2 1

Note that Fb has infinite dimensional positive and negative eigenspaces.

1.1.1.5. If H is a separable Hilbert space, then we can consider the space

K1

of

all selfadjoint Fredholm operators F on H such that 1 − F

2

is compact and F has

infinite dimensional positive and negative eigenspaces. We equip this space with

the smallest topology such that for all ψ ∈ H the families K1 F → Fψ ∈ H, and

the family K1 F → 1 − F 2 are norm continuous.

One can show that K1 has the homotopy type of the classifying space of the

complex K-theory functor K1 (see [21]).

If M is closed and odd-dimensional, then our family (Db)b∈B gives rise to a

continuous map F : B → K1 and therefore to a homotopy class

index((Db)b∈B) = [F ] ∈ [B,

K1]

=

K1(B)

.

1.1.1.6. Let H be a Z/2Z-graded separable Hilbert space. We consider the space

K0

of all selfadjoint Fredholm operators F on H which are odd and such that F

2

−1

is compact. In order to define the topology we consider

K0

as a subspace of

K1

with the induced topology.

Again one can show that

K0

has the homotopy type of the classifying space of

the complex K-theory functor

K0

(see [21]).

If M is even-dimensional and we set Fb := Db(Db

2

+

1)1/2

as before, then

Fb ∈

K0.

The family (Fb)b∈B gives rise to a continuous map F : B →

K0

and

therefore to a homotopy class

index((Db)b∈B) = [F ] ∈ [B,

K0]

=

K0(B)

.

1.1.1.7. Observe that this definition of the index of a family of generalized Dirac

operators involves an unitary identification of H with

L2(M,

V ). By Kuiper’s theo-

rem the space of such unitary identifications is contractible so that the construction

is actually independent of the choice.

In fact, the scalar product on the Hilbert space L2(M, V ) in general also de-

pends on b ∈ B since the volume measure depends on the Riemannian metric on

M which is determined by the symbol of D. So we must in fact choose is a trivial-

ization of the bundle of Hilbert spaces (L2(M, V, ., .

b

))b∈B which exists and is

again unique up to homotopy by Kuiper’s theorem.

Arrived at this point we see that the construction above has the following

generalization. In the situation above the family of generalized Dirac operators

(Db)b∈B is a family of fiber-wise differential operators on the trivial fiber bundle

B × M → B. It is now straight forward to generalize the construction of the index

to the case of a family of fiber-wise generalized Dirac operators on a merely locally

trivial bundle E → B with fiber M.

1.1.1.8. In the even-dimensional case there is another interpretation of the index in

terms of an index bundle. Let (Db)b∈B be a family of fiber-wise generalized Dirac

operators on a fiber bundle E → B with even-dimensional closed fibers and compact

base B. After a perturbation of the family we can assume that dim(ker(Db)) is

independent of b ∈ B. In this case the family of vector spaces (ker(Db))b∈B forms

a Z/2Z-graded vector bundle ker(D) over the base B.

If one considers

K0(B)

as the Grothendieck group generated by isomorphism

classes of vector bundles over B, then the class [ker(D)] ∈

K0(B)

corresponds to the

index index((Db)b∈B) ∈

K0(B)

as defined in 1.1.1.6 under the usual identification

of the two pictures of the

K0-functor.