{
  "id": "1807.03994",
  "version": "v1",
  "published": "2018-07-11T08:31:54.000Z",
  "updated": "2018-07-11T08:31:54.000Z",
  "title": "An upper bound for topological complexity",
  "authors": [
    "Michael Farber",
    "Mark Grant",
    "Gregory Lupton",
    "John Oprea"
  ],
  "comment": "20 pages",
  "categories": [
    "math.AT"
  ],
  "abstract": "In arXiv:1711.10132 a new approximating invariant ${\\mathsf{TC}}^{\\mathcal{D}}$ for topological complexity was introduced called $\\mathcal{D}$-topological complexity. In this paper, we explore more fully the properties of ${\\mathsf{TC}}^{\\mathcal{D}}$ and the connections between ${\\mathsf{TC}}^{\\mathcal{D}}$ and invariants of Lusternik-Schnirelmann type. We also introduce a new $\\mathsf{TC}$-type invariant $\\widetilde{\\mathsf{TC}}$ that can be used to give an upper bound for $\\mathsf{TC}$, $$\\mathsf{TC}(X)\\le {\\mathsf{TC}}^{\\mathcal{D}}(X) + \\left\\lceil \\frac{2\\dim X -k}{k+1}\\right\\rceil,$$ where $X$ is a finite dimensional simplicial complex with $k$-connected universal cover $\\tilde X$. The above inequality is a refinement of an estimate given by Dranishnikov.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-11T08:31:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55M30",
      "55P99"
    ],
    "keywords": [
      "topological complexity",
      "upper bound",
      "finite dimensional simplicial complex",
      "lusternik-schnirelmann type",
      "type invariant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}