{
  "id": "1008.4186",
  "version": "v3",
  "published": "2010-08-25T01:58:25.000Z",
  "updated": "2010-11-25T00:14:49.000Z",
  "title": "$S^2$-bundles over 2-orbifolds",
  "authors": [
    "Jonathan A. Hillman"
  ],
  "comment": "We have completed the determination of which $S^2$-orbifold bundles are geometric, and also computed the second Wu class for each such manifold. Further minor changes have been made (principally to the section on surgery) in the second revision",
  "journal": "J. London Math. Soc. 87 (2013), 69--86",
  "doi": "10.1112/jlms/jds038",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $M$ be a closed 4-manifold with $\\pi_2(M)\\cong{Z}$. Then $M$ is homotopy equivalent to either $CP^2$, or the total space of an orbifold bundle with general fibre $S^2$ over a 2-orbifold $B$, or the total space of an $RP^2$-bundle over an aspherical surface. If $\\pi=\\pi_1(M)\\not=1$ there are at most two such bundle spaces with given action $u:\\pi\\to{Aut}(\\pi_2(M))$. The bundle space has the geometry $\\mathbb{S}^2\\times\\mathbb{E}^2$ (if $\\chi(M)=0$) or $\\mathbb{S}^2\\times\\mathbb{H}^2$ (if $\\chi(M)<0$), except when $B$ is orientable and $\\pi$ is generated by involutions, in which case the action is unique and there is one non-geometric orbifold bundle.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-11-25T00:14:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57N13"
    ],
    "keywords": [
      "total space",
      "bundle space",
      "non-geometric orbifold bundle",
      "general fibre",
      "homotopy equivalent"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.4186H"
    }
  }
}