Ganea and Whitehead definitions for the tangential Lusternik-Schnirelmann category of foliations

TL;DR

This paper extends Ganea and Whitehead definitions to the tangential Lusternik-Schnirelmann category of foliated manifolds within stratified spaces, establishing their equivalence with classical invariants for closed manifolds with $C^1$-foliations.

Contribution

It develops Ganea and Whitehead definitions for the tangential category of foliated manifolds in stratified spaces and compares them to an open set-based invariant.

Findings

The definitions apply to stratified spaces with a class of subsets.

The three invariants coincide for closed manifolds with $C^1$-foliations.

The invariants are homotopical in this setting.

Abstract

This work solves the problem of elaborating Ganea and Whitehead definitions for the tangential category of a foliated manifold. We develop these two notions in the category $\Tops$ of stratified spaces, that are topological spaces $X$ endowed with a partition $\cF$ and compare them to a third invariant defined by using open sets. More precisely, these definitions apply to an element $(X,\cF)$ of $\Tops$ together with a class $\cA$ of subsets of $X$; they are similar to invariants introduced by M. Clapp and D. Puppe. If $(X,\cF)\in\Tops$, we define a transverse subset as a subspace $A$ of $X$ such that the intersection $S\cap A$ is at most countable for any $S\in \cF$. Then we define the Whitehead and Ganea LS-categories of the stratified space by taking the infimum along the transverse subsets. When we have a closed manifold, endowed with a $C^1$-foliation, the three previous definitions, with $\cA$ the class of transverse subsets, coincide with the tangential category and are homotopical invariants.