{
  "id": "0705.2823",
  "version": "v1",
  "published": "2007-05-19T15:46:18.000Z",
  "updated": "2007-05-19T15:46:18.000Z",
  "title": "Cohomology of affine Artin groups and applications",
  "authors": [
    "Filippo Callegaro",
    "Davide Moroni",
    "Mario Salvetti"
  ],
  "comment": "21 pages, 4 figures",
  "categories": [
    "math.AT"
  ],
  "abstract": "The result of this paper is the determination of the cohomology of Artin groups of type A_n, B_n and \\tilde{A}_{n} with non-trivial local coefficients. The main result is an explicit computation of the cohomology of the Artin group of type B_n with coefficients over the module \\Q[q^{\\pm 1},t^{\\pm 1}]. Here the first (n-1) standard generators of the group act by (-q)-multiplication, while the last one acts by (-t)-multiplication. The proof uses some technical results from previous papers plus computations over a suitable spectral sequence. The remaining cases follow from an application of Shapiro's lemma, by considering some well-known inclusions: we obtain the rational cohomology of the Artin group of affine type \\tilde{A}_{n} as well as the cohomology of the classical braid group {Br}_{n} with coefficients in the n-dimensional representation presented in \\cite{tong}. The topological counterpart is the explicit construction of finite CW-complexes endowed with a free action of the Artin groups, which are known to be K(\\pi,1) spaces in some cases (including finite type groups). Particularly simple formulas for the Euler-characteristic of these orbit spaces are derived.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-05-19T15:46:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20J06",
      "20F36"
    ],
    "keywords": [
      "affine artin groups",
      "cohomology",
      "application",
      "finite type groups",
      "non-trivial local coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0705.2823C"
    }
  }
}