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 $\mathbb{Z}/L\mathbb{Z}$-cover of the surface. If the surface has one marked point, then the answer is $\Q^{τ(L)}$, 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 $H_L=H_1({\Sigma }_g;\mathbb{Z}/L\mathbb{Z})$. 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 $\Q$.

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

References in corpus (5)

• The second rational homology group of the moduli space of curves with level structures
• Cohomology of mapping class groups and the abelian moduli space
• The first cohomology of the mapping class group with coefficients in algebraic functions on the SL(2, C) moduli space
• Improved homological stability for the mapping class group with integral or twisted coefficients
• Separating twists and the Magnus representation of the Torelli group

Cited by in corpus (3)

• Normal Functions and the Geometry of Moduli Spaces of Curves
• The second rational homology group of the moduli space of curves with level structures
• Higher-order signature cocycles for subgroups of mapping class groups and homology cylinders