{
  "id": "1807.07976",
  "version": "v1",
  "published": "2018-07-20T18:00:20.000Z",
  "updated": "2018-07-20T18:00:20.000Z",
  "title": "On the discontinuity of the $π_1$-action",
  "authors": [
    "Jeremy Brazas"
  ],
  "comment": "13 pages",
  "categories": [
    "math.AT",
    "math.GN"
  ],
  "abstract": "We show the classical $\\pi_1$-action on the $n$-th homotopy group can fail to be continuous for any $n$ when the homotopy groups are equipped with the natural quotient topology. In particular, we prove the action $\\pi_1(X)\\times\\pi_n(X)\\to\\pi_n(X)$ fails to be continuous for a one-point union $X=A\\vee \\mathbb{H}_n$ where $A$ is an aspherical space such that $\\pi_1(A)$ is a topological group and $\\mathbb{H}_n$ is the $(n-1)$-connected, n-dimensional Hawaiian earring space $\\mathbb{H}_n$ for which $\\pi_n(\\mathbb{H}_n)$ is a topological abelian group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-20T18:00:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55Q52",
      "14F35"
    ],
    "keywords": [
      "discontinuity",
      "th homotopy group",
      "natural quotient topology",
      "n-dimensional hawaiian earring space",
      "topological abelian group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}