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The Bott cofiber sequence in deformation K-theory and simultaneous similarity in U(n)

Published online by Cambridge University Press:  01 March 2009

TYLER LAWSON
TYLER LAWSON*
Affiliation:
Department of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A. e-mail: tlawson@math.umn.edu
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Abstract

We show that there is a homotopy cofiber sequence of spectra relating Carlsson's deformation K-theory of a group G to its “deformation representation ring,” analogous to the Bott periodicity sequence relating connective K-theory to ordinary homology. We then apply this to study simultaneous similarity of unitary matrices.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 2 , March 2009 , pp. 379 - 393
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[1]Bökstedt, M., Hsiang, W. C. and Madsen, I.The cyclotomic trace and algebraic K-theory of spaces. Invent. Math. 111 (3) (1993), 465–539.CrossRefGoogle Scholar
[2]Carlsson, G. Structured stable homotopy theory and the descent problem for the algebraic K-theory of fields. Preprint, http://math.stanford.edu/~gunnar/ (2003).Google Scholar
[3]Dold, A. and Thom, R.Quasifaserungen und unendliche symmetrische Produkte. Ann. of Math. (2) 67 (1958), 239–281.Google Scholar
[4]Friedland, S.Simultaneous similarity of matrices. Adv. Math. 50 (3) (1983), 189–265.CrossRefGoogle Scholar
[5]Friedlander, E. M. and Mazur, B.Filtrations on the homology of algebraic varieties. Mem. Amer. Math. Soc. 110 (529): (1994), x+110. With an appendix by Daniel Quillen.Google Scholar
[6]Harris, B.Bott periodicity via simplicial spaces. J. Algebra 62 (2) (1980), 450–454.Google Scholar
[7]Lawson, T.Completed representation ring spectra of nilpotent groups. Algebr. Geom. Topol. 6 (2006), (electronic), 253–286.Google Scholar
[8]Lawson, T.The product formula in unitary deformation K-theory. K-Theory 37 (4) (2006), 395–422.CrossRefGoogle Scholar
[9]Heui Park, D. and Yupp Suh, D.Linear embeddings of semialgebraic G-spaces. Math. Z. 242 (4) (2002), 725–742.CrossRefGoogle Scholar
[10]Ramras, D. A. Yang-mills theory over surfaces and the Atiyah–Segal theorem. To appear.Google Scholar
[11]Ramras, D. A.Excision for deformation K-theory of free products. Algebr. Geom. Topol. 7 (2007), 2239–2270.CrossRefGoogle Scholar
[12]Schwede, S.Stable homotopical algebra and Γ-spaces. Math. Proc. Camb. Phil. Soc. 126 (2) (1999), 329–356.Google Scholar
[13]Segal, G.Categories and cohomology theories. Topology 13 (1974), 293–312.CrossRefGoogle Scholar