{
  "id": "1809.09021",
  "version": "v1",
  "published": "2018-09-24T15:54:27.000Z",
  "updated": "2018-09-24T15:54:27.000Z",
  "title": "Topological complexity of a fibration",
  "authors": [
    "Petar Pavešić"
  ],
  "categories": [
    "math.AT"
  ],
  "abstract": "We study certain topological problems that are inspired by applications to autonomous robot manipulation. Consider a continuous map $f\\colon X\\to Y$, where $f$ can be a kinematic map from the configuration space $X$ to the working space $Y$ of a robot arm or a similar mechanism. Then one can associate to $f$ a number $\\mathrm{TC}(f)$, which is, roughly speaking, the minimal number of continuous rules that are necessary to construct a complete manipulation algorithm for the device. Examples show that $\\mathrm{TC}(f)$ is very sensitive to small perturbations of $f$ and that its value depends heavily on the singularities of $f$. This fact considerably complicates the computations, so we focus here on estimates of $\\mathrm{TC}(f)$ that can be expressed in terms of homotopy invariants of spaces $X$ and $Y$, or that are valid if $f$ satisfy some additional assumptions like, for example, being a fibration. Some of the main results are the derivation of a general upper bound for $\\mathrm{TC}(f)$, invariance of $\\mathrm{TC}(f)$ with respect to deformations of the domain and codomain, proof that $\\mathrm{TC}(f)$ is a FHE-invariant, and the description of a cohomological lower bound for $\\mathrm{TC}(f)$. Furthermore, if $f$ is a fibration we derive more precise estimates for $\\mathrm{TC}(f)$ in terms of the Lusternik-Schnirelmann category and the topological complexity of $X$ and $Y$. We also obtain some results for the important special case of covering projections.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-24T15:54:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55M99",
      "70B15",
      "68T40"
    ],
    "keywords": [
      "topological complexity",
      "complete manipulation algorithm",
      "general upper bound",
      "important special case",
      "kinematic map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}