Bott PeriodicityTopological KK-theory

3 Topological KK-theory

We will end by saying a bit more about the functor KUKU defined above. Pullback of vector bundle makes it a contravariant functor on the category of finite CW complexes. We can extend this to a functor on all CW complexes by defining it to be the functor represented by BU×ZB\mathrm{U}\times \mathbb {Z}, but its values on infinite complexes have less straightforward descriptions. 1 For technical reasons, we actually want a reduced version of this — on a based space XX, we have

KUundefined(X)=[X,BU×Z]. \widetilde{KU}(X) = [X, B\mathrm{U}\times \mathbb {Z}]_*.

This corresponds to virtual vector bundles that are rank zero on the base point component. This is really not that important and not worth worrying about.

This functor KUundefined\widetilde{KU} behaves like the degree 00 part of a (reduced) cohomology theory. For example, it satisfies an appropriate form of Mayer–Vietoris. So we will write it as KU0KU^0 instead. The goal is the manufacture a (generalized) cohomology KUKU whose degree 00 part is this KU0KU^0 we already have. This is called (complex) topological KK-theory, and is of utmost importance in algebraic topology.

We first do it for negative degrees, which is easy. If hh^* is a (reduced) cohomology theory, then Mayer–Vietoris implies we always have

hn(ΣX)=hn1(X). h^n(\Sigma X) = h^{n - 1}(X).

So for n0n \geq 0, we can simply define

KUn(X)=KU0(ΣnX). KU^{-n}(X) = KU^0(\Sigma ^n X).

The functor KUn(X)KU^{-n}(X) is then represented by Ωn(BU×Z)\Omega ^n (B\mathrm{U}\times \mathbb {Z}).

The key fact is that Bott periodicity tells us Ω2(BU×Z)BU×Z\Omega ^2 (B\mathrm{U}\times \mathbb {Z}) \cong B\mathrm{U}\times \mathbb {Z}. So another way to state Bott periodicity is that

Theorem 13 (Complex Bott periodicity)

There is a canonical isomorphism

KUk(X)KUk2(X) KU^k(X) \cong KU^{k - 2}(X)

whenever both are defined.

Once we know this, we can simply define the remaining groups by

KUn(X)={KU0(X)n evenKU1(X)n odd. KU^n(X) = \begin{cases} KU^0(X) & n\text{ even}\\ KU^{-1}(X) & n\text{ odd}. \end{cases}

We then know automatically that this satisfies properties like Mayer–Vietoris, and hence is a generalized cohomology theory.

For completeness, we state the corresponding real result as well.

Theorem 14 (Real Bott periodicity)

There is a canonical isomorphism

KO0(Σ8X)KO0(X). KO^0(\Sigma ^8 X) \cong KO^0(X).