{
  "id": "2002.02388",
  "version": "v1",
  "published": "2020-02-06T17:30:23.000Z",
  "updated": "2020-02-06T17:30:23.000Z",
  "title": "Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups",
  "authors": [
    "Wacław Marzantowicz",
    "Łukasz Patryk Michalak"
  ],
  "comment": "20 pages",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "abstract": "In this work we present a construction which gives a correspondence between epimorphisms $\\varphi \\colon \\pi_1(W) \\to F_r$, from the fundamental group of a compact manifold $W$ onto the free group of rank $r$, and systems of framed non-separating hypersurfaces in $W$. In consequence, any such $\\varphi$, which corresponds to a system of hypersurfaces without boundary, can be represented by the Reeb epimorphism of a Morse function $f\\colon W \\to \\mathbb{R}$, i.e. by the epimorphism induced by the quotient map $W \\to \\mathcal{R}(f)$ onto the Reeb graph of $f$. We study properties and natural relations between these three objects. In particular, from this point of view we discuss the problem of classification up to (strong-)equivalence of epimorphisms onto free groups and we provide a purely geometrical-topological proof of the solution of this problem for surface groups which was given earlier by Grigorchuk, Kurchanov and Zieshang by using other methods.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-06T17:30:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "57M15",
      "57R90"
    ],
    "keywords": [
      "free group",
      "reeb graph",
      "hypersurfaces",
      "morse function",
      "reeb epimorphism"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}