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

Kuperberg, Greg; Kuperberg, Krystyna

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.

Reviewer: C.Apreutesei (Iaşi)

MSC:

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

Keywords:

minimal sets; symbolic dynamics; leaves; dynamical system; 3-sphere; periodic trajectory; limit sets; plugs