Transverse Lusternik–Schnirelmann category of foliated manifolds

The purpose of this paper is to develop a transverse notion of Lusternik–Schnirelmann category in the field of foliations. Our transverse category, denoted by cat⋔(M, F), is an invariant of the foliated homotopy type which is finite on compact manifolds. It coincides with the classical notion when the foliation is by points. We prove that for any foliated manifold cat M⩽cat L cat⋔(M, F), where L is a leaf of maximal category, thus generalizing a result of Varadarajan for fibrations. Also we prove that cat⋔(M, F) is bounded below by the index of k∗Hb+(M), the latter being the image in HDR(M) of the algebra of basic cohomology in positive degrees. In the second part of the paper we prove that cat⋔(M, F) is a lower bound for the number of critical leaves of any basic function provided that F is a foliation satisfying certain conditions of Palais–Smale type. As a consequence, we prove that the result is true for compact Hausdorff foliations and for foliations of codimension one. This generalizes the classical result of Lusternik and Schnirelmann about the number of critical points of a smooth function.