{
  "id": "1006.0119",
  "version": "v3",
  "published": "2010-06-01T11:55:46.000Z",
  "updated": "2010-07-19T16:35:58.000Z",
  "title": "The topological fundamental group and free topological groups",
  "authors": [
    "Jeremy Brazas"
  ],
  "comment": "33 pages",
  "journal": "Topology Appl. 158 (2011) pp. 779-802",
  "categories": [
    "math.AT",
    "math.GN"
  ],
  "abstract": "The topological fundamental group $\\pi_{1}^{top}$ is a homotopy invariant finer than the usual fundamental group. It assigns to each space a quasitopological group and is discrete on spaces which admit universal covers. For an arbitrary space $X$, we compute the topological fundamental group of the suspension space $\\Sigma(X_+)$ and find that $\\pi_{1}^{top}(\\Sigma(X_+))$ either fails to be a topological group or is the free topological group on the path component space of $X$. Using this computation, we provide an abundance of counterexamples to the assertion that all topological fundamental groups are topological groups. A relation to free topological groups allows us to reduce the problem of characterizing Hausdorff spaces $X$ for which $\\pi_{1}^{top}(\\Sigma(X_+))$ is a Hausdorff topological group to some well known classification problems in topology.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-07-19T16:35:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "topological fundamental group",
      "free topological group",
      "homotopy invariant finer",
      "admit universal covers",
      "path component space"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1006.0119B"
    }
  }
}