A cohomological lower bound for the transverse LS category of a foliated manifold

Abstract

Let $\mathcal{F}$ be a compact Hausdorff foliation on a compact manifold. Let ${E_2^{>0,\bullet}}=\bigoplus\{E_2^{p,q} : p>0,q\geq0\}$ be the subalgebra of cohomology classes with positive transverse degree in the $E_2$ term of the spectral sequence of the foliation. We prove that the saturated transverse Lusternik-Schnirelmann category of $\mathcal {F}$ is bounded below by the length of the cup product in ${E_2^{>0,\bullet}}$. Other cohomological bounds are discussed.