{
  "id": "2011.00295",
  "version": "v1",
  "published": "2020-10-31T15:40:52.000Z",
  "updated": "2020-10-31T15:40:52.000Z",
  "title": "On spectral sequence for the action of genus 3 Torelli group on the complex of cycles",
  "authors": [
    "Alexander A. Gaifullin"
  ],
  "comment": "53 pages",
  "categories": [
    "math.GT",
    "math.AT",
    "math.GR"
  ],
  "abstract": "The Torelli group of a genus $g$ oriented surface $S_g$ is the subgroup $\\mathcal{I}_g$ of the mapping class group $\\mathrm{Mod}(S_g)$ consisting of all mapping classes that act trivially on the homology of $S_g$. One of the most intriguing open problems concerning Torelli groups is the question of whether the group $\\mathcal{I}_3$ is finitely presented or not. A possible approach to this problem relies upon the study of the second homology group of $\\mathcal{I}_3$ using the spectral sequence $E^r_{p,q}$ for the action of $\\mathcal{I}_3$ on the complex of cycles. In this paper we obtain a partial result towards the conjecture that $H_2(\\mathcal{I}_3;\\mathbb{Z})$ is not finitely generated and hence $\\mathcal{I}_3$ is not finitely presented. Namely, we prove that the term $E^3_{0,2}$ of the spectral sequence is infinitely generated, that is, the group $E^1_{0,2}$ remains infinitely generated after taking quotients by images of the differentials $d^1$ and $d^2$. If one proceeded with the proof that it also remains infinitely generated after taking quotient by the image of $d^3$, he would complete the proof of the fact that $\\mathcal{I}_3$ is not finitely presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-31T15:40:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F34",
      "57M07",
      "20J05"
    ],
    "keywords": [
      "spectral sequence",
      "open problems concerning torelli groups",
      "intriguing open problems concerning torelli",
      "mapping class",
      "second homology group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}