{
  "id": "0712.3637",
  "version": "v3",
  "published": "2007-12-21T07:56:11.000Z",
  "updated": "2013-10-30T02:08:11.000Z",
  "title": "On Lusternik-Schnirelmann category of SO(10)",
  "authors": [
    "Norio Iwase",
    "Kai Kikuchi",
    "Toshiyuki Miyauchi"
  ],
  "comment": "28 pages, 4 figures",
  "categories": [
    "math.AT"
  ],
  "abstract": "Let $G$ be a compact connected Lie group and $p : E\\to \\Sigma A$ be a principal G-bundle with a characteristic map $\\alpha : A\\to G$, where $A=\\Sigma A_{0}$ for some $A_{0}$. Let $\\{K_{i}{\\to} F_{i-1}{\\hookrightarrow} F_{i} \\,|\\, 1{\\le} i {\\le} n,\\, F_{0}{=} \\{\\ast\\} \\; F_{1}{=} \\Sigma{K_{1}} \\; \\text{and}\\; F_{n}{\\simeq} G \\}$ be a cone-decomposition of $G$ of length $m$ and $F'_{1}=\\Sigma{K'_{1}} \\subset F_{1}$ with $K'_{1} \\subset K_{1}$ which satisfy $F_{i}F'_{1} \\subset F_{i+1}$ up to homotopy for any $i$. Our main result is as follows: we have $\\operatorname{cat}(X) \\le m{+}1$, if firstly the characteristic map $\\alpha$ is compressible into $F'_{1}$, secondly the Berstein-Hilton Hopf invariant $H_{1}(\\alpha)$ vanishes in $[A, \\Omega F'_1{\\ast}\\Omega F'_1]$ and thirdly $K_{m}$ is a sphere. We apply this to the principal bundle $\\mathrm{SO}(9)\\hookrightarrow\\mathrm{SO}(10)\\to S^{9}$ to determine L-S category of $\\mathrm{SO}(10)$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-10-30T02:08:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55M30",
      "55P05",
      "55R10",
      "57T10"
    ],
    "keywords": [
      "lusternik-schnirelmann category",
      "characteristic map",
      "determine l-s category",
      "compact connected lie group",
      "berstein-hilton hopf invariant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0712.3637I"
    }
  }
}