On the classification of prolongations up to Engel homotopy

TL;DR

This paper investigates the classification of Cartan and Lorentz prolongations of Engel structures up to Engel homotopy, showing it reduces to formal data for large turning numbers and analyzing the homotopy type of the space of Cartan prolongations.

Contribution

It demonstrates that classification reduces to formal data for large turning numbers and characterizes the homotopy type of Cartan prolongations in the overtwisted case.

Findings

Classification reduces to formal data with large turning numbers.

Connected components of Cartan prolongations are fully described in the overtwisted case.

Addresses the open question of a full $h$-principle for Engel structures.

Abstract

In [CPPP] it was shown that Engel structures satisfy an existence $h$-principle, and the question of whether a full $h$-principle holds was left open. In this note we address the classification problem, up to Engel deformation, of Cartan and Lorentz prolongations. We show that it reduces to their formal data as soon as the turning number is large enough. Somewhat separately, we study the homotopy type of the space of Cartan prolongations, describing completely its connected components in the overtwisted case.