The second rational homology group of the moduli space of curves with level structures

TL;DR

This paper proves that for certain subgroups of the mapping class group of genus g surfaces, the second rational homology is one-dimensional for g ≥ 5, impacting the understanding of the moduli space of curves.

Contribution

It establishes the second rational homology group of these subgroups as one-dimensional, extending to punctured and bordered surfaces, and relates to the rational Picard groups of moduli space covers.

Findings

$H_2( ext{Gamma}; \\Q) \\cong \\Q$ for $g \\geq 5$

Rational Picard groups of associated moduli space covers are equal to \\Q

Results extend to punctured and bordered surfaces

Abstract

Let $\Gamma$ be a finite-index subgroup of the mapping class group of a closed genus $g$ surface that contains the Torelli group. For instance, $\Gamma$ can be the level $L$ subgroup or the spin mapping class group. We show that $H_2(\Gamma;\Q) \cong \Q$ for $g \geq 5$. A corollary of this is that the rational Picard groups of the associated finite covers of the moduli space of curves are equal to $\Q$. We also prove analogous results for surface with punctures and boundary components.