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Generic flows on 3-manifolds. (English) Zbl 1323.57015

The flows mentioned in the title are smooth, nowhere vanishing flows on closed oriented \(3\)-manifolds with non-empty boundary, restricted by a genericity condition on the boundary that goes back to B. Morin. These flows are completely described by special flows called streams, as well as classified up to topological flow equivalence and vector field homotopy with fixed configuration on the boundary. The classification is achieved by means of certain \(2\)-dimensional polyhedra called stream spines locally modeled on unions of quadrants in the coordinate planes in euclidean \(3\)-space, together with a technical local compatibility condition. Due to the type of genericity, a stream spine gives rise to a unique \(3\)-manifold flow with a stream on it and, given a stream, one finds two canonical but isomorphic stream spines. An important step in the classification is that if two equivalent flows are each induced by a stream spine then the latter are related by so-called sliding moves.– Theorem 1.4 is misstated, but line 9 from below on p.153 contains its essence.

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
37C10 Dynamics induced by flows and semiflows
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
57M20 Two-dimensional complexes (manifolds) (MSC2010)
57R25 Vector fields, frame fields in differential topology
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
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References:

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