{
  "id": "1902.10012",
  "version": "v1",
  "published": "2019-02-26T15:50:34.000Z",
  "updated": "2019-02-26T15:50:34.000Z",
  "title": "Alternative versions of the Johnson homomorphisms and the LMO functor",
  "authors": [
    "Anderson Vera"
  ],
  "comment": "62 pages, several figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $\\Sigma$ be a compact connected oriented surface with one boundary component and let $\\mathcal{M}$ denote the mapping class group of $\\Sigma$. By considering the action of $\\mathcal{M}$ on the fundamental group of $\\Sigma$ it is possible to define different filtrations of $\\mathcal{M}$ together with some homomorphisms on each term of the filtration. The aim of this paper is twofold. Firstly we study a filtration of $\\mathcal{M}$ introduced recently by Habiro and Massuyeau, whose definition involves a handlebody bounded by $\\Sigma$. We shall call it the \"alternative Johnson filtration\", and the corresponding homomorphisms are referred to as \"alternative Johnson homomorphisms\". We provide a comparison between the alternative Johnson filtration and two previously known filtrations: the original Johnson filtration and the Johnson-Levine filtration. Secondly, we study the relationship between the alternative Johnson homomorphisms and the functorial extension of the Le-Murakami-Ohtsuki invariant of $3$-manifolds. We prove that these homomorphisms can be read in the tree reduction of the LMO functor. In particular, this provides a new reading grid for the tree reduction of the LMO functor.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-26T15:50:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M27",
      "57M05",
      "57S05"
    ],
    "keywords": [
      "lmo functor",
      "alternative versions",
      "alternative johnson homomorphisms",
      "alternative johnson filtration",
      "tree reduction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 62,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}