Exotic Dehn twists on sums of two contact 3-manifolds

TL;DR

This paper introduces the first examples of exotic contactomorphisms of infinite order in contact 3-manifolds, using Dehn twists and advanced invariants, revealing new phenomena in contact topology.

Contribution

It provides the first known examples of infinite order exotic contactomorphisms on sums of contact 3-manifolds, combining novel invariants and an h-principle approach.

Findings

Existence of exotic contactomorphisms with infinite order

Development of a generalized contact invariant in monopole Floer homology

Discovery of new phenomena in overtwisted 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.