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Foliations with few non-compact leaves

Elmar Vogt

Algebraic & Geometric Topology 2 (2002) 257–284

arXiv: math.GT/0205036

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
Mathematical Subject Classification 2000
Primary: 57R30
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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