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

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.

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Correspondence to V. Z. Grines, E. Ya. Gurevich, V. S. Medvedev or E. V. Zhuzhoma.

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Grines, V.Z., Gurevich, E.Y., Medvedev, V.S. et al. On Topology of Manifolds Admitting a Gradient-Like Flow with a Prescribed Non-Wandering Set. Sib. Adv. Math. 29, 116–127 (2019). https://doi.org/10.3103/S1055134419020020

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Keywords

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