Existence and classification of the Cartan $(2,3,5)$-distribution

TL;DR

This paper establishes a purely topological criterion for the existence of Cartan (2,3,5)-distributions on 5-manifolds and classifies these structures up to homotopy as formal Cartan distributions.

Contribution

It provides the first purely topological condition for the existence of Cartan (2,3,5)-distributions and classifies these structures up to homotopy.

Findings

Derived a necessary and sufficient topological condition for existence.

Classified Cartan (2,3,5)-distributions up to homotopy.

Established a framework for understanding these distributions in topological terms.

Abstract

The Cartan $(2,3,5)$-distribution is a tangent distribution of rank~$2$ on a $5$-dimensional manifold satisfying certain generic conditions. The necessary and sufficient condition for a manifold to admit such a structure is established in this paper. The condition obtained is purely topological. In addition, the classification of such structures, up to homotopy as formal Cartan distributions, is obtained.