## Algebraic & Geometric Topology

### Excision for deformation $K$–theory of free products

Daniel Ramras

#### 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 $G∗H$ (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.

#### Article information

Source
Algebr. Geom. Topol., Volume 7, Number 4 (2007), 2239-2270.

Dates
Revised: 30 November 2007
Accepted: 15 November 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796787

Digital Object Identifier
doi:10.2140/agt.2007.7.2239

Mathematical Reviews number (MathSciNet)
MR2366192

Zentralblatt MATH identifier
1127.19003

Subjects
Secondary: 55P45: $H$-spaces and duals

#### Citation

Ramras, Daniel. Excision for deformation $K$–theory of free products. Algebr. Geom. Topol. 7 (2007), no. 4, 2239--2270. doi:10.2140/agt.2007.7.2239. https://projecteuclid.org/euclid.agt/1513796787

#### References

• R C Alperin, Notes: $PSL_2(\mathbb{Z}) = \mathbb{Z}_2*\mathbb{Z}_3$, Amer. Math. Monthly 100 (1993) 385–386
• M F Atiyah, Characters and cohomology of finite groups, Inst. Hautes Études Sci. Publ. Math. (1961) 23–64
• M F Atiyah, G B Segal, Equivariant $K$–theory and completion, J. Differential Geometry 3 (1969) 1–18
• G Carlsson, Derived representation theory and the algebraic $K$–theory of fields (2003) Available at \setbox0\makeatletter\@url http://math.stanford.edu/~gunnar/ {\unhbox0
• P Gabriel, M Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 35, Springer, New York (1967)
• A Hatcher, Algebraic topology, Cambridge University Press, Cambridge (2002)
• H Hironaka, Triangulations of algebraic sets, from: “Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, CA, 1974)”, (R Hartshorne, editor), Amer. Math. Soc., Providence, R.I. (1975) 165–185
• T Lawson, Derived Representation Theory of Nilpotent Groups, PhD thesis, Stanford University (2004)
• T Lawson, The Bott cofiber sequence in deformation $K$–theory (2006) Available at \setbox0\makeatletter\@url http://www-math.mit.edu/~tlawson/ {\unhbox0
• J P May, The spectra associated to permutative categories, Topology 17 (1978) 225–228
• D McDuff, G Segal, Homology fibrations and the “group-completion” theorem, Invent. Math. 31 (1975/76) 279–284
• D A Ramras, Yang–Mills theory over surfaces and the Atiyah–Segal theorem
• D A Ramras, Stable Representation Theory of Infinite Discrete Groups, PhD thesis, Stanford University (2007) Available at \setbox0\makeatletter\@url http://www.math.vanderbilt.edu/~ramrasda/ {\unhbox0
• G Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. (1968) 105–112
• G Segal, Categories and cohomology theories, Topology 13 (1974) 293–312
• H Whitney, Elementary structure of real algebraic varieties, Ann. of Math. $(2)$ 66 (1957) 545–556