Algebraic & Geometric Topology

Yang–Mills theory over surfaces and the Atiyah–Segal theorem

Daniel A Ramras

Abstract

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 $K∗def(π1Σ)≅K−∗(Σ)$ 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.

Article information

Source
Algebr. Geom. Topol., Volume 8, Number 4 (2008), 2209-2251.

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796932

Digital Object Identifier
doi:10.2140/agt.2008.8.2209

Mathematical Reviews number (MathSciNet)
MR2465739

Zentralblatt MATH identifier
1240.19005

Citation

Ramras, Daniel A. Yang–Mills theory over surfaces and the Atiyah–Segal theorem. Algebr. Geom. Topol. 8 (2008), no. 4, 2209--2251. doi:10.2140/agt.2008.8.2209. https://projecteuclid.org/euclid.agt/1513796932

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