{
  "id": "0709.3998",
  "version": "v1",
  "published": "2007-09-25T18:12:19.000Z",
  "updated": "2007-09-25T18:12:19.000Z",
  "title": "Face enumeration - from spheres to manifolds",
  "authors": [
    "Ed Swartz"
  ],
  "comment": "44 pages, 8 figures",
  "categories": [
    "math.CO",
    "math.GT"
  ],
  "abstract": "We prove a number of new restrictions on the enumerative properties of homology manifolds and semi-Eulerian complexes and posets. These include a determination of the affine span of the fine $h$-vector of balanced semi-Eulerian complexes and the toric $h$-vector of semi-Eulerian posets. The lower bounds on simplicial homology manifolds, when combined with higher dimensional analogues of Walkup's 3-dimensional constructions \\cite{Wal}, allow us to give a complete characterization of the $f$-vectors of arbitrary simplicial triangulations of $S^1 \\times S^3, \\C P^2,$ $ K3$ surfaces, and $(S^2 \\times S^2) # (S^2 \\times S^2).$ We also establish a principle which leads to a conjecture for homology manifolds which is almost logically equivalent to the $g$-conjecture for homology spheres. Lastly, we show that with sufficiently many vertices, every triangulable homology manifold without boundary of dimension three or greater can be triangulated in a 2-neighborly fashion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-09-25T18:12:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13F55",
      "52B05"
    ],
    "keywords": [
      "face enumeration",
      "arbitrary simplicial triangulations",
      "higher dimensional analogues",
      "simplicial homology manifolds",
      "conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0709.3998S"
    }
  }
}