Notes on the Atiyah-Singer index theorem

Catarina C. Carvalho,
Department of Mathematics,
Instituto Superior Tecnico,
Technical University of Lisbon
email: ccarv@math.ist.utl.pt
January 19, 2007

Contents
1 The Fredholm index

2 Pseudodifferential operators

3 Examples

4 Symbol class

11

5 The index formula in K-theory

15

6 The index formula in cohomology

19

7 The index in K-theory for C -algebras

21

The Atiyah-Singer Index Theorem

Introduction
In these notes, we make a review of the classic index theorem from Atiyah and
Singer, which computes the Fredholm index of an elliptic pseudodifferential
operator on a compact smooth manifold without boundary depending only
on topological invariants associated to its symbol.
The approach followed here is the one given in [3], the so-called embedding proof, which is K-theoretic in nature and makes use of push-forward
maps and their functoriality properties. Atiyah and Singer showed that each
elliptic pseudodifferential operator on a closed manifold M defines, through
its symbol, a class in the K-theory of T M , the tangent space of M , such
that K 0 (T M ) is exhausted by these classes (in fact, this was one of the
main motivations for the development of topological K-theory by Atiyah
and Hirzebruch). The Fredholm index depends only on this symbol class, so
that it can be regarded as a K-theory map. The main point in the line of
proof given in [3] is to take an embedding i : M Rn and to consider an
associated map in K-theory
i! : K 0 (T M ) K 0 (T Rn ) = K 0 (R2n ),
the push-forward map, which is shown to behave functorially with respect to
the Fredholm index. This fact enables one to reduce the proof of the index
formula to Rn , where the index coincides with the Bott Periodicity map in
K-theory.
It should be noted that there are many different approaches to prove the
classic index formula. The proof presented here has the advantage of applying
to operators equivariant with respect to an action of a compact Lie group, and
it generalises easily to the case of families [5]. Atiyah and Singers first proof
[2] relied on the cobordism invariance of the index, following Hirzebruchs
proof of the signature theorem (see also Palais book [15] for an extensive
review). On the analytic side, Atiyah, Bott and Patodi gave yet another
proof of the index formula using an asymptotic expansion of the kernel of the
heat operator [6] (this had an immediate generalization to an index formula
on compact manifolds with boundary [7], see also [14]). On a very different
line, Connes gave a proof of the Atiyah-Singer index theorem totally within
the framework of noncommutative geometry in[9], using the tangent groupoid
of a manifold and deformations of C -algebras. There is also a remarkable
generalization of the index formula for longitudinally elliptic operators on
foliations [10], using an analogue of the push-forward map defined above.
We now review the contents of this paper. We start with recalling the
main results on Fredholm operators in Section 1. In Section 2, we develop the
theory of pseudodifferential operators on manifolds and their symbols, which

The Atiyah-Singer Index Theorem

are functions on the cotangent space. Section 3 is devoted to examples: we

consider the de Rham operator, the Hodge-* operator and the Doulbeault
operator, for which the Fredholm index coincides with well-known topological invariants. In Section 4, we make a review of topological K-theory and
the K-theory symbol class is defined. The Fredholm index is then regarded
as a homomorphism of K-groups. The proof of Atiyah and Singers index formula in K-theory is sketched in Section 5, following an axiomatic approach.
The better known index formula in cohomology is then deduced in Section
6. Finally, in Section 7 we define the K-groups associated to a C -algebra
and place the Fredholm index in the context of K-theory for C -algebras,
showing that it coincides with the connecting map of a suitable long exact
sequence in K-theory. This is the starting point of noncommutative index
theory.
Remark : These notes were written based on working seminars given at
the University of Amsterdam, The Netherlands, and at Instituto Superior
Tecnico, Lisbon, Portugal.

The Fredholm index

For separable, complex Hilbert spaces H1 and H2 , we call an operator P :

H1 H2 a Fredholm operator if its kernel and cokernel are finite dimensional
(where ker P = {u H1 : P u = 0} and coker P = H2 / Im P .) In this case,
the Fredholm index of P is defined by
ind P = dim ker P dim coker P.
Of course, if H1 and H2 are finite dimensional, then every operator is Freholm
with index 0. Less trivial examples of Fredholm operators are given by
operators of the form I + K, where I : H1 H1 is the identity and K
is a compact operator. Typically, we will be dealing with bounded operators
on Hilbert spaces of sections of vector bundles.
We let B(H1 , H2 ) be the class of bounded operators and F(H1 , H2 ) the
class of bounded Fredholm operators from H1 to H2 (if H1 = H2 , we write
F(H1 , H2 ) = F(H1 )). If P F(H1 , H2 ) then P has closed range and, if
P : H2 H1 is the Hilbert space adjoint, then ker P
= coker P , coker P
=
ker P . Hence P is also Fredholm, with
ind(P ) = ind(P ).
By Atkinsons theorem, if P is bounded then P is Fredholm if and only
if there exist operators Q : H2 H1 and Q0 : H1 H2 such that QP I
and P Q0 I are compact operators on H1 and H2 , respectively.

The Atiyah-Singer Index Theorem

Using this characterization, one can see that if H3 is also an Hilbert space
and P F(H1 , H2 ), Q F(H2 , H3 ) then QP F(H1 , H3 ) and
ind(QP ) = ind(P ) + ind(Q).
On the other hand, we can also see that F(H1 , H2 ) is open, and more over,
that the Fredholm index remains invariant under homotopy, that is, it is
constant on each connected component. The index induces then a map
[F(H1 , H2 )] Z.

(1)

In particular, if P F(H1 , H2 ) and K is a compact operator, then P + K

is also Fredholm and we have
ind(P + K) = ind(P )
(since the ideal of compact operators is connected).
If [F] denotes the set of homotopy classes of bounded Fredholm operators
H H, where H is an infinite dimensional Hilbert space, we have now that
[F] is a group and that the Fredholm index is a group homomorphism. This
map is clearly surjective and one can check also that if two operators have the
same index then they are homotopic, so that we have in fact an isomorphism
ind : [F] Z.
Note that, in this case, F coincides with the group of units in the Calkin
algebra B(H)/K(H).
There are many well-known invariants given by Fredholm indices of operators: the Euler characteristic and the signature of a manifold, for instance,
coincide with indices of Dirac type operators . Also, the winding number of
a curve coincides with the index of the induced Toepliz operator.
The homotopy invariance of the index famously led Gelfand to suggest
that there might be a way of computing it from invariants of the space. As
Atiyah and Singer showed, the algebraic topology best suited to tackle this
problem is K-theory and the class of operators turns out to be the class of
pseudodifferential operators.

Pseudodifferential operators

We review here a few facts of the theory of pseudodifferential operators on

manifolds (see for instance [11, 16, 18]). For U Rn open, m Z, we define
a class S m (U Rn ) of a C (U Rn ) satisfying
|x a(x, )| CK (1 + ||)m|| ,

The Atiyah-Singer Index Theorem

n
for multi-indices , and x K U compact, R
P. We 1assume that
a is classical, that is, it has an infinite expansion as
amk , with amk
m
positively homogeneous of degree m k in . We call S (U Rn ) the class
of symbols of order m
An element a S m (U Rn ) defines an operator A : Cc (U ) C (U )
given by
Z
1
a(x, )eix u
b()d.
(2)
Au(x) = a(, D)u(x) :=
(2)n Rn

An operator of the form A = a(, D), with a S m (U Rn ) is said to be a

pseudodifferential operator of order m on U .
The class of pseudodifferential operators of order m is denoted by m (U ).
It is clear that j (U ) m (U ) for j < m; if P m (U ) for all m Z, then
P is said to be of order , P (U ).
Of course, any differential operator is also pseudodifferential, and its symbol is given by the characteristic polynomial. Another example is given by
regularizing operators, that is, operators given by a smooth kernel, which are
pseudodifferential of order . In fact, parametrices of elliptic differential
operators, that is, inverses modulo regularizing operators, are pseudodifferential and this was one of the motivations to consider such classes.
The principal symbol m (A) of a pseudodifferential operator A of order m
is defined as the class of a in S m (U Rn )/S m1 (U Rn ), that is, the leading
term in the expansion of a as a classical symbol. It is positively homogeneous
of order m on and smooth for 6= 0. For any m, the class of such symbols
can be identified with C (U S n1 ), where S n1 = { Rn : kk = 1}.
Now we define pseudodifferential operators on sections of vector bundles. Locally, these are just vector valued functions on open sets of Rn .
Let M be a smooth Riemannian manifold, with a positive smooth measure, and let E, F be complex vector bundles over M , endowed with hermitian structures. We define a pseudodifferential operator operator of order m
P : Cc (M ; E) C (M ; F ) requiring that on every coordinate chart of M
trivializing E and F , such P yields a matrix of pseudodifferential operators
of order m on U . Conversely, given order m pseudodifferential operators on a
covering of M by coordinate charts, we can use a partition of unity to define
a pseudodifferential operator of order m on M . (Note that the difference
between two local representations yields a regularizing operator).
Remark 2.1 Since pseudodifferential operators are not local, we need to
check the pseudodifferential operator property on each chart, which is not
1

This means that a

PN

k=0

amk S m1N (U Rn ).

The Atiyah-Singer Index Theorem

very practical. If we have a priori that P is pseudolocal, which reduces to the

kernel of P being smooth outside the diagonal, then it suffices to consider an
atlas for M trivializing E, F .
The local principal symbols defined on U Rn transform to define globally defined functions on the cotangent bundle T M . We obtain a function on T M with values in Hom(E, F ), that is a section m (P ) : T M
Hom( E, F ), with : T M M the projection, such that m (x, )
Hom(Ex Fx ). To our purposes, m (P ) is best regarded as a bundle map
m (P ) : E F.

(3)

The globally defined symbol is smooth outside the zero-section and positively
homogeneous of degree m on the fibers. We denote the class of such sections
by S m (T M ; E, F )/S m1 (T M ; E, F ), and this space can be identified, for
any m, with the space of smooth sections on S M = {(x, ) T M : kk =
1k}, the sphere bundle of T M . Note that, again, a local representation only
defines m modulo regularizing operators. Using partitions of unity, it is easy
to see that the symbol map defined by
m : m (M ; E, F ) S m (T M ; E, F )/S m1 (T M ; E, F ),

P 7 m (P ),
(4)

is surjective and ker m = m1 (M ; E, F ).

From now on, we assume that M is a compact manifold, without boundary. Using the Hermitian structure in E, we define an inner product in
C (M ; E)
Z
< u(x), v(x) >E dx.

(u, v) :=
M

Let L2 (M ; E) be the completion of C (M ; E) with respect to the norm

induced by the inner-product. We can also define Sobolev spaces H s (M ; E),
for s R, using partitions of unit and the Sobolev norms in Rn .
Theorem 2.2 Let P m (M ; E, F ). Then P can be extended as a bounded
operator H s (M ; E, F ) H sm (M ; E, F ), for any s R. Moreover, if m <
0, then P is compact.
(The compactness part of the result above is a consequence of the Rellich
lemma.) In particular, an operator of order 0 is bounded L2 (M ; E, F )
L2 (M ; E, F ). Also, if P (M ; E, F ), then P is compact and regularizing, so that in fact (M ; E, F ) coincides with the class of operators given
by smooth kernels. One can prove also that the kernel of any pseudodifferential operator is smooth outside the diagonal.

The Atiyah-Singer Index Theorem

We will now be concerned with properties of the principal symbol map

(4). Recall that given a bounded operator P : C (M ; E) C (M ; F ), its
adjoint is the operator P : C (M ; F ) C (M ; E) such that (P u, v) =
(u, P v) for u C (M ; F ), v C (M ; E). If P m (M ; E, F ) then
P m (M ; F, E) and

m (P ) = m
(P ),
where m denotes the principal symbol map. Also, if P m (M ; E, F ) and
0
0
Q m (M ; F, G) then QP m+m (M ; E, G) and
m+m0 (QP ) = m0 (Q)m (P ).
Let P m (M ; E, F ). Then P is elliptic if its symbol is invertible
outside the zero-section, that is if m (P )(x, ) is an isomorphism for 6= 0
(in particular, E and F have the same fiber-dimension).
We have the following:
Theorem 2.3 If P m (M ; E, F ) is elliptic, then P is a Fredholm operator. Moreover, its index only depends on the homotopy class of the principal
symbol m (P ).
Proof. Let P be elliptic, with principal symbol p S m (T M ; E, F ); then
there is q S m (T M ; F, E) with pq = I. From the symbolic calculus, there
is then a pseudodifferential operator Q of order m such that 0 (P QI) = 0,
that is, P Q I 1 (M ; E) and hence P Q I is a compact operator.
For the second part, if m (P ) = m (Q), then also m (Pt ) = m (P ), with
Pt = tP + (1 t)Q, 0 t 1, so that each Pt is elliptic, hence Fredholm.
From the homotopy invariance of the index, we have then ind(P ) = ind(Q).

We claim now that in what concerns the index of elliptic pseudodifferential
operators , it suffices to consider operators of order 0. In fact, given an
arbitrary pseudodifferential operator P of order m, we can always find an
elliptic pseudodifferential operator Q of order 0, namely, an operator with
principal symbol given by



m (P )(x, )
= m (P ) x,
,
0 (Q)(x, ) :=
kkm
kk
such that ind(Q) = ind(P ). We will denote from now on by (M ; E, F ) the
class of pseudodifferential operators of order 0. Note that (M ; E, F )
B(L2 (M ; E), L2 (M ; F ). Endowing (M ; E, F ) with the induced topology,
and the space of symbols C (S M ; E, F ) with the sup-norm topology, one

The Atiyah-Singer Index Theorem

proves that the symbol map is bounded. If E = F , then (M ; E) and

S 0 (M ; E) are *-algebras, and the symbol map leads to the following exact
sequence of C -algebras:

0 K(M ; E) (M, E) C(S M ; E) 0

(5)

where denotes the extended symbol map. As we shall see in .7 in the

framework of K-theory for C -algebras, the Fredholm index is closely related
to the above sequence.

Examples

We will now see some examples of elliptic operators and their indices (for a
more detailed account, check [13, 15]). Throughout this section, M will be
a compact oriented manifold.
Example 3.1 Let (T M ) denote the space of smooth complex forms on M
regarded as an Hermitian vector bundle over M and denote by C ( (T M ))
the space of smooth sections. Consider the operator
d : C ( (T M )) C ( (T M ))
given by exterior derivative on forms, and let d be its formal adjoint; these
are differential operators of order 1 and 1, respectively. We let D := d + d
and D2 = dd + d d is the so-called Hodge laplacian. Self-adjointness of D
yields that ker D = ker D2 so that ker D coincides with the space H = H p of
harmonic forms on M . It is a fundamental result from Hodge theory that, if
M is compact, H p is isomorphic to the p-th de Rham cohomology group,that
p
is, to HdR
(M ) := ker dp / Im dp1 , where dp : p (T M ) p+1 (T M ).
On the other hand, one can see that for (x, ) T M , e ( (Tx M )),
1 (D)(x, )e = e i e,

0 (D2 )(x, ) = kk2 .

where i is given by contraction with . Hence, both D and D2 are elliptic,

p
and therefore Fredholm. In particular, we get that the spaces H p
= HdR (M )
are finite dimensional.2 Note that, since D is self-adjoint, ind(D) = 0. As
we will see in the next two examples, we obtain operators with interesting
indices if we consider gradings on (T M ).
2

p
The numbers i := dim HdR
(M ) are called Betti numbers.

The Atiyah-Singer Index Theorem

Example 3.2 Consider the grading (T M ) = even (T M ) odd (T M ).

Then D as in the previous example defines an operator
D : C (even (T M )) C (odd (T M )),
which is again elliptic. Clearly,
L ker Dp coincides with the even-dimensional
harmonic
forms, hence with p even HdR (M ) and, in the same way, ker D =
L
p
p odd HdR (M ), so that the Fredholm index is given by
X
X
p
p
ind(D) =
dim HdR
(M )
dim HdR
(M ) = (M ),
p even

p odd

the Euler characteristic of M .

In the even-dimensional case, the Atiyah-Singer index theorem reduces
to the well-known fact that the Euler characteristic coincides with the Euler
number (the Euler class of T M computed on the fundamental class of the
manifold).
If dim M = 2, this result is known as the Gauss-Bonnet theorem and can
be expressed by
Z Z
KdA = 2(M ),
M

where is K is the Gaussian curvature. One can also show using the (general)
Atiyah-Singer index formula that the index of any differential operator on
an odd-dimensional manifold is 0, generalizing the well-known fact that the
Euler characteristic of an odd dimensional manifold is 0.
Example 3.3 Let now dim M = 4k. Define a grading (T M ) = + (T M )
(T M ) as follows: recall that the Hodge-* operator
: p (T M ) 4kp (T M ), (e1 ... ep ) = ep+1 ... e4k
satisfies 2 = (1)p . If we let C be the complex volume element C :=
p(p1)
(1)k+ 2 on p (T M ), we have that C2 = 1. We let (T M ) be the
1 eigenspaces of C . Consider again the operator D = d + d . In this case,
C D = DC so that
D : C (+ (T M )) C ( (T M )).
L p
is well defined.
Again,
D
is
elliptic,
hence
Fredholm,
and
ker
D
=
p H+ ,
L p
p
p

ker D = p H , with H := H C ( (T M )). One can check that the

Fredholm index is
X
ind(D) =
dim H+p Hp = dim H+2k dim H2k = sign(M ).
p

10

The Atiyah-Singer Index Theorem

(Note that C is invariant on H p H 4kp for p < 2k, so that the only
contributions for the index that do not cancel out are the ones corresponding
to H 2k ). The signature of
definition, the signature of the
R M , sign(M ), is by 2k
bilinear form (, ) := M , , H . Since < , >= (, ),
where ( , ) denotes the inner product on C ( (T M )), we see that H+2k and
H2k are generated by {e1 , ..., em } and {f1 , ..., fn }, respectively, such that <
ei , ej >= ij , < fi , fj >= ij , < ei , fj >= 0, so that sign(M ) = dim H+2k
dim H2k = ind(D).
The index formula in this case was found by Hirzebruch, who showed that
b
the signature is a topological invariant, and that it coincides with the L-genus
of M :
b M ).
sign(M ) = L(T
Example 3.4 This example is the (twisted) complex analogue of Example
3.2. Let M be a complex manifold and p,q T M be the complex valued (p, q)differential forms that is, the subspace of r T M , r = p + q generated by
forms given locally by
w = dzi1 ... dzip dz j1 ... dz jq ,
where z1 , ..., zm are holomorphic coordinates. We have a differential operator
: p,q T M p,q+1 T M,

(hw)

X h
dz i w
z i

with h C (M ).
Now let E be an holomorphic vector bundle over M , with an hermitian
structure, and let p,q T M E be the complex valued (p, q)-differential forms

with values in E. Then can be extended to p,q T M E and can be

defined with respect to the induced hermitian structure on p,q T M E. Con
sider D = + . We have as in Example 3.2 that D is elliptic, with ker D =
ker D2 , the harmonic E-valued forms. Doulbeaults theorem now states that
the spaces of E-valued harmonic (p, q)-forms is isomorphic to H p,q (M ; E),
the cohomology groups ker p,q / Im p,q1 . In particular, H p,q (M ; E) is finite dimensional
we define the holomorphic Euler characteristic of E
P and
q
by (E) := (1) dim H 0,q (M ; E). (Note that hp,q (E) := dim H p,q (M ; E)
depend only on the complex structures of M and E, and so does (E).)
Considering a grading 0,q T M E = 0,even T M E 0,odd T M E,
then D : C (0,even T M E) C (0,odd T M E) is well defined and
elliptic, and its Fredholm index is given by
X
ind(D) =
(1)q dim H 0,q (M ; E) = (E).

The Atiyah-Singer Index Theorem

11

The Atiyah and Singer theorem states that

Z
(E) =
ch(E) T d(TC M )
M

with ch(E) the Chern character and T d(TC M ) the Todd class of TC M (see
.6). It shows in particular that the Euler characteristic of E is indeed a
topological invariant of M and E. The result above extends the RiemmanRoch-Hirzebruch theorem, known only for projective algebraic manifolds, to
compact complex manifolds. In (real) dimension 2, with E a line bundle, it
reduces to the Riemann-Roch theorem for Riemann surfaces.
The examples given up to now are all instances of generalized Dirac operators (in the framework of Clifford modules). The Dirac operator will be
considered in the next example.
Example 3.5 Let M be a spin manifold of dim M = 4k and S be its complex
spinor bundle, with D its associated Dirac operator (see [13]). As in Example
3.3, we have a volume element C : S S such that C2 = 1, and if we let S
be the 1- eigenspaces of C then D : C (S + ) C(S ) is well defined.
One has that D is elliptic, hence Fredholm. Atiyah and Singer showed that
b )
ind(D) = A(M
b ) is the A-genus
b
where A(M
of M . This is a priori a rational number and
in general it is not an integer. The above formula shows that, however, for
compact spin 4k-manifolds that is indeed the case.
In all the examples above, the formula for the Fredholm index was given
depending on cohomology classes. The right framework for proving such
formulas is, however, K-theory, and we will see in the next sections how this
can be established. The step from K-theory to cohomology is then not so
hard (.6).

Symbol class

In this section, we define a class in K-theory associated to the symbol of a

general elliptic pseudodifferential operator and show that the Fredholm index
is well defined on such a class. We start with a short review of topological
K-theory (see [1]).
Let X be a compact space. Then the set V (X) of isomorphism classes of
complex vector bundles over X is an abelian semigroup with direct sum. We

12

The Atiyah-Singer Index Theorem

let K 0 (X) be the Grothendieck group of V (X), that is, the group of formal
differences
K 0 (X) = {[E] [F ] : E, F vector bundles over X}
where [E], [F ] are stable isomorphism classes, that is [E] = [F ] if and only
if there exists G with E G
= F G. We get that [E] [F ] = [E 0 ] [F 0 ] if
and only if there exist a vector bundle G such that
E E0 G
= F F 0 G.
In this way, we get a contravariant functor from the category of compact
spaces to the category of abelian groups. As a trivial example, we see that,
since two vector spaces are isomorphic if and only if have the same dimension,
K 0 ({x0 }) = Z.
If X is a locally compact space, let X + be its one-point compactification.
If j : + X + is the inclusion then j : K 0 (X + ) K 0 (+) = Z and we
define
K 0 (X) = ker j K 0 (X + ).
The elements of K 0 (X) are now given by formal differences [E][F ] of vector
bundles E and F over X that are trivial and isomorphic at infinity, that is,
outside a compact subset of X. We again get a functor from the category
of locally compact spaces with proper maps to abelian groups. It is easy to
check that it is homotopy invariant.
Note that since every vector bundle over a compact space can be complemented,3 we can always write an arbitrary element of K 0 (X) as [E] [n ],
where n is the trivial n-dimensional bundle over X and E is trivial outside
a compact.
Let U X be open. Then we have a map
j : K 0 (U ) K 0 (X)

(6)

induced by the collapsing map j : X + X + /(X + U ). Also, given Y X

closed,if i : Y X is the inclusion, we have an exact sequence given by
j

K 0 (X\Y ) K 0 (X) K 0 (Y ).

(7)

Define now
K n (X) = K 0 (X Rn )
3

In fact, every vector bundle over a not necessarily compact manifold has a complement
in some trivial bundle - for each E, one always has a finite cover trivializing E.

13

The Atiyah-Singer Index Theorem

The exact sequence (7) fits into an long exact sequence infinite to the left
j

...K 2 (Y ) K 1 (X\Y ) K 1 (X) K 1 (Y )

(8)

K (X\Y ) K (X) K (Y ).
where is the connecting map. The main feature of K-theory is the fundamental Bott periodicity theorem that states that there in fact only two
K-groups: K 0 and K 1 .
Theorem 4.1 (Bott) K n+2 (X)
= K n (X).
The above isomorphism is given by cup product with so called Bott class
K 0 (R2 ):
K 0 (X R2 ) K 0 (X),
where = 1 2 , with i , i = 1, 2, the projections onto the first and
second component, respectively. The Bott class is given by
:= [H 1 ] []
H 1 the dual Hopft bundle over P 1 (C)
= R2 and the trivial bundle. For
closed Y X, using the isomorphism K 2 (Y )
= K 0 (Y ), the exact sequence
(8) becomes a cyclic 6-term exact sequence.
For a locally compact space X, there is an alternative description of
0
K (X) that is of interest to index theory. An element [E] [F ] K 0 (X),
with E, F vector bundles over X that are isomorphic (and trivial) on a
neighborhood of infinity, represents a class [E, F, ], with : E F a
smooth homomorphism that is bijective outside a compact set. Two triples
are said to be equivalent [E, F, ] = [E 0 , F 0 , 0 ] if and only if there are vector
bundles G and H such that
(E, F, ) (G, G, IG )
= (E 0 , F 0 , 0 ) (H, H, IH )
that is, if E G
= E 0 H, F G
= F 0 H and these isomorphisms behave
well with respect to IG and 0 IH . The set of equivalence classes of
triples is a group under the direct sum (the identity is given by [P, P, Ip ])
and the map
[E] [F ] [E, F, ]
is well defined and an injective group homomorphism. We can see that this
map is also surjective, at least when X is a manifold, since in this case any
vector bundle over X can be complemented in some n , (for the general
case, see [1, 17]). We have then [E, F, ] = [H, n , ], with H = E G,

14

The Atiyah-Singer Index Theorem

F G = n and = IG . Moreover, we can assume that is the

identity outside a compact, so that H can be extended to X + and in this
case, [H] [n ] K 0 (X).
Now let M be a compact manifold, E and F vector bundles over M , and
P be an elliptic, order 0 pseudodifferential operator with symbol : E
F . Since P is elliptic, is an isomorphism outside the zero-section, that
is, outside a compact subset of T M , so that
[(P )] := [ E, F, ] K 0 (T M )
is well defined. We call [(P )] the symbol class of P .
In .1 we saw that the Fredholm index is homotopy invariant, and in .2
that, for elliptic pseudodifferential operators, it only depends on the symbol
class (Theorem 2.3). Now we have the following.
Proposition 4.2 The index of an elliptic pseudodifferential operator is well
defined on the K-theory class of its symbol.
Proof. If two operators P , Q have the same symbol class,
[ E, F, (P )] = [ E 0 , F 0 , (Q)],
then there exist G, H vector bundles over T M such that
( E G, F G, (P ) IG )
= ( E 0 H, F 0 H, (Q) IH )
( (EG0 ), (F G0 ), (P ) IG0 )
= ( (E 0 H0 ), (F 0 H0 ), (Q) IH0 ),
writing G = G0 , H = H0 , with G0 , H0 vector bundles over M . If now
IG0 denotes the identity operator C (M ; G0 ) C (M ; G0 ), then P IG0
is also an elliptic operator and clearly ind(P IG0 ) = ind(P ). We have
also ind(Q) = ind(Q IH0 ). On the other hand it is easy to see, using the
homotopy invariance of the index, that the indices of P IG0 and Q IH0
coincide, so that ind(P ) = ind(Q).

Now we check that, for a compact manifold M , K 0 (T M ) is exhausted
by symbol classes of pseudodifferential operators on M ,4 . Since the symbol
map is surjective, it suffices to check that every class in K 0 (T M ) is of the
form [ E, F, ], where is positively homogeneous of degree 0 and an
isomorphism outside a compact. (We identify here the categories of smooth
and topological vector bundles.)
4

This is not true if we consider only differential operators.

15

The Atiyah-Singer Index Theorem

Let [E, n , ] K 0 (T M ) such that is the identity on T M K, for

some compact K. Note that there is an isomorphism f : E
= E0 , for E0 :=
E|M , and we can assume that f is the identity on E|M L = (M L) Cn ,
where L (K) is open, relatively compact. We have then
[E, n , ] = [ E0 , n , f 1 ],
where f 1 is an isomorphism outside K (the identity on T M 1 (L)).
If we deform f 1 to be positively homogeneous, we get the symbol of a
pseudodifferential operator of order 0.
We conclude then that the Fredholm index defines an homomorphism of
K-groups:
an-ind : K 0 (T M ) Z, [(P )] 7 ind(P ).
(9)
The index formula computes this map depending on topological invariants
of the manifold M and on the symbol class of P .

The index formula in K-theory

Let E be a complex vector bundle over a locally compact space X; then

K 0 (E) is a K 0 (X)-module, with v a := v (a). A fundamental result
by Thom states that K 0 (E) is K-oriented, that is, that there exists a class
E K 0 (E) that generates K 0 (E) as a K 0 (X)-module:
E : K 0 (X) K 0 (E),

a 7 E a

is the Thom isomorphism. When E is the trivial bundle E = C, this is

the Bott isomorphism theorem. The class E is induced by the complex by
(E) defined as the exterior algebra of E (as a vector bundle over E).
If X is compact, we have, in our alternative definition of K 0 (E), that
E := [ even (E), odd (E), ] K 0 (E),
where e (w) = e w e (w), for e E, e its dual element, and w
even (Ee ).
Let M be a compact manifold and i : M Rn a smooth embedding,
with normal bundle N . We denote also by i : T M R2n the induced
embedding on tangent bundles, which has normal bundle N N . Now,
N N can be given the structure of a complex bundle so that we have the
Thom isomorphism
: K 0 (T M ) K 0 (N N ).

(10)

16

The Atiyah-Singer Index Theorem

On the other hand, recall that the normal bundle can be identified with an
open tubular neighborhood of T M in R2n so that we have a map
h : K 0 (N N ) K 0 (R2n )

(11)

associated to the open inclusion (as in (6)). We define the push-forward map
as
i! : K 0 (T M ) K 0 (R2n ), i! := h .
(12)
This map is independent of the tubular neighborhood chosen. The transitivity of the Thom isomorphism yields that (i j)! = i! j! , for two embeddings
i, j. Note that considering the inclusion of a point i : P Rn , we get that
i! = , the Bott isomorphism. We have finally:
Theorem 5.1 (Atiyah-Singer) Let P be an elliptic pseudodifferential operator on a compact manifold M without boundary, and let K 0 (T M )
denote its symbol class. Then
ind(P ) = 1 i! ().
where i! is the push-forward map induced by an embedding i : M Rn and
denotes the Bott isomorphism.
(We can check directly that the left-hand side, the so-called topological
index, is independent of the embedding i.)
Remark 5.2 Note that the topological index is also well defined when M
is not compact. In fact, one can also consider index maps on non-compact
manifolds (see also the excision property (13) below). In this case, one computes the Fredholm index of operators that are multiplication at infinity, so
that the symbols are constant on the fibers outside a compact and still define a K-theory class. In this context, the index theorem on Rn for elliptic
operators that are multiplication at infinity is
ind(P ) = 1 ()
where is the Bott isomorphism.
A proof of Theorem 5.1 can be found in [3]; we will sketch it below. It
relies on two properties of the index map that characterize it uniquely:
1. if M = {x0 } then ind is the identity;
2. if i : M X is an embedding, then indX i! = indM .

17

The Atiyah-Singer Index Theorem

In fact, if we have a family of maps fM : K 0 (T M ) Z satisfying 1. and 2.,

and i : M Rn is an embedding, then the following diagram commutes:
1

!
K 0 (T M )
K 0 (T Rn ) K 0 ({pt})

f{pt} =idy
fM y
fR n y

Z).

and we must have f = 1 i! .

Now, while 1. is trivially checked for the index map (9), property 2. is
indeed the core of this proof of the index theorem. Once we have set that
the index is functorial with respect to push-forward maps, we reduce the
computation of the index of arbitrary elliptic operators to those on spheres
(in fact, on a point).
Note that for an open embedding i : U M , we get i! = h, as in (11).
Also, if i : M V is the zero-section embedding in a complex vector bundle
V , then i! coincides with the Thom isomorphism V . The proof of 2. is
made in two steps: first one establishes excision, showing that, if i1 , i2 are
open embeddings U M and, h1 , h2 : K 0 (T U ) K 0 (T M ) are the induced
extension maps, then
ind h1 = ind h2 .
(13)
Note that then the index is well defined also on open subsets ind : K 0 (T U )
Z, by picking a compactification of M 5 . We get by definition, ind h = ind
and in this case
ind i! = ind h = ind .
We must study then the behavior of the index with respect to the Thom
isomorphism. For this, Atiyah and Singer established the so called multiplicativity property of the index. In fact, the multiplicativity property will
yield 2. when i : M V is the zero section embedding in a real vector
bundle over M , and the fact that the index is invariant with respect to the
Thom isomorphism will stem as a consequence. In its simplest form, the
multiplicativity property is
indM F (ab) = indM (a) indF (b)

(14)

where a K 0 (T M ), b K 0 (T F ) and ab K 0 (T (M F )) is defined through

the external product. Ones need however a stronger version of (14) , which
makes use of equivariant K-theory. Briefly, if G is a compact Lie group
5

This map gives coincides with the Fredholm index for operators on U that are multiplication outside a compact.

18

The Atiyah-Singer Index Theorem

acting on M , we consider G-vector bundles over M and define KG0 (M ) in

a similar way to the non-equivariant case. Most results follow trivially, and
we have again a Bott isomorphism (this takes some extra care - see [17])
: KG0 (X) KG0 (X Rn ). Note that the KG -group of a point is now
R(G), the representation ring of G. The index theorem (5.1) also holds in
this case, where now we compute the index of G-equivariant operators and
ind : KG0 (T M ) R(G).
0
Atiyah and Singer defined a product KG0 (T X) KGH
(T F ) KG0 (T Y ),
with Y = P H F for some principal H-bundle P . The multiplicativ0
ity property now states that, for a KG0 (T X), b KGH
(T F ) such that
F
indGH (b) R(G),
F
indYG (ab) = indM
G (a) indGH (b).

(15)

The point is that if j : P Rn inclusion of a point, and j! : R(O(n))

KO(n) (T Rn ), then indO(n) j! (1) = 1. Hence, if V is a real vector bundle over
M , with V = P O(n) Rn , then for a KG0 (T M ), and b = j! (1) KO(n) (T Rn )
we get
indV (ab) = ind(a) indO(n) (b) = ind(a).
and ab = i! (a), with i : M V the zero-section. Using excision, we get 2.
in general.
The proof of the index theorem is then reduced to checking the excision
and multiplicativity properties for the Fredholm index, by finding suitable
operators, together with showing that indO(n) j! (1) = 1.
At this point, we should note that there are many different approaches to
prove the index theorem (we will see one in the context of noncommutative
geometry in ??). One of the advantages of the embedding proof presented
here is that it generalizes easily to families of operators (this is proved in [5]).
Remark 5.3 Let P = (Pt )tY be a continuous family of elliptic pseudodifferential operator Pt : C (X, E) C (X, F ), parametrized by some Hausdorff
space Y , that is, such that in local representations the coefficients are jointly
continuous in t Y and x X. 6 If dim ker Pt , and dim coker Pt are locally
constant then ker P and coker P are vector bundles over Y and we get a class
[ker P ] [coker P ] K 0 (Y ). In general, the dimensions will not be locally
constant, but we can stabilize P so that we still get a well defined class in
K 0 (Y ); this will be, by definition, the (analytic) index of the family P . One
6

Actually, one considers more general families: we allow X and E, F to twist over Y ,
considering fiber bundles X , with structure group Diff(X) and fiber X, and E, F with
structure groups Diff(E; X) and Diff(F ; X) and fibers E and F (where Diff(E; X) are
diffeomorphisms of E that carry fibers to fibers linearly).

19

The Atiyah-Singer Index Theorem

can check that this index is invariant under homotopy, using a fundamental
result by Atiyah that states that K 0 (Y ) = [Y, F]. Moreover, taking a map
i : Y X Y Rn that restricts to embeddings it : X Rn , for each
t Y , we get a push-forward map i! : K 0 (Y T X) K 0 (Y R2n ). Using
the Bott isomorphism Y to identify K 0 (Y R2n ) with K 0 (Y ), one can then
show that
ind(P ) = Y1 i! .
for P a family of elliptic operators.

The index formula in cohomology

We now give an equivalent formulation of Theorem 5.1 using cohomology;

this was shown in [4]. For a locally compact space X, let H (X) denote its
rational cohomology with compact supports.7
To each complex vector bundle E over a manifold M , let c(E) := c1 (E) +
... + cn (E) H ev (M ) denote its total Chern class. We can write formally
c(E) = nk=1 (1 + xk ) such that xk H 2 (M ) and ck (E) is the symmetric
function on the xk s, and define
ch(E) := ex1 + ...exn = n +

n
X
j=1

xj + ...

1 X k
x + ...
k! j=1 j

We have ch(E) H ev (M ) satisfying ch(E F ) = ch(E) + ch(F ) and ch(E

F ) = ch(E)ch(F ). The Chern character is defined as the ring homomorphism
ch : K 0 (M ) H ev (M ), [E] [F ] 7 ch(E) ch(F ).
If M is compact, ch defines a rational isomorphism K (M ) Q
= H (M ).
Now let M and E be oriented, and let : H (M ) H (E) denote the
Thom isomorphism in cohomology, : K (M ) K (E) denote the Thom
isomorphism in K-theory. There is a class (E) H ev (M ) such that for
a K (M ),
1 ch (a) = ch(a)(E),
We find that (E) = (1)n Td(E)1 , where E denotes the conjugate bundle
of E and Td denotes the Todd class.
Finally, recall that for an oriented manifold M , the fundamental class is
the homology class [M ] Hn (M ; Z) defined by the orientation that generates
7

Since we are dealing only with manifolds, it does not matter what cohomology theory
we pick; one can think of real coefficients and de Rham cohomology.

20

The Atiyah-Singer Index Theorem

R
Hn (M ; Z). In real coefficients, u[M ] = M u, for u H n (M ; R). We give the
tangent space T M the orientation of an almost complex manifold, identifying
T (T M )
= T M T M
= T M C.
Theorem 6.1 (Atiyah-Singer) Let P be an elliptic pseudodifferential operator on a compact manifold M without boundary, and let K 0 (T M )
denote its symbol class, : T M M the projection. Then
ind(P ) = (1)n ch() Td(TC M )[T M ].
More suggestively, one can write, borrowing the notation from de Rham
cohomology,
Z
n
ind(P ) = (1)
ch() Td(TC M ).
TM

The proof of the above formula can be found in [4], and goes roughly as
follows. The main point is noting that for trivial even-dimensional bundles,
the Thom isomorphisms and the Chern character do commute, so that we
have that, for v K 0 (T P ) = Z
ch (v)[T Rn ] = ch(v)[T Rn ] = ch(v)[T P ] = v,
writing the Thom isomorphism as V (a)[V ] = a[X], for V X an oriented
vector bundle. Hence, the inverse of the Bott isomorphism is given by
1 (w) = ch(w)[T Rn ],
for w K 0 (pt) = Z, and we have for a K 0 (T M ), and i : M Rn an
embedding,
ind(a) = 1 i! (a) = ch(i! (a))[T Rn ] = ch(h (a))[T Rn ].
Using naturality of ch with respect to the extension map h, we get finally
ind(a) = ch((a))[T N ] = (ch(a)(T N ))[T N ] = ch(a)(T N )[T M ].
Since (T N T (T M )) = (T Rn ) = 1, we have
(T N ) = (T (T M ))1 = (1)n Td( T M C),
where we identified T (T M )
= T M C
= T M C. Atiyah and Singers
formula then follows.
For the so-called Hodge operator in Example 3.2, the right-hand side is
just the Euler class; for the signature operator of Example 3.3 one gets Hirzebruchs L-genus, and the signature theorem follows. For the Dirac operator

The Atiyah-Singer Index Theorem

21

b
on spin manifolds, we get the A-genus.
In particular, we get that the L-genus
b genus of spin manifolds.
is an integer, and the same for the AFor the Doulbeault operator of Example 3.4, the theorem above generalizes to compact manifolds Hirzebruchs Rieman-Roch theorem for projective
algebraic varieties: the holomorphic Euler characteristic of a holomorphic
bundle E is given by
(M ; E) = ch(E) Td(T M )[M ].

The index in K-theory for C -algebras

In this section, we will place the index in the framework of noncommutative

geometry. For a start, we have that the Fredholm index of elliptic pseudodifferential operators coincides with the connecting map in a long exact
sequence in K-theory for C -algebras. We will see different forms of associating K-theory classes to elliptic operators, and define the Fredholm index
using asymptotic morphisms and deformations of C -algebras.
We start with reviewing a few definitions and results from K-theory for

C -algebras (see [8, 12]). It is well known, from Gelfand-Naimark theory,

that for compact spaces X, Y , we have X
= Y iff C(X)
= C(Y ); we will
define a covariant functor K0 from the category of C -algebras to that of
abelian groups such that K0 (C(X))
= K 0 (X).
Note that since every vector bundle E over a compact space X can be
complemented in a trivial bundle, we can always identify E with the image
pn of a homomorphism p : n n such that p2 = p, that is, of an idempotent map in the algebra C(X) Mn (C), where Mn (C) denotes the n n
complex matrices. Conversely, given such a map, its image is locally trivial,
hence it defines a vector bundle over X. In this respect, vector bundles over
X can be replaced by projections on matrix algebras over C(X).
Remark 7.1 This leads to the Serre-Swan theorem that states that the category of vector bundles over a compact space X is equivalent to that of finitely
generated projective C(X)-modules, associating to a vector bundle E the module of smooth sections C (X; E) (which is finitely generated and projective
since for a trivial bundle we get a finite rank module).
One checks that two vector bundles are isomorphic, pn
= qm , iff the
associated idempotents are algebraically equivalent, that is, if there are u, v
such that p = uv and q = vu.
Let A be a C -algebra and assume for now that A has an identity. We
will be considering projections in A, that is, self-adjoint idempotents. In this

22

The Atiyah-Singer Index Theorem

context, two projections p, q are algebraically equivalent iff there is u A

such that
p = u u, q = uu ;
such an u is said to be a partial isometry.8 It is useful sometimes to use
different equivalence relations on projections. Two projections p, q are unitarily equivalent, p u q if there is an unitary u such that q = upu , and
homotopic, p h q, if there is a continuous path of projections from p to q.
We have
p h q p u q p q in A;




p 0
q 0
pq
u
in M2 (A);
0 0
0 0




p 0
q 0
p u q
h
in M2 (A).
0 0
0 0
Let Pn (A) denote the set of projections in Mn (A) and P (A) := lim Pn (A),
with inclusions Mn (A) Mk (A), n < k in the upper left corner. All three
equivalence relations defined above coincide in P (A). Take P (A)/ , the
set of equivalence classes of projections; if


p 0
p q :=
,
0 q
direct sum is well defined on equivalence classes and we have an abelian
semigroup. We define K0 (A) to be the Grothendieck group associated to
P (A)/ , so that elements of K0 (A) are given by formal differences
{[p] [q] : p, q Pn (A), n N},
and [p] = [q] iff there is r Pn (A) with p r q r, or equivalently p 1k
q 1k , where 1k is the identity in Mk (A). Clearly, K0 (A) = K0 (Mn (A)), for
any n N.
Every -homomorphism : A B, for unital A, B, induces a semigroup
homomorphism : (P (A)/ ) (P (B)/ ). We obtain in this way a
covariant functor from the category of unital C -algebras to the category of
abelian groups. One easily checks that it is homotopy invariant.
Of course, if A is commutative, that is, A = C(X) for some compact
space X, then K0 (A) = K 0 (X). For instance, if A = C, then K0 (C) =
K0 (Mn (C)) = Z. An explicit isomorphism is induced by the map p 7
8

Any idempotent is algebraically equivalent to a projection

23

The Atiyah-Singer Index Theorem

dim pCn , for p P (Mn (C)). For a separable Hilbert space H, we have
also that p q iff dim pH n = dim qH n , for p, q Pn (B(H)) (identified with
P (B(H n ))), so that (P (B(H))/ )
= N . Note that n + = m + =
, hence n = m in K0 (B(H)), that is, we have K0 (B(H)) = 0.
To define K0 also for non-unital algebras, we let A+ be the unitization of
the C -algebra A, that is, A+ := AC, with (a, )(b, ) := (ab+a+b, )
and unit 1+ = (0, 1), so that A+ = {a + 1+ : a A, C}. Let
: A+ C be the projection; then
K0 (A) := ker K0 (A+ ).
The elements of K0 (A) are now given by formal differences [p] [q], where
p, q Pn (A+ ) are such that p q Mn (A). Noting that [q] + [1n q] = [1n ],
with 1n the identity in Mn (A+ ), we have that [p] [q] = [p] [1n ] + [1n q] =
[p (1n q)] [1n ], so that one can always write arbitrary elements of K0 (A)
as formal differences
K0 (A) = {[p] [1n ] : p Pn (A+ ), p 1n Mn (A)}.
A -homomorphism : A B extends to an unit preserving -homomorphism
: A+ B + , and we get again a homotopy invariant functor from the category of C -algebras to that of abelian groups. The functor K0 is half-exact:
given an exact sequence 0 I A B 0, we have that
+

K0 (I) K0 (A) K0 (B)

is exact. Also, the functor K0 preserves direct limits, in the sense that
K0 (lim An )
= lim K0 (An ).

This yields an important stability property: let K denote the space of compact operators on some Hilbert space H; then we can write K = lim Mn (C)
and we have that
K0 (K A) = K0 (lim Mn (A))
= K0 (A).

In particular, K0 (K)
= K0 (C)
= Z. An explicit isomorphism is given in this
case p Pn (K) 7 dim pH n (note that a compact projection has finite rank).
Now we define the K1 -group: for a unitary C -algebra A, let Gln (A) denote the group of invertible elements in Mn (A), and Gl (A) := lim Gln (A),
with maps Gln (A) Glk (A), k > n, given by u 7 u 1kn . We say that
u v iff there is k N such that u 1kn h v 1mk , where u Gln (A),
v Glm (A). For a an arbitrary C -algebra A, define now
K1 (A) := Gl (A+ )/ ,

24

The Atiyah-Singer Index Theorem

with [u] + [v] := [u v]. From the Whitehead lemma [uv] = [vu] = [u v],
so that [uu ] = [1n ] = 0 and K1 (A) is a group. We again get a half-exact,
homotopy invariant functor from C -algebras to abelian groups, and one can
check that it is also stable: K1 (A K)
= K1 (A). If A is unital, K1 (A) =
Gl (A)/ . We could have defined K1 with unitaries: if we let Un (A) be the
group of unitary elements in Mn (A), and U (A) := lim Un (A), as above,
then Un (A) is a retract of Gln (A), through polar decomposition, so that
K1 (A)
= U (A)/ .
Since the unitary group of in Mk (Mn (C)) is connected, for any k, n, we
have that K1 (Mn (C)) = 0, for all n N. As a consequence of stability, we
have now K1 (K)
= K1 (C)
= Z. Similarly, since Un (B(H)) is connected, one
has also that K1 (B(H)) = 0.
Letting SA := C0 (R, A) denote the suspension of A, one shows that
K1 (A)
= K0 (SA). Noting that C0 (R, C0 (X))
= C0 (X R), we see that, in
fact, for locally compact X, K1 (C0 (X)) = K0 (X R) = K 1 (X).
Defining Kn (A) := K0 (S n A) we have that given an exact sequence 0
I A B 0, there is a long exact sequence

...K2 (B) K1 (I)

K1 (A)
K1 (B)

(16)

K0 (I) K0 (A) K0 (B).

where : Ki+1 (B) Ki (I) is the so-called connecting map. For [u] K1 (B),
with u Gln (B), one can always lift u u1 to an element w Gl2n (A)
(that is, with (w) = u). The map : K1 (B) K0 (I) is given by
[u] := [w1n w1 ] [1n ] K0 (I).

(17)

As in topological K-theory, the crucial result here is the Bott periodicity

theorem that states that
K2 (A) = K0 (S 2 A)
= K0 (A).
The group K0 (C0 (R2 )) is generated by the Bott element




1
zz z
1 0
,
:= [p]
, with p(z) =
z 1
0 0
1 + zz
identifying R2 with C. (p corresponds to the tautological bundle H 1 , generating K 0 (S 2 )). The Bott isomorphism can be written as A : K0 (A) K2 (A)
with
(x) = x,

25

The Atiyah-Singer Index Theorem

where [p] [q] := [p q] K0 (C(R2 ) A). As in the commutative case, the

long exact sequence above becomes a cyclic 6-term exact sequence.
Consider now an Hilbert space H and the exact sequence

(18)

0 K(H) B(H) B(H)/K(H) 0

where K(H) are compact operators and B(H)/K(H) is the Calkin algebra.
Since K0 (B(H)) = K0 (B(H)) = 0 and K0 (K(H)) = Z, we have that the
connecting map in the K-theory sequence induced by (18) is an isomorphism
: K1 (B(H)/K(H)) Z.
Now let P B(H) be Fredholm; then it is invertible in B(H)/K(H), hence it
defines an element [(P )]1 in K1 (B(H)/K(H)). We have that the Fredholm
index coincides with the connecting map in this sequence:
ind(P ) = [(P )]1 K0 (K)
= Z.
To check this, note first that we can always write P = |P |V , where |P | =
1
(P P ) 2 > 0 and V is a partial isometry, so that V is Fredholm and ind(V ) =
ind(P ). One checks that (V ) is a unitary and
 [(V )] = [(P )]. Now

V
1VV
, and from (17) we have
(V ) (V ) lifts to w =

1V V
V
1

[(P )] = [w1n w ] [1n ] =



VV
0
0
1 V V





1 0
0 0


.

Since V V and 1V V are the projections onto Im V and ker V , respectively,

so that 1 V V is the projection onto (Im V ) = ker V , we get then

 

0
0
1VV 0
[(P )] =

= [1V V ][1V V ].
0
0
0 1 V V
Under the isomorphism K0 (K(H)) Z, [p] [q] 7 dim pH dim qH, we
have as claimed
[(P )] = dim ker V dim ker V = ind(V ) = ind(P ).
More generally, naturality of the connecting map yields that for any subalgebra E B(H) that fits into an exact sequence
0 K(H) E C(X) 0

26

The Atiyah-Singer Index Theorem

(that is, such that the commutators in E are compact), the connecting map
will give the Fredholm index. Now recall that for pseudodifferential operators
of order 0 on a compact manifold M we do have an exact sequence

0 K (M ) C(S M ) 0,
and have now that for an elliptic pseudodifferential operator P , with invertible symbol (P ) C(S M ),
ind(P ) = [(P )]1 ,
where [(P )]1 K1 (C(S M )) is the class defined by (P ). This result
also holds for operators acting on vector valued functions, where [(P )]
K1 (C(S M ) Mk (C)) = K1 (C M ), and for operators in (M ; E), by complementing E in a trivial bundle.
The class [(P )]1 K1 (C(S M )) relates to the symbol class as defined in
.4, [(P )] K 0 (T M ) = K0 (C0 (T M )), by the connecting map associated
to the exact sequence 0 C0 (T M ) C(B M ) C(S M ) 0, where
B M is the ball bundle, that is [(P )] = [(P )]1 , : K1 (C(S M ))
K0 (C( T M )).

References
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1973.

The Atiyah-Singer Index Theorem

27

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