Tangential category of foliations

We present techniques to construct tangential homotopies of subsets of foliated manifolds and use these to obtain bounds and explicit computations for the tangential Lusternik-Schnirelmann category of foliations. For example, we show that this number is not greater than the dimension of the foliation, that it is an upper semi-continuous function on the space of p-dimensional foliations of a given manifold, and that it is equal to the dimension of the foliation for all codimension 1 foliations without holonomy on compact nilmanifolds.