2023 Theses Doctoral

Monopoles and Dehn twists on contact 3-manifolds

Muñoz Echániz, Juan Álvaro

In this dissertation, we study the isotopy problem for a certain three-dimensional contactomorphism which is supported in a neighbourhood of an embedded 2-sphere with standard characteristic foliation. The diffeomorphism which underlies it is the Dehn twist on the sphere, and therefore its square becomes smoothly isotopic to the identity. The main result of this dissertation gives conditions under which any iterate of the Dehn twist along a non-trivial sphere is not contact isotopic to the identity.

This provides the first examples of exotic contactomorphisms with infinite order in the contact mapping class group, as well as the first examples of exotic contactomorphisms of 3-manifolds with b_1 = 0. The proof crucially relies on the construction of an invariant for families of contact structures in monopole Floer homology which generalises the Kronheimer--Mrowka--Ozsváth--Szabó contact invariant, together with the nice interaction between this families invariant and the U map in Floer homology.