Graph complexes and the symplectic character of the Torelli group

Abstract

The mapping class group of a closed surface of genus $g$ is an extension of the Torelli group by the symplectic group. This leads to two natural problems: (a) compute (stably) the symplectic decomposition of the lower central series of the Torelli group and (b) compute (stably) the Poincar\'e polynomial of the cohomology of the mapping group with coefficients in a symplectic representation $V$. Using ideas from graph cohomology, we give an effective computation of the symplectic decomposition of the quadratic dual of the lower central series of the Torelli group, and assuming the later is Kozsul, it provides a solution to the first problem. This, together with Mumford's conjecture, proven by Madsen-Weiss, provides a solution to the second problem. Finally, we present samples of computations, up to degree 13.