{
  "id": "1601.03723",
  "version": "v1",
  "published": "2016-01-14T20:36:29.000Z",
  "updated": "2016-01-14T20:36:29.000Z",
  "title": "Fake $13$-projective spaces with cohomogeneity one actions",
  "authors": [
    "Chenxu He",
    "Priyanka Rajan"
  ],
  "comment": "39 pages, one appendix",
  "categories": [
    "math.DG"
  ],
  "abstract": "We show that some embedded standard $13$-spheres in Shimada's exotic $15$-spheres have $\\mathbb{Z}_2$ quotient spaces, $P^{13}$s, that are fake real $13$-dimensional projective spaces, i.e., they are homotopy equivalent, but not diffeomorphic to the standard $\\mathbb{R}\\mathrm{P}^{13}$. As observed by F. Wilhelm and the second named author in [RW], the Davis $\\mathsf{SO}(2)\\times \\mathsf{G}_2$ actions on Shimada's exotic $15$-spheres descend to the cohomogeneity one actions on the $P^{13}$s. We prove that the $P^{13}$s are diffeomorphic to well-known $\\mathbb{Z}_2$ quotients of certain Brieskorn varieties, and that the Davis $\\mathsf{SO}(2)\\times \\mathsf{G}_2$ actions on the $P^{13}$s are equivariantly diffeomorphic to well-known actions on these Brieskorn quotients. The $P^{13}$s are octonionic analogues of the Hirsch-Milnor fake $5$-dimensional projective spaces, $P^{5}$s. K. Grove and W. Ziller showed that the $P^{5}$s admit metrics of non-negative curvature that are invariant with respect to the Davis $\\mathsf{SO}(2)\\times \\mathsf{SO}(3)$-cohomogeneity one actions. In contrast, we show that the $P^{13}$s do not support $\\mathsf{SO}(2)\\times \\mathsf{G}_2$-invariant metrics with non-negative sectional curvature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-14T20:36:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C20",
      "53C30"
    ],
    "keywords": [
      "cohomogeneity",
      "dimensional projective spaces",
      "shimadas exotic",
      "diffeomorphic",
      "quotient spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160103723H"
    }
  }
}