Open books and exact symplectic cobordisms

TL;DR

This paper constructs exact symplectic cobordisms between contact manifolds associated with open books, extends known results to higher dimensions, and shows the absence of local filling obstructions in open book bindings.

Contribution

It introduces new constructions of symplectic cobordisms for open books with equal pages and extends Eliashberg's cobordism results to higher dimensions.

Findings

Existence of exact symplectic cobordisms between contact manifolds from open books.

Strong fillings for contact manifolds from doubled open books.

The complement of an open book binding has no local filling obstructions.

Abstract

Given two open books with equal pages we show the existence of an exact symplectic cobordism whose negative end equals the disjoint union of the contact manifolds associated to the given open books, and whose positive end induces the contact manifold associated to the open book with the same page and concatenated monodromy. Using similar methods we show the existence of strong fillings for contact manifolds associated with doubled open books, a certain class of fiber bundles over the circle obtained by performing the binding sum of two open books with equal pages and inverse monodromies. From this we conclude, following an outline by Wendl, that the complement of the binding of an open book cannot contain any local filling obstruction. Given a contact $3$-manifold, according to Eliashberg there is a symplectic cobordism to a fibration over the circle with symplectic fibers. We extend this result to higher dimensions recovering a recent result by D\"orner--Geiges--Zehmisch. Our cobordisms can also be thought of as the result of the attachment of a generalized symplectic $1$-handle.