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

TL;DR

This paper computes the first homology group of the mapping class group with coefficients in the homology of abelian covers, revealing explicit structures depending on the surface's features and the level L.

Contribution

It provides explicit calculations of the first homology of the level L mapping class group with coefficients in the homology of universal abelian covers, extending understanding of these groups.

Findings

For surfaces with one marked point, the homology is ^{ au(L)}.

For surfaces with one boundary component, the homology is .

The homology of the level L subgroup is given by the rational group ring [H_L].

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 $\Q$.