On the degree-two part of the associated graded of the lower central series of the Torelli group

Abstract

We consider the associated graded ${⨁}_{k\ge 1}{\Gamma }_k\mathcal{I}/{\Gamma }_{k+1}\mathcal{I}$ of the lower central series $\mathcal{I}={\Gamma }_1\mathcal{I}\supset {\Gamma }_2\mathcal{I}\supset {\Gamma }_3\mathcal{I}\supset \cdots$ of the Torelli group $\mathcal{I}$ of a compact oriented surface. Its degree-one part is well-understood by D. Johnson's seminal works on the abelianization of the Torelli group. The knowledge of the degree-two part $({\Gamma }_2\mathcal{I}/{\Gamma }_3\mathcal{I})\otimes \mathbb{Q}$ with rational coefficients arises from works of S. Morita on the Casson invariant and R. Hain on the Malcev completion of $\mathcal{I}$. Here, we prove that the abelian group ${\Gamma }_2\mathcal{I}/{\Gamma }_3\mathcal{I}$ is torsion-free, and we describe it as a lattice in a rational vector space. As an application, the group $\mathcal{I}/{\Gamma }_3\mathcal{I}$ is computed, and it is shown to embed in the group of homology cylinders modulo the surgery relation of $Y_3$-equivalence.

28 pages. Final version