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 endowed with a partition 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 in the case of topological spaces. If (X,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 . When we have a closed manifold, endowed with a C-foliation, the three previous definitions, with A the class of transverse subsets, coincide with the tangential category and are homotopical invariants.