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Mat. Sb., 2010, Volume 201, Number 4, Pages 33–98 (Mi msb7557)  

This article is cited in 6 scientific papers (total in 6 papers)

Framed Morse functions on surfaces

E. A. Kudryavtseva, D. A. Permyakov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: Let $M$ be a smooth, compact, not necessarily orientable surface with (maybe empty) boundary, and let $F$ be the space of Morse functions on $M$ that are constant on each component of the boundary and have no critical points at the boundary. The notion of framing is defined for a Morse function $f\in F$. In the case of an orientable surface $M$ this is a closed 1-form $\alpha$ on $M$ with punctures at the critical points of local minimum and maximum of $f$ such that in a neighbourhood of each critical point the pair $(f,\alpha)$ has a canonical form in a suitable local coordinate chart and the 2-form $df\wedge\alpha$ does not vanish on $M$ punctured at the critical points and defines there a positive orientation. Each Morse function on $M$ is shown to have a framing, and the space $F$ endowed with the $C^\infty$-topology is homotopy equivalent to the space $\mathbb F$ of framed Morse functions. The results obtained make it possible to reduce the problem of describing the homotopy type of $F$ to the simpler problem of finding the homotopy type of $\mathbb F$. As a solution of the latter, an analogue of the parametric $h$-principle is stated for the space $\mathbb F$.
Bibliography: 41 titles.

Keywords: Morse functions, framed Morse functions, equivalence of functions, compact surface, $C^\infty$-topology.

DOI: https://doi.org/10.4213/sm7557

Full text: PDF file (1099 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2010, 201:4, 501–567

Bibliographic databases:

UDC: 515.164.174+515.164.22+515.122.55
MSC: 57R45, 58D15
Received: 18.03.2009 and 02.07.2009

Citation: E. A. Kudryavtseva, D. A. Permyakov, “Framed Morse functions on surfaces”, Mat. Sb., 201:4 (2010), 33–98; Sb. Math., 201:4 (2010), 501–567

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. E. A. Kudryavtseva, “The Topology of Spaces of Morse Functions on Surfaces”, Math. Notes, 92:2 (2012), 219–236  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. E. A. Kudryavtseva, “Special framed Morse functions on surfaces”, Moscow University Mathematics Bulletin, 67:4 (2012), 151–157  mathnet  crossref  mathscinet
    3. E. A. Kudryavtseva, “Connected components of spaces of Morse functions with fixed critical points”, Moscow University Mathematics Bulletin, 67:1 (2012), 1–10  mathnet  crossref  mathscinet
    4. E. A. Kudryavtseva, “On the homotopy type of spaces of Morse functions on surfaces”, Sb. Math., 204:1 (2013), 75–113  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. D. A. Permyakov, “Abelian subgroups generated by Dehn twists in homeomorphism group”, Moscow University Mathematics Bulletin, 68:1 (2013), 42–47  mathnet  crossref
    6. Kudryavtseva E.A., “Topology of the spaces of functions with prescribed singularities on surfaces”, Dokl. Math., 93:3 (2016), 264–266  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
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