Tangential category of foliations (Q1811543)

The notion of tangential Ljusternik-Schnirelman category for foliations was introduced by Helen Colman in her thesis. The authors obtain some estimates for tangential category and prove that the tangential category is upper semicontinuous with respect to foliations. As an application, we mention here the following result (Theorem 7.2): Let \(\mathcal F\) be a \(C^2\)-foliation of codimension 1 on a closed manifold \(M\) of dimension \(n\geq 3\). Then the tangential category of \(\mathcal F\) is at most 1 if and only if the leaves of \(F\) are the fibers of a homotopy sphere bundle over \(S^1\).