LS-category of compact Hausdorff foliations

Abstract

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, . . . , ∞}. 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 thatWe 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.

Transverse LS category for Riemannian foliations

“…Theorem 1.4 follows from Propositions 8.1 and 8.4. As a further consequence, we obtain estimates for cat ∩ | (M, F ) which extend those proven in [16] for compact Hausdorff foliations. PROPOSITION 8.1.…”

“…Also, the transverse LS category is an invariant of foliated homotopy. The transverse saturated category cat ∩ | (M, F ) has been further studied by various authors [13,14,16,36,38,43,65].…”

“…This paper can be viewed as a sequel to the work [16] by the first author with Hellen Colman. The results of this paper also extends the results of [14].…”

A cohomological lower bound for the transverse LS category of a foliated manifold

“…A foliation F on the compact manifold M is said to be compact Hausdorff if every leaf is compact and the space of leaves is Hausdorff [11]. Several interesting results and computations of the saturated transverse LS category cat s ∩ | (M, F ) in this setting have been obtained by H. Colman, S. Hurder and P. G. Walczak [6,15]. Recall that cat s ∩ | (M, F ) is defined [8] by considering transversely categorical open sets which are saturated (i.e.…”

“…The transverse Lusternik-Schnirelmann category cat ∩ | (M, F ) of a foliated manifold (M, F ) was introduced in H. Colman's thesis [4,8]. This notion (and the analogous one of saturated transverse category) has been studied by several authors in the last years [5,6,14,15,17,26].…”