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.
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BibTeX
@misc{faes2026spstructure,
title = {{On the Sp-structure of the torsion of the Lie algebra of homology cylinders}},
author = {Quentin Faes and Gwénaël Massuyeau and Masatoshi Sato},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26409},
eprint = {2609.26409},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05760163v1},
}
RIS
TY - UNPB
TI - On the Sp-structure of the torsion of the Lie algebra of homology cylinders
AU - Quentin Faes
AU - Gwénaël Massuyeau
AU - Masatoshi Sato
PY - 2026
DA - 2026/09/22
PB - HAL
DO - 10.48550/arxiv.2609.26409
UR - https://hal.science/hal-05760163v1
AB - 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.
ER -