Monopoles and families of contact structures

Beyond the tight/overtwisted dichotomy, 3-dimensional contact topology would appear to be dominated by flexibility: a central result of Eliashberg and Mishachev says that the contactomorphism group of the standard contact 3-ball has the homotopy type of the diffeomorphism group. In contrast with this, I will discuss how monopole Floer homology imposes constraints on the behaviour of families of contact structures on 3-manifolds. Applications include detecting exotic contactomorphisms given by certain “Dehn twists” on embedded spheres.