{
  "id": "1706.03322",
  "version": "v1",
  "published": "2017-06-11T08:11:42.000Z",
  "updated": "2017-06-11T08:11:42.000Z",
  "title": "The face numbers of homology spheres",
  "authors": [
    "Kai Fong Ernest Chong",
    "Tiong Seng Tay"
  ],
  "comment": "31 pages",
  "categories": [
    "math.CO",
    "math.AC",
    "math.RA"
  ],
  "abstract": "The $g$-theorem is a momentous result in combinatorics that gives a complete numerical characterization of the face numbers of simplicial convex polytopes. The $g$-conjecture asserts that the same numerical conditions given in the $g$-theorem also characterizes the face numbers of all simplicial spheres, or even more generally, all simplicial homology spheres. In this paper, we prove the $g$-conjecture for simplicial $\\mathbb{R}$-homology spheres. A key idea in our proof is a new algebra structure for polytopal complexes. Given a polytopal $d$-complex $\\Delta$, we use ideas from rigidity theory to construct a graded Artinian $\\mathbb{R}$-algebra $\\Psi(\\Delta,\\nu)$ of stresses on a PL realization $\\nu$ of $\\Delta$ in $\\mathbb{R}^d$, where overlapping realized $d$-faces are allowed. In particular, we prove that if $\\Delta$ is a simplicial $\\mathbb{R}$-homology sphere, then for generic PL realizations $\\nu$, the stress algebra $\\Psi(\\Delta,\\nu)$ is Gorenstein and has the weak Lefschetz property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-11T08:11:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E45",
      "05E40",
      "13A02",
      "13E10",
      "13J30",
      "52B70",
      "52C25"
    ],
    "keywords": [
      "face numbers",
      "weak lefschetz property",
      "simplicial convex polytopes",
      "simplicial homology spheres",
      "generic pl realizations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}