moduli of flat bundles on Riemann surfaces

30 March 2009
Soren Galatius
Let G be a compact semisimple Lie group. A classical paper of Atiyah and Bott (from 1982) studies the moduli space of flat G-bundles on a fixed Riemann surface S. Their approach completely determines the integral homology of this moduli space, using Morse theoretic methods. In the case where G is U(n), this moduli space is homotopy equivalent to the moduli space of holomorphic vector bundles on S which are "semi-stable". Previous work of Harder and Narasimhan determined the Betti numbers of this moduli space using the Weil conjectures. 20 years later, a Madsen and Weiss determined the homology of the moduli space of Riemann surfaces, in the limit where the genus of the surface goes to infinity. My talk will combine these two spaces: I will describe the homology of the moduli space of Riemann surfaces S, equipped with a flat G-bundle E -> S, where we allow both the flat bundle and the surface to vary. I will start by reviewing parts of the Atiyah-Bott and Madsen-Weiss papers. Our main theorem will then be a rather easy consequence. This is joint work with Nitu Kitchloo and Ralph Cohen.