Compact foliations with finite transverse LS category
by Steven Hurder and Pawel Walczak

ABSTRACT:

We prove that if $\F$ is a foliation of a compact manifold with all leaves compact submanifolds, and the transverse category $\catt(M,\F)$ is finite, then the leaf space $M/\F$ is compact Hausdorff. The proof is surprisingly delicate, and is based on some new observations about the geometry of compact foliations. Colman proved in \cite{Colman1998,CM2000} that the transverse category of a compact Hausdorff foliation is always finite, so we obtain a new characterization of the compact Hausdorff foliations among the compact foliations as those with $\catt(M,\F)$ finite.