{
  "id": "0804.2153",
  "version": "v3",
  "published": "2008-04-14T12:01:57.000Z",
  "updated": "2011-02-26T05:46:18.000Z",
  "title": "On Walkup's class ${\\cal K}(d)$ and a minimal triangulation of $(S^3 \\times \\rotatebox{90}{\\ltimes} S^1)^{\\#3}$",
  "authors": [
    "Bhaskar Bagchi",
    "Basudeb Datta"
  ],
  "comment": "To appear in `Discrete Mathematics'",
  "journal": "Discrete Math. 311 (2011), 989--995",
  "categories": [
    "math.GT",
    "math.CO"
  ],
  "abstract": "For $d \\geq 2$, Walkup's class ${\\cal K}(d)$ consists of the $d$-dimensional simplicial complexes all whose vertex-links are stacked $(d-1)$-spheres. Kalai showed that for $d \\geq 4$, all connected members of ${\\cal K}(d)$ are obtained from stacked $d$-spheres by finitely many elementary handle additions. According to a result of Walkup, the face vector of any triangulated 4-manifold $X$ with Euler characteristic $\\chi$ satisfies $f_1 \\geq 5f_0 - {15/2} \\chi$, with equality only for $X \\in {\\cal K}(4)$. K\\\"{u}hnel observed that this implies $f_0(f_0 - 11) \\geq -15\\chi$, with equality only for 2-neighborly members of ${\\cal K}(4)$. K\\\"{u}hnel also asked if there is a triangulated 4-manifold with $f_0 = 15$, $\\chi = -4$ (attaining equality in his lower bound). In this paper, guided by Kalai's theorem, we show that indeed there is such a triangulation. It triangulates the connected sum of three copies of the twisted sphere product $S^3 \\times {-2.8mm}_{-} S^1$. Because of K\\\"{u}hnel's inequality, the given triangulation of this manifold is a vertex-minimal triangulation. By a recent result of Effenberger, the triangulation constructed here is tight. Apart from the neighborly 2-manifolds and the infinite family of $(2d+ 3)$-vertex sphere products $S^{d-1} \\times S^1$ (twisted for $d$ odd), only fourteen tight triangulated manifolds were known so far. The present construction yields a new member of this sporadic family. We also present a self-contained proof of Kalai's result.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-02-26T05:46:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57Q15",
      "57R05"
    ],
    "keywords": [
      "walkups class",
      "elementary handle additions",
      "dimensional simplicial complexes",
      "vertex sphere products",
      "kalais result"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0804.2153B"
    }
  }
}