{
  "id": "1207.5599",
  "version": "v2",
  "published": "2012-07-24T06:40:05.000Z",
  "updated": "2013-05-16T10:57:04.000Z",
  "title": "On stellated spheres and a tightness criterion for combinatorial manifolds",
  "authors": [
    "Bhaskar Bagchi",
    "Basudeb Datta"
  ],
  "comment": "Revised version, 22 pages. arXiv admin note: substantial text overlap with arXiv:1102.0856",
  "categories": [
    "math.GT",
    "math.CO"
  ],
  "abstract": "We introduce the $k$-stellated spheres and consider the class ${\\cal W}_k(d)$ of triangulated $d$-manifolds all whose vertex links are $k$-stellated, and its subclass ${\\cal W}^{\\ast}_k(d)$ consisting of the $(k+1)$-neighbourly members of ${\\cal W}_k(d)$. We introduce the mu-vector of any simplicial complex and show that, in the case of 2-neighbourly simplicial complexes, the mu-vector dominates the vector of its Betti numbers componentwise; the two vectors are equal precisely for tight simplicial complexes. We are able to estimate/compute certain alternating sums of the components of the mu-vector of any 2-neighbourly member of ${\\cal W}_k(d)$ for $d\\geq 2k$. As one consequence of this theory, we prove a lower bound theorem for such triangulated manifolds, as well as determine the integral homology type of members of ${\\cal W}^{\\ast}_k(d)$ for $d \\geq 2k+2$. As another application, we prove that, when $d \\neq 2k+1$, all members of ${\\cal W}^{\\ast}_k(d)$ are tight. We also characterize the tight members of ${\\cal W}^{\\ast}_k(2k + 1)$ in terms of their $k^{\\rm th}$ Betti numbers. These results more or less answer a recent question of Effenberger, and also provide a uniform and conceptual tightness proof for all except two of the known tight triangulated manifolds. We also prove a lower bound theorem for triangulated manifolds in which the members of ${\\cal W}_1(d)$ provide the equality case. This generalises a result (the $d=4$ case) due to Walkup and Kuehnel. As a consequence, it is shown that every tight member of ${\\cal W}_1(d)$ is strongly minimal, thus providing substantial evidence in favour of a conjecture of Kuehnel and Lutz asserting that tight triangulated manifolds should be strongly minimal.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-05-16T10:57:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57Q15",
      "52B05"
    ],
    "keywords": [
      "stellated spheres",
      "tightness criterion",
      "combinatorial manifolds",
      "lower bound theorem",
      "tight triangulated manifolds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1207.5599B"
    }
  }
}