Tight Contact Structures on Contact Mapping Tori and their Folded Sums

Abstract

It is known that the folded sum of two contact mapping tori whose fibers are compact exact symplectic manifolds having a common convex boundary (called the ``fold'') admits a cooriented contact structure compatible with the obvious fibration map onto the circle. Here we first provide an alternative bundle-theoretical construction of such a ``folded'' contact structure based on a gluing process near the fold. Moreover, we prove that in any odd dimension $2n+1\geq 7$ a folded contact structure on a folded sum of two contact mapping tori is tight if the induced contact form on the (common) contact fold admits no contractible Reeb orbit. In particular, any contact mapping torus of an odd dimension $2n+1\geq 7$ is tight if the induced contact form on the convex boundary of a fiber admits no contractible Reeb orbit.