  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 of G. 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 $G*H$ (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.
