{
  "id": "2303.01019",
  "version": "v2",
  "published": "2023-03-02T07:17:58.000Z",
  "updated": "2023-09-12T01:50:17.000Z",
  "title": "Vietoris thickenings and complexes are weakly homotopy equivalent",
  "authors": [
    "Patrick Gillespie"
  ],
  "comment": "17 pages",
  "categories": [
    "math.AT",
    "math.GN",
    "math.MG"
  ],
  "abstract": "Characterizing the homotopy types of the Vietoris--Rips complexes of a metric space $X$ is in general a difficult problem. The Vietoris--Rips metric thickening, a metric space analogue of the Vietoris--Rips complex, was introduced as a potentially more amenable object of study with several advantageous properties, yet the relationship between its homotopy type and that of the Vietoris--Rips complex was not fully understood. We show that for any metric space $X$ and threshold $r>0$, the natural bijection between the (open) Vietoris--Rips complex and Vietoris--Rips metric thickening is a weak homotopy equivalence.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-09-12T01:50:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55N31",
      "51F99",
      "55P10"
    ],
    "keywords": [
      "weakly homotopy equivalent",
      "vietoris thickenings",
      "vietoris-rips complex",
      "homotopy type",
      "vietoris-rips metric thickening"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}