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

TL;DR

This paper establishes a cohomological lower bound for the transverse LS category of a compact Hausdorff foliation on a compact manifold, linking it to the cup product length in a specific spectral sequence cohomology algebra.

Contribution

It introduces a new cohomological lower bound for the saturated transverse LS category based on the cup product length in the E2 spectral sequence cohomology.

Findings

Bound on transverse LS category via cup product length

Discussion of other cohomological bounds

Application to compact Hausdorff foliations

Abstract

Let $\mathcal{F}$ be a compact Hausdorff foliation on a compact manifold. Let ${E_2^{>0,\bullet}}=\oplus\{E_2^{p,q}\colon p>0,q\geq 0\}$ 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.