Generalized counterexamples to the Seifert conjecture. (English) Zbl 0856.57026

Ann. Math. (2) 143, No. 3, 547-576 (1996); correction ibid. 144, No. 2, 239-268 (1996).
The famous Seifert problem asserts that every dynamical system on the 3-sphere \(S^3\) with no singular points has a periodic trajectory. The main purpose of this note is to establish that a foliation of any codimension of any manifold can be modified in a real analytic or piecewise-linear fashion so that all minimal sets have codimension 1. The dynamical systems on \(S^3\) are investigated.
Theorem: The sphere \(S^3\) has an analytic dynamical system such that all limit sets are 2-dimensional. In particular, it has no circular trajectories.
In PL category, the authors prove the following theorem:
A 1-foliation of a manifold of dimension at least 3 can be modified in a PL fashion so there are no closed leaves but all minimal sets are 1-dimensional. Moreover, if the manifold is closed, then there is an aperiodic PL modification with only one minimal set, and the minimal set is 1-dimensional.
The method: all counterexamples to the Seifert conjecture described here are based on constructions of aperiodic plugs. The exposition is clear and all the fundamental concepts are defined.


57R30 Foliations in differential topology; geometric theory
57N12 Topology of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
53C12 Foliations (differential geometric aspects)
Full Text: DOI arXiv