{
  "id": "1405.4579",
  "version": "v1",
  "published": "2014-05-19T02:09:24.000Z",
  "updated": "2014-05-19T02:09:24.000Z",
  "title": "On Representation of the Reeb Graph as a Sub-Complex of Manifold",
  "authors": [
    "Marek Kaluba",
    "Wacław Marzantowicz",
    "Nelson Silva"
  ],
  "comment": "18 pages",
  "categories": [
    "math.GT"
  ],
  "abstract": "The Reeb graph $\\mathcal{R}(f) $ is one of the fundamental invariants of a smooth function $f\\colon M\\to \\mathbb{R} $ with isolated critical points. It is defined as the quotient space $M/_{\\!\\sim}$ of the closed manifold $M$ by a relation that depends on $f$. Here we construct a $1$-dimensional complex $\\Gamma(f)$ embedded into $M$ which is homotopy equivalent to $\\mathcal{R}(f)$. As a consequence we show that for every function $f$ on a manifold with finite fundamental group, the Reeb graph of $f$ is a tree. If $\\pi_1(M)$ is an abelian group, or more general, a discrete amenable group, then $\\mathcal{R}(f)$ contains at most one loop. Finally we prove that the number of loops in the Reeb graph of every function on a surface $M_g$ is estimated from above by $g$, the genus of $M_g$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-19T02:09:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57N65",
      "57R70",
      "57M50",
      "58K65"
    ],
    "keywords": [
      "reeb graph",
      "representation",
      "sub-complex",
      "finite fundamental group",
      "discrete amenable group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.4579K"
    }
  }
}