#### Volume 7, issue 4 (2007)

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Excision for deformation $K$–theory of free products

### Daniel A Ramras

Algebraic & Geometric Topology 7 (2007) 2239–2270
 arXiv: math.KT/0703463
##### 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.

##### Keywords
deformation $K$–theory, excision, group completion
Primary: 19D23
Secondary: 55P45
##### Publication
Received: 30 June 2007
Revised: 30 November 2007
Accepted: 15 November 2007
Published: 26 December 2007
##### Authors
 Daniel A Ramras Dept of Mathematics 1326 Stevenson Center Vanderbilt University Nashville TN 37240 USA http://www.math.vanderbilt.edu/~ramrasda