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

TL;DR

This paper uses Morse theory on the Yang-Mills functional to establish a homotopy-theoretic analogue of the Atiyah-Segal theorem for surface groups, linking deformation K-theory to the K-theory of surfaces.

Contribution

It proves an isomorphism between deformation K-theory of surface groups and the K-theory of the surface, extending the Atiyah-Segal theorem to infinite discrete groups.

Findings

Homotopy equivalence between deformation K-theory and surface K-theory.

Provides new insights into the stable moduli space of flat connections.

Extends classical theorems to infinite discrete groups.

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(\Gamma) 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 $\K (\Gamma)$ (the homotopy-theoretical analogue of $R(\Gamma)$). Our main theorem provides an isomorphism in homotopy $\K_*(\pi_1 \Sigma)\isom K^{-*}(\Sigma)$ for all compact, aspherical surfaces $\Sigma$ 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.