Overtwisted (2,3,5)-structures and the h-principle

Abstract

A smooth distribution is a smooth subbundle of the tangent bundle. Locally, smooth distributions are spanned by vector fields to which one can apply the Lie bracket. Intuitively, one can view a distribution as the ``allowed directions of motion’’, and the Lie bracket as a way of measuring whether at a point one can move in a direction by moving along the distribution. If we can move in any direction, we call a distribution bracket-generating. This thesis focusses on bracket-generating distributions called (2,3,5)-structures. These are maximally non-integrable 2-distributions on 5-manifolds.

The h-principle is a useful tool for classifying families of distributions up to homotopy. In 1969 Gromov proved a powerful result which shows that the h-principle holds for many types of (bracket-generating) distributions, on open manifolds. Therefore, a natural question to ask is, what about closed manifolds? In this thesis we define a special class of (2,3,5)-structures called overtwisted (2,3,5)-structures, and we prove that the h-principle holds for this family of distributions on closed manifolds.