{
  "id": "1310.2768",
  "version": "v2",
  "published": "2013-10-10T10:59:21.000Z",
  "updated": "2013-11-14T16:35:40.000Z",
  "title": "Triangular homotopy equivalences",
  "authors": [
    "Spiros Adams-Florou"
  ],
  "comment": "8 pages, 1 figure, part of my PhD thesis, minor TeX corrections to abstract",
  "categories": [
    "math.AT"
  ],
  "abstract": "A map $f:X\\to Y$ to a simplicial complex $Y$ is called a $Y$-triangular homotopy equivalence if it has a homotopy inverse $g$ and homotopies $h_1:f\\circ g\\simeq \\mathrm{id}_Y$, $h_2:g\\circ f\\simeq \\mathrm{id}_X$ such that for all simplices $\\sigma\\in Y$, $f|_\\sigma:f^{-1}(\\sigma) \\to \\sigma$ is a homotopy equivalence with inverse $g|_\\sigma:\\sigma \\to f^{-1}(\\sigma)$ and homotopies $h_1|_\\sigma$ and $h_2|_\\sigma$. In this paper we prove that for all pairs $X,Y$ of finite-dimensional locally finite simplicial complexes there is an $\\epsilon(X,Y)>0$ such that any $\\epsilon$-controlled homotopy equivalence $f:X\\to Y$ for $\\epsilon<\\epsilon(X,Y)$ is homotopic to a $Y$-triangular homotopy equivalence. Conversely, we conjecture that it is possible to `subdivide' a $Y$-triangular homotopy equivalence by finding a homotopic $(Sd\\, Y)$-triangular homotopy equivalence, consequently a $Y$-triangular homotopy equivalence would be homotopic to an $\\epsilon$-controlled homotopy equivalence for all $\\epsilon>0$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-11-14T16:35:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57R67",
      "57Qxx"
    ],
    "keywords": [
      "triangular homotopy equivalence",
      "controlled homotopy equivalence",
      "finite-dimensional locally finite simplicial complexes",
      "homotopy inverse"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.2768A"
    }
  }
}