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

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 $\mathrm{\Gamma }$ to the complex K-theory of the classifying space $B\mathrm{\Gamma }$. 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 $\K (\Gamma)$(thehomotopy-theoreticalanalogueof$R(\Gamma)$). Our main theorem provides an isomorphism in homotopy ${\text{K}}_{\ast }({\pi }_1\mathrm{\Sigma })\text{isom}{K}^{-\ast }(\mathrm{\Sigma })$ for all compact, aspherical surfaces $\mathrm{\Sigma }$ 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.