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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?

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  • $\begingroup$ 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. $\endgroup$
    – yeshengkui
    Commented Nov 22, 2009 at 10:08
  • $\begingroup$ Let me amend what I said. F is, say, the space of Fredholm operators, and X is a compact manifold on which G acts smoothly. Also, I really only care about finite groups. $\endgroup$
    – Michael Ortiz
    Commented Nov 22, 2009 at 14:13
  • $\begingroup$ K-theory is a generalized cohomology theory, so it is represented by a spectrum. Correct me if I am wrong, but I think that the "classifying space" that Michael Ortiz is referring to is this spectrum. $\endgroup$
    – Kevin H. Lin
    Commented Dec 24, 2009 at 3:21

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Atiyah gave a very precise answer to this question: an equivariant vector bundle V/M is K-orientable (satisfies the Thom isomorphism for K-theory, i.e. K(B(V),S(V)) is a rank-one free module over K(M)) if and only if V has an equivariant Spin_c-structure. Atiyah's proof definitely involves analysis (the family index). The proof by Kasparov is a little more technical, since the definition of the product in Kasparov's KK theory is rather technical. (It is a generalization of the family index theorem of Atiyah and Singer.) But all of these proofs come down to the same thing: the study of families of equivariant Fredholm operators.

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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.

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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.

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  • $\begingroup$ Yes, it's the Gysin map. I don't really know anything about the equivariant K-theory case, and I haven't thought about it much, but for the case of non-equivariant K-theory this should be very simple. One can take a standard exposition of the Gysin sequence for ordinary singular or de Rham cohomology (say Milnor-Stasheff or Bott-Tu), and it should be straightforward to generalize it to K-theory or any other generalized cohomology theory. Of course you will need things to be K-oriented rather than oriented, etc. $\endgroup$
    – Kevin H. Lin
    Commented Dec 24, 2009 at 3:26

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