LS-category of compact Hausdorff foliations

Abstract:

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 $\operatorname {cat}_{\cap {\mkern -9mu}\mid }M$ 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 $\operatorname {cat}_{\cap {\mkern -9mu}\mid }(M)$ 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 \[ \max \{\operatorname {cat}(M/{\mathcal {F}}), \operatorname {cat}_{\cap {\mkern -9mu}\mid }(\mathcal {E})\} \leq \operatorname {cat}_{\cap {\mkern -9mu}\mid }(M) \leq \operatorname {cat}_{\cap {\mkern -9mu}\mid }(\mathcal {E}) + q.\] 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 compact Hausdorff foliation can be arbitrarily large, though the category of the leaf spaces is constant.