LS-category of compact Hausdorff foliations
by Hellen Colman and Steven Hurder

ABSTRACT:

The transverse Lusternik-Schnirelmann category of foliations was introduced by Colman in her thesis. This reduces to the classical concept when the foliation is by points. Transverse category is an invariant of the foliated homotopy type.

We study this new invariant for a compact manifold M endowed with a compact-Hausdorff foliation F. In this context our category is finite and is a lower bound for the number of critical leaves of any basic function.

If the foliation is a fibration, the transverse category coincides with the category of the leaf space. In general, it is just a lower bound. We construct a number of examples for which the transverse category for a compact-Hausdorff foliation can be arbitrarily large, though the category of the leaf spaces is constant.

Consider the set E of exceptional leaves (leaves with non-trivial holonomy) with the foliation induced by F. We prove that

Examples show that both the lower and upper bounds are realized.