{
  "id": "1604.01701",
  "version": "v1",
  "published": "2016-04-06T17:36:24.000Z",
  "updated": "2016-04-06T17:36:24.000Z",
  "title": "Cohomology of automorphism groups of free groups with twisted coefficients",
  "authors": [
    "Oscar Randal-Williams"
  ],
  "comment": "20 pages. Supersedes arXiv:1012.1433",
  "categories": [
    "math.AT",
    "math.GT",
    "math.RT"
  ],
  "abstract": "We compute the groups $H^*(\\mathrm{Aut}(F_n); M)$ and $H^*(\\mathrm{Out}(F_n); M)$ in a stable range, where $M$ is obtained by applying a Schur functor to $H_\\mathbb{Q}$ or $H^*_\\mathbb{Q}$, respectively the first rational homology and cohomology of $F_n$. For reasons which are not conceptually clear, taking coefficients in $H_\\mathbb{Q}$ and its related modules behaves in a far less trivial way than taking coefficients in $H^*_\\mathbb{Q}$ and its related modules. The answer may be described in terms of stable multiplicities of irreducibles in the plethysm $\\mathrm{Sym}^k \\circ \\mathrm{Sym}^l$ of symmetric powers. We also compute the stable integral cohomology groups of $\\mathrm{Aut}(F_n)$ with coefficients in $H$ or $H^*$, respectively the first integral homology and cohomology of $F_n$, and compute the stable cohomology with coefficients in Schur functors of $H$ or $H^*$ modulo small primes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-06T17:36:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F28",
      "20J06",
      "57R20"
    ],
    "keywords": [
      "free groups",
      "automorphism groups",
      "twisted coefficients",
      "schur functor",
      "modulo small primes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160401701R"
    }
  }
}