Equivariant Kasparov theory of finite groups via Mackey functors

  • Ivo Dell'Ambrogio

    Bielefeld University, Germany


Let GG be any finite group. In this paper we systematically exploit general homological methods in order to reduce the computation of GG-equivariant KK-theory to topological equivariant K-theory. The key observation is that the functor on KKG\mathsf{KK}^G that assigns to a GG-C*-algebra AA the collection of its K-theory groups {KH(A):HG}\{ K^H_*(A) : H\leqslant G \} admits a lifting to the abelian category of Z/2\mathbb{Z}/2-graded Mackey modules over the representation Green functor for GG; moreover, this lifting is the universal exact homological functor for the resulting relative homological algebra in KKG\mathsf{KK}^G. It follows that there is a spectral sequence abutting to KKG(A,B)\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 ABA\otimes B. Both spectral sequences behave nicely if AA belongs to the localizing subcategory of KKG\mathsf{KK}^G generated by the algebras C(G/H)C(G/H) for all subgroups HGH\leqslant 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