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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
Article
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Abstract

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.

Keywords

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

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Copyright information

© Allerton Press, Inc. 2019

Authors and Affiliations

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

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