Ukrainian Mathematical Journal

, Volume 64, Issue 9, pp 1350–1369 | Cite as

Homotopic types of right stabilizers and orbits of smooth functions on surfaces

  • S. I. Maksimenko

Let M be a connected smooth compact surface and let P be either the number line \( \mathbb{R} \) or a circle S 1. For a subset XM, by \( \mathcal{D} \)(M, X) we denote a group of diffeomorphisms of M fixed on X. We consider a special class \( \mathcal{F} \) of smooth mappings f:MP with isolated singularities containing all Morse mappings. For each mapping f\( \mathcal{F} \), we consider certain submanifolds XM “adapted” to f in a natural way and study the right action of the group \( \mathcal{D} \)(M, X) on C ∞( M, P). The main results of the paper describe the homotopic types of the connected components of stabilizers \( \mathcal{S} \)(f) and the orbits \( \mathcal{O} \)(f) of all mappings f\( \mathcal{F} \) and generalize the results of the author in this field obtained earlier.


Smooth Function Exact Sequence Mapping Class Group Morse Function Regular Component 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    S. Maksymenko, “Homotopy types of stabilizers and orbits of Morse functions on surfaces,” Ann. Global Anal. Geom., 29, No. 3, 241–285 (2006).MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    S. Maksymenko, “Homotopy dimension of orbits of Morse functions on surfaces,” Trav. Math., 18, 39–44 (2008).MathSciNetGoogle Scholar
  3. 3.
    S. Maksymenko, “Functions with isolated singularities on surfaces,” in: Geometry and Topology of Functions on Manifolds, Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, 7, No. 4 (2010), pp. 7–66.Google Scholar
  4. 4.
    S. Maksymenko, “Functions on surfaces and incompressible subsurfaces,” Meth. Funct. Anal. Topol., 16, No. 2, 167–182 (2010).MathSciNetzbMATHGoogle Scholar
  5. 5.
    F. Sergeraert, “Un théorème de fonctions implicites sur certains espaces de Fréchet et quelques applications,” Ann. Sci. École Norm. Super., 5, No. 4, 599–660 (1972).MathSciNetzbMATHGoogle Scholar
  6. 6.
    S. Smale, “Diffeomorphisms of the 2-sphere,” Proc. Amer. Math. Soc., 10, 621–626 (1969).MathSciNetGoogle Scholar
  7. 7.
    J. S. Birman, “Mapping class groups and their relationship to braid groups,” Commun. Pure Appl. Math., 22, 213–238 (1969).MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    C. J. Earle and J. Eells, “A fibre bundle description of Teichmüller theory,” J. Different. Geom., 3, 19–43 (1969).MathSciNetzbMATHGoogle Scholar
  9. 9.
    C. J. Earle and A. Schatz, “Teichmüller theory for surfaces with boundary,” J. Different. Geom., 4, 169–185 (1970).MathSciNetzbMATHGoogle Scholar
  10. 10.
    A. Gramain, “Le type d’homotopie du groupe des difféomorphismes d’une surface compacte,” Ann. Sci. École Norm. Super., 6, No. 4, 53–66 (1973).MathSciNetzbMATHGoogle Scholar
  11. 11.
    R. T. Seeley, “Extension of C functions defined in a half space,” Proc. Amer. Math. Soc., 15, 625–626 (1964).MathSciNetzbMATHGoogle Scholar
  12. 12.
    S. Maksymenko, “Smooth shifts along trajectories of flows,” Topol. Appl., 130, No. 2, 183–204 (2003).MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • S. I. Maksimenko
    • 1
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKievUkraine

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