Clifford Algebras and Bott PeriodicityThe Atiyah–Jänich theorem

2.3 The Atiyah–Jänich theorem

Equipped with Kuiper's theorem, we can finally prove what we wanted to.

Theorem (Atiyah, Jänich)

Let HH be an infinite-dimensional real Hilbert space, and F\mathcal{F} the space of all Fredholm operators on HH. Then for every compact space XX, there is a natural isomorphism

idx:[X,F]KO(X). \operatorname{idx}: [X, \mathcal{F}] \to KO(X).

The same holds in the complex case with KOKO replaced by KUKU.

For f:XFf: X \to \mathcal{F}, we would like to define idx(f)\operatorname{idx}(f) by setting the fiber at each xXx \in X to be the formal difference between kerf(x)\ker f(x) and cokerf(x)\operatorname{coker}f(x). In general, this does not give a vector bundle, but it is a general fact (which we shall not prove) that we can homotope ff so that it does, and the difference [kerf][cokerf][\ker f] - [\operatorname{coker}f] is independent of the choice of homotopy. In particular, it depends only on the homotopy class of ff.

[Proof (cf. [1])] First observe the following trivial lemma:

Let f:MGf: M \to G be a surjective homomorphism from a monoid to an abelian group with trivial kernel, i.e. f1({0})={0}f^{-1}(\{ 0\} ) = \{ 0\} . Then ff is in fact injective, hence an isomorphism.

So it suffices to show our map is surjective and has trivial kernel.