Twisted cohomology of mapping class groups with infinite dimensional coefficients and rational cohomology of Torelli groups

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The Torelli group is a subgroup of the mapping class group of a closed surface, and is defined as the kernel of the action of the mapping class group to the integral first homology group of the surface. The purpose of this research is to determine whether the Morita-Mumford classes vanish in the Torelli group or not. We cannot determine it, however, we showed that most of them vanish in some subgroup of the Torelli group.The level 2 mapping class group is the subgroup of the mapping class group defined as the kernel of the action to the first homology group of the surface with coefficient in the cyclic group of order 2. We gave a minimal generating set of this group, and determined its abelianization, and posted a preprint.