Transverse Lusternik-Schnirelmann category of Riemannian foliations (Q1878520)

The transverse Lusternik-Schnirelmann category of a foliation is an invariant of foliated homotopy type that has been introduced by H. Colman and E. Macias. Let \((M,{\mathcal F})\) and \((M',{\mathcal F}')\) be foliated manifolds. A homotopy \(H : M\times M'\) is foliated if for each \(t\in I\), the map \(H_t\) sends each leaf of \(\mathcal {\mathcal F}\) into another leaf of \({\mathcal F}'\). An open subset \(U\subset M\) is called transversely categorical if there is a foliated homotopy \(H : U\times I\to M\) such that \(H_0: U \to M\) is the inclusion map and the image of \(H_1\) is contained in a single leaf of \(\mathcal F\). The transverse LS category is the least integer \(k\) such that \(M\) may be covered by \(k\) open saturated transversely categorical subsets in \(M\). In this paper, the author shows that the category is infinite if there exists a non-compact leaf that verifies a certain condition. Explicit examples are given.