{
  "id": "2104.03751",
  "version": "v1",
  "published": "2021-04-08T13:11:35.000Z",
  "updated": "2021-04-08T13:11:35.000Z",
  "title": "A characterization of normal 3-pseudomanifolds with at most two singularities",
  "authors": [
    "Biplab Basak",
    "Raju Kumar Gupta",
    "Sourav Sarkar"
  ],
  "comment": "25 pages, 1 figure",
  "categories": [
    "math.GT",
    "math.CO"
  ],
  "abstract": "Characterizing face-number related invariants of a given class of simplicial complexes has been a central topic in combinatorial topology. In this regards, one of the most well-known invariant is $g_2$. Kalai's relative lower bound [9] for $g_2$ says that if $K$ is a normal $d$-pseudomanifold with $d \\geq 3,$ then $g_2(K) \\geq g_2(lk (v))$ for any vertex $v$ of $K.$ In [6], two combinatorial tools - `vertex folding' and `edge folding' were defined. Let $K$ be a normal 3-pseudomanifold with at most two singularities and $t$ be a vertex of $K$ such that $g_2(lk (t)) \\geq g_2(lk(v))$ for any other vertex $v$. They proved that if $g_2(K) = g_2(lk (t))$ then $K$ is obtained from a triangulation of $3$-sphere by a sequence of vertex folding and edge folding. This leads to a natural question - what will be the maximum value of $n\\in\\mathbb{N}$, for which $g_2{(K)} \\leq g_2(lk (t)) + n$ implies $K$ is such combinatorial normal $3$-pseudomanifold? In this article we give the complete answer of this question. Let $K$ be a normal $3$-pseudomanifold with at most two singularities (in case of two singularities, we take one singularity is $\\mathbb{RP}^2$). We prove that if $g_2{(K)} \\leq g_2(lk (t)) + 9$ then $K$ is obtained from a triangulation of $3$-sphere by a sequence of vertex folding and edge folding. Further, we prove that the upper bound is sharp for such combinatorial normal $3$-pseudomanifolds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-08T13:11:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57Q05",
      "57Q25",
      "57Q15",
      "05E45"
    ],
    "keywords": [
      "singularity",
      "combinatorial normal",
      "edge folding",
      "vertex folding",
      "pseudomanifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}