Equivariant K-theory and the Chern character for discrete groups

TL;DR

This paper explores the relationship between equivariant K-theory and the K-theory of a quotient space for discrete group actions, providing an example where the expected isomorphism does not hold.

Contribution

It presents a specific example demonstrating that the isomorphism between equivariant K-theory and quotient space K-theory can fail, challenging prior assumptions.

Findings

Counterexample to the isomorphism conjecture

Shows limitations of the Chern character in this context

Highlights differences in K-theory for certain group actions

Abstract

Let $X$ be a compact Hausdorff space, let $\Gamma$ be a discrete group that acts continuously on $X$ from the right, define $\widetilde{X} = \{(x,\gamma) \in X \times \Gamma : x\cdot\gamma= x\}$, and let $\Gamma$ act on $\widetilde{X}$ via the formula $(x,\gamma)\cdot\alpha = (x\cdot\alpha, \alpha^{-1}\gamma\alpha)$. Results of P. Baum and A. Connes, along with facts about the Chern character, imply that $K^i_\Gamma(X) \otimes \mathbb{C} \cong K^i(\widetilde{X}\slash\Gamma) \otimes \mathbb{C}$ for $i = 0, -1$. In this note, we present an example where the groups $K^i_\Gamma(X)$ and $K^i(\widetilde{X}\slash\Gamma)$ are not isomorphic.