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

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

2009-07-10

arxiv.org/abs/0907.1718v3

Mathematics

Geometric Topology

32 pages, 10 figures; numerous corrections and simplifications; to appear in J. Topol. Anal

Scientific paper

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 $\Q$.

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