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Transverse Lusternik-Schnirelmann category of foliated manifolds. (English) Zbl 0972.55002

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.

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
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