{
  "id": "math/9611220",
  "version": "v1",
  "published": "1996-11-19T00:00:00.000Z",
  "updated": "1996-11-19T00:00:00.000Z",
  "title": "Cohomology at infinity and the well-rounded retract for general Linear Groups",
  "authors": [
    "Avner Ash",
    "Mark W. McConnell"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\bold G$ be a reductive algebraic group defined over $\\Q$, and let $\\Gamma$ be an arithmetic subgroup of $\\bold G(\\Q)$. Let $X$ be the symmetric space for $\\bold G(\\R)$, and assume $X$ is contractible. Then the cohomology (mod torsion) of the space $X/\\Gamma$ is the same as the cohomology of $\\Gamma$. In turn, $X/\\Gamma$ will have the same cohomology as $W/\\Gamma$, if $W$ is a ``spine'' in $X$. This means that $W$ (if it exists) is a deformation retract of $X$ by a $\\Gamma$-equivariant deformation retraction, that $W/\\Gamma$ is compact, and that $\\dim W$ equals the virtual cohomological dimension (vcd) of $\\Gamma$. Then $W$ can be given the structure of a cell complex on which $\\Gamma$ acts cellularly, and the cohomology of $W/\\Gamma$ can be found combinatorially.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1996-11-19T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "general linear groups",
      "cohomology",
      "well-rounded retract",
      "equivariant deformation retraction",
      "cell complex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1996math.....11220A"
    }
  }
}