# Equivariant Kasparov theory of finite groups via Mackey functors

### Ivo Dell'Ambrogio

Bielefeld University, Germany

## 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 $KK_{G}$ that assigns to a $G$-C*-algebra $A$ the collection of its K-theory groups ${K_{∗}(A):H⩽G}$ admits a lifting to the abelian category of $Z/2$-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 $KK_{G}$. It follows that there is a spectral sequence abutting to $KK_{∗}(A,B)$, 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 $A⊗B$. Both spectral sequences behave nicely if $A$ belongs to the localizing subcategory of $KK_{G}$ generated by the algebras $C(G/H)$ for all subgroups $H⩽G$.

## Cite this article

Ivo Dell'Ambrogio, Equivariant Kasparov theory of finite groups via Mackey functors. J. Noncommut. Geom. 8 (2014), no. 3, pp. 837–871

DOI 10.4171/JNCG/172