О топологической классификации потоков с гетероклиническими кривыми на четырехмерных многоообразиях

Математика и теоретические компьютерные науки. 2025. Т. 3. № 3. С. 20–42.

Gurevich E.

A topological classification of smooth, structurally stable flows on four-dimensional closed manifolds is obtained, the wandering set of which contains isolated trajectories connecting saddle equilibria (heteroclinic curves). For dimensional reasons, the heteroclinic curves of such flows belong to the intersection of invariant manifolds of saddles of adjacent Morse indices. We assume that the non-wandering set of the flows under consideration consists of exactly one source, one sink, and an arbitrary number of saddles, the unstable manifolds of which have dimensions equal to 1 and 2. In this paper, we obtain necessary and sufficient conditions for the topological equivalence of such flows and present a realization algorithm for a representative of each topological equivalence class. In particular, it is shown that in the class of flows under consideration on the sphere S4, there exists exactly one topological equivalence class of flows with a single heteroclinic curve and a countable set of topologically nonequivalent flows with three heteroclinic curves. The latter result contrasts with the three-dimensional situation, where for a similar class of flows there are only finitely many equivalence classes for each number of heteroclinic curves.

Keywords: топологическая классификация, topological classification, gradient-like flow, гетероклиническая кривая, heteroclinic curve, функция Морса, градиентно-подобный поток, Morse function