{
  "id": "1007.4398",
  "version": "v3",
  "published": "2010-07-26T08:33:07.000Z",
  "updated": "2011-06-19T20:59:05.000Z",
  "title": "Connected components of spaces of Morse functions with fixed critical points",
  "authors": [
    "Elena A. Kudryavtseva"
  ],
  "comment": "12 pages with 2 figures, in Russian, to be published in Vestnik Moskov. Univ., a typo in theorem 1 is corrected",
  "journal": "Vestnik Moskov. Univ. Mat. Mekh. (2012) No.1, 3-12",
  "categories": [
    "math.GT",
    "math.DG"
  ],
  "abstract": "Let $M$ be a smooth closed orientable surface and $F=F_{p,q,r}$ be the space of Morse functions on $M$ having exactly $p$ critical points of local minima, $q\\ge1$ saddle critical points, and $r$ critical points of local maxima, moreover all the points are fixed. Let $F_f$ be the connected component of a function $f\\in F$ in $F$. By means of the winding number introduced by Reinhart (1960), a surjection $\\pi_0(F)\\to{\\mathbb Z}^{p+r-1}$ is constructed. In particular, $|\\pi_0(F)|=\\infty$, and the Dehn twist about the boundary of any disk containing exactly two critical points, exactly one of which is a saddle point, does not preserve $F_f$. Let $\\mathscr D$ be the group of orientation preserving diffeomorphisms of $M$ leaving fixed the critical points, ${\\mathscr D}^0$ be the connected component of ${\\rm id}_M$ in $\\mathscr D$, and ${\\mathscr D}_f\\subset{\\mathscr D}$ the set of diffeomorphisms preserving $F_f$. Let ${\\mathscr H}_f$ be the subgroup of ${\\mathscr D}_f$ generated by ${\\mathscr D}^0$ and all diffeomorphisms $h\\in{\\mathscr D}$ which preserve some functions $f_1\\in F_f$, and let ${\\mathscr H}_f^{\\rm abs}$ be its subgroup generated ${\\mathscr D}^0$ and the Dehn twists about the components of level curves of functions $f_1\\in F_f$. We prove that ${\\mathscr H}_f^{\\rm abs}\\subsetneq{\\mathscr D}_f$ if $q\\ge2$, and construct an epimorphism ${\\mathscr D}_f/{\\mathscr H}_f^{\\rm abs}\\to{\\mathbb Z}_2^{q-1}$, by means of the winding number. A finite polyhedral complex $K=K_{p,q,r}$ associated to the space $F$ is defined. An epimorphism $\\mu:\\pi_1(K)\\to{\\mathscr D}_f/{\\mathscr H}_f$ and finite generating sets for the groups ${\\mathscr D}_f/{\\mathscr D}^0$ and ${\\mathscr D}_f/{\\mathscr H}_f$ in terms of the 2-skeleton of the complex $K$ are constructed.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-06-19T20:59:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58E05",
      "57M50",
      "58K65",
      "46M18"
    ],
    "keywords": [
      "connected component",
      "fixed critical points",
      "morse functions",
      "dehn twist",
      "winding number"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "ru",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.4398K"
    }
  }
}