Compact leaves of foliations

Abstract

This paper is a survey of theorems and problems about the existence of compact leaves in foliations of a manifold M with a countable base. Question 1. For which manifolds M does every C codimension q foliation of M have a compact leaf? (Here q and r are given integers, 0 < q < m = dim M and 0 ^ r ^ oo.) As an example of a partial answer, Novikov's theorem states that every C codimension one foliation of S has a compact leaf which is a torus [N, Theorem 7.1]. On the other hand, the Seifert conjecture that every codimension two foliation of S has a compact leaf (i.e., a circle) has been shown to be false in the C case [Sl], but remains open for C foliations when r ^ 2. It is unknown whether Novikov's theorem can be extended to sufficiently smooth codimension one foliations of any higher dimensional manifolds, but it does not extend to C° foliations on any manifold of dimension five or more (see Theorem 6, below). We explore what is known about the above question for codimensions greater than one in §1, and for codimension one in §2,