{
  "id": "1610.00330",
  "version": "v1",
  "published": "2016-10-02T18:33:07.000Z",
  "updated": "2016-10-02T18:33:07.000Z",
  "title": "On powers of the Euler class for flat circle bundles",
  "authors": [
    "Sam Nariman"
  ],
  "comment": "Accepted for publication by Journal of Topology and Analysis",
  "categories": [
    "math.GT",
    "math.AT"
  ],
  "abstract": "Apparently a lost theorem of Thurston states that the cube of the Euler class $e^3\\in H^6(BDiff^{\\delta}_{\\omega}(S^1);\\mathbb{Q})$ is zero where $Diff^{\\delta}_{\\omega}(S^1)$ is the analytic orientation preserving diffeomorphisms of the circle with the discrete topology. This is in contrast with Morita's theorem that the powers of the Euler class are nonzero in $H^*(BDiff^{\\delta}(S^1);\\mathbb{Q})$ where $Diff^{\\delta}(S^1)$ is the orientation preserving $C^{\\infty}$- diffeomorphisms of the circle with the discrete topology. The purpose of this short note is to prove that the powers of the Euler class $e^k \\in H^*(BDiff^{\\delta}_{\\omega}(S^1);\\mathbb{Z})$ in fact are nonzero in cohomology with integer coefficients. We also give a short proof of Morita's theorem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-02T18:33:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55R10",
      "57R32",
      "57R50",
      "58D05"
    ],
    "keywords": [
      "euler class",
      "flat circle bundles",
      "moritas theorem",
      "discrete topology",
      "analytic orientation preserving diffeomorphisms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}