On Newstead's Mayer-Vietoris argument in characteristic 2

TL;DR

This paper adapts Newstead's Mayer-Vietoris approach to compute mod two Betti numbers of framed flat U(2) connection moduli spaces in characteristic two, providing conjectural formulas and exploring cohomology ring structures.

Contribution

It introduces a method to determine Betti numbers in characteristic two, extending Newstead's original approach and offering conjectural recursive formulas.

Findings

Derived conjectural recursive formulas for mod two Betti numbers.

Partially verified the formulas through computations.

Discussed the cohomology ring structure of unframed moduli spaces.

Abstract

Consider the moduli space of framed flat $U(2)$ connections with fixed odd determinant over a surface. Newstead combined some fundamental facts about this moduli space with the Mayer-Vietoris sequence to compute its betti numbers over any field not of characteristic two. We adapt his method in characteristic two to produce conjectural recursive formulae for the mod two betti numbers of the framed moduli space which we partially verify. We also discuss the interplay with the mod two cohomology ring structure of the unframed moduli space.