A compactly generated pseudogroup which is not realizable

TL;DR

This paper presents a specific example of a compactly generated pseudogroup of smooth transformations on the real line that cannot be realized as the holonomy pseudogroup of any codimension 1 foliation on a compact manifold, highlighting limitations in foliation realizability.

Contribution

It introduces a novel example of a pseudogroup that is compactly generated but not realizable as a foliation holonomy, expanding understanding of foliation dynamics.

Findings

Identifies a pseudogroup not realizable as a foliation holonomy

Provides a description of foliations with Reeb component dynamics

Highlights limitations of foliation realization on compact manifolds

Abstract

We exhibit a pseudogroup of smooth local transformations of the real line which is compactly generated, but not realizable as the holonomy pseudogroup of a foliation of codimension 1 on a compact manifold. The proof relies on a description of all foliations with the same dynamic as the Reeb component.