{
  "id": "math/0611004",
  "version": "v2",
  "published": "2006-10-31T22:53:03.000Z",
  "updated": "2016-05-28T17:34:13.000Z",
  "title": "An infinite-dimensional phenomenon in finite-dimensional metric topology",
  "authors": [
    "A. Dranishnikov",
    "S. Ferry",
    "S. Weinberger"
  ],
  "categories": [
    "math.GT",
    "math.AT"
  ],
  "abstract": "We show that there are homotopy equivalences $h:N\\to M$ between closed manifolds which are induced by cell-like maps $p:N\\to X$ and $q:M\\to X$ but which are not homotopic to homeomorphisms. The phenomenon is based on construction of cell-like maps that kill certain $\\mathbb L$-classes. The image space in these constructions is necessarily infinite-dimensional. In dimension $>6$ we classify all such homotopy equivalences. As an application, we show that such homotopy equivalences are realized by deformations of Riemannian manifolds in Gromov-Hausdorff space preserving a contractibility function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-10-31T22:53:03.000Z",
      "title": "Cell-like maps and topological structure groups on manifolds",
      "abstract": "We show that there are homotopy equivalences $h:N\\to M$ between closed manifolds which are induced by cell-like maps $p:N\\to X$ and $q:M\\to X$ but which are not homotopic to homeomorphisms. The phenomenon is based on construction of cell-like maps that kill certain $\\bL$-classes. In dimension $>5$ we identify all such homotopy equivalences to $M$ with a torsion subgroup $\\CS^{CE}(M)$ of the topological structure group $\\CS(M)$. In the case of simply connected $M$ with finite $\\pi_2(M)$, the subgroup $\\CS^{CE}(M)$ coincides with the odd torsion in $\\CS(M)$. For general $M$, the group $\\CS^{CE}(M)$ admits a description in terms of the second stage of the Postnikov tower of $M$. As an application, we show that there exist a contractibility function $\\rho$ and a precompact subset (\\mathcal{C}) of Gromov-Hausdorff space such that for every $\\epsilon>0$ there are nonhomeomorphic Riemannian manifolds with contractibility function $\\rho$ which lie within $\\epsilon$ of each other in (\\mathcal{C}).",
      "comment": null,
      "journal": null,
      "doi": null,
      "authors": [
        "A. Dranishnikov",
        "S. Ferry"
      ]
    },
    {
      "version": "v2",
      "updated": "2016-05-28T17:34:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57N65",
      "53C23",
      "53C20",
      "57R65",
      "57N60"
    ],
    "keywords": [
      "topological structure group",
      "cell-like maps",
      "contractibility function",
      "homotopy equivalences",
      "nonhomeomorphic riemannian manifolds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....11004D"
    }
  }
}