Characteristic Classes for the Degenerations of Two-Plane Fields in Four Dimensions

TL;DR

This paper investigates the degenerations of two-plane fields in four-dimensional manifolds, focusing on Engel distributions, and establishes that their degeneration loci represent Chern classes of the distribution.

Contribution

It proves that the surfaces where Engel conditions fail correspond to Chern classes, linking geometric degenerations to topological invariants.

Findings

Degeneration loci are finite unions of surfaces.

These surfaces represent Chern classes of the distribution.

Engel structures imply the manifold is parallelizable.

Abstract

There is a remarkable type of field of two-planes special to four dimensions known as an Engel distributions. They are the only stable regular distributions besides the contact, quasi-contact and line fields. If an arbitrary two-plane field on a four-manifold is slightly perturbed then it will be Engel at generic points. On the other hand, if a manifold admits an oriented Engel structure then the manifold must be parallelizable and consequently the alleged Engel distribution must have a degeneration loci -- a point set where the Engel conditions fails. By a theorem of Zhitomirskii this locus is a finite union of surfaces. We prove that these surfaces represent Chern classes associated to the distribution.