{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26409",
  "author_total": 3,
  "authors": [
    {
      "name": "Quentin Faes"
    },
    {
      "name": "Gwénaël Massuyeau"
    },
    {
      "name": "Masatoshi Sato"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0hfa3mcbrb7eb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26409",
  "fulltext_url": null,
  "id": "0hfa3mcbrb7eb",
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  "landing_url": "https://hal.science/hal-05760163v1",
  "language": "en",
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
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  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GT"
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  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.26409",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0hfa3mcbrb7eb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:39Z",
      "native_id": "hal-05760163",
      "source": "hal",
      "url": "https://hal.science/hal-05760163v1"
    },
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26409",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26409"
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  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On the Sp-structure of the torsion of the Lie algebra of homology cylinders",
  "title_text": "On the Sp-structure of the torsion of the Lie algebra of homology cylinders",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/0hfa3mcbrb7eb",
  "venue": "HAL",
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}