{
  "id": "1409.4319",
  "version": "v1",
  "published": "2014-09-15T16:47:26.000Z",
  "updated": "2014-09-15T16:47:26.000Z",
  "title": "Finiteness of the homotopy types of right orbits of Morse functions on surfaces",
  "authors": [
    "Sergiy Maksymenko"
  ],
  "comment": "10 pages, 2 figures",
  "categories": [
    "math.GT",
    "math.AT"
  ],
  "abstract": "Let $M$ be a connected orientable surface, $P$ be either a real line $\\mathbb{R}$ or a circle $S^1$, and $f:M \\to P$ be a Morse map. Denote by $\\mathcal{D}_{\\mathrm{id}}$ the group of diffeomorphisms of $M$ isotopic to the identity. This group acts from the right on the space of smooth maps $C^{\\infty}(M,P)$ and one can define the stabilizer $\\mathcal{S} = \\{h \\in \\mathcal{D}_{\\mathrm{id}} \\mid f \\circ h = f\\}$ and the orbit $\\mathcal{O} = \\{f \\circ h \\mid h \\in \\mathcal{D}_{\\mathrm{id}} \\}$ of $f$ with respect to that action. Then $\\mathcal{S}$ acts on the Kronrod-Reeb graph $\\Gamma$ of $f$ and we denote by $G$ the group of all automorphisms of $\\Gamma$ induced by elements from $\\mathcal{S}$. Previously the author shown that $\\mathcal{O}$ has a homotopy type of some CW-complex of dimension $2k-1$, where $k$ is the total number of critical points of $f$. Moreover, if $f$ is generic and $M$ differs from $2$-sphere and projective plane, then $\\mathcal{O}$ is homotopy equivalent to some torus. In the present paper we refine these facts by proving the following theorem: if $M$ is distinct from the $2$-sphere and $2$-torus then there exists a free action of $G$ on a $p$-dimensional torus $T^p$ for some $p\\geq0$ such that the orbit $\\mathcal{O}$ (endowed with $C^{\\infty}$ topology) is homotopy equivalent to the factor space $T^p/G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-15T16:47:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57S05",
      "20E22",
      "20F16"
    ],
    "keywords": [
      "homotopy type",
      "right orbits",
      "morse functions",
      "finiteness",
      "homotopy equivalent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.4319M"
    }
  }
}