Is there any "deep" relation between the localization theorem of equivariant cohomology and the localization theorem of equivariant K-theory

Question

First let's consider equivariant cohomology: if a compact Lie group $G$ acts on a compact manifold $M$. We have the equivariant cohomology $ H_G(M)$ defined as the cohomology of the cochain complex $((\mathbb{C}[\mathfrak{g^*}]\otimes \Omega^{\bullet}{M})^G, d_G)$ (for the definition of equivariant cohomology we can look at chapter 1 and 4 of Guillemin and Sternberg's book "Supersymmetry and Equivariant de Rham Theory"). $K< G$ is a closed subgroup, Let $M^K$ be the points of $M$ which has isotropy groups conjugated to $K$, obviously $M^K$ is a $G$-submanifold of $M$ and let $~i: M^K \rightarrow M$ denote the inclusion map. we have a version of localization theorem, see Guillemin and Sternberg's book "Supersymmetry and Equivariant de Rham Theory" chapter 11, especially Theorem 11.4.3 in page 178. In more details :

Consider the equivariant cohomology $ H_G(M)$ and $H_G(M^K)$ as $ S( \mathfrak{g^* })^G $ modules. Then the pullback map $$ i^*: H^ * _G(M)\rightarrow H^ *_G(M^K) $$ is an isomorphism after localizing at some certain prime ideals of $ S( \mathfrak{g^* })^G $.

On the other hand, we have the equivariant K-theory $K_G(M)$ and we also have the localization theorem in this side, see Segal "Equivariant K-theory" (1967) section 4, proposition 4.1, which also claims that Then the pullback map $$ i^*: K^ * _G(M)\rightarrow K^ *_G(M^K) $$ is an isomorphism after localizing at some certain prime ideals of $R(G)$, the representation ring of $G$.

We notice the similarity of the above two version of localization theorems. Nevertheless equivariant cohomology and equivariant K-theory are different. The first is the cohomology of a differential graded algebra and the latter is the Grothedieck group of modules of the cross product algebra $G \ltimes C(M)$.

My question is: is there any deep relation between them? Are they valid because of the same reason?

This post imported from StackExchange MathOverflow at 2014-11-20 20:24 (UTC), posted by SE-user Zhaoting Wei

Comments

The localization theorem for equivariant K-theory is valid even when $G$ is a finite group. On the other hand, I think (correct me if I am wrong) that the localisation theorem for equivariant cohomology needs $G$ to have positive dimension to have a non-trivial content. This suggests that there is a conceptual difference between them.

This post imported from StackExchange MathOverflow at 2014-11-20 20:24 (UTC), posted by SE-user Damian Rössler

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Thank you for your comments Damian! Yes, as far as I know the localization theorem is true essentially for compact abelian Lie groups.

This post imported from StackExchange MathOverflow at 2014-11-20 20:24 (UTC), posted by SE-user Zhaoting Wei

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I'm still hoping someone who understands this better than me answers this, but... Yes - there is (at least in some cases) a "Chern map" taking (equivariant) vector bundles to their (equivariant) Chern classes, and this should be the relation you are looking for.

This post imported from StackExchange MathOverflow at 2014-11-20 20:24 (UTC), posted by SE-user Alexander Woo

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@Alexander Thank you for your comment! Yes I think Chern character map may be the answer and there are a lot of interesting theory on it (for example the paper by J. Block and E. Getzler "Equivariant cyclic homology and equivariant differential forms"). I will think more carefully about this.

This post imported from StackExchange MathOverflow at 2014-11-20 20:24 (UTC), posted by SE-user Zhaoting Wei