{
  "id": "2111.10568",
  "version": "v2",
  "published": "2021-11-20T11:53:37.000Z",
  "updated": "2022-04-18T18:09:16.000Z",
  "title": "On the structure of the top homology group of the Johnson kernel",
  "authors": [
    "Igor A. Spiridonov"
  ],
  "comment": "22 pages",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "abstract": "The Johnson kernel is the subgroup $\\mathcal{K}_g$ of the mapping class group ${\\rm Mod}(\\Sigma_{g})$ of a genus $g$ oriented closed surface $\\Sigma_{g}$ generated by all Dehn twists about separating curves. In this paper we study the structure of the top homology group ${\\rm H}_{2g-3}(\\mathcal{K}_g, \\mathbb{Z})$. For any collection of $2g-3$ disjoint separating curves on $\\Sigma_{g}$ one can construct the corresponding abelian cycle in the group ${\\rm H}_{2g-3}(\\mathcal{K}_g, \\mathbb{Z})$; such abelian cycles will be called simplest. In this paper we describe the structure of $\\mathbb{Z}[{\\rm Mod}(\\Sigma_{g})/ \\mathcal{K}_g]$-module on the subgroup of ${\\rm H}_{2g-3}(\\mathcal{K}_g, \\mathbb{Z})$ generated by all simplest abelian cycles and find all relations between them.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-04-18T18:09:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F34",
      "20F36",
      "57M07",
      "20J05"
    ],
    "keywords": [
      "johnson kernel",
      "homology group",
      "simplest abelian cycles",
      "dehn twists",
      "disjoint separating curves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}