{
  "id": "0910.3685",
  "version": "v2",
  "published": "2009-10-19T20:19:32.000Z",
  "updated": "2010-07-08T17:47:07.000Z",
  "title": "The Topological Fundamental Group and Hoop Earring Spaces",
  "authors": [
    "Jeremy Brazas"
  ],
  "comment": "9 pages This paper has been withdrawn by the author due to much more general results appearing in the paper \"The topological fundamental group and free topological groups\" which includes the results in this paper. This paper, includes a sketch of the main proof (citing a thesis in progress) whereas a full proof is given in the mentioned paper.",
  "categories": [
    "math.AT",
    "math.GT"
  ],
  "abstract": "The topological fundamental group $\\pi_{1}^{top}$ is a topological invariant that assigns to each space a quasi-topological group and is discrete on spaces which are well behaved locally. For a totally path-disconnected, Hausdorff, unbased space $X$, we compute the topological fundamental group of the \"hoop earring\" space of $X$, which is the reduced suspension of $X$ with disjoint basepoint. We do so by factorizing the quotient map $\\Omega(\\Sigma X_{+},x)\\to \\pi_{1}^{top}(\\Sigma X_{+},x)$ through a free topological monoid with involution $M(X)$ such that the map $M(X)\\shortrightarrow \\pi_{1}^{top}(\\Sigma X_{+},x)$ is also a quotient map. $\\pi_{1}^{top}(\\Sigma X_{+},x)$ is T1 and an embedding $X\\shortrightarrow \\pi_{1}^{top}(\\Sigma X_{+},x)$ illustrates that $\\pi_{1}^{top}(\\Sigma X_{+},x)$ is not a topological group when $X$ is not regular. These hoop earring spaces provide a simple class of counterexamples to the claim that $\\pi_{1}^{top}$ is a functor to the category of topological groups.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-07-08T17:47:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "topological fundamental group",
      "hoop earring spaces",
      "quotient map",
      "topological group",
      "disjoint basepoint"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0910.3685B"
    }
  }
}