Foliations with all non-closed leaves on non-compact surfaces

Abstract

Let $X$ be a connected non-compact $2$-dimensional manifold possibly with boundary and $\Delta$ be a foliation on $X$ such that each leaf $\omega\in\Delta$ is homeomorphic to $\mathbb{R}$ and has a trivially foliated neighborhood. Such foliations on the plane were studied by W. Kaplan who also gave their topological classification. He proved that the plane splits into a family of open strips foliated by parallel lines and glued along some boundary intervals. However W. Kaplan's construction depends on a choice of those intervals, and a foliation is described in a non-unique way. We propose a canonical cutting by open strips which gives a uniqueness of classifying invariant. We also describe topological types of closures of those strips under additional assumptions on $\Delta$.

Key words: Foliation, non-compact surface, fiber bundles.

Full Text

Article Information

Title	Foliations with all non-closed leaves on non-compact surfaces
Source	Methods Funct. Anal. Topology, Vol. 22 (2016), no. 3, 266-282
MathSciNet	MR3554653
zbMATH	06742111
Milestones	Received 30/05/2016
Copyright	The Author(s) 2016 (CC BY-SA)

Authors Information

Sergiy Maksymenko
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Teresh\-chenkivs'ka, Kyiv, 01601, Ukraine

Eugene Polulyakh
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Teresh\-chenkivs'ka, Kyiv, 01601, Ukraine

Citation Example

Sergiy Maksymenko and Eugene Polulyakh, Foliations with all non-closed leaves on non-compact surfaces, Methods Funct. Anal. Topology 22 (2016), no. 3, 266-282.

BibTex

@article {MFAT884,
    AUTHOR = {Maksymenko, Sergiy and Polulyakh, Eugene},
     TITLE = {Foliations with all non-closed leaves on non-compact surfaces},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {22},
      YEAR = {2016},
    NUMBER = {3},
     PAGES = {266-282},
      ISSN = {1029-3531},
  MRNUMBER = {MR3554653},
 ZBLNUMBER = {06742111},
       URL = {http://mfat.imath.kiev.ua/article/?id=884},
}

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