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Smooth functions on 2-torus whose Kronrod-Reeb graph contains a cycle

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Abstract

Let $f:M\to \mathbb{R}$ be a Morse function on a connected compact surface $M$, and $\mathcal{S}(f)$ and $\mathcal{O}(f)$ be respectively the stabilizer and the orbit of $f$ with respect to the right action of the group of diffeomorphisms $\mathcal{D}(M)$. In a series of papers the first author described the homotopy types of connected components of $\mathcal{S}(f)$ and $\mathcal{O}(f)$ for the cases when $M$ is either a $2$-disk or a cylinder or $\chi(M)<0$. Moreover, in two recent papers the authors considered special classes of smooth functions on $2$-torus $T^2$ and shown that the computations of $\pi_1\mathcal{O}(f)$ for those functions reduces to the cases of $2$-disk and cylinder.

In the present paper we consider another class of Morse functions $f:T^2\to\mathbb{R}$ whose KR-graphs have exactly one cycle and prove that for every such function there exists a subsurface $Q\subset T^2$, diffeomorphic with a cylinder, such that $\pi_1\mathcal{O}(f)$ is expressed via the fundamental group $\pi_1\mathcal{O}(f|_{Q})$ of the restriction of $f$ to $Q$.

This result holds for a larger class of smooth functions $f:T^2\to \mathbb{R}$ having the following property: for every critical point $z$ of $f$ the germ of $f$ at $z$ is smoothly equivalent to a homogeneous polynomial $\mathbb{R}^2\to \mathbb{R}$ without multiple factors.

Key words: Diffeomorphism, Morse function, homotopy type.


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Article Information

TitleSmooth functions on 2-torus whose Kronrod-Reeb graph contains a cycle
SourceMethods Funct. Anal. Topology, Vol. 21 (2015), no. 1, 22-40
MathSciNet 3407918
zbMATH 06533465
MilestonesReceived 25/11/2014
CopyrightThe Author(s) 2015 (CC BY-SA)

Authors Information

S. Maksymenko
Topology Department, Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

B. Feshchenko
Topology Department, Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine


Citation Example

Sergiy Maksymenko and Bohdan Feshchenko, Smooth functions on 2-torus whose Kronrod-Reeb graph contains a cycle, Methods Funct. Anal. Topology 21 (2015), no. 1, 22-40.


BibTex

@article {MFAT764,
    AUTHOR = {Maksymenko, Sergiy and Feshchenko, Bohdan},
     TITLE = {Smooth functions on 2-torus whose Kronrod-Reeb graph contains a cycle},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {21},
      YEAR = {2015},
    NUMBER = {1},
     PAGES = {22-40},
      ISSN = {1029-3531},
  MRNUMBER = {3407918},
 ZBLNUMBER = {06533465},
       URL = {http://mfat.imath.kiev.ua/article/?id=764},
}


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