I will describe the structure of the cobordism category with morphisms being oriented 2-manifolds endowed with flat principal connections for bundles with compact connected structucture group. We use techniques introduced by Madsen-Weiss in their proof of the Mumford conjecture, as well as classical techniques of Atiyah-Bott. The talk will be motivated by applications to the cohomology ring of the universal moduli space of semistable bundles over closed surfaces. This is joint work with R.Cohen and S.Galatius. |