{
  "id": "1606.02550",
  "version": "v1",
  "published": "2016-06-08T13:26:08.000Z",
  "updated": "2016-06-08T13:26:08.000Z",
  "title": "Face numbers and the fundamental group",
  "authors": [
    "Satoshi Murai",
    "Isabella Novik"
  ],
  "comment": "13 pages",
  "categories": [
    "math.CO",
    "math.AC",
    "math.AT"
  ],
  "abstract": "We resolve a conjecture of Kalai asserting that the $g_2$-number of any simplicial complex $\\Delta$ that represents a connected normal pseudomanifold of dimension $d\\geq 3$ is at least as large as ${d+2 \\choose 2}m(\\Delta)$, where $m(\\Delta)$ denotes the minimum number of generators of the fundamental group of $\\Delta$. Furthermore, we prove that a weaker bound, $h_2(\\Delta)\\geq {d+1 \\choose 2}m(\\Delta)$, applies to any $d$-dimensional pure simplicial poset $\\Delta$ all of whose faces of co-dimension $\\geq 2$ have connected links. This generalizes a result of Klee. Finally, for a pure relative simplicial poset $\\Psi$ all of whose vertex links satisfy Serre's condition $(S_r)$, we establish lower bounds on $h_1(\\Psi),\\ldots,h_r(\\Psi)$ in terms of the $\\mu$-numbers introduced by Bagchi and Datta.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-08T13:26:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E45",
      "57Q15",
      "57M05",
      "13F55"
    ],
    "keywords": [
      "fundamental group",
      "face numbers",
      "vertex links satisfy serres condition",
      "dimensional pure simplicial poset",
      "pure relative simplicial poset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}