Topological Classification of Polar Flows on Four-dimensional Manifolds

Regular and Chaotic Dynamics. 2025. Vol. 30. No. 2. P. 254–278.

Elena Ya. Gurevich, Ilya A. Saraev

S.~Smale has shown that any closed  smooth manifold admits a gradient-like flow, which is a structurally stable flow with a finite non-wandering set. Polar flows are a specific type of gradient-like flows characterized by the simplest non-wandering set for the given manifold, consisting of exactly one source, one sink, and a finite number of saddle equilibria. We describe the topology of  four-dimensional closed manifolds that admit polar flows without heteroclinic intersections, as well as all classes of topological equivalence of polar flows on each manifold. In particular, we demonstrate that there exists a countable set of non-equivalent flows with a given number $k \geq 2$ of saddle equilibria on each manifold, that  contrasts with the situation in lower-dimensional analogues.

Ключевые слова: топологическая классификация, topological classification, gradient-like flows, градиентно-подобные потоки, структурно устойчивые потоки, structurally stable flows

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