Exotic Dehn twists on sums of two contact3-manifolds

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.We make essential use of an invariant for families of contact structures which generalizes 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 exhibit new exotic 1-parametric phenomena in overtwisted contact 3-manifolds.53D35, 57K33, 57R58 1. Introduction 1571 2. Background 1579 3. Contact Dehn twists on spheres 1583 4. Monopole Floer homology and families of contact structures 1596 5.The space of standard convex spheres in a tight contact 3-manifold 1602 6. Families of contact structures on sums of contact 3-manifolds 1607 7. Exotic phenomena in overtwisted contact 3-manifolds 1613 References 1615 1 IntroductionThroughout this article all 3-manifolds are closed, oriented and connected unless otherwise noted, and all contact structures on 3-manifolds are co-oriented and positive. Main resultA fundamental problem in contact topology is to understand the isotopy classes of contact diffeomorphisms, usually called "contactomorphisms", of a contact manifold.The following is a longstanding open question in all dimensions: