\(C^ 2\) counterexamples to the Seifert conjecture. (English) Zbl 0669.57011

Harrison, J.

Topology 27, No. 3, 249-278 (1988).

By P. Schweitzer’s counterexample to the famous Seifert conjecture, there exists a non zero \(C^ 1\) vector field on the 3-sphere without closed integral curve. Starting with Schweitzer’s method and investing her own new ideas the author is able to present us a counterexample that is of class \(C^ 2\). Schweitzer uses a \(C^ 1\) Denjoy vector field in a thickened punctured torus as a device to reomove existing closed integral curves. On the torus, Denjoy’s vector field cannot be made \(C^ 2\). But it can be smoothed - as the author shows - in three dimensions, thereby decreasing the smoothness of the torus! The main new arguments concern dimension 2 and are technically intricate and involved. The bulk of the paper deals with the construction of a \(C^{3-\epsilon}\) self- diffeomorphism of the closed annulus, for 1-\(\epsilon\) sufficiently small, with certain properties. One of the ingredients is a result of the author on Denjoy fractals (to appear in Topology).

Reviewer: D.Erle

MSC:

57R30	Foliations in differential topology; geometric theory
37C10	Dynamics induced by flows and semiflows
57R50	Differential topological aspects of diffeomorphisms

Keywords:

flow plug; clerical collar; quasi-circle; quasi-structure; diamond circle; periodic orbit; Hausdorff dimension; Whitney extension; Seifert conjecture; vector field on the 3-sphere without closed integral curve; Denjoy vector field