Compact foliations with finite transverse LS category

TL;DR

This paper establishes that for compact foliations with all leaves compact, a finite transverse saturated category implies the leaf space is compact Hausdorff, providing a new characterization of such foliations.

Contribution

It proves a new criterion linking finite transverse saturated category to the leaf space being compact Hausdorff in compact foliations.

Findings

Finite transverse saturated category implies compact Hausdorff leaf space.

Provides a new characterization of compact Hausdorff foliations.

Introduces new geometric observations about compact foliations.

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. Colman proved in her 1998 doctoral thesis that 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.