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Globaltheoryof elliptic operators

By M.F.
§1.

ATIYAH|

Fredholm operators and K-theory.

In recent years the problem of computing the index of an elliptic differential operator on a closed manifold (or on a manifold with boundary) has
been successfully solved [3] by the use of a new branch of algebraic topology
Further investigation has brought to light many fundamental(K-theory).
connections between elliptic operators and K-theory and in this lecture I want|to present a new view-point on these matters.
The best place to start is with abstract functional analysis which provides
the natural meeting ground of algebraic topology and partial differential equations. So let H be complex Hilbert space (infinite-dimensional and separable)
and let us recall that a Fredholm operator T on His a bounded linear operator|~~ on H such that .
T and

(1.1)

T* havefinite-dimensional

null-spaces (or kernels)

and

|

(1.2)

T has closed range (so that Im (T)=(Ker T*)L).

The

index

of

T is then

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by

defined|index T'=dim Ker T—dim Ker T*.
Thebasic properties of the index of Fredholm

operators areindex (T+ A)=index T;
if T is Fredholm, and if S is a bounded operator
near in norm to T, then S is also Fredholm and

(1.4)

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sufficiently ||index S=index T'.
If we topologise
logy,

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If Tis Fredholm and A is compact, then T+ A is Fredholm and

(13)

then

-

& (the space of Fredholm operators on H) by the norm

(1.4) asserts

topo-

that

index: ¥F—Z|

is a continuous map
and
Proc.

not

hard

to show

to the integers Z.
that

two

It is trivial that this map

operators with the same index

areis surjectivein theof the Int. Symposium on Functional Analysis, Tokyo. University of Tokyo Press (1969), 21-30

same

|

22

~

component

of #.

Thus

1.5)

M.F.

ATiYAH-

the index

induces a bijection|t(F)—Z

where 7, denotes the set of path-components.
|
Suppose now that X is a compact space and that 7:
X—& is a continuous
map, so that 7, is a family of Fredholm operators depending continuously on
the parameter x € X. If dim Ker T, is independent of x the family of vector
spaces Ker T, forms a vector bundle Ker T over X and similarly for Ker T*.
We can then define the index of the family by

EE

index T=[Ker T]—[Ker T*] € K(X)

where K(X) is the abelian group generated by the semi-group of vector bundles
over X (up to isomorphism).
If Ker T, has a variable dimension we can compose with a projection (see [1] or [2]) to reduce to the previous case. It|turns out (see [1]) that T—index T defines a bijectionof the set [X, F] (ofhomotopy classes of maps X— &) onto K(X ). The composition of operators
in & corresponds to the addition in K(X) and adjoints correspond to negatives.

For X a point we then have

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K(poin) = n(F)=Z.

|

The group K(X) is a contravariant functor of X, depending only on thehomotopy of X. It has many of the formal properties of cohomology, andone can prove [1] that|KX)
®zR = H"(X; R):

where R is the real field and H® denotes the direct sum of all even-dimensional cohomology groups. In fact if we replace the algebra of operators on
H

by

[4],

a von
the

Now,
H*(X),

Neumannfactorcorrespondingof typeK-groups

Il, andturn out

defineto be"

Fredholm

actually

operators

isomorphic

as is well-known, a topological space not only has cohomology

it alsohashomologygroups

H,(X).

These

are

as in

to He".

groupscovariant functorsof

X and are, in a certain sense, dual of cohomology.
If we take real coefficientsand make some reasonable assumptions on X (for example that it is a compact
polyhedron)”, then H,(X; R) is in fact the dual of the finite-dimensional vector
space

HX;

R).

one can defineForintegercoefficients theH (X; Z) by putting

relation

where
~

1)

DyX

2)

is a Spanier-Whitehead N-dual

See a forthcoming

Forschung

des

These

Landeswill be made

but

Z)

of X:

we embedXin some

Eucli-

article by I.M. Singer published by Arbeitsgemeinschaft

Nordrhein-Westfalen,

assumptions

complicated|

H(X,Z)=HYYDyX;|

is more|from

Diisseldorf,

now

on.

fiir

1969.

|

|

dean

Globalspace RY!

RY*1—X.

By

theoryand take DyX

the same

deviceof elliptic operators

to be a deformation
wecandefine®

a|23

retract of the complement

covariant

we putK-functor,

namely||K(X)=K(D,yX).

||

In the next two sections I shall explain how this covariant—or homology type—K-functor is related to elliptic operators.
|
NoTE.
Partly for reasons of symmetry and partly to help to distinguishthe two different K-functors we shall now write K° instead of K. Thus K°§ 2.

Elliptic operators.

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oo

is contravariant, K, is covariant.

oo

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|

I shall now recall a few basic facts concerning linear elliptic differential
operators on closed manifolds.
If P is such an operator (acting on scalarfunctions,vector-valued

functionsor moregenerallyon

sectionsofavectorbundle) it has an orderm and if we introduce the Sobolev spaces H® (functions f all of whose partial derivatives up to order s are in L?) we can regard
P as a bounded operator P: H s_ Hs-™
The ellipticity of P implies ‘that it
has a parametrix (inverse modulo lower order) and hence that it is a Fred-holm operator.

It therefore has an index which (because of the regularity ofsolutions of elliptic equations)is independent of s.

shows

be

how

the

index

of P cancalculatedThe

in terms

index theoremof theof [3]

geometrical

data

provided by P and the underlying manifold X.
|
Of course for a differential operator (other than the trivial case of a multiplication) the order m is a positive integer. However using the larger classof pseudo-differential operators [3] one can reduce to the case of order 0, ob- taining

An

what

used

elliptic

to becalledoperator(ofoperator on Hilbert space.

singularorder

integralzero)

operators.

therefore gives rise to a Fredholm

There is however

more structurein theelliptic

operator which has been ignored on passing to the Hilbert space. To reinstatethis further structure we must make use of the fact that our Hilbert spacesarenot just abstract vector spaces but are in fact function spaces. Thus
they not only admit multiplication by complex scalars but also by complex~ scalar-valued functions on X. A differential operator does not of course commute with multiplication by functions, but it does commute modulo a lower~~. order operator.
|seThus if P is a differential operator of order m ‘with C= coefficients andif

f is a C~
3)function,

There are also otherthecommutator(equivalent)

Pf—fP is a differential operator ofdefinitions,see [7].

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24

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M.F. ATIYAH

order m—1 with C= coefficients.
zero one even has the following

commutator Pf—fP

For pseudo-differential operators P of orderresult: for any continuous function f, theis a compact operator.

We have now arrived at a property of pseudo-differential eperators which
can be abstracted out and applied to general topological spaces. Thus let X
be any compact Hausdorff space, C(X) the ring of continuous complex-valued

functions

on

X,

and

let H,, H, be two Hilbert spaces which are continuous

C(X)-modules (that is, we have uniformly continuous *-algebra homomorphismsof C(X)
linearinto thealgebrasofboundedoperators on H, and H,).

operatorAbounded|oo

P:H —H,will be called an(2.1)

operator

for any fe C(X)

on X if|the commutator

Pf—fP

is a compact

operator.

P will be called an elliptic operator on X if, in addition, it is a Fredholm
operator. There is then another operator Q on X such that QP—I and PQ—I
are both compact (where I denotes the identity operator).

The

set

of all elliptic

operators on X will be denoted by Ell (X).

This

definition abstracts out the interaction between elliptic operators and multiplication by functions.
It does not attempt to give any abstract analogue ofthe local representation (or symbol) of a differential operator.
to thisquestion§3.

later.

a.

We shall returni

From elliptic operators to K,.

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In this section I shall show how to define a map
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-

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Ell(X)— K(X),that is we shall associate to each elliptic operator on X (in the sense of §2)
an element of K(X).
Moreover every element of K(X) can be shown toarise in this way so that we can think of elliptic operators on X as repre-

sentative “cycles” for the “homology ” group K,(X).
in §4.
When

operator

one has|This will be explained|X

is a point,

and

hence

an

elliptic

definesan||operator on X is justelement

of

K°point).

CaK, (point) = K°(point)anabstract

But

Fredholm

rather formally||and so we get our map Ell (point)— K,(point).
In fact, since K°point)= Z,
given by the index, our map just assigns to each operator its index.
Consider next the dependence of Ell (X) on X. If f: X—Y is a continuousGlobaltheoryof elliptic operators| 25

map of compact spaces, we get a homomorphism of rings f*: C(Y)— C(X).
If Hy, H, are Hilbert space modules for C(X) they can then be viewed, usingf*,as C(Y)-modules.

onan elliptic operator

In this way

oo

fx:

Ell(X)—

~~

|

oo

|

oo

the

underlying— K (point))= Z
Ell (point

Co

covariantly

on

can

o

EN(X)—ENl(Y),so that elliptic operators depend
particular, if Y is a point,X

P: H,—H, on

be viewed as an elliptic operator on Y, thus f induces|
space.

Inoo

is given by P— index P.

The main construction we need is one which defines a “ cap-product” beMore precisely, given P< Ell (X) and atween a “cycle” and a “cocycle”.
For
vector bundle VV on X we shall define a new element Pn VeEll(X).

variable

P and

fixed

V

will then extend by lin-

V—index (PV)the mapoooo|earity to a homomorphism|

K(X)—Z.

this,

and

Thus

our

remove

we

shall

oo

®2Q
— K(X)

EX)
refineoo.

which becomes an isomorphism after tensoring with the rationals.
Co|
construction will certainly give a map

To|we want, because one has a homomorphism

Hom (KX), Z)
K\(X)—oo

EE|

Ell (X)— Hom, (K%X), Z).

This is nearly what@Q,afurther|I

In this way (varying P) we will obtain a map

oo

|

|

is neededargumentandreturn to this later. For the moment we concentrate on defining our “ cap_|||
product ”.
Given a vector bundle VV on X one can always find (see [1]) a complementary bundle, that is a vector bundle W on X so that V@ W is isomorphic totrivial bundle X X C¥. This is the same as saying that there is a cona
tinuousmap

T: X—Proj(C¥)

~

|

o

of X into the projections in C¥|

so that V is isomorphic to the bundleof kernels of T.

oo

Given

a T andsuch

H, on X we ‘shall define a new elliptic operator Q
an elliptic operator P: H,—
on X. First we define projection operators T,, T, on H{', HY respectively®

asfollows.

4)

Let ¢, +, ey be the standard basis of C7”, then for each xe X,

HY=H,®C¥ fori=1,2.

oo

.

26

~

T(x) is given

by

oo

M.F.

|

ATivaH||T(x)e; =X

TH(x)e;N|oowhere the T(x) are continuous functions of x. Since H, is a C(X)-module
we can multiply any u € H, by the function T¥ to get T%ue H,,
We cantherefore define a bounded operator 7, on HY by|

T(u®e)=STu®e.
T, is defined similarly.
andso theNotethat T, and T, are C(X)-module

decompositions

homomorphisms||

HY =TH¥DPA-T)H¥

|

DE

|

HY =T,HY®(1—-THHY
commutefe CX),

with the action of C(X). Moreover, since [P, f] is compact for any
we deduce that T,P—PT, is compact®.
We now consider the opera-

tor Q=T,PT, as an operator T,HY —T,HY.

|

For any fe C(X)

[Q, f1="T.LP, FIT,:ooand so is compact.
If P’ is a parametrix for P (an inverse
operators) then Q’= T.P'T, is a parametrix for Q:

modulocompact

QQ=T.P'T,T,PT,= TP PT, +T P(T,B—-BT)T,_||= T, modulo

compact

QQ’ = T, similarly.

|

operators.

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|

oo

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|

Thus

Q
Ell (X) as required.
The operator ( is the cap-product we set out to define. Since it depends
on the choice of the familyT of projections and not just on the bundle V (ofkernels of T) it is better to write Q =P T, rather than PV.
Howeverthe various choices ofT (and the associated integer N) turn out to be equivalent in an appropriate

sense.

In particular index (PNT)

not on T and so T—index (PNT)

3.1)|
REMARK.

|

on T.
and

V and

still defines (for fixed P) a map

K(X)—~Z.

oo||

The Hilbert spaces T,HY and T,HY depend only on V and not

In fact, the space of continuous sections of V is a C(X )-module
|
|
i
oo
|
i

|

TiH{

If P actually commuted
5)depends on

Wedenote

=H;

Qeay

M(V),

1=1,2.

with C(X'), then Q wouldby P: HY— HYsimplythe natural extension PRI

|

M(V),
ATL

| )

be PX I, where
of P.

I is

Global~ the identity on M(V).
~theory of elliptic operators

27

Since P does not actually commute with C(X), @ can-|not be definedso simply, hence the need to introduce the complementary
bundle W and the projections T.
|oo|
We next generalise this cap-product construction as follows. In addition
to our space X take another compact space Y (which will play the role of aparameter space). Given P< Ell (X) and a vector bundle V on XXY (with adefining map T: XXY —ProjC?) we perform the above construction over eachpoint ye Y. We get a continuous family of elliptic operators @(y) and hence(ignoring the C(X)-structures) a continuous family of abstract Fredholm operators,” in other words we get an element of K%Y).
For fixed P and varying

V we get in this way a homomorphism
3.2)

oo

|

KY XXY)—KY)

which generalises (3.1).
||||
Although the homomorphism (3.1) does not determine an element of K(X)
(only an elementof K(X)®,Q) the homomorphisms (3.2)—for general Y—do
First we remark that general cohomologicaldetermine an element of K(X).

theory shows that one always has a pairing|(XX Y)—K*(Y)K(X)QK

3.3)

Moreover if Y=DX is a dual of X, as in §1,
called the “slant product”.
ue K%(Xx DX), and multiplication by gz inelement”
fundamental
a
is
there
(3.3) induces the duality isomorphism

|

Ny

|

|

|

BN

K(X)—-K'DX).

|

Hence applying the homoA general reference for these remarks is [7].
morphism (3.2) to the element pe K%XXDX) we obtain finally the required
Thus we have defined a naturalelement associated to P in K%(DX)=K,(X).
map
B
|

|

Ell (X)—

2

G4)

BR

K(X)

Since (3.4) is obtained by assigning to each P< Ell (X) the index
as required.
of an associated family of Fredholm operators, we shall call the map (3.4) theoo
K-index. Thus for Pe Ell(X), we have6)

Actually

theHilbertspaces

on

which

these

operators

act

also

vary

with

y but

this is not a serious point: using Kuiper’s Theorem [6] we can make the Hilbert spaces
constant.
|||
Recalling that D,yX is a deforma7) The element pz may be defined as follows.
tion retract of R2N+1—X, we define f: Xx DX—S2¥ by taking f(x,y) to be point
Then p= f*(a)where the directed line xy meets a very large sphere S2¥NC R2N+1.
|
element.
Bott
basic
the
is
where a = K9(S2¥)

28

|

a.
~

oo
In theindex,

and

next

M.F.

ATivyAaH|~ K-index
Pe K(X).

sectionwein particular,

shall

what

discusshow

its relation

oo

much

is to the

is

known

index

about

the

theorems forK-

opera-

tors on differential manifolds.

~
:

§4.

Operators

on manifolds.

Fora manifold,ordinary

caré

duality.

|homology

Corresponding

andcohomology

relations exist

in

are

K-theory.

related
For

by

our

Poinpresent

purposes these are best put in the following form.
||~
Let TX, NX denote respectively the tangent bundle and normal bundle®
of a compact differentiable manifold X. Let K%TX), K°(NX) denote the Kgroups with compact support (obtained by forming one-point compactifications

as explained in [3]).

Then there is a natural isomorphism

(4.1)

@:

K(TX)—On the other hand, as a simple

consequenceK,, we have a natural isomorphism

(4.2)

|

|

KNX).
of the definition

|

given

in

§1 of

|

¢: K(X)— KNX).

oo~
Now let Diff-Ell (X) denote the class of pseudo-differential operators oforder zero acting on C= vector bundles as in [3]. Our motivation for introducing Ell (X) in §2 was precisely because there is an inclusion
|

|

oo

~

Diff-Ell(X)C Ell (X).

For Pe Diff-Ell (X) one has a symbol o(P), given from the local expression
for P, and this symbol defines an element [o(P)] K%TX) called the symbol- class of P (see [3, §6)).
||
We then have the following main theorem.
|
THEOREM (4.3). If P is an elliptic pseudo-differential operator of order
zero

on

a compactC=

manifold,

we

have

|

¢(K-index (P))= ¢[a(P)].

||In other words if we use the isomorphisms (4.1) and (4.2) to identify K(X)and K°TX), the K-index is “equal” to the symbol class. The significanceof this is of course that the symbol class is defined by local data while the Kindex

is defined

globally.

Theorem (4.3) gives an equality in the group KNX), or equivalently inK(X).
If we apply the homomorphism K(X)— K,(point)=Z, (4.3) gives an|

8)Relative to someC>embeddingof X in R”, with

n= dim X mod 2.

|
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Global theory of elliptic operators

29

equality between integers which is precisely the formula for index P given in[3]. Theorem (4.3) is thus a generalisation of the main theorem of [3] andit is proved by essentially the same methods.
In fact, it is really a con-sequenceof the index theorem for families of elliptic operators.”

As a consequence of (4.3) it follows that, for a manifold X,

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CElL(X)— K(X)is surjective (since even Diff-Ell (X)— K(X) is surjective). On the other hand,a result of Conner-Floyd [5] implies that, for any space X and any element& e K(X), there exists a C=-manifold M, an element » € K,(M) and a continuous map f: M—
X so that £=f.(»).
Since we have commutative diagram

Co

EM)
|
it then follows that
(4.4)

BE

—ElI(X)

KM)

—

|

|

K(X),

oo
Ell (X)—

K(X)

1s surjective.

||

This result justifies us in thinking of elements of Ell (X) as representative
cycles for the “homology ” group K(X).
|||A natural question which arises is whether it is possible to describe explicitly the equivalence relations which must be imposed on our “cycles” toproduce the “homology”.
There are some obviously necessary relations
including homotopy, but whether these are also sufficient seems to be a dif-|ficult question.

-

What I have outlined here is really just the beginning of an attempt at
developing a significant theory of operators on spaces other than C*-manifolds.
I hope that more can be done in particular on spaces such as piece-wise linear"manifolds or complex analytic spaces where there is some further structure.
The main problem would then be to find some substitute for the symbol interms of local data.

|

|

oo

oo

References
[1]
[2]
~

|
[3]

|

M.F. Atiyah,
K-Theory, Benjamin, 1967.
|||
Algebraic Topology and Operators in Hilbert Space, Lectures in
M.F. Atiyah,
Modern Analysis, Lecture Notes in Mathematics No. 103, Springer, 1969.
M.F. Atiyah and I.M. Singer, The index of elliptic operators I, Ann. of Math,© 9) See a forthcoming
elliptic operators IV.”:—

87 (1968), 484-530.

paper

by

M.F.

AtiyahandI.M.

Singer, « The index of
|
:

|

30

~ M.F. ATiYAH[4]

M. Breuer, Fredholm theories in von Neumann algebras I, Math. Ann., 178 (1968),
243-354.
|
P. Conner and E.E. Floyd,
The Relation of Cobordism to K-Theories, Lecture[5]|

Notes in Mathematics No. 28, Springer, 1966.

[6]

N. Kuiper,

Contractibility of the unitary group in Hilbert space, Topology, 3(1964), 19-30.
G.W. Whitehead,(1962), 227-283.

[7]

.

|
:

Co
So

|
homology
oo|Generalized|

MATHEMATICAL

24 ST. GILE’S,||
PRINCETON,Trans.

INSTITUTE,

A
Math. Soc.,Amer.

OXFORD

UNIVERSITYOXFORD, GREAT BRITAIN
AND

THE INSTITUTE||
theories,

FOR ADVANCED

N. J. 08540,

oo|U.S. A.

STUDY

102