Opening Closed Leaves of Foliations

Abstract

Novikov proved that every C codimension one foliation of S has a closed leaf ([5, Theorems 6.1 and 7.1]). In higher codimension, the situation is quite different. According to Schweitzer, if a manifold M has a C foliation of codimension q ^ 3 with 0 ^ r ^ oo, then it possesses a foliation with no closed leaves ([6, Theorem D]). To get the same result in codimension q — 2, Schweitzer uses the celebrated Denjoy CMoroidal containing a proper minimal set with no closed trajectories (see [1]). Since that phenomenon cannot occur in a C on a surface (see [1]) his methods give only C results when q = 2. In [3] the author topologically embeds the Denjoy C vector field in a C vector field defined on a punctured, thickened torus, N = (T\D) x / to obtain a C flow plug. Much as in Schweitzer, but with an alternate exposition, this plug can be modified and used to open closed leaves of C foliations. Let M be a C smooth, paracompact manifold without boundary of dimension k ^ 3, and ^ a C r foliation of M. A leaf of $F is closed, if it is closed as a subset of M.