Compact foliations with finite transverse LS category

Abstract

We prove that if 𝐹 is a foliation of a compact manifold 𝑀 with all leaves compact submanifolds, and the transverse saturated category of 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. The transverse saturated 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 finite transverse saturated category.