On the connectedness of the space of codimension one foliations on a closed 3-manifold

TL;DR

This paper proves that the space of smooth codimension one foliations on a closed 3-manifold is topologically equivalent to the space of all smooth plane fields, establishing a bijection between their connected components.

Contribution

It demonstrates that the inclusion map from the space of integrable plane fields to all plane fields induces a bijection on connected components, clarifying the topology of foliation spaces.

Findings

The space of foliations and plane fields have the same connected component structure.

Every plane field can be deformed into a foliation without changing the component.

The inclusion map between these spaces is a bijection on connected components.

Abstract

We study the topology of the space of smooth codimension one foliations on a closed 3-manifold. We regard this space as the space of integrable plane fields included in the space of all smooth plane fields. It has been known since the late 60's that every plane field can be deformed continuously to an integrable one, so the above inclusion induces a surjective map between connected components. We prove that this map is actually a bijection.