Nonperiodic leaves of codimension one foliations

TL;DR

This paper constructs examples of 5-manifolds that can be leaves in certain foliations but not in others, revealing new phenomena in the topology of foliated manifolds and minimal sets with nonhomeomorphic leaves.

Contribution

It provides the first known examples of 5-manifolds that are leaves in some foliations but not in others, demonstrating complex leaf topology in codimension one foliations.

Findings

Existence of 5-manifolds as leaves in some $C^1$ and $C^ ext{inf}$ foliations

Examples of minimal invariant sets with nonhomeomorphic leaves in certain foliations

First known examples of this kind in foliation theory

Abstract

In this work we exhibit examples of $5$-manifolds that are not homeomorphic to any leaf of any $C^2$ codimension one foliation of any compact $6$-manifold but are homeomorphic to (proper) leaves of some $C^1$ codimension one foliations and also to (proper) leaves of some $C^\inf$ codimension $2$ foliations. As far as we know, this is the first example of this nature. In addition, it is shown examples of $C^{r}$ codimension one foliations, $r\in[0,2)$, with a minimal invariant set whose leaves are pairwise nonhomeomorphic.