{
  "id": "1012.3235",
  "version": "v1",
  "published": "2010-12-15T06:42:31.000Z",
  "updated": "2010-12-15T06:42:31.000Z",
  "title": "A triangulation of $\\CC P^3$ as symmetric cube of $S^2$",
  "authors": [
    "Bhaskar Bagchi",
    "Basudeb Datta"
  ],
  "comment": "29 pages",
  "journal": "Discrete Comput Geom 48 (2012), 310--329",
  "categories": [
    "math.AT",
    "math.CO"
  ],
  "abstract": "The symmetric group $S_3$ acts on $S^2 \\times S^2 \\times S^2$ by coordinate permutation, and the quotient space $(S^2 \\times S^2 \\times S^2)/S_3$ is homeomorphic to the complex projective space $\\CC P^3$. In this paper, we construct an 124-vertex simplicial subdivision $(S^2 \\times S^2 \\times S^2)_{124}$ of the 64-vertex standard cellulation $S^2_4 \\times S^2_4 \\times S^2_4$ of $S^2 \\times S^2 \\times S^2$, such that the $S_3$-action on this cellulation naturally extends to an action on $(S^2 \\times S^2 \\times S^2)_{124}$. Further, the $S_3$-action on $(S^2 \\times S^2 \\times S^2)_{124}$ is \"good\", so that the quotient simplicial complex $(S^2 \\times S^2 \\times S^2)_{124}/S_3$ is a 30-vertex triangulation $\\CC P^3_{30}$ of $\\CC P^3$. In other words, we construct a simplicial realization $(S^2 \\times S^2 \\times S^2)_{124} \\to \\CC P^3_{30}$ of the branched covering $S^2 \\times S^2 \\times S^2 \\to \\CC P^3$. Finally, we apply the BISTELLAR program of Lutz on $\\CC P^3_{30}$, resulting in an 18-vertex 2-neighbourly triangulation $\\CC P^3_{18}$ of $\\CC P^3$. The automorphism group of $\\CC P^3_{18}$ is trivial. It may be recalled that, by a result of Arnoux and Marin, any triangulation of $\\CC P^3$ requires at least 17 vertices. So, $\\CC P^3_{18}$ is close to vertex-minimal, if not actually vertex-minimal. Moreover, no explicit triangulation of $\\CC P^3$ was known so far.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-12-15T06:42:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57Q15",
      "57R05",
      "57M60"
    ],
    "keywords": [
      "triangulation",
      "symmetric cube",
      "quotient simplicial complex",
      "vertex-minimal",
      "coordinate permutation"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.3235B"
    }
  }
}