{
  "id": "1909.06729",
  "version": "v1",
  "published": "2019-09-15T04:19:29.000Z",
  "updated": "2019-09-15T04:19:29.000Z",
  "title": "$g$-vectors of manifolds with boundary",
  "authors": [
    "Isabella Novik",
    "Ed Swartz"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We extend several $g$-type theorems for connected, orientable homology manifolds without boundary to manifolds with boundary. As applications of these results we obtain K\\\"uhnel-type bounds on the Betti numbers as well as on certain weighted sums of Betti numbers of manifolds with boundary. Our main tool is the completion $\\hat\\Delta$ of a manifold with boundary $\\Delta$; it is obtained from $\\Delta$ by coning off the boundary of $\\Delta$ with a single new vertex. We show that despite the fact that $\\hat{\\Delta}$ has a singular vertex, its Stanley--Reisner ring shares a few properties with the Stanley--Reisner rings of homology spheres. We close with a discussion of a connection between three lower bound theorems for manifolds, PL-handle decompositions, and surgery.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-15T04:19:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E45",
      "52B05",
      "13F55"
    ],
    "keywords": [
      "betti numbers",
      "lower bound theorems",
      "pl-handle decompositions",
      "main tool",
      "type theorems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}