{
  "id": "1210.1564",
  "version": "v3",
  "published": "2012-10-04T19:57:56.000Z",
  "updated": "2013-08-06T12:06:46.000Z",
  "title": "Group cohomology and control of p-fusion",
  "authors": [
    "David Benson",
    "Jesper Grodal",
    "Ellen Henke"
  ],
  "comment": "15 pages; v3: title and exposition shortened, and a short section on Lannes' T-functor removed, following comments by referees. Final version, to appear in Invent. Math",
  "doi": "10.1007/s00222-013-0489-5",
  "categories": [
    "math.AT",
    "math.GR"
  ],
  "abstract": "We show that if an inclusion of finite groups H < G of index prime to p induces a homeomorphism of mod p cohomology varieties, or equivalently an F-isomorphism in mod p cohomology, then H controls p-fusion in G, if p is odd. This generalizes classical results of Quillen who proved this when H is a Sylow p-subgroup, and furthermore implies a hitherto difficult result of Mislin about cohomology isomorphisms. For p=2 we give analogous results, at the cost of replacing mod p cohomology with higher chromatic cohomology theories. The results are consequences of a general algebraic theorem we prove, that says that isomorphisms between p-fusion systems over the same finite p-group are detected on elementary abelian p-groups if p odd and abelian 2-groups of exponent at most 4 if p=2.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-08-06T12:06:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20J06",
      "20D20",
      "20J05",
      "55P60"
    ],
    "keywords": [
      "group cohomology",
      "higher chromatic cohomology theories",
      "elementary abelian p-groups",
      "general algebraic theorem",
      "cohomology varieties"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.1564B"
    }
  }
}