Twenty-five years ago Eliashberg established a dichotomy of contact structures on 3-manifolds as either tight or overtwisted. The overtwisted manifolds were motivated by the property that they satisfy certain flexibility principals. 1. Every co-oriented 2-plane field on a 3-manifold is homotopic to an overtwisted contact structure. 2. Any two overtwisted contact structures which are homotopic as 2-plane fields are isotopic as contact structures. (More generally, there is a homotopy equivalence between the space of contact structures and the space of 2-plane fields when you restrict each space to only include those with a particular overtwisted disk.) Recently, this theorem has been generalized to higher dimensions by Borman, Eliashberg, and Murphy , who explained their work at a recent workshop. The next few posts will be an attempt to share the intuition they gave, to work through some of the steps of their paper, to provide some historical background, and to discuss some related constructions and interesting open questions.
I also want to mention, that we are happy to give author permissions to young mathematicians who want to contribute blog posts. Thanks to Roger for agreeing to contribute on this topic, and welcome to the blog.