{
  "id": "2008.11548",
  "version": "v1",
  "published": "2020-08-26T13:24:18.000Z",
  "updated": "2020-08-26T13:24:18.000Z",
  "title": "Thick isotopy property and the mapping class groups of Heegaard splittings",
  "authors": [
    "Daiki Iguchi"
  ],
  "comment": "8 pages, 1 figure",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $M$ be a closed orientable $3$-manifold and $S$ a Heegaard surface of $M$. The space of Heegaard surfaces $\\mathcal{H}(M,S)$ is defined to be the space of left cosets $\\mathrm{Diff}(M)/\\mathrm{Diff}(M,S)$. We prove that the fundamental group $\\pi_{1}(\\mathcal{H}(M,S))$ is finitely generated if and only if any element of $\\pi_{1}(\\mathcal{H}(M,S))$ can be represented by a \"thick\" isotopy. As an application, we prove that the mapping class group of a strongly irreducible Heegaard splitting of a closed hyperbolic $3$-manifold is finitely generated.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-26T13:24:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M60",
      "57M50"
    ],
    "keywords": [
      "mapping class group",
      "thick isotopy property",
      "heegaard splitting",
      "heegaard surface",
      "fundamental group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}