One dimensional non-Hausdorff manifolds and foliations of the plane

TL;DR

This paper explores the relationship between foliations of the plane and non-Hausdorff 1-dimensional manifolds, highlighting their significance in topology and dynamical systems, and provides an English translation of a foundational 1957 article.

Contribution

It offers an accessible English translation of Haefliger and Reeb's foundational 1957 work, clarifying the connection between plane foliations and non-Hausdorff leaf spaces.

Findings

Established the link between plane foliations and non-Hausdorff 1-manifolds.

Influenced subsequent research in dynamical systems and 3-manifold foliations.

Served as a foundational reference in foliation theory.

Abstract

The original article titled "Vari\'et\'es (non s\'epar\'ees) \`a une dimension et structures feuillet\'ees du plan" was published in 1957 in French in L'Enseignement Math\'ematique. It establishes a beautiful connection between foliations of the plane and non-Hausdorff $1$-dimensional manifolds arising naturally as leaf spaces of the foliations. Since its appearance, this theory has paved the way for several results concerning dynamical systems and foliations of the plane and $2$-manifolds. Haefliger and Reeb's article inspires many results in the theory of foliation of $3$-manifolds as well: we refer the interested reader to Danny Calegari's book on the topic. Haefliger and Reeb's article also has been applied to areas outside topological dynamics. This article has been referenced in $43$ papers to-date and has helped build many nice results. In the literature, the main theorem of this article has been commonly referred to as Haefliger-Reeb theory or "a classical result by Haefliger and Reeb". However, to the best of our knowledge, no English version of this article is currently available. We have kept the article entirely unchanged except to add figures and correct a few typographical errors. We hope that this translation will be useful for the broader audience.