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Automorphisms of Kronrod–Reeb graphs of Morse functions on compact surfaces

  • Anna Kravchenko
  • Sergiy MaksymenkoEmail author
Research Article

Abstract

Let M be a connected orientable compact surface, \(f:M\rightarrow {\mathbb {R}}\) be a Morse function, and Open image in new window be the group of diffeomorphisms of M isotopic to the identity. Denote by Open image in new window the subgroup of Open image in new window consisting of diffeomorphisms “preserving” f, i.e., the stabilizer of f with respect to the right action of Open image in new window on the space Open image in new window of smooth functions on M. Let also \({\mathbf {G}}(f)\) be the group of automorphisms of the Kronrod–Reeb graph of f induced by diffeomorphisms belonging to Open image in new window . This group is an important ingredient in determining the homotopy type of the orbit of f with respect to the above action of Open image in new window and it is trivial if f is “generic”, i.e., has at most one critical point at each level set \(f^{-1}(c)\), \(c\in {\mathbb {R}}\). For the case when \(M\) is distinct from 2-sphere and 2-torus we present a precise description of the family \({\mathbf {G}}(M,{\mathbb {R}})\) of isomorphism classes of groups \({\mathbf {G}}(f)\), where f runs over all Morse functions on M, and of its subfamily \({\mathbf {G}}^\mathrm{smp}(M,{\mathbb {R}}) \subset {\mathbf {G}}(M,{\mathbb {R}})\) consisting of groups corresponding to simple Morse functions, i.e., functions having at most one critical point at each connected component of each level set. In fact, \({\mathbf {G}}(M,{\mathbb {R}})\) (respectively, \({\mathbf {G}}^\mathrm{smp}(M,{\mathbb {R}})\)) coincides with the minimal family of isomorphism classes of groups containing the trivial group and closed with respect to direct products and also with respect to wreath products “from the top” with arbitrary finite cyclic groups (respectively, with the group \({\mathbb {Z}}_2\) only).

Keywords

Morse function Kronrod–Reeb graph Wreath product 

Mathematics Subject Classification

20E22 57M60 22F50 

Notes

Acknowledgements

The authors sincerely thank the anonymous referee for careful reading of the manuscript and constructive remarks.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Mechanics and MathematicsTaras Shevchenko National University of KyivKyivUkraine
  2. 2.Topology Laboratory of Algebra and Topology DepartmentInstitute of Mathematics of National Academy of Sciences of UkraineKyivUkraine

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