{
  "id": "0911.0085",
  "version": "v3",
  "published": "2009-10-31T14:33:33.000Z",
  "updated": "2010-12-21T21:22:22.000Z",
  "title": "Saturated fusion systems as idempotents in the double Burnside ring",
  "authors": [
    "Kari Ragnarsson",
    "Radu Stancu"
  ],
  "comment": "Added citation to Puig; clarified discussion of Miller's conjecture on the homotopy characterization of p-local finite groups",
  "categories": [
    "math.AT",
    "math.GR"
  ],
  "abstract": "We give a new, unexpected characterization of saturated fusion systems on a p-group S in terms of idempotents in the p-local double Burnside ring of S that satisfy a Frobenius reciprocity relation, and reformulate fusion-theoretic phenomena in the language of idempotents. Interpreting our results in stable homotopy, we answer a long-standing question on stable splittings of classifying spaces of finite groups, and generalize the Adams--Wilkerson criterion for recognizing rings of invariants in the cohomology of an elementary abelian p-group. This work is partly motivated by a conjecture of Haynes Miller which proposes retractive transfer triples as a purely homotopy-theoretic model for p-local finite groups. We take an important step toward proving this conjecture by showing that a retractive transfer triple gives rise to a p-local finite group when two technical assumptions are made, thus reducing the conjecture to proving those two assumptions.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-12-21T21:22:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D20",
      "19A22",
      "55R35",
      "55P42"
    ],
    "keywords": [
      "saturated fusion systems",
      "double burnside ring",
      "idempotents",
      "p-local finite group",
      "retractive transfer triple"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0911.0085R"
    }
  }
}