Finite type 3-manifold invariants and the structure of the Torelli group. I

Abstract

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 [Mo2, Mo3] of the structure of the Torelli group. Our results generalize those of S. Morita [Mo2, Mo3] and complement the recent calculation, due to R. Hain [Ha2], 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.