On the Sp-structure of the torsion of the Lie algebra of homology cylinders

Abstract

Let $Σ$ be a compact oriented surface. We investigate the graded Lie algebra of homology cylinders over $Σ$, as introduced by M. Goussarov and K. Habiro. To analyze its torsion, we refine a strategy initiated by Y. Nozaki, M. Suzuki, and the third author, which relies on the reduction modulo $1$ of the LMO functor and on clasper calculus. Specifically, we develop general tools for studying, under the standard action of the symplectic group, the Sp-module structure of the torsion in the odd-degree component of this Lie algebra. We show that this torsion part surjects onto an $2$-torsion Sp-module, which is explicitly described in terms of Jacobi diagrams. As an application, we provide an intrinsic description of the Sp-module given by the degree-three component of the Lie algebra of homology cylinders. A further motivation is to understand torsion phenomena in the associated graded of the lower central series of the Torelli group of $Σ$: in this direction, we exhibit an explicit Sp-module onto which the torsion of the Torelli Lie algebra surjects in degree three.