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.