{
  "id": "1805.06727",
  "version": "v1",
  "published": "2018-05-17T12:36:45.000Z",
  "updated": "2018-05-17T12:36:45.000Z",
  "title": "Realization of a graph as the Reeb graph of a Morse function on a manifold",
  "authors": [
    "Łukasz Patryk Michalak"
  ],
  "categories": [
    "math.GT"
  ],
  "abstract": "We investigate the problem of the realization of a given graph as the Reeb graph $\\mathcal{R}(f)$ of a smooth function $f\\colon M\\rightarrow \\mathbb{R}$ with finitely many critical points, where $M$ is a closed manifold. We show that for any $n\\geq2$ and any graph $\\Gamma$ admitting the so called good orientation there exist an $n$-manifold $M$ and a Morse function $f\\colon M\\rightarrow \\mathbb{R} $ such that its Reeb graph $\\mathcal{R}(f)$ is isomorphic to $\\Gamma$, extending previous results of Sharko and Masumoto-Saeki. We prove that Reeb graphs of simple Morse functions maximize the number of cycles. Furthermore, we provide a complete characterization of graphs which can arise as Reeb graphs of surfaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-17T12:36:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58K05",
      "57M15",
      "58K65"
    ],
    "keywords": [
      "reeb graph",
      "realization",
      "smooth function",
      "simple morse functions maximize",
      "complete characterization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}