Category and compact leaves
by Steven Hurder

ABSTRACT:

The transverse saturated Lusternik-Schnirelmann category of foliations was introduced by H. Colman, and is an invariant of the foliated homotopy type. The transverse saturated category provides a lower bound for the number of critical leaves of any basic function. If the foliation is a fibration, the transverse saturated category coincides with the category of the leaf space. If F is a compact Hausdorff foliation of a compact manifold M, then the transverse saturated category is always finite. One of the open problems is to understand for what other classes of foliations must be finite. In this paper we prove that if M is a compact foliated manifold, and F is a foliation of M with finite transverse category then F has a compact leaf. In contrast, we show that if F is expansive on some non-trivial minimal set, then the transverse category is infinite. We also provide a variety of examples of foliations with non-compact leaves having finite transverse category.