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.