{
  "id": "math/0401006",
  "version": "v1",
  "published": "2004-01-02T20:44:10.000Z",
  "updated": "2004-01-02T20:44:10.000Z",
  "title": "Geometrically constructed bases for homology of partition lattices of types A, B and D",
  "authors": [
    "Anders Björner",
    "Michelle L. Wachs"
  ],
  "comment": "29 pages, 4 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "We use the theory of hyperplane arrangements to construct natural bases for the homology of partition lattices of types A, B and D. This extends and explains the \"splitting basis\" for the homology of the partition lattice given in [Wa96], thus answering a question asked by R. Stanley. More explicitly, the following general technique is presented and utilized. Let A be a central and essential hyperplane arrangement in R^d. Let R_1,...,R_k be the bounded regions of a generic hyperplane section of A. We show that there are induced polytopal cycles \\rho_{R_i} in the homology of the proper part \\bar{L_A} of the intersection lattice such that {\\rho_{R_i}}_{i=1,...,k} is a basis for \\tilde H_{d-2}(\\bar{L_A}). This geometric method for constructing combinatorial homology bases is applied to the Coxeter arrangements of types A, B and D, and to some interpolating arrangements.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-01-02T20:44:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E25",
      "52C35",
      "52C40"
    ],
    "keywords": [
      "partition lattice",
      "geometrically constructed bases",
      "generic hyperplane section",
      "essential hyperplane arrangement",
      "constructing combinatorial homology bases"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......1006B"
    }
  }
}