On the rational cohomology of moduli spaces of curves with level structures

TL;DR

This paper studies the rational cohomology of moduli spaces of curves with level structures, specifically computing low-degree cohomology groups for spin curve moduli spaces of genus g.

Contribution

It provides explicit calculations of the first few rational cohomology groups of these moduli spaces, advancing understanding of their topological structure.

Findings

Determined $H^k(ar{rak{S}}_g, Q)$ for $g ge 2$ and $k le 3$

Established new results on the topology of moduli spaces of spin curves

Enhanced understanding of the cohomological properties of moduli spaces with level structures

Abstract

We investigate low degree rational cohomology groups of smooth compactifications of moduli spaces of curves with level structures. In particular, we determine $H^k(\sgbar, \Q)$ for $g \ge 2$ and $k \le 3$, where $\sgbar$ denotes the moduli space of spin curves of genus $g$.