Abelian covers of surfaces and the homology of the level L mapping class group
نویسنده
چکیده
We calculate the first homology group of the mapping class group with coefficients in the first rational homology group of the universal abelian Z/LZ-cover of the surface. If the surface has one marked point, then the answer is Qτ(L), where τ(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 HL = H1(Σg;Z/LZ). If the surface has one marked point, then the answer isQ[HL], the rational group ring of HL. If the surface instead has one boundary component, then the answer is Q.
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