Abstract: The classification problem for bracket–generating distributions in differentiable manifolds
from a homotopic viewpoint has been tackled for certain cases.
Gromov classified contact structures up to homotopy in open manifolds. Afterwards, a subclass of this type of structures called overtwisted were classified in closed manifolds up to isotopy, first by Eliashberg in dimension 3 and later on by Borman, Eliashberg and Murphy in all dimensions. McDuff classified even-contact structures in even-dimensional manifolds, showing that there exists a full h-principle.
More recently, Casals, del Pino, Pérez and Presas proved an existenceh−principle for Engel structures in smooth 4−manifolds. On the other hand, del Pino and Vogel showed that there exists a full h−principle when restricted to the subclass of overtwisted Engel structures.
In this talk we will discuss the classification problem for (4,6)−bracket–generating structures through convex integration. This is work in progress with Alvaro del Pino (Utrecht University).