{
  "id": "1812.08742",
  "version": "v1",
  "published": "2018-12-20T18:18:18.000Z",
  "updated": "2018-12-20T18:18:18.000Z",
  "title": "Homological stability for classical groups",
  "authors": [
    "David Sprehn",
    "Nathalie Wahl"
  ],
  "categories": [
    "math.AT",
    "math.KT"
  ],
  "abstract": "We prove a slope 1 stability range for the homology of the symplectic, orthogonal and unitary groups with respect to the hyperbolic form, over any fields other than $F_2$, improving the known range by a factor 2 in the case of finite fields. Our result more generally applies to the automorphism groups of vector spaces equipped with a possibly degenerate form (in the sense of Bak, Tits and Wall). For finite fields of odd characteristic, and more generally fields in which -1 is a sum of two squares, we deduce a stability range for the orthogonal groups with respect to the Euclidean form, and a corresponding result for the unitary groups. In addition, we include an exposition of Quillen's unpublished slope 1 stability argument for the general linear groups over fields other than $F_2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-20T18:18:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "homological stability",
      "classical groups",
      "stability range",
      "unitary groups",
      "finite fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}