈ K(TR1).

Now both of these elements are defined by maps

TR1 \K → C \ 0,

where K is some compact set. For σ, the K is the rectangle 0 ≤ x ≤ π, |ξ| ≤ 1, while for

j!(1), it is the disc |x + iξ| ≤ 1. On the rectangle, σ is eix on top and 1 elsewhere, while

j!(1) is x+ iξ on the unit circle. An elementary homotopy then deforms σ into −j!(1) and

the verification of axiom (B2”) is complete.


7.1.3 Multiplication

Proposition 7.1.4. (B3): Let X be a compact manifold, H a compact topological group

and F a compact H-manifold. If a ∈ K(TX) and b ∈ KH(TF ) then

a−indY ab = (a−indXa) · (a−indFHb),

provided that a−indFHb is a multiple of the trivial representation 1 ∈ R(H). Here the

product ab is the product constructed in Chapter 6 equation (6.5).

Proof. Let Y = P ×H F be a fibre bundle over X, where P → X is a principal H-bundle.

First represent a by a homogeneous complex [π∗E, π∗F ;α] of degree 1 over TX. Then

construct an elliptic operator A ∈ ΨCO1 over X with principal symbol σ(A) = α. Take

a partition of unity φ2i subordinate to an open cover Ui which trivializes P and thus

Y . Define operators Aj := φjAφj and let Yj = p−1(Uj) ∼= Uj × F . These operators can

be lifted to Y by letting Aj′(u ⊗ v) = Aj

′(u) ⊗ v for the restriction u|Uj . We then define

A′ :=∑

j Aj ′ ∈ ΨCO1(Y ). Note that σ(Aj) =

∑φ2jσ(A) and taking symbols commutes

with lifting so

σ(A′) = α′

where α′ is the lift of α.

Next we represent the element b ∈ KH(TF ) by an H-invariant operator B ∈ ΨCO1(F )

with principal symbol β such that [β] = b. We then lift this as before to get an operator B′

on P × F . The operator B′ inherits the H-invariance of B and as such can be considered

to live on Y = P ×H F .

We now write the product ab in terms of the operators A′ and B′. We obtain the

operator D on Y given by

D =

(A′ −B′∗

B′ A′∗

), with σ(D) =

(α 1 −1 β∗

1 β α∗ 1

).

We note that D is not necessarily in ΨCO, however it is in the closure ΨCO by Lemma

4.3.5. This is sufficient for our purposes as the index is locally constant on Fred(Ls+d, Ls)

(see Lemma 5.1.1) and thus on Opm. Recalling the expression for the product of two

complexes of length one (see Proposition 2.2.9), we note that σ(D) corresponds to the

element ab ∈ K(TY ) represented by the complex

0→ E0 G0 ⊕ E1 G1 θ→ E0 G1 ⊕ E1 G0 → 0


and so we must calculate the index of D. To do this, we consider the following two

diagonalized operators (noting that A′ and B′ commute)

D∗D =

(A′∗A′ +B′∗B′ 0

0 AA′∗ +B′B′∗

)=

(P0 0

0 Q0

),

DD∗ =

(A′A′∗ +B′∗B′ 0

0 A′∗A+B′B′∗

)=

(P1 0

0 Q1

)

where

Ker(D) =Ker(D∗D) = Ker(P0)⊕Ker(Q0),

Ker(D∗) =Ker(DD∗) = Ker(P1)⊕Ker(Q1)

Thus we have

Index(D) = (dim Ker(P0)− dim Ker(P1)) + (dim Ker(Q0)− dim Ker(Q1)).

Considering the operator P0 = A′∗A′ +B′∗B′, we note that

〈P0u, u〉 = 〈A′u,A′u〉+ 〈B′u,B′u〉,

and so Ker(P0) = Ker(A′) ∩ Ker(B′). Since B′ extends B to the fibres of Y , we see that

Ker(B′) is the space of smooth sections of the vector bundle KB = P ×H Ker(B) over X.

Then A′ induces an operator C on sections of KB with σ(C) = α⊗ Id(KB), and it follows

that σ(C) = a[KB] ∈ K(TX), where [KB] is the class of KB in K(X). Replacing A′ by

A′∗ we obtain the analogous result for P1 and C∗, which gives us

dim Ker(P0)− dim Ker(P1) = dim Ker(C)− dim Ker(C∗)

=Index(C) = a−indX(a[KB]).

Similarly, taking LB = P ×H Ker(B∗), we obtain

dim Ker(Q0)− dim Ker(Q1) = dim Ker(C)− dim Ker(C∗) = a−indX(a[LB])

Combining these results gives

IndexD = a−indX(a · ([KB]− [LB])).


If in addition a−indFHb = [Ker(B)] − [Ker(B∗)] ∈ R(H) is an integer (i.e. a multiple of

the trivial representation of H on C), then [KB] − [LB] = a−indFHb · 1 is a multiple of

the trivial bundle 1, and since a−indX is a homomorphism we obtain the desired product

formula IndexYD = (a−indXa) · (a−indFHb).

Having proved that the analytic index satisfies these three axioms we appeal to Propo-

sition 6.2.3 and the proof of Theorem 7.0.1 is complete.

Chapter 8

An Example: The Dirac Operator on

S2

We will now build an example which displays many of the concepts which we have

discussed. The base space for this example will be S2 and our operator will be the Dirac

operator which acts on sections of the spinor bundle. The Dirac operator is fundamental

both to the theory and applications of the Atiyah-Singer index theorem, although it does

not appear in the proof which we have presented here. The importance of the Dirac opera-

tor lies in the fact that it can be used to generate the symbol class of any elliptic operator.

This is done by twisting the operator with another bundle via the K-theory multiplication

map. Dirac operators are also compatible with the product operator construction given in

the proof of Proposition 7.1.4.

8.1 An atlas for S2

First we set up coordinates. We may describe S2 by two charts UN and US which omit

the north and south poles respectively. Local coordinates on the north and south charts

respectively are given by

z = e−iφ cot θ/2 and ζ = eiφ tan θ/2. (8.1)

On the overlap we relate these charts by ζ = 1/z, note that this just reflects everything

across the equator interchanging the north and south poles.

111

112 CHAPTER 8. AN EXAMPLE: THE DIRAC OPERATOR ON S2

If we let q(z) := 1 + zz then our metric g and area form Ω are given by

g = dθ2 + sin2 θdφ2 = 4q(z)−2dzdz = 4q(ζ)−2dζdζ, and (8.2)

Ω = sin θdθ ∧ dφ = 2iq(z)−2dz ∧ dz = 2iq(ζ)−2dζ ∧ dζ (8.3)

8.2 The bundles on S2

Recall from Example 2.4.1 that reduced K-theory of the sphere is K(Sn) ∼= Z and is

generated by the element (1 − [H]). We can use the following projection to generate line

bundles representing the nontrivial classes in K(S2) = Z⊕ Z.

pm :=1

1 + (zz)m

((zz)m zm

zm 1

), p−m :=

1

1 + (zz)m

((zz)m zm

zm 1

)

Note that these projections are globally well-defined on S2. For every point z ∈ S2 they

project a complex line from C2 which we consider to be the fibre over z. In other words

these projections define the bundles Em with fibres Emz = pm(z)C2.

8.2.1 Sections

The charts we have defined provide a trivialization for the above bundles, so we can

write down sections explicitly by considering them over each chart. Define two local sections

on UN and US by

σmN(z) :=1√

1 + (zz)m

(zm

1

), σmS(ζ) :=

1√1 + (ζζ)m

(1

ζm

).

Over each chart these sections pick a vector from the fibre of Em over z and ζ respectively.

Over each chart sections can then be defined by functions fN : UN → C and fS : US → Cby taking fNσmN and fSσmS. In order to form a global section we take a pair of such

sections which are related in the appropriate way on the overlap UN ∩ US. The relation

which they must satisfy is known as a gauge transformation and in our case it is given

by

fN(z) = (z/z)m/2fS(z−1).

8.2. THE BUNDLES ON S2 113

8.2.2 The spinor bundle

Using the bundles Em constructed above we can define the spinor bundle S = S+⊕S−,

where S+ = E1 and S− = E−1. The importance of the spinor bundle lies in the fact that

it carries an action γ of cotangent vectors. This action is know as the Clifford action and

will be described, for our specific case, in the next section. Note that the spinor bundle

can only be constructed on manifolds which satisfy a certain topological condition, this

condition for example implies orientability (see Section 2.1 [Fr]).

From the above discussion we see that a section of the spinor bundle, known as a spinor,

is represented by a pair of functions on each chart ψ±N(z) and ψ±S (z), related on UN ∩ USby

ψ+N(z) =

√z/zψ+

S (z−1), ψ−N(z) =√z/zψ−S (z−1). (8.4)

8.2.3 The spinor bundle connection

Our operator will be defined in terms of a connection on the spinor bundle known as

the spin connection. The spin connection 5S is determined by the Leibniz rule and the

action γ

5S(γ(α)ψ) = γ(5gα)ψ + γ(α)5s ψ (8.5)

We will unpack this definition starting with 5g.

The connection 5g is the Levi-Civita connection on the cotangent bundle. It is deter-

mined by

5gq∂z

(dz

q) = z

dz

q, 5g

q∂z(dz

q) = −z dz

q

5g

q∂z(dz

q) = −zdz

q, 5g

q∂z(dz

q) = z

dz

q

The Clifford action γ(α)ψ is an action of the 1-form α on the spinor ψ. To construct

the action we will use the following matricies

γ1 :=

(0 1

1 0

), γ2 :=

(0 −ii 0

)(8.6)

and define the complex combinations γ± := 12(γ1 ± iγ2). The grading operator for the

spinor bundle S = E1 ⊕ E−1 is then given by

γ3 := −iγ1γ2 =

(1 0

0 −1

).


The bundles E± give the eigenspinors of γ3. Note that γ±γ3 = ∓γ±. Finally, we take

γ(dz) := q(z)γ+ and γ(dz) := q(z)γ−.

The spin connection is then given by

5Sq∂z = q∂z +

1

2zγ3,5S

q∂z= q∂z −

1

2zγ3, (8.7)

which is consistent with 8.5 and 8.6. These operators commute with the grading operator

γ3 and thus act on each factor S± of the spinor bundle by

5±q∂z = q∂z ±1

2z,5±

q∂z= q∂z ∓

1

2z, (8.8)

8.3 The Dirac operator

The Dirac operator D := −iγ(dxj)5s∂j

on S2 may be written in complex coordinates

as

D = −iγ(dz)5s∂z −iγ(dz)5s

∂z= −iγ(dζ)5s

∂ζ−iγ(dζ)5s

∂z.

We may write the principal symbol of the Dirac operator as two local sections. Letting

ξ = ξ1dz + ξ2dz we have

σD(z, ξ) = −iγ(dz)ξ1 − iγ(dz)ξ2

= − i2q(z)

(0 1− i

1 + i 0

)ξ1 −

i

2q(z)

(0 1 + i

1− i 0

)ξ2

on UN . As q(z) cannot equal zero this is invertible at all points of UN . Similarly, σ is

invertible on US, so D is elliptic.

The action γ determines both the bundle S and the operator D on this bundle. This

provides an explicit example of the tight relationship between the vector bundle, operator

and principal symbol. As we shall see later the Dirac operator gives a canonical example

of this relationship.

We will now write the Dirac operator in a slightly different form. From the equation

8.7 for the spin connection we get

D = −iγ+5Sq∂z −iγ

−5Sq∂z

= −iγ+(q∂z +1

2zγ3)− iγ−(q∂z +

1

2zγ3)

= −i(q∂z −1

2z)γ+ − i(q∂z −

1

2z)γ−.

8.3. THE DIRAC OPERATOR 115

We introducing the first order differential operator

δz := (1 + zz)∂z −1

2z = q∂z −

1

2z = q3/2 · ∂z · q−1/2

and it’s conjugation δz := q∂z − 12z. We may then express D as

D = −i(δzγ+ + δzγ−) = −i

(0 δz

δz 0

). (8.9)

8.3.1 Extension to Hilbert space

We now extend our operator to L2 spaces. The operator δ : Γ(S+) → Γ(S−) is skew-

adjoint

〈φ+, δzψ−〉L2(S+,S−) = −〈δzφ+, ψ−〉L2(S+,S−),

on the Hilbert space L2(S+,S−), in view of δz = q3/2 · ∂z · q−1/2.

The scalar product of spinors is then given by

〈ψ1, ψ2〉 = 〈ψ+1 , ψ

+2 〉+ 〈ψ−1 , ψ−2 〉 :=

∫S2

(ψ+1 ψ

+2 + ψ−1 ψ

−2 )Ω.

Our operator D thus extends to a self-adjoint operator on this space of spinors, which

we call H := L2(S2, S). Moreover, γ3 extends to a grading operator on H for which

H± = L2(S2, S±), and it is immediate that Dγ3 = −γ3D.

8.3.2 Spinor Harmonics and the spectrum of D

We will compute the spectrum of this operator. We construct eigenspinors based on

two observations. The first is a calculation with the δ operator:

δz(q−lzr(−z)s) = (l +

1

2− r)q−lzr(−z)s+1 + rq−lzr−1(−z)s, (8.10)

−δz(q−lzr(−z)s) = (l +1

2− s)q−lzr+1(−z)s + sq−lzr(−z)s−1, (8.11)

where r, s ∈ N. The first can be checked directly, the second follows by complex con-

jugation. One sees from these equations that suitable combinations of the functions

q−lzr(−z)s, with l and (r − s) held fixed, will form eigenvectors for the operator D on

account of it’s presentation in (8.9). The second observation is that compatibility with


the gauge transforms of spinors (8.4) imposes restrictions on the exponents l, r, s. If

φ(z) :=∑

r,s≥ a(r, s)q−lzr(−z)s, then

(z/z)1/2φ(z−1) = (−1)l+12

∑r,s≥0

a(r, s)q−lzl−12−r(−z)s,

so that φ ∈ S+ iff l− 12∈ N, and a(r, s) 6= 0 only for r = 0, 1, . . . , l− 1

2and s = 0, 1, . . . , l+ 1

2.

To have φ ∈ S−, interchange the restrictions on r and s.

If m := r − s± 12, the corresponding restriction is m = −l,−l + 1, . . . , l − 1, l.

We can now display the spinor harmonics. They form two families Y +lm and Y −lm, corre-

sponding to upper and lower spinor components; they are indexed by

l ∈ N = 1

2,3

2,5

2, . . .,m ∈ −l,−l + 1, . . . , l − 1, l,

and the formulas are

Y +lm(z) := Clmq

−l∑

r−s=m− 12

(l − 1

2

r

)(l + 1

2

s

)zr(−z)s,

Y −lm(z) := Clmq−l

∑r−s=m+ 1

2

(l + 1

2

r

)(l − 1

2

s

)zr(−z)s,

(8.12)

where the normalization constants Clm are defined as

Clm := (−1)l−m√

2l + 1

4π

√(l +m)!(l −m)!

(l + 12)!(l − 1

2)!.

Eigenspinors.

The coefficients in (8.12) are chosen to satisfy

δzY−lm = (l +

1

2)Y +

lm and δzY+lm = −(l +

1

2)Y −lm

in view of (8.10). If we then normalize spinors by

DY 1lm :=

1√2

(Y +lm

iY −lm

), DY 2

lm :=1√2

(−Y +

lm

iY −lm

)

we get an orthonormal family of eigenspinors for the Dirac operator:

DY 1lm = (l +

1

2)Y 1

lm, DY2lm = −(l +

1

2)Y 2

lm

8.4. TWISTED SPINOR BUNDLES 117

where the eigenvalues are nonzero integers and each eigenvalue ±(l + 12) has multiplic-

ity (2l + 1). In fact these are all the eigenvalues of D; for that, we need the following

completeness result (see [GMNRS]):∑l,m

Y ±lm(z)Y ±lm(z′) = δ(φ− φ′)δ(cos θ − cos θ′),

(note that these δ’s are Dirac deltas, not our operator).

Consequently, the spinors Y 1lm, Y

2lm : l ∈ N + 1

2,m ∈ −l, . . . , l form an orthonormal

basis for the Hilbert space H = L2(S2, S).

The Spectrum.

We have thus computed the spectrum of the Dirac operator:

sp(D) = ±(l +1

2) : l ∈ N +

1

2 = Z \ 0,

with the aforementioned multiplicities (2l+ 1). Thus D is a Fredholm operator with index

zero.

8.4 Twisted Spinor Bundles

In this section will will apply the multiplication map of Section 2.2.4 in order to generate

representatives of all the other symbol classes.

To define other elliptic operators on S2, we apply the multiplication map, twisting the

spinor bundle and Dirac operator by tensoring it with some other bundle E. The resulting

Clifford action on the twisted bundle S ⊗ E is given by

γ(α)(ψ ⊗ σ) := (γ(α)ψ)⊗ σ for ψ ∈ S, σ ∈ E

We call S ⊗ E, with the above action a twisted spinor bundle.

We will use our example on S2 to show that new Dirac operators can be created by

twisting the fundamental one.

The bundles E−m, which we defined earlier, give representatives for all of the non-trivial

isomorphism classes of Vect(S2) so we shall tensor the spinor bundle with these. In this

case we have S ⊗ Em ∼= Em+1 ⊕ Em−1.


8.4.1 Twisted Dirac operators

The Clifford action induced on S⊗E can be used to define a twisted Dirac operator

as follows. The half-spinor bundles S± = E± have connections 5± given by (8.8). Now

Em ∼= E1 ⊗ . . .⊗ E1 (m times) if m > 0 and Em ∼= E−1 ⊗ . . .⊗ E−1 (|m| times) if m < 0,

so we can define a connection 5m on Em by

5m(s1 ⊗ . . .⊗ s|m|) :=

|m|∑j=1

s1 ⊗ . . .⊗5±(sj)⊗ . . .⊗ s|m|.

From (8.8) and the Leibnitz rule it follows that

5mq∂z = q∂z +

1

2mz,5m

q∂z= q∂z +

1

2mz.

On S ⊗ E we define the twisted spin connection 5S:= 5S ⊗ 1 + 1⊗5m. We obtain

5S

q∂z = q∂z +1

2z(m+ γ3), 5S

q∂z = q∂z +1

2z(m+ γ3).

The corresponding dirac operator is then

Dm = −iγ+5S

q∂z − iγ−5S

q∂z

= −i(q∂z +1

2(m− 1)z)γ+ − i(q∂z −

1

2(m+ 1)z)γ−

= −i(δz +1

2mz)γ+ − i(δz −

1

2mz)γ−,

In matrix form this is

Dm = −i

(0 D−m

D+m 0

)= −i

(0 δz + 1

2mz

δz − 12mz 0

)

8.4.2 The index of the twisted Dirac operator

Notice that

D+m = −iq(m+3)/2 · ∂z · q(−(m+1)/2,

so that a half-spinor ψ+ lies in Ker(D+m) if and only in ψ+

N(z) = q(z)(m+1)/2a(z) where

a is an entire holomorphic function. The gauge transformation rule (8.4) shows that the

function ψ+S (z−1) = (z/z)(m+1)/2ψ+

N(z) is regular at z = ∞ only if either a = 0, or m < 0

and a(z) is a polynomial of degree < |m|. Thus dim Ker(D+m) = |m| if m < 0 and equals 0

8.4. TWISTED SPINOR BUNDLES 119

for m ≥ 0. A similar argument shows that dim Ker(D−m) = m if m > 0 and is 0 for m ≤ 0.

We conclude that D+m is a Fredholm operator on Hm, whose index is

Index(D+m) := dim Ker(D+

m)− dim Ker(D−m) = −m

Thus we see that the index of this operator depends only on the class of the twisting

bundle Em. Moreover, each of the twisted Dirac operators for m ∈ Z must be in a different

symbol class as the index is constant on the symbol class. Thus we see that all of the

non-trivial symbol classes in K(TS2) ∼= K(Ss) = Z ⊕ Z can be represented by twisted

Dirac operators. This property is true in general and shows the significance of the Dirac

operators to index theory; they can be used to represent all of the K-theory classes.


Chapter 9

Summary

In this thesis we have developed both the topological and analytic basis for the K-

theoretic proof of the Atiyah-Singer index theorem. We have presented some basic K-

theory, seen an important characterisation of K(X) in terms of complexes and defined the

Thom isomorphism. Pseudo-differential operators have been developed and, in analogy

with differential operators principal symbols and ellipticity have been defined. The prin-

cipal symbol was seen to be globally well-defined and used to relate ellitpic operators to

K-theory.

We developed the analytic properties of elliptic operators, showing that elliptic opera-

tors are invertible modulo compact operators. This result was used to show that elliptic

operators are Fredholm operators and have a well-defined index. The structure of the

space of Fredholm operators was investigated and it was shown that the index is locally

constant on Fred. This result allowed us to define the analytic index, the index of a class

in K(T ∗X).

We presented the proof of the Atiyah-Singer index theorem as given in [AS1]. The

topological index was and we gave axioms which characterise it. These axioms were devel-

oped into sufficient conditions for a general index function to be equal to the topological

index. These conditions prove easier to verify for the analytic index. The the conditions

were then verified for the analytic index, proving the Atiyah-Singer index theorem.

Finally, an example was given which illustrated many of the phenomena associated with

the index theorem.

121

122 CHAPTER 9. SUMMARY

Bibliography

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[AS] M. F. Atiyah and I. M. Singer, The index of elliptic operators on compact man-

ifolds, Bull. Amer. Math. Soc. Vol. 69, No. 3, 1963, pp. 422-433.

[AS1] M. F. Atiyah and I. M. Singer, The index of elliptic operators I, The Annals of

Mathematics, Second Series, Vol. 87, No. 3, (May, 1968), pp. 484-530.

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The Atiyah-Singer index theorem