Algebraic and Geometric Topology 2 (2002), paper no. 12, pages 257-284.

Foliations with few non-compact leaves

Elmar Vogt

Abstract. 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

AMS subject classification. Primary: 57R30.

DOI: 10.2140/agt.2002.2.257

E-print: arXiv:math.GT/0205036

Submitted: 23 July 2001. (Revised: 3 April 2002.) Accepted: 4 April 2002. Published: 16 April 2002.

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Elmar Vogt
2, Mathematisches Institut, Freie Universitaet Berlin
Arnimallee 3, 14195 Berlin, Germany
Email: vogt@math.fu-berlin.de

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