University of Toronto's Symplectic Geometry Seminar

Abstract:

In this thesis, we study the topology of the moduli space of flat G-bundles over a nonorientable surface $\Sigma$, where $G$ is a compact Lie group. This moduli space may be identified with the space of homomorphisms $Hom(\pi_1(\Sigma),G)$ modulo conjugation by $G$. In particular, we compute the (rational) equivariant cohomology ring for $Hom(\pi_1(\Sigma),G)$ and its image under localization to the fixed point set of a maximal torus when $G = SU(2)$, $SO(3)$ and $U(2)$, and use this to compute the ordinary cohomology groups of the quotient $Hom(\pi_1,G)/G$. A key property in these cases is that the conjugation action is equivariantly formal. More generally, for $G$ a connected compact group, $T\subset G$ a maximal torus, we determine a description of $H_T(Hom(\pi_1(\Sigma),G))$ as a subring of $H_T(Hom(\pi_1(\Sigma),T))$ which is valid whenever this equivariant formality property holds.