On topological classification of flows with heteroclinic curves on four-dimensional manifolds
https://doi.org/10.26907/2949-3919.2025.3.20-42
Abstract
We obtain a topological classification of smooth structurally stable flows on four-dimensional closed manifolds whose wandering set contains isolated trajectories connecting saddle equilibria (heteroclinic curves). For dimensional reasons, heteroclinic curves of such flows belong to the intersection of invariant manifolds of saddles of neighboring 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 dimension of whose unstable manifolds is equal to 1 and 2. We obtain necessary and sufficient conditions for the topological equivalence of such flows and present an algorithm for realizing a representative in each class of topological equivalence. In particular, we show that in the considered class of flows on the sphere S 4 there exists exactly one class of topological equivalence 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
Supporting agencies: Research is partially support by RSF (grant No 23-71-30008).
About the Author
E. Ya. Gurevich
HSE University
Russian Federation
Russian Federation
Elena Yakovlevna Gurevich
25/12 Bolshaya Pecherskaya str., Nizhny Novgorod 603155
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For citations:
Gurevich E.Ya. On topological classification of flows with heteroclinic curves on four-dimensional manifolds. Mathematics and Theoretical Computer Science. 2025;3(3):20-42. (In Russ.) https://doi.org/10.26907/2949-3919.2025.3.20-42
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