ABELIAN COVERS OF SURFACES AND THE HOMOLOGYOF THE LEVEL L MAPPING CLASS GROUP

摘要

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-cover of the surface. If the surface has one marked point, then the answer is Q~t(L), where t(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(Σ_g;Z/L). If the surface has one marked point, then the answer is Q[H_L], the rational group ring of HL. If the surface instead has one boundary component, then the answer is Q.