Excision for deformation $K$–theory of free products

Abstract

Associated to a discrete group $G$, one has the topological category of finite dimensional (unitary) $G$–representations and (unitary) isomorphisms. Block sums provide this category with a permutative structure, and the associated $K$–theory spectrum is Carlsson’s deformation $K$–theory $K_{def}\left(G\right)$. The goal of this paper is to examine the behavior of this functor on free products. Our main theorem shows the square of spectra associated to $G\ast H$ (considered as an amalgamated product over the trivial group) is homotopy cartesian. The proof uses a general result regarding group completions of homotopy commutative topological monoids, which may be of some independent interest.