Exotic Dehn twists on sums of two contact 3-manifolds

Abstract: We exhibit the first examples of exotic contactomorphisms with infinite order as elements of the contact mapping class group. These are given by certain Dehn twists on the separating sphere in a connected sum of two closed contact 3-manifolds. We detect these by a combination of hard and soft techniques. On the one hand, we make essential use of an invariant for families of contact structures which generalises the Kronheimer--Mrowka contact invariant in monopole Floer homology. We then exploit an h-principle for families of convex spheres in tight contact 3-manifolds, from which we establish a parametric version of Colin's decomposition theorem. As a further application, we also exhibit new exotic 1-parametric phenomena in overtwisted contact 3-manifolds.