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Functions on surfaces and incompressible subsurfaces


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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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

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


@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},
  MRNUMBER = {MR2667811},
       URL = {},

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