Algebraic & Geometric Topology

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

Daniel A Ramras

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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.

Article information

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55N15: $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19- XX} 58E15: Application to extremal problems in several variables; Yang-Mills functionals [See also 81T13], etc.
Secondary: 58D27: Moduli problems for differential geometric structures 19L41: Connective $K$-theory, cobordism [See also 55N22]

Atiyah–Segal theorem deformation $K$–theory flat connection Yang–Mills theory


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.

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