Colman, Hellen; Macias-Virgós, Enrique Transverse Lusternik-Schnirelmann category of foliated manifolds. (English) Zbl 0972.55002 Topology 40, No. 2, 419-430 (2001). Let \((M,{\mathcal F}),(M',{\mathcal F}')\) be foliated manifolds. A \(C^\infty\)-homotopy \(H:M\times \mathbb{R}\to M'\) is said to be foliated if for all \(t\in \mathbb{R}\) the map \(H_t\) sends each leaf of \({\mathcal F}\) into another leaf \(L'\) of \({\mathcal F}'\). The transverse category of \((M,{\mathcal F})\) is the least number \(\text{cat}_\pitchfork (M,{\mathcal F})\) of transversely categorical open sets required to cover \(M\). The authors prove:(1) Transverse category is an invariant of foliated homotopy type and it is finite on compact manifolds;(2) \(\text{cat} M\leq\text{cat} L\cdot \text{cat}_\pitchfork (M,{\mathcal F})\), where \(L\) is a leaf of maximal category,(3) \(\text{cat}_\pitchfork (M,{\mathcal F})\geq \text{nil} K^*\cdot H^+_b(M)\), where the latter is the index of the image in \(H_{DR}(M)\) of the basic cohomology in positive degrees.Let \(f:M\to \mathbb{R}\) be a basic function. The authors also deduce (under certain conditions) that the transverse category gives a lower bound for the number of critical leaves of \(f\). Some examples are considered. Reviewer: Costache Apreutesei (Iaşi) Cited in 1 ReviewCited in 10 Documents MSC: 55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects) 57R30 Foliations in differential topology; geometric theory 53C12 Foliations (differential geometric aspects) 57R32 Classifying spaces for foliations; Gelfand-Fuks cohomology Keywords:foliation; critical points; Lyusternik-Shnirel’man category; basic cohomology PDFBibTeX XMLCite \textit{H. Colman} and \textit{E. Macias-Virgós}, Topology 40, No. 2, 419--430 (2001; Zbl 0972.55002) Full Text: DOI