{
  "id": "1908.00039",
  "version": "v1",
  "published": "2019-07-31T18:29:48.000Z",
  "updated": "2019-07-31T18:29:48.000Z",
  "title": "Linear homology in a nutshell",
  "authors": [
    "Jonathan Fine"
  ],
  "comment": "LaTeX, 21 pages, no figures",
  "categories": [
    "math.CO",
    "math.KT",
    "math.RT"
  ],
  "abstract": "In 1985 Bayer and Billera defined a flag vector $f(X)$ for every convex polytope $X$, and proved some fundamental properties. The flag vectors $f(X)$ span a graded ring $\\mathcal{R}=\\bigoplus_{d\\geq0}\\mathcal{R}_d$. Here $\\mathcal{R}_d$ is the span of the $f(X)$ with $\\dim X=d$. It has dimension the Fibonacci number $F_{d+1}$. This paper introduces and explores the conjecture, that $\\mathcal{R}$ has a counting basis $\\{e_i\\}$. If true then the equation $f(X) = \\sum g_i(X)e_i$ conjecturally provides a formula for the Betti numbers $g_i(X)$ of a new homology theory. As the $g_i(X)$ are linear functions of $f(X)$, we call the new theory linear homology. Further, assuming the conjecture each $g_i$ will have a rank $r\\geq0$. The rank zero part of linear homology will be (middle perversity) intersection homology. The higher rank $g_i$ measure successively more complicated singularities. In dimension $d$ we will have $\\dim\\mathcal{R}_d$ linearly independent Betti numbers. This paper produces a basis $\\{e_i\\}$ for $\\mathcal{R}$, that is conjecturally a counting basis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-31T18:29:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "flag vector",
      "rank zero part",
      "linearly independent betti numbers",
      "theory linear homology",
      "counting basis"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}