A note on open 3-manifolds supporting foliations by planes

TL;DR

This paper investigates the structure of open 3-manifolds with certain foliations, showing that such manifolds with closed-plane foliations have free fundamental groups and characterizing those with abelian fundamental groups.

Contribution

It establishes that open manifolds with closed-plane foliations have free fundamental groups and fully describes the structure of 3-manifolds with abelian fundamental groups supporting such foliations.

Findings

Fundamental group of manifolds with closed-plane foliations is free.

3-manifolds with abelian fundamental group of rank > 1 are homeomorphic to torus times real line.

Complete description of foliations on these 3-manifolds.

Abstract

We show that if $N$, an open connected $n$-manifold with finitely generated fundamental group, is $C^{2}$ foliated by closed planes, then $\pi_{1}(N)$ is a free group. This implies that if $\pi_{1}(N)$ has an Abelian subgroup of rank greater than one, then $\mathcal{F}$ has at least a non closed leaf. Next, we show that if $N$ is three dimensional with fundamental group abelian of rank greater than one, then $N$ is homeomorphic to $\mathbb{T}^2\times \mathbb{R}.$ Furthermore, in this case we give a complete description of the foliation.