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

Jean-Paul Doeraene, Enrique Macias-Virgós, Daniel Tanré

TL;DR

This work formalizes the tangential Lusternik-Schnirelmann category of foliations within the category of stratified spaces by developing Ganea and Whitehead definitions in ${\mathcal S}$-$\mathbf Top$, and introducing an open-set invariant ${\rm Ocat}$ based on transverse subsets. It proves that the Whitehead and Ganea constructions coincide in the stratified setting and that, under suitable hypotheses, ${\rm Ocat}$ bounds and interrelates with the other invariants, yielding a robust, homotopy-invariant framework. For foliations on closed $C^1$ manifolds, the paper establishes the key result that the tangential LS-category ${\rm cat}_{\mathcal F}(M)$ coincides with ${\rm Ocat}(M,\mathcal F_M)$, ${\rm Gcat}(M,\mathcal F_M)$, and ${\rm Whcat}(M,\mathcal F_M)$, thereby unifying multiple perspectives on tangential complexity. The methodology hinges on translating foliation transverse data into stratified categorical notions, leveraging a stratified fibration-category structure, and employing transverse subsets to align tangential and stratified invariants, with concrete illustrations such as the Reeb foliation.

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.

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

TL;DR

This work formalizes the tangential Lusternik-Schnirelmann category of foliations within the category of stratified spaces by developing Ganea and Whitehead definitions in -, and introducing an open-set invariant based on transverse subsets. It proves that the Whitehead and Ganea constructions coincide in the stratified setting and that, under suitable hypotheses, bounds and interrelates with the other invariants, yielding a robust, homotopy-invariant framework. For foliations on closed manifolds, the paper establishes the key result that the tangential LS-category coincides with , , and , thereby unifying multiple perspectives on tangential complexity. The methodology hinges on translating foliation transverse data into stratified categorical notions, leveraging a stratified fibration-category structure, and employing transverse subsets to align tangential and stratified invariants, with concrete illustrations such as the Reeb foliation.

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 of stratified spaces, that are topological spaces endowed with a partition and compare them to a third invariant defined by using open sets. More precisely, these definitions apply to an element of together with a class of subsets of ; they are similar to invariants introduced by M. Clapp and D. Puppe. If , we define a transverse subset as a subspace of such that the intersection is at most countable for any . 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 -foliation, the three previous definitions, with the class of transverse subsets, coincide with the tangential category and are homotopical invariants.

Paper Structure

This paper contains 10 sections, 7 theorems, 7 equations.

Table of Contents

  1. The category of stratified spaces
  2. Lusternik-Schnirelmann category of stratified pairs
  3. Ganea construction: the invariant ${\rm Gcat}\,(X,A,{\mathcal{F}})$
  4. Whitehead construction: the invariant ${\rm Whcat}\,(X,A,{\mathcal{F}}_X)$
  5. Open LS-category: the invariant ${\rm Ocat}\,(X,A,{\mathcal{F}}_X)$
  6. Comparison of the three invariants
  7. Lusternik-Schnirelmann category of stratified spaces
  8. Tangential LS-category of foliations
  9. Definition of tangential LS-category
  10. Equivalence of the four invariants

Key Result

Theorem A

The category ${\mathcal{S}}$-$\bf Top$ is a fibration category with -- fibrations, the stratified maps $f\colon (X,{\mathcal{F}}_X)\to (Y,{\mathcal{F}}_Y)$ having the ${\mathcal{S}}$-homotopy lifting property; -- weak equivalences, the stratified maps $f\colon (X,{\mathcal{F}}_X)\to (Y,{\mathcal{F}}

Theorems & Definitions (32)

  • Definition 1
  • Theorem A
  • proof
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • ...and 22 more