{
  "id": "2211.10499",
  "version": "v1",
  "published": "2022-11-18T20:26:45.000Z",
  "updated": "2022-11-18T20:26:45.000Z",
  "title": "Fundamental groups of reduced suspensions are locally free",
  "authors": [
    "Jeremy Brazas",
    "Patrick Gillespie"
  ],
  "comment": "15 pages",
  "categories": [
    "math.AT"
  ],
  "abstract": "In this paper, we analyze the fundamental group $\\pi_1(\\Sigma X,\\overline{x_0})$ of the reduced suspension $\\Sigma X$ where $(X,x_0)$ is an arbitrary based Hausdorff space. We show that $\\pi_1(\\Sigma X,\\overline{x_0})$ is canonically isomorphic to a direct limit $\\varinjlim_{A\\in\\mathscr{P}}\\pi_1(\\Sigma A,\\overline{x_0})$ where each group $\\pi_1(\\Sigma A,\\overline{x_0})$ is isomorphic to a finitely generated free group or the infinite earring group. A direct consequence of this characterization is that $\\pi_1(\\Sigma X,\\overline{x_0})$ is locally free for any Hausdorff space $X$. Additionally, we show that $\\Sigma X$ is simply connected if and only if $X$ is sequentially $0$-connected at $x_0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-18T20:26:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55Q52",
      "08A65"
    ],
    "keywords": [
      "fundamental group",
      "reduced suspension",
      "locally free",
      "hausdorff space",
      "direct consequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}