The moduli space of flat SU(2)-bundles over a nonorientable surface

TL;DR

This paper investigates the topology of the moduli space of flat SU(2)-bundles over nonorientable surfaces, computing its cohomology by analyzing the equivariant properties of the homomorphism space.

Contribution

It provides the first detailed computation of the (rational) equivariant cohomology ring for these moduli spaces, revealing their topological structure.

Findings

Computed the rational equivariant cohomology ring of Hom(_1(X),SU(2))

Determined the ordinary cohomology groups of the quotient space

Established that the conjugation action is equivariantly formal

Abstract

We study the topology of the moduli space of flat SU(2)-bundles over a nonorientable surface X. This moduli space may be identified with the space of homomorphisms Hom(\pi_1(X),SU(2)) modulo conjugation by SU(2). In particular, we compute the (rational) equivariant cohomology ring of Hom(\pi_1(X),SU(2)) and use this to compute the ordinary cohomology groups of the quotient Hom(\pi_1(X),SU(2))/SU(2). A key property is that the conjugation action is equivariantly formal.