{
  "id": "1302.5822",
  "version": "v2",
  "published": "2013-02-23T16:18:51.000Z",
  "updated": "2013-10-20T13:01:03.000Z",
  "title": "On the second nilpotent quotient of higher homotopy groups, for hypersolvable arrangements",
  "authors": [
    "Daniela Anca Macinic",
    "Daniel Matei",
    "Stefan Papadima"
  ],
  "comment": "11 pages, updated references",
  "categories": [
    "math.AT",
    "math.CO"
  ],
  "abstract": "We examine the first non-vanishing higher homotopy group, $\\pi_p$, of the complement of a hypersolvable, non--supersolvable, complex hyperplane arrangement, as a module over the group ring of the fundamental group, $\\Z\\pi_1$. We give a presentation for the $I$--adic completion of $\\pi_p$. We deduce that the second nilpotent $I$--adic quotient of $\\pi_p$ is determined by the combinatorics of the arrangement, and we give a combinatorial formula for the second associated graded piece, $\\gr^1_I \\pi_p$. We relate the torsion of this graded piece to the dimensions of the minimal generating systems of the Orlik--Solomon ideal of the arrangement $\\A$ in degree $p+2$, for various field coefficients. When $\\A$ is associated to a finite simple graph, we show that $\\gr^1_I \\pi_p$ is torsion--free, with rank explicitly computable from the graph.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-10-20T13:01:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C35",
      "55Q52",
      "16S37",
      "20C07"
    ],
    "keywords": [
      "second nilpotent quotient",
      "hypersolvable arrangements",
      "first non-vanishing higher homotopy group",
      "finite simple graph",
      "complex hyperplane arrangement"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1302.5822A"
    }
  }
}