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

Daniel A Ramras

Algebraic & Geometric Topology 8 (2008) 2209–2251

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.

Atiyah–Segal theorem, deformation $K$–theory, flat connection, Yang–Mills theory
Mathematical Subject Classification 2000
Primary: 55N15, 58E15
Secondary: 58D27, 19L41
Received: 14 May 2008
Revised: 17 October 2008
Accepted: 26 October 2008
Published: 4 December 2008
Daniel A Ramras
Vanderbilt University
Department of Mathematics
1326 Stevenson Center
Nashville, TN 37240