{
  "id": "2209.07195",
  "version": "v1",
  "published": "2022-09-15T10:11:18.000Z",
  "updated": "2022-09-15T10:11:18.000Z",
  "title": "On Topological Homotopy Groups and Relation to Hawaiian Groups",
  "authors": [
    "Ameneh Babaee",
    "Behrooz Mashayekhy",
    "Hanieh Mirebrahimi",
    "Hamid Torabi",
    "Mahdi Abdullahi Rashid nad Seyyed Zeynal Pashaei"
  ],
  "journal": "BABAEE A, MASHAYEKHY B, MIREBRAHIMI H, TORABI H, RASHID M, PASHAEI S (2020). On topological homotopy groups and relation to Hawaiian groups. Hacettepe Journal of Mathematics and Statistics, 49(4), 1437 - 1449. 10.15672/hujms.565367",
  "doi": "10.15672/hujms.565367",
  "categories": [
    "math.AT"
  ],
  "abstract": "By generalizing the whisker topology on the $n$th homotopy group of pointed space $(X, x_0)$, denoted by $\\pi_n^{wh}(X, x_0)$, we show that $\\pi_n^{wh}(X, x_0)$ is a topological group if $n \\ge 2$. Also, we present some necessary and sufficient conditions for $\\pi_n^{wh}(X,x_0)$ to be discrete, Hausdorff and indiscrete. Then we prove that $L_n(X,x_0)$ the natural epimorphic image of the Hawaiian group $\\mathcal{H}_n(X, x_0)$ is equal to the set of all classes of convergent sequences to the identity in $\\pi_n^{wh}(X, x_0)$. As a consequence, we show that $L_n(X, x_0) \\cong L_n(Y, y_0)$ if $\\pi_n^{wh}(X, x_0) \\cong \\pi_n^{wh}(Y, y_0)$, but the converse does not hold in general, except for some conditions. Also, we show that on some classes of spaces such as semilocally $n$-simply connected spaces and $n$-Hawaiian like spaces, the whisker topology and the topology induced by the compact-open topology of $n$-loop space coincide. Finally, we show that $n$-SLT paths can transfer $\\pi_n^{wh}$ and hence $L_n$ isomorphically along its points.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-15T10:11:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55Q05",
      "55Q20",
      "55P65",
      "55Q52"
    ],
    "keywords": [
      "topological homotopy groups",
      "hawaiian group",
      "whisker topology",
      "th homotopy group",
      "loop space coincide"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}