Counterexamples to the Seifert conjecture. (English) Zbl 0924.58086

Kuperberg, Krystyna

Doc. Math., Extra Vol. ICM Berlin 1998, vol. II, 831-840 (1998).

Summary: Since H. Seifert proved in 1950 the existence of a periodic orbit for a vector field on the 3-dimensional sphere \(S^3\) which forms small angles with the fibers of the Hopf fibration, several examples of aperiodic vector fields on \(S^3\) have been produced as well as results showing that in some situations a compact orbit must exist. This paper surveys presently known types of vector fields without periodic orbits on \(S^3\) and on other manifolds.

MSC:

37C10	Dynamics induced by flows and semiflows
37C85	Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)

Keywords:

3-dimensional sphere \(S^3\); Hopf fibration; aperiodic vector fields on \(S^3\); periodic orbits; dynamical system; plug; minimal set; PL foliation