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

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 $H_2(Γ;\Q) \cong \Q$ for $g\ge 5$. 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.

27 pages, 4 figures, mild revision. To appear in Adv. Math

References in corpus (5)

• The Picard group of the moduli space of curves with level structures
• An infinite presentation of the Torelli group
• Probing mapping class groups using arcs
• The abelianization of the level L mapping class group
• Abelian covers of surfaces and the homology of the level L mapping class group

Cited by in corpus (12)

• Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t=-1
• The Picard group of the moduli space of curves with level structures
• The Torelli group and congruence subgroups of the mapping class group
• The complex of partial bases for F_n and finite generation of the Torelli subgroup of Aut(F_n)
• On the second homology group of the Torelli subgroup of Aut(F_n)
• The Picard group of the moduli space of r-Spin Riemann surfaces
• Equivariant cohomology of moduli spaces of genus three curves with level two structure
• Abelian covers of surfaces and the homology of the level L mapping class group
• Prym curves with a vanishing theta-null
• Borel stability for congruence subgroups
• Galois coverings of moduli spaces of curves and loci of curves with symmetry
• On the rational cohomology of moduli spaces of curves with level structures