Ganea and Whitehead definitions for the tangential Lusternik–Schnirelmann category of foliations

This work solves the problem of elaborating Ganea and Whitehead definitions for the tangential category of a foliated manifold. We develop these two notions in the category S - Top of stratified spaces, that are topological spaces X endowed with a partition F and compare them to a third invariant defined by using open sets. More precisely, these definitions apply to an element ( X , F ) of S - Top together with a class A of subsets of X; they are similar to invariants introduced by M. Clapp and D. Puppe. , F ) ∈ S - Top , we define a transverse subset as a subspace A of X such that the intersection S ∩ A is at most countable for any S ∈ F . Then we define the Whitehead and Ganea LS-categories of the stratified space by taking the infimum along the transverse subsets. When we have a closed manifold, endowed with a C 1 -foliation, the three previous definitions, with A the class of transverse subsets, coincide with the tangential category and are homotopical invariants.