## Yang–Mills theory over surfaces and the Atiyah-Segal theorem (2008)

Citations: | 7 - 5 self |

### BibTeX

@MISC{Ramras08yang–millstheory,

author = {Daniel A. Ramras},

title = {Yang–Mills theory over surfaces and the Atiyah-Segal theorem},

year = {2008}

}

### OpenURL

### Abstract

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 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 homotopytheoretical analogue of R(Γ)). Our main theorem provides an isomorphism in homotopy K ∗ def (π1Σ) ∼ = K ∗ (Σ) for all compact, aspherical surfaces Σ and all ∗> 0. Combining this result with work of Lawson, we obtain homotopy theoretical information about the stable moduli space of flat connections over surfaces. 1.