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 $Gamma$ to the complex K-theory of the classifying space $BGamma$. 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 Carlssons 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.

The Yang-Mills stratification for surfaces revisited

We revisit Atiyah and Botts study of Morse theory for the Yang-Mills functional over a Riemann surface, and establish new formulas for the minimum codimension of a (non-semi-stable) stratum. These results yield the exact connectivity of the natural map (C_{min} E)//G(E) --> Map^E (M, BU(n)) from the homotopy orbits of the space of central Yang-Mills connections to the classifying space of the gauge group G(E). All of these results carry over to non-orientable surfaces via Ho and Lius non-orientable Yang-Mills theory. A somewhat less detailed version of this paper (titled On the Yang-Mills stratification for surfaces) will appear in the Proceedings of the AMS.

The stable moduli space of flat connections over a surface

We compute the homotopy type of the moduli space of flat, unitary connections over aspherical surfaces, after stabilizing with respect to the rank of the underlying bundle. Over the orientable surface M^g, we show that this space has the homotopy type of the infinite symmetric product of M^g, generalizing a well-known fact for the torus. Over a non-orientable surface, we show that this space is homotopy equivalent to a disjoint union of two tori, whose common dimension corresponds to the rank of the first (co)homology group of the surface. Similar calculations are provided for products of surfaces, and show a close analogy with the Quillen-Lichtenbaum conjectures in algebraic K-theory. The proofs utilize Tyler Lawsons work in deformation K-theory, and rely heavily on Yang-Mills theory and gauge theory.

Orbit categories, classifying spaces, and generalized homotopy fixed points

We give a new description of Rosenthals generalized homotopy fixed point spaces as homotopy limits over the orbit category. This is achieved using a simple categorical model for classifying spaces with respect to families of subgroups.

The prequantum line bundle on the moduli space of flat $SU(N)$ connections on a Riemann surface and the homotopy of the large $N$ limit

We show that the prequantum line bundle on the moduli space of flat $SU(2)$ connections on a closed Riemann surface of positive genus has degree 1. It then follows from work of Lawton and the second author that the classifying map for this line bundle induces a homotopy equivalence between the stable moduli space of flat $SU(N)$ connections, in the limit as $N$ tends to infinity, and $mathbb{C}P^infty$. Applications to the stable moduli space of flat unitary connections are also discussed.