The templates of non-singular Smale flows on three-manifolds
The paper analyzes non-singular Smale flows (structurally stable flows with \(1\)-dimensional invariant sets, without singularities) on 3-manifolds. Connections are established between two types of neighborhoods of hyperbolic \(1\)-dimensional basic sets which have proved useful in the study of Smale flows on \(3\)-manifolds: ``thickened templates'' and ``filtrating neighborhoods'' [\textit{F. Béguin} and \textit{C. Bonatti}, Topology 41, No. 1, 119--162 (2002; Zbl 1102.37307); \textit{J. Franks}, Topology 24, 265--282 (1985; Zbl 0609.58039)]. A relationship is established in terms of template moves, surgeries, and more general topological operations. This theory is then applied to the realization problem of non-singular Smale flows on \(3\)-manifolds and related questions, such as determining the topological types of filtrating neighborhoods modeled by a given template on a given manifold. In particular, the author proves that any template can model a basic set of a non-singular Smale flow on \(nS^1\times S^2\) for some positive integer \(n\).
scientific article; zbMATH DE number 6076569
scientific article