{
  "id": "0901.0589",
  "version": "v3",
  "published": "2009-01-06T02:49:25.000Z",
  "updated": "2012-05-18T03:56:58.000Z",
  "title": "First cohomology groups of the automorphism group of a free group with coefficients in the abelianization of the IA-automorphism group",
  "authors": [
    "Takao Satoh"
  ],
  "comment": "26 pages",
  "categories": [
    "math.AT",
    "math.GR"
  ],
  "abstract": "We compute a twisted first cohomology group of the automorphism group of a free group with coefficients in the abelianization $V$ of the IA-automorphism group of a free group. In particular, we show that it is generated by two crossed homomorphisms constructed with the Magnus representation and the Magnus expansion due to Morita and Kawazumi respectively. As a corollary, we see that the first Johnson homomorphism does not extend to the automorphism group of a free group as a crossed homomorphism for the rank of the free group is greater than 4.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-05-18T03:56:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F28"
    ],
    "keywords": [
      "free group",
      "ia-automorphism group",
      "coefficients",
      "abelianization",
      "twisted first cohomology group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}