Foliations with few non-compact leaves

Let $(F)$ be a foliation of codimension 2 on a compact manifold with at least one non-compact leaf. We show that then $(F)$ must contain uncountably many non-compact leaves. We prove the same statement for oriented $p$ –dimensional foliations of arbitrary codimension if there exists a closed $p$ form which evaluates positively on every compact leaf. For foliations of codimension 1 on compact manifolds it is known that the union of all non-compact leaves is an open set [A Haefliger, Varietes feuilletes, Ann. Scuola Norm. Sup. Pisa 16 (1962) 367–397].

Keywords

non-compact leaves, Seifert fibration, Epstein hierarchy, foliation cycle, suspension foliation

Mathematical Subject Classification 2000

Primary: 57R30

References

Forward citations

Publication

Received: 23 July 2001

Revised: 3 April 2002

Accepted: 4 April 2002

Published: 16 April 2002

Authors

Elmar Vogt
2, Mathematisches Institut Freie Universität Berlin Arnimallee 3 14195 Berlin Germany

Volume 2, issue 1 (2002)

Elmar Vogt