Abstract:In their paper "The Yang-Mills Equations over Riemann Surfaces" Atiyah and Bott described a method for using Morse theory and symplectic reduction to study the topology of the moduli space of semi stable bundles over a compact Riemann surface. Later, in his paper "The Self-Duality Equations on a Riemann Surface", Hitchin defined the moduli space of semistable Higgs bundles over a compact Riemann surface and described this as a hyperkahler quotient.
Here I will describe a version of Morse theory which has been developed for spaces with mild singularities, which allows us to study the topology of the moduli space of rank 2 semistable Higgs bundles over a compact Riemann surface by a method in the spirit of Atiyah-Bott's original approach. The surjectivity of the hyperkahler Kirwan map for non-fixed determinant Higgs bundles then follows naturally from the Morse theory methods used.