Universal moduli spaces of surfaces with flat connections and cobordism theory

TL;DR

This paper studies the homotopy and homology of universal moduli spaces of flat connections and holomorphic bundles on surfaces, revealing their stable cohomology structure and cobordism category relations.

Contribution

It establishes the stable homotopy type and homology of these moduli spaces, linking them to infinite loop spaces and characteristic classes, extending to arbitrary compact Lie groups.

Findings

Stable homotopy type depends on surface genus.

Stable cohomology generated by Mumford-Morita-Miller classes and H^*(BG).

Homotopy category of cobordisms with flat G-bundles characterized.

Abstract

Given a semisimple, compact, connected Lie group G with complexification G^c, we show there is a stable range in the homotopy type of the universal moduli space of flat connections on a principal G-bundle on a closed Riemann surface, and equivalently, the universal moduli space of semistable holomorphic G^c-bundles. The stable range depends on the genus of the surface. We then identify the homology of this moduli space in the stable range in terms of the homology of an explicit infinite loop space. Rationally this says that the stable cohomology of this moduli space is generated by the Mumford-Morita-Miller kappa-classes, and the ring of characteristic classes of principal G-bundles, H^*(BG). We then identify the homotopy type of the category of one-manifolds and surface cobordisms, each equipped with a flat G-bundle. We also explain how these results may be generalized to arbitrary compact connected Lie groups. Our methods combine the classical techniques of Atiyah and Bott, with the new techniques coming out of Madsen and Weiss's proof of Mumford's conjecture on the stable cohomology of the moduli space of Riemann surfaces.