Abelian covers of surfaces and the homology of the level L mapping class group

Abstract

We calculate the first homology group of the mapping class group with coefficients in the first rational homology group of the universal abelian $\Z / L \Z$-cover of the surface. If the surface has one marked point, then the answer is $\Q^{\tau(L)}$, where $\tau(L)$ is the number of positive divisors of $L$. If the surface instead has one boundary component, then the answer is $\Q$. We also perform the same calculation for the level $L$ subgroup of the mapping class group. Set $H_L = H_1(\Sigma_g;\Z/L\Z)$. If the surface has one marked point, then the answer is $\Q[H_L]$, the rational group ring of $H_L$. If the surface instead has one boundary component, then the answer is $\Q$.