2.3 The Atiyah–Jänich theorem
Equipped with Kuiper's theorem, we can finally prove what we wanted to.
Let be an infinite-dimensional real Hilbert space, and the space of all Fredholm operators on . Then for every compact space , there is a natural isomorphism
The same holds in the complex case with replaced by .
For , we would like to define by setting the fiber at each to be the formal difference between and . In general, this does not give a vector bundle, but it is a general fact (which we shall not prove) that we can homotope so that it does, and the difference is independent of the choice of homotopy. In particular, it depends only on the homotopy class of .
Let be a surjective homomorphism from a monoid to an abelian group with trivial kernel, i.e. . Then is in fact injective, hence an isomorphism.
So it suffices to show our map is surjective and has trivial kernel.
To show the kernel is trivial, by Kuiper's theorem, it suffices to show that every map with index is homotopic to a map with image in .
If , we can homotope so that . Thus, by definition of the -groups, we can find some large such that
Thus, by homotoping to vanish further on a trivial subbundle isomorphic to , we may assume that there is an isomorphism . Then we have a homotopy
going from to an isomorphism. Then we are done.
To show surjectivity, since every element in is of the form , it suffices to show that we can produce maps with index and .
The case of is easy — take the map to constantly be .
To construct the one with , first pick some bundle such that for some .
Now consider the bundle , which is isomorphic to . Then take the Fredholm operator to be