Equivariant Kasparov theory of finite groups via Mackey functors

Abstract

Let $G$ be any finite group. In this paper we systematically exploit general homological methods in order to reduce the computation of $G$-equivariant KK-theory to topological equivariant K-theory. The key observation is that the functor on $\mathsf{KK}^G$ that assigns to a $G$-C*-algebra $A$ the collection of its K-theory groups $\{ K^H_*(A) : H\leqslant G \}$ admits a lifting to the abelian category of $\mathbb{Z}/2$-graded Mackey modules over the representation Green functor for $G$; moreover, this lifting is the universal exact homological functor for the resulting relative homological algebra in $\mathsf{KK}^G$. It follows that there is a spectral sequence abutting to $\mathsf{KK}^G_*(A,B)$, whose second page displays Ext groups computed in the category of Mackey modules. Due to the nice properties of Mackey functors, we obtain a similar Künneth spectral sequence which computes the equivariant K-theory groups of a tensor product $A\otimes B$. Both spectral sequences behave nicely if $A$ belongs to the localizing subcategory of $\mathsf{KK}^G$ generated by the algebras $C(G/H)$ for all subgroups $H\leqslant G$.