Equivariant Kasparov theory of finite groups via Mackey functors

TL;DR

This paper introduces a homological approach to compute G-equivariant KK-theory for finite groups by leveraging Mackey functors, leading to new spectral sequences with improved convergence properties.

Contribution

It develops a universal coefficient and Kuenneth spectral sequence for G-equivariant Kasparov theory using Mackey functors, simplifying computations for finite groups.

Findings

New spectral sequences for G-equivariant KK-theory

Enhanced convergence for a broad class of G-C*-algebras

Reduction of KK-theory computations to topological K-theory

Abstract

Let G be a finite group. 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 assigning to a separable G-C*-algebra the collection of all its equivariant K-theory groups lifts naturally to a homological functor taking values in the abelian tensor category of Mackey modules over the classical representation Green functor for G. This fact yields a new universal coefficient and a new Kuenneth spectral sequence for the G-equivariant Kasparov category, whose convergence behavior is nice for all G-C*-algebras in a certain bootstrap class.