{
  "id": "1403.7832",
  "version": "v2",
  "published": "2014-03-30T23:20:51.000Z",
  "updated": "2015-03-03T02:56:47.000Z",
  "title": "Arc complexes, sphere complexes and Goeritz groups",
  "authors": [
    "Sangbum Cho",
    "Yuya Koda",
    "Arim Seo"
  ],
  "comment": "16 pages, 4 figures; minor changes, typos corrected",
  "categories": [
    "math.GT"
  ],
  "abstract": "We show that if a Heegaard splitting is obtained by gluing a splitting of Hempel distance at least 4 and the genus-1 splitting of $S^2 \\times S^1$, then the Goeritz group of the splitting is finitely generated. To show this, we first provide a sufficient condition for a full subcomplex of the arc complex for a compact orientable surface to be contractible, which generalizes the result by Hatcher that the arc complexes are contractible. We then construct infinitely many Heegaard splittings, including the above-mentioned Heegaard splitting, for which suitably defined complexes of Haken spheres are contractible.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-30T23:20:51.000Z",
      "abstract": "We provide a sufficient condition for a full subcomplex of the arc complex for a compact orientable surface to be contractible, which generalizes the result by Hatcher that the arc complexes are contractible. As an application, we construct infinitely many Heegaard splittings whose sphere complexes are contractible, including the genus-$2$ Heegaard splitting of $S^2 \\times S^1$. Further, if a Heegaard splitting is obtained by gluing a splitting of Hempel distance at least $4$ and the genus-$1$ splitting of $S^2 \\times S^1$, we show that the Goeritz group of the splitting is finitely generated.",
      "comment": "16 pages, 4 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-03-03T02:56:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57N10",
      "57M60"
    ],
    "keywords": [
      "goeritz group",
      "sphere complexes",
      "arc complexes",
      "heegaard splitting",
      "full subcomplex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.7832C"
    }
  }
}