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

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 $\Gamma$ to the complex K-theory of the classifying space $B\Gamma$. 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 $\K (Γ)$ (the homotopy-theoretical analogue of $R(\Gamma )$). Our main theorem provides an isomorphism in homotopy $\K_*(π_1 Σ)\isom K^{-*}(Σ)$ for all compact, aspherical surfaces $\Sigma$ and all $\ast >0$. Combining this result with work of Tyler Lawson, we obtain homotopy theoretical information about the stable moduli space of flat unitary connections over surfaces.

43 pages. Changes in v4: improved results in Section 7, simplified arguments in the Appendix, various minor revisions

References in corpus (2)

• Rings, modules, and algebras in infinite loop space theory
• Excision for deformation K-theory of free products

Cited by in corpus (3)

• Fundamental Groups of Character Varieties: Surfaces and Tori
• A finite dimensional approach to the strong Novikov conjecture
• Bad Representations and Homotopy of Character Varieties