{
  "id": "0809.4629",
  "version": "v4",
  "published": "2008-09-26T14:11:36.000Z",
  "updated": "2011-12-13T17:29:46.000Z",
  "title": "Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant",
  "authors": [
    "Gwenael Massuyeau"
  ],
  "comment": "46 pages, some figures. Minor modifications in this fourth version",
  "journal": "Bull. Soc. Math. France 140:1 (2012), 101-161",
  "categories": [
    "math.GT",
    "math.AT",
    "math.QA"
  ],
  "abstract": "Let S be a compact connected oriented surface with one boundary component, and let P be the fundamental group of S. The Johnson filtration is a decreasing sequence of subgroups of the Torelli group of S, whose k-th term consists of the self-homeomorphisms of S that act trivially at the level of the k-th nilpotent quotient of P. Morita defined a homomorphism from the k-th term of the Johnson filtration to the third homology group of the k-th nilpotent quotient of P. In this paper, we replace groups by their Malcev Lie algebras and we study the \"infinitesimal\" version of the k-th Morita homomorphism, which is shown to correspond to the original version by a canonical isomorphism. We provide a diagrammatic description of the k-th infinitesimal Morita homomorphism and, given an expansion of the free group P that is \"symplectic\" in some sense, we show how to compute it from Kawazumi's \"total Johnson map\". Besides, we give a topological interpretation of the full tree-reduction of the LMO homomorphism, which is a diagrammatic representation of the Torelli group derived from the Le-Murakami-Ohtsuki invariant of 3-manifolds. More precisely, a symplectic expansion of P is constructed from the LMO invariant, and it is shown that the tree-level of the LMO homomorphism is equivalent to the total Johnson map induced by this specific expansion. It follows that the k-th infinitesimal Morita homomorphism coincides with the degree [k,2k[ part of the tree-reduction of the LMO homomorphism. Our results also apply to the monoid of homology cylinders over S.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2011-12-13T17:29:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M27",
      "57R19",
      "57R50",
      "20F28",
      "20F38",
      "20F40"
    ],
    "keywords": [
      "lmo invariant",
      "lmo homomorphism",
      "total johnson map",
      "k-th nilpotent quotient",
      "tree-level"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0809.4629M"
    }
  }
}