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

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 H2(Γ;Q)≅Q for g⩾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.