  Using the recently developed theory of finite type invariants of integral
homology 3-spheres we study the structure of the Torelli group of a closed
surface. Explicitly, we construct (a) natural cocycles of the Torelli group
(with coefficients in a space of trivalent graphs) and cohomology classes of
the abelianized Torelli group; (b) group homomorphisms that detect (rationally)
the nontriviality of the lower central series of the Torelli group. Our results
are motivated by the appearance of trivalent graphs in topology and in
representation theory and the dual role played by the Casson invariant in the
theory of finite type invariants of integral homology 3-spheres and in Morita's
study of the structure of the Torelli group Our results generalize those of S.
Morita {Mo1,Mo2} and complement the recent calculation, due to R. Hain, of the
I-adic completion of the rational group ring of the Torelli group. We also give
analogous results for two other subgroups of the mapping class group.
