Homotopy classes of total foliations and bi-contact structures on three-manifolds

TL;DR

This paper constructs total foliations on all compact orientable three-manifolds within any homotopy class of plane fields with zero Euler class, and extends these results to bi-contact structures.

Contribution

It provides a universal construction of total foliations and bi-contact structures on three-manifolds for specified homotopy classes.

Findings

Total foliations exist on all compact orientable three-manifolds.

Construction applies to any homotopy class with vanishing Euler class.

Results extend to bi-contact structures.

Abstract

On every compact and orientable three-manifold, we construct total foliations (three codimension 1 foliations that are transverse at every point). This construction can be performed on any homotopy class of plane fields with vanishing Euler class. As a corollary we obtain similar results on bi-contact structures.