Open Access

Actions of finite groups and smooth functions on surfaces


Let $f:M\to \mathbb{R}$ be a Morse function on a smooth closed surface, $V$ be a connected component of some critical level of $f$, and $\mathcal{E}_V$ be its atom. Let also $\mathcal{S}(f)$ be a stabilizer of the function $f$ under the right action of the group of diffeomorphisms $\mathrm{Diff}(M)$ on the space of smooth functions on $M,$ and $\mathcal{S}_V(f) = \{h\in\mathcal{S}(f)\,| h(V) = V\}.$ The group $\mathcal{S}_V(f)$ acts on the set $\pi_0\partial \mathcal{E}_V$ of connected components of the boundary of $\mathcal{E}_V.$ Therefore we have a homomorphism $\phi:\mathcal{S}(f)\to \mathrm{Aut}(\pi_0\partial \mathcal{E}_V)$. Let also $G = \phi(\mathcal{S}(f))$ be the image of $\mathcal{S}(f)$ in $\mathrm{Aut}(\pi_0\partial \mathcal{E}_V).$ Suppose that the inclusion $\partial \mathcal{E}_V\subset M\setminus V$ induces a bijection $\pi_0 \partial \mathcal{E}_V\to\pi_0(M\setminus V).$ Let $H$ be a subgroup of $G.$ We present a sufficient condition for existence of a section $s:H\to \mathcal{S}_V(f)$ of the homomorphism $\phi,$ so, the action of $H$ on $\partial \mathcal{E}_V$ lifts to the $H$-action on $M$ by $f$-preserving diffeomorphisms of $M$. This result holds for a larger class of smooth functions $f:M\to \mathbb{R}$ having the following property: for each 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 linear factors.

Key words: Diffeomorphism, Morse function.

Full Text

Article Information

TitleActions of finite groups and smooth functions on surfaces
SourceMethods Funct. Anal. Topology, Vol. 22 (2016), no. 3, 210-219
MathSciNet   MR3554649
zbMATH 06742107
Milestones  Received 20/05/2016
CopyrightThe Author(s) 2016 (CC BY-SA)

Authors Information

Bohdan Feshchenko
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka, Kyiv, 01601, Ukraine

Export article

Save to Mendeley

Citation Example

Bohdan Feshchenko, Actions of finite groups and smooth functions on surfaces, Methods Funct. Anal. Topology 22 (2016), no. 3, 210-219.


@article {MFAT883,
    AUTHOR = {Feshchenko, Bohdan},
     TITLE = {Actions of finite groups and smooth functions on surfaces},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {22},
      YEAR = {2016},
    NUMBER = {3},
     PAGES = {210-219},
      ISSN = {1029-3531},
  MRNUMBER = {MR3554649},
 ZBLNUMBER = {06742107},
       URL = {},


  1. A. V. Bolsinov and A. T. Fomenko, Some actual unsolved problems in topology of integrable Hamiltonian systems, Topological classification in theory of Hamiltonian systems, Factorial, 1999, pp. 5-23.
  2. Yu. Brailov, Algebraic properties of symmetries of atoms, Topological classification in theory of Hamiltonian systems, Factorial, 1999, pp. 24-40.
  3. Yu. A. Brailov and E. A. Kudryavtseva, Stable topological nonconjugacy of Hamiltonian systems on two-dimensional surfaces, Moscow Univ. Math. Bull. 54 (1999), no. 2, 20-27.  MathSciNet
  4. E. N. Dancer, Degenerate critical points, homotopy indices and Morse inequalities, J. Reine Angew. Math. 350 (1984), 1-22.  MathSciNet CrossRef
  5. E. A. Kudryavtseva, Realization of smooth functions on surfaces as height functions, Mat. Sb. 190 (1999), no. 3, 29-88.  MathSciNet CrossRef
  6. E. A. Kudryavtseva and A. T. Fomenko, Symmetry groups of nice Morse functions on surfaces, Dokl. Akad. Nauk 446 (2012), no. 6, 615-617.  MathSciNet CrossRef
  7. E. A. Kudryavtseva and A. T. Fomenko, Each finite group is a symmetry group of some map (an “Atom”-bifurcation), Moscow Univ. Math. Bull. 68 (2013), no. 3, 148-155.  MathSciNet CrossRef
  8. Sergey Maksymenko, Smooth shifts along trajectories of flows, Topology Appl. 130 (2003), no. 2, 183-204.  MathSciNet CrossRef
  9. Sergiy Maksymenko, Homotopy types of stabilizers and orbits of Morse functions on surfaces, Ann. Global Anal. Geom. 29 (2006), no. 3, 241-285.  MathSciNet CrossRef
  10. Sergiy Maksymenko, Connected components of partition preserving diffeomorphisms, Methods Funct. Anal. Topology 15 (2009), no. 3, 264-279.  MathSciNet
  11. Sergiy Maksymenko, Deformations of functions on surfaces by isotopic to the identity diffeomorphisms, 2013  arXiv:1311.3347
  12. A. A. Oshemkov, Morse functions on two-dimensional surfaces. Coding of singularities, Trudy Mat. Inst. Steklov. 205 (1994), no. Novye Rezult. v Teor. Topol. Klassif. Integr. Sistem, 131-140.  MathSciNet
  13. A. O. Prishlyak, Topological equivalence of smooth functions with isolated critical points on a closed surface, Topology Appl. 119 (2002), no. 3, 257-267.  MathSciNet CrossRef
  14. R. T. Seeley, Extension of $C^\infty $ functions defined in a half space, Proc. Amer. Math. Soc. 15 (1964), 625-626.  MathSciNet

All Issues