Siberian Advances in Mathematics

, Volume 29, Issue 2, pp 116–127 | Cite as

On Topology of Manifolds Admitting a Gradient-Like Flow with a Prescribed Non-Wandering Set

  • V. Z. GrinesEmail author
  • E. Ya. GurevichEmail author
  • V. S. MedvedevEmail author
  • E. V. ZhuzhomaEmail author


We study relations between the structure of the set of equilibrium points of a gradient-like flow and the topology of the support manifold of dimension 4 and higher. We introduce a class of manifolds that admit a generalized Heegaard splitting. We consider gradient-like flows such that the non-wandering set consists of exactly μ node and ν saddle equilibrium points of indices equal to either 1 or n — 1. We show that, for such a flow, there exists a generalized Heegaard splitting of the support manifold of genius \(g=\frac{\nu-\mu+2}{2}\). We also suggest an algorithm for constructing gradientlike flows on closed manifolds of dimension 3 and higher with prescribed numbers of node and saddle equilibrium points of prescribed indices.


gradient-like flows on manifolds Heegaard splitting relations between dynamics and topology 


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  1. 1.
    A. A. Andronov, and L. S. Pontryagin, “Syste'mes grossiers,” Dokl. Akad. Nauk SSSR 14, 247 (1937) [in Russian].zbMATHGoogle Scholar
  2. 2.
    C. Bonatti, V. Grines, V. Medvedev, and E. Pecou, “Three-manifolds admitting Morse-Smale diffeomor phisms without heteroclinic curves,” Topology Appl. 117, 335 (2002).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    R. J. Daverman, and G. A. Venema, Embeddings in Manifolds (Amer. Math. Soc., Providence, RI, 2009).Google Scholar
  4. 4.
    V. Z. Grines, E. A. Gurevich, and O. V. Pochinka, “Topological classification of Morse-Smale diffeomorphisms without heteroclinic intersections,” Probl. Mat. Anal. 79, 73 (2015) [J. Math. Sci. 208, 81 (2015)].zbMATHGoogle Scholar
  5. 5.
    V. Z. Grines, E. V. Zhuzhoma, and V. S. Medvedev, “New relations for Morse-Smale systems with trivially embedded one-dimensional separatrices,” Mat. Sb. 194, no. 7, 25 (2003) [Sb. Math. 194, 979 (2003)].MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    V. Z. Grines, E. V. Zhuzhoma, and V. S. Medvedev, “On the structure of the ambient manifold for Morse-Smale systems without heteroclinic intersections,” Trudy Mat. Inst. Steklov 297, 201 (2017) [Proc. Steklov Inst. Math. 297, 179 (2017)].MathSciNetzbMATHGoogle Scholar
  7. 7.
    M. V. Hirsch, Differential Topology (Springer-verlag, New York-Heidelberg-Berlin, 1976).CrossRefzbMATHGoogle Scholar
  8. 8.
    L. V. Keldysh, “Topological imbeddings in Euclidean space,” Trudy Mat. Inst. Steklov 81, 3 (1966) [Proc. Steklov Inst. Math. 81, 1 (1966)].zbMATHGoogle Scholar
  9. 9.
    M. A. Kervaire, and J. W. Milnor, “Groups of homotopy spheres. I,” Ann. Math. (2) 77, 504 (1963).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    J. M. Lee, Introduction to Smooth Manifolds (Springer-Verlag, New York, 2012).CrossRefGoogle Scholar
  11. 11.
    Y. Matsumoto, An Introduction to Morse Theory (Amer. Math. Soc., Providence, RI, 2002).Google Scholar
  12. 12.
    J. W. Milnor, Morse Theory. Based on Lecture Notes by M. Spivak and R. Wells (Princeton Univ. Press, Princeton, NJ, 1963).zbMATHGoogle Scholar
  13. 13.
    J. W. Milnor, Topology from the Differentiable Viewpoint (The Univ. Press Virginia, Charlottesville, 1965).zbMATHGoogle Scholar
  14. 14.
    S. Smale, “Morse inequalities for a dynamical system,” Bull. Amer. Math. Soc. 66, 43 (1960).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    S. Smale, “On gradient dynamical systems,” Ann. Math. (2) 74, 199 (1961).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    S. Smale, “Differentiable dynamical systems,” Bull. Amer. Math. Soc. 73, 747 (1967).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev, and L. O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics. Part I (World Scientific, Singapore, 1998).CrossRefzbMATHGoogle Scholar
  18. 18.
    V. S. Medvedev, and E. V. Zhuzhoma, “Morse-Smale systems with few non-wandering points,” Topology Appl. 160, 498 (2013).MathSciNetCrossRefzbMATHGoogle Scholar

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© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsNizhniĭ NovgorodRussia

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