Transverse Lusternik{Schnirelmann category of foliated manifolds

The purpose of this paper is to develop a transverse notion of Lusternik{Schnirelmann category in the eld of foliations. Our transverse category, denoted cat\j (M;F), is an invariant of the foliated homotopy type which is nite on compact manifolds. It coincides with the classical notion when the foliation is by points. We prove that for any foliated manifold catM catL cat\j (M;F), where L is a leaf of maximal category, thus generalizing a result of Varadarajan for brations. Also we prove that cat\j (M;F) is bounded below by the index of k H b (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\j (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 Hausdor foliations and for codimension one foliations. This generalizes the classical result of Lusternik and Schnirelmann about the number of critical points of a smooth function.