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

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

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

Bibliographic databases:

UDC: 515.164.174+515.164.22+515.122.55
MSC: 57R45, 58D15

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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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
2. E. A. Kudryavtseva, “Special framed Morse functions on surfaces”, Moscow University Mathematics Bulletin, 67:4 (2012), 151–157
3. E. A. Kudryavtseva, “Connected components of spaces of Morse functions with fixed critical points”, Moscow University Mathematics Bulletin, 67:1 (2012), 1–10
4. E. A. Kudryavtseva, “On the homotopy type of spaces of Morse functions on surfaces”, Sb. Math., 204:1 (2013), 75–113
5. D. A. Permyakov, “Abelian subgroups generated by Dehn twists in homeomorphism group”, Moscow University Mathematics Bulletin, 68:1 (2013), 42–47
6. Kudryavtseva E.A., “Topology of the spaces of functions with prescribed singularities on surfaces”, Dokl. Math., 93:3 (2016), 264–266
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