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

Andrew Putman

doi:10.1016/j.aim.2011.10.017

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arXiv · 0809.4477

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

Andrew Putman

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Abstract

Let $Γ$ be a finite-index subgroup of the mapping class group of a closed genus $g$ surface that contains the Torelli group. For instance, $Γ$ can be the level $L$ subgroup or the spin mapping class group. We show that $H_2(Γ;\Q) \cong \Q$ 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.

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Andrew Putman. 2011-10-28. The second rational homology group of the moduli space of curves with level structures. https://doi.org/10.1016/j.aim.2011.10.017

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