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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 Kdef(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 GH (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
Mathematical Subject Classification 2000
Primary: 19D23
Secondary: 55P45
References
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