{
  "id": "2211.14141",
  "version": "v1",
  "published": "2022-11-25T14:40:47.000Z",
  "updated": "2022-11-25T14:40:47.000Z",
  "title": "A natural pseudometric on homotopy groups of metric spaces",
  "authors": [
    "Jeremy Brazas",
    "Paul Fabel"
  ],
  "comment": "16 pages",
  "categories": [
    "math.AT",
    "math.MG"
  ],
  "abstract": "For a path-connected metric space $(X,d)$, the $n$-th homotopy group $\\pi_n(X)$ inherits a natural pseudometric from the $n$-th iterated loop space with the uniform metric. This pseudometric gives $\\pi_n(X)$ the structure of a topological group and when $X$ is compact, the induced pseudometric topology is independent of the metric $d$. In this paper, we study the properties of this pseudometric and how it relates to previously studied structures on $\\pi_n(X)$. Our main result is that the pseudometric topology agrees with the shape topology on $\\pi_n(X)$ if $X$ is comapact and $UV^{n-1}$ or if $X$ is an inverse limit of finite polyhedra with retraction bonding maps.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-25T14:40:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55Q52",
      "55P55",
      "54E35",
      "54C56"
    ],
    "keywords": [
      "natural pseudometric",
      "th iterated loop space",
      "th homotopy group",
      "uniform metric",
      "induced pseudometric topology"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}