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.