  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.
