×

A volume-preserving counterexample to the Seifert conjecture. (English) Zbl 0859.57017

Summary: We prove that every 3-manifold possesses a \(C^1\), volume-preserving flow with no fixed points and no closed trajectories. The main construction is a volume-preserving version of the Schweitzer plug. We also prove that every 3-manifold possesses a volume-preserving, \(C^\infty\) flow with discrete closed trajectories and no fixed points (as well as a PL flow with the same geometry), which is needed for the first result. The proof uses a Dehn-twisted Wilson-type plug which also preserves volume.

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57R30 Foliations in differential topology; geometric theory
PDFBibTeX XMLCite
Full Text: DOI arXiv EuDML