Question

Let F be a smooth classifying space for K-theory (ordinary or equivariant). If X is a smooth compact manifold and W is a real vector space of dimension n, there is an integration map from the compactly supported K-theory of X x W to that of X. It can be represented on the level of classifying maps by regarding a map f from X x W into F as a map from X into the nth loop space of F.

Does a similar picture exist in equivariant K-theory, where X is a G-space and W is a group representation? Of course it does if W is simply a trivial representation. But what if W is some other representation? Is there an integration map, and can one represent it on the level of classifying maps? Is there a good reference for this?

Answer 1

I don't know about the homotopy-theoretic picture (and would like to learn more about it), but maybe the following also helps: If I understand you correctly, the integration maps of K-theory are what is often called the Gysin/shriek maps (the case you mention would be the Gysin map induced by the zero section of a trivial G-vector bundle). I think they only exist if the map is K-oriented; in that case they can be implemented by right multiplication with functorial KK-elements. The canonical reference for the non-equivariant case would be Connes-Skandalis and for the equivariant case Kasparov-Skandalis.

Answer 2

It sounds like what you are after is a theory of genuine equivariant K-theory (Which exists!)

If you think of a cohomology theory as a sequence of functors to abelian groups together with some properties and suspension isomorphism, then a genuine G-equivariant cohomology theory can be similarly thought of as a sequence of functors (indexed on the representation ring of G) together with "suspension isomorphisms" where you suspend by any G-representation, i.e. take the G-sphere with is the one point compactification of W (with its G-action) and smash with it. This is only the rough picture. The real theory is somewhat technical and develops the theory of genuine G-spectra. There were some comments about this here with some references. The punchline is that yes K-theory is an example of a genuine G-spectrum.

The next step is that you need to identify your K-theory with compact supports as the reduced K-theory of the suspension. I don't think this is too hard. Then your integration map reduces to the suspension isomorphism:

$\tilde{K}^*G( \Sigma^W X) \cong \tilde{K}^{ -W}_G(X)$

As Michael and Kevin mention, under a suitable equivariant K-theory orientation hypothesis there will be similar integration maps for more general vector bundles, but that's another story.

Answer 3

What do you mean by "smooth classifying space for K-theory"? The "integration" for equivarient case may be not not so easy if X is just a G-space.