The cohomology of rank two stable bundle moduli: mod two nilpotency & skew Schur polynomials

TL;DR

This paper calculates the cup product structure in the cohomology of rank two stable bundle moduli spaces over Riemann surfaces, linking it to skew Schur polynomials and analyzing mod two cohomology properties.

Contribution

It introduces a new formula for cup product pairings using Zagier's methods and relates it to skew Schur polynomials, providing insights into mod two cohomology and invariants under mapping class group actions.

Findings

Computed cup product pairings in integral cohomology

Determined nilpotency degree of a key generator in mod two cohomology

Described low genus mod two cohomology rings and their invariants

Abstract

We compute cup product pairings in the integral cohomology ring of the moduli space of rank two stable bundles with odd determinant over a Riemann surface using methods of Zagier. The resulting formula is related to a generating function for certain skew Schur polynomials. As an application, we compute the nilpotency degree of a distinguished degree two generator in the mod two cohomology ring. We then give descriptions of the mod two cohomology rings in low genus, and describe the subrings invariant under the mapping class group action.