LS-category of compact Hausdorff foliations

The transverse (saturated) Lusternik-Schnirelmann category of foliations, introduced by the first author, is an invariant of foliated homotopy type with values in $\{1,2, \ldots, \infty\}$. A foliation with all leaves compact and Hausdorff leaf space $M/\mathcal{F}$ is called compact Hausdorff. The transverse saturated category $\operatorname{cat}_{\mathbin{\cap{\mkern-9mu}\mid}\,\,}M$ of a compact Hausdorff foliation is always finite.

In this paper we study the transverse category of compact Hausdorff foliations. Our main result provides upper and lower bounds on the transverse category $\operatorname{cat}_{\mathbin{\cap{\mkern-9mu}\mid}\,\,}(M)$ in terms of the geometry of $\mathcal{F}$ and the Epstein filtration of the exceptional set $\mathcal{E}$. The exceptional set is the closed saturated foliated space which is the union of the leaves with non-trivial holonomy. We prove that

We give examples to show that both the upper and lower bounds are realized, so the estimate is sharp. We also construct a family 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.