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2008 Yang–Mills theory over surfaces and the Atiyah–Segal theorem
Daniel A Ramras
Algebr. Geom. Topol. 8(4): 2209-2251 (2008). DOI: 10.2140/agt.2008.8.2209


In this paper we explain how Morse theory for the Yang–Mills functional can be used to prove an analogue for surface groups of the Atiyah–Segal theorem. Classically, the Atiyah–Segal theorem relates the representation ring R(Γ) of a compact Lie group Γ to the complex K–theory of the classifying space BΓ. For infinite discrete groups, it is necessary to take into account deformations of representations, and with this in mind we replace the representation ring by Carlsson’s deformation K–theory spectrum Kdef(Γ) (the homotopy-theoretical analogue of R(Γ)). Our main theorem provides an isomorphism in homotopy Kdef(π1Σ)K(Σ) for all compact, aspherical surfaces Σ and all >0. Combining this result with work of Tyler Lawson, we obtain homotopy theoretical information about the stable moduli space of flat unitary connections over surfaces.


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Daniel A Ramras. "Yang–Mills theory over surfaces and the Atiyah–Segal theorem." Algebr. Geom. Topol. 8 (4) 2209 - 2251, 2008.


Received: 14 May 2008; Revised: 17 October 2008; Accepted: 26 October 2008; Published: 2008
First available in Project Euclid: 20 December 2017

zbMATH: 1240.19005
MathSciNet: MR2465739
Digital Object Identifier: 10.2140/agt.2008.8.2209

Primary: 55N15, 58E15
Secondary: 19L41, 58D27

Rights: Copyright © 2008 Mathematical Sciences Publishers


Vol.8 • No. 4 • 2008
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