{
  "id": "1607.08270",
  "version": "v1",
  "published": "2016-07-27T21:30:06.000Z",
  "updated": "2016-07-27T21:30:06.000Z",
  "title": "Approximating certain cell-like maps by homeomorphisms",
  "authors": [
    "Robert D. Edwards"
  ],
  "comment": "22 pages, 4 figures. This is one of three unpublished but highly influential articles on the topology of manifolds written by Robert D. Edwards in the 1970's. This article is being posted now because an electronic version has recently been completed",
  "categories": [
    "math.GT"
  ],
  "abstract": "Given a proper map f : M $\\rightarrow$ Q, having cell-like point-inverses, from a manifold-without-boundary M onto an ANR Q, it is a much-studied problem to find when f is approximable by homeomorphisms, i.e., when the decomposition of M induced by f is shrinkable (in the sense of Bing). If dimension M $\\geq$ 5, J. W. Cannon's recent work focuses attention on whether Q has the disjoint disc property (which is: Any two maps of a 2-disc into Q can be homotoped by an arbitrarily small amount to have disjoint images; this is clearly a necessary condition for Q to be a manifold, in this dimension range). This paper establishes that such an f is approximable by homeomorphisms whenever dimension M $\\geq$ 5 and Q has the disjoint disc property. As a corollary, one obtains that given an arbitrary map f : M $\\rightarrow$ Q as above, the stabilized map f $\\times$ id($\\mathbb{R}^{2}$) : M $\\times$ $\\mathbb{R}^{2}$ -> Q $\\times$ $\\mathbb{R}^{2}$ is approximable by homeomorphisms. The proof of the theorem is different from the proofs of the special cases in the earlier work of myself and Cannon, and it is quite self-contained. This work provides an alternative proof of L. Siebenmann's Approximation Theorem, which is the case where Q is given to be a manifold.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-27T21:30:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57N15"
    ],
    "keywords": [
      "homeomorphisms",
      "cell-like maps",
      "disjoint disc property",
      "work focuses attention",
      "siebenmanns approximation theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}