Contact Symplectic Fibrations and Fiber Connected Sum

Abstract

We consider certain type of fiber bundles with odd dimensional compact contact base, exact symplectic fibers, and the structure group contained in the group of exact symplectomorphisms of the fiber. We call such fibrations "contact symplectic fibrations". By a result of Hajduk-Walczak, some of these admit contact structures which are "compatible" (in a certain sense) with the corresponding fibration structures. We show that this result can be extended to get a compatible contact structure on any contact symplectic fibration, and also that isotopic contact structures on the base produce isotopic contact structures on the total space.. Moreover, we prove that the fiber connected summing of two contact symplectic fibrations along their fibers results in another contact symplectic fibration which admits a compatible contact structure agreeing with the original ones away from the region where we take fiber connected sum, and whose base is the contact connected sum of the original contact bases.