Let be any finite group. In this paper we systematically exploit general homological methods in order to reduce the computation of -equivariant KK-theory to topological equivariant K-theory. The key observation is that the functor on that assigns to a -C*-algebra the collection of its K-theory groups admits a lifting to the abelian category of -graded Mackey modules over the representation Green functor for ; moreover, this lifting is the universal exact homological functor for the resulting relative homological algebra in . It follows that there is a spectral sequence abutting to , 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 . Both spectral sequences behave nicely if belongs to the localizing subcategory of generated by the algebras for all subgroups .
Cite this article
Ivo Dell'Ambrogio, Equivariant Kasparov theory of finite groups via Mackey functors. J. Noncommut. Geom. 8 (2014), no. 3, pp. 837–871DOI 10.4171/JNCG/172