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Functions on surfaces and incompressible subsurfaces

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Abstract

Let $M$ be a smooth connected compact surface, $P$ be either a real line $\mathbb R$ or a circle $S^1$. Then we have a natural right action of the group $D(M)$ of diffeomorphisms of $M$ on $C^\infty(M,P)$. For $f\in C^\infty(M,P)$ denote respectively by $S(f)$ and $O(f)$ its stabilizer and orbit with respect to this action. Recently, for a large class of smooth maps $f:M\to P$ the author calculated the homotopy types of the connected components of $S(f)$ and $O(f)$. It turned out that except for few cases the identity component of $S(f)$ is contractible, $\pi_i O(f)=\pi_i M$ for $i\geq3$, and $\pi_2 O(f)=0$, while $\pi_1 O(f)$ it only proved to be a finite extension of $\pi_1D_{Id}M\oplus\mathbb Z^{l}$ for some $l\geq0$. In this note it is shown that if $\chi(M)<0$, then $\pi_1O(f)=G_1\times\cdots\times G_n$, where each $G_i$ is a fundamental group of the restriction of $f$ to a subsurface $B_i\subset M$ being either a $2$-disk or a cylinder or a Mobius band. For the proof of main result incompressible subsurfaces and cellular automorphisms of surfaces are studied.


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Article Information

TitleFunctions on surfaces and incompressible subsurfaces
SourceMethods Funct. Anal. Topology, Vol. 16 (2010), no. 2, 167-182
MathSciNet MR2667811
CopyrightThe Author(s) 2010 (CC BY-SA)

Authors Information

Sergiy Maksymenko
Institute of Mathematics of NAS of Ukraine, Tereshchenkivska st. 3, Kyiv, 01601 Ukraine


Citation Example

Sergiy Maksymenko, Functions on surfaces and incompressible subsurfaces, Methods Funct. Anal. Topology 16 (2010), no. 2, 167-182.


BibTex

@article {MFAT553,
    AUTHOR = {Maksymenko, Sergiy},
     TITLE = {Functions on surfaces and incompressible subsurfaces},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {16},
      YEAR = {2010},
    NUMBER = {2},
     PAGES = {167-182},
      ISSN = {1029-3531},
       URL = {http://mfat.imath.kiev.ua/article/?id=553},
}


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