{
  "id": "1408.2612",
  "version": "v2",
  "published": "2014-08-12T03:29:34.000Z",
  "updated": "2014-08-20T14:40:58.000Z",
  "title": "Structure of the fundamental groups of orbits of smooth functions on surfaces",
  "authors": [
    "Sergiy Maksymenko"
  ],
  "comment": "18 pages, 9 figures. v2 improved exposition and figures",
  "categories": [
    "math.GT",
    "math.AT"
  ],
  "abstract": "Let $M$ be a smooth compact connected surface, $P$ be either the real line $\\mathbb{R}$ or the circle $S^1$ and $f:M\\to P$ be a Morse map. Denote by $\\mathcal{S}(f)$ and $\\mathcal{O}(f)$ the corresponding stabilizer and orbit of $f$ with respect to the right action of the group $\\mathcal{D}(M)$ of diffeomorphisms of $M$. In a series of papers the author described homotopy types of $\\mathcal{S}(f)$ and computed higher homotopy groups of $\\mathcal{O}(f)$. The present paper describes the structure of the remained fundamental group $\\pi_1 \\mathcal{O}(f)$ for the case when $M$ is orientable and differs from $2$-sphere and $2$-torus. The result holds as well for a larger class of smooth maps $f:M\\to P$ having the following property: the germ of $f$ at each of its critical points is smoothly equivalent to a homogeneous polynomial $\\mathbb{R}^2\\to\\mathbb{R}$ without multiple factors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-12T03:29:34.000Z",
      "abstract": "Let $M$ be a smooth compact connected surface, $P$ be either the real line $\\mathbb{R}$ or the circle $S^1$, $f:M\\to P$ be a Morse map, $\\mathcal{S}(f)$ and $\\mathcal{O}(f)$ be respectively the stabilizer and the orbit of $f$ with respect to the right action of the group $\\mathcal{D}(M)$ of diffeomorphisms of $M$. In a series of papers the author described homotopy types of $\\mathcal{S}(f)$ and computed higher homotopy groups of $\\mathcal{O}(f)$. This paper describes the structure of the fundamental group $\\pi_1 \\mathcal{O}(f)$ for the case when $M$ is orientable and differs from $2$-sphere and $2$-torus. The result holds as well for a larger class of smooth maps $f:M\\to P$ having the following property: at each of its critical points the germ of $f$ is smoothly equivalent to a homogeneous polynomial $\\mathbb{R}^2\\to\\mathbb{R}$ without multiple factors.",
      "comment": "18 pages, 9 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-08-20T14:40:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57S05",
      "20E22",
      "58B05"
    ],
    "keywords": [
      "fundamental group",
      "smooth functions",
      "higher homotopy groups",
      "smooth compact connected surface",
      "result holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.2612M"
    }
  }
}