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 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.
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