I'm looking for an accessible reference for the definition of the fundamental class in equivariant K-theory.

The setup I'm interested in is the following: suppose $V$ is a vector space equipped with the $G$-action and $M \subset V$ is a closed $G$-invariant subvariety. From the equivariant cohomology point of view, the equivariant Poincare dual of $M$ is the class in $$\operatorname{eP}(M)\in H_G^* (pt).$$

Can the K-theoretic fundamental class of $M$ be described as a class in $K_G (pt)\cong R(G)?$ And if yes, then how is it defined?

  • 1
    $\begingroup$ There is an equivariant Poincare duality theorem in K-theory; I don't have a reference offhand, but Blackadar's book on K-theory might have it. In this framework the equivariant Dirac operator on a spin$^c$ manifold plays the role of the fundamental class in equivariant K-homology, and its equivariant index (an element of $R(G)$) is its Poincare dual. This all works in the case where $M$ and $G$ are compact; I'm not sure offhand how much it generalizes. $\endgroup$ Nov 30 '17 at 16:10

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