LS-category of compact-Hausdorff foliations

The transverse (saturated) Lusternik-Schnirelmann category of foliations, introduced by the first author in [5, 9], is an invariant of foliated homotopy type with values in {1, 2,...,∞}. A foliation with all leaves compact and Hausdorff leaf space M/F is called compact Hausdorff. The transverse saturated category cat∩ | 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 cat∩ | (M) in terms of the geometry of F and the Epstein filtration of the exceptional set E. The exceptional set is the closed saturated foliated space which is the union of the leaves with non-trivial holonomy. We prove that max{cat(M/F), cat∩ | (E)} ≤ cat∩ | (M) ≤ cat∩ | (E) + q 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 compac