Volume 2, issue 1 (2002)

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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 $\left(F\right)$ be a foliation of codimension 2 on a compact manifold with at least one non-compact leaf. We show that then $\left(F\right)$ 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
Primary: 57R30