Exotic Dehn twists on contact 3-manifolds
Abstract: It is a fundamental problem in contact topology to understand the homotopy type of the group of contact diffeomorphisms (or ""contactomorphisms"") of a contact manifold, and how this compares to that of the group of diffeomorphisms of the underlying smooth manifold. Little is known, in general, about this problem. Most results available concern the group of connected components and the tendency is to follow the h-principle: roughly speaking, contactomorphisms which are smoothly isotopic are also contact isotopic.
In contrast with this picture, I will describe the first example of an ""exotic"" contactomorphism (i.e. smootly but not contact isotopic to the identity, roughly speaking) which has infinite order in the contact mapping class group (i.e. its iterates remain exotic). This is a contactomorphism of certain connected sums of two contact 3-manifolds which is supported on a small neighbourhood of the the separating sphere. The essential tool that is used to detect this phenomenon is a new invariant of families of contact structures with values in the monopole Floer homology of the underlying 3-manifold, which generalises the contact invariant introduced by Kronheimer, Mrowka, Ozsváth and Szabó. This is based on joint work with Eduardo Fernández.