{
  "id": "2109.11106",
  "version": "v1",
  "published": "2021-09-23T02:18:44.000Z",
  "updated": "2021-09-23T02:18:44.000Z",
  "title": "Motion planning in polyhedral products of groups and a Fadell-Husseini approach to topological complexity",
  "authors": [
    "Jorge Aguilar-Guzmán",
    "Jesús González"
  ],
  "comment": "15 pages",
  "categories": [
    "math.AT"
  ],
  "abstract": "We compute the topological complexity of a polyhedral product $\\mathcal{Z}$ defined in terms of an LS-logarithmic family of locally compact connected CW topological groups. The answer is given by a combinatorial formula that involves the LS category of the polyhedral-product factors. As a by-product, we show that the Iwase-Sakai conjecture holds true for $\\mathcal{Z}$. The proof methodology uses a Fadell-Husseini viewpoint for the monoidal topological complexity (MTC) of a space, which, under mild conditions, recovers Iwase-Sakai's original definition. In the Fadell-Husseini context, the stasis condition -- MTC's raison d'\\^etre -- can be encoded at the covering level. Our Fadell-Husseini-inspired definition provides an alternative to the MTC variant given by Dranishnikov, as well as to the ones provided by Garc\\'ia-Calcines, Carrasquel-Vera and Vandembroucq in terms of relative category.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-23T02:18:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55M30",
      "55M15",
      "68T40"
    ],
    "keywords": [
      "topological complexity",
      "polyhedral product",
      "fadell-husseini approach",
      "motion planning",
      "iwase-sakai conjecture holds true"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}