{
  "id": "2004.05677",
  "version": "v1",
  "published": "2020-04-12T18:59:46.000Z",
  "updated": "2020-04-12T18:59:46.000Z",
  "title": "The order complex of $PGL_2(p^{2^n})$ is contractible when $p$ is odd",
  "authors": [
    "Emilio Pierro"
  ],
  "comment": "3 pages",
  "categories": [
    "math.CO",
    "math.AT",
    "math.GR"
  ],
  "abstract": "Given a group $G$, its lattice of subgroups $\\mathcal{L}(G)$ can be viewed as a simplicial complex in a natural way. The inclusion of $1_G, G \\in \\mathcal{L}(G)$ implies that $\\mathcal{L}(G)$ is contractible, and so we study the topology of the order complex $\\widehat{\\mathcal{L}(G)} := \\mathcal{L}(G) \\setminus \\{1_G,G\\}$. In this short note we consider the homotopy type of $\\widehat{\\mathcal{L}(G)}$ where $G \\cong PGL_2(p^{2^n})$, $p \\geq 3$, $n \\geq 1$ and show that $\\widehat{\\mathcal{L}(G)}$ is contractible. This is consistent with a conjecture of Shareshian on the homotopy type of order complexes of finite groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-12T18:59:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "order complex",
      "contractible",
      "homotopy type",
      "finite groups",
      "simplicial complex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 3,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}