Open books and exact symplectic cobordisms

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 Dörner-Geiges-Zehmisch. Our cobordisms can also be thought of as the result of the attachment of a generalized symplectic 1-handle.

Bourgeois contact structures: Tightness, fillability and applications

“…The first ingredient is the construction of a strong symplectic cobordism between Bourgeois contact structures; this is done in Section 3.1. More precisely, Theorem 3.1 is a "stabilized" version of the analogous result for open books, which was proven (independently) in [2,35]; see Figure 1. We point out that, while the symplectic form on the strong cobordism of Theorem 3.1 is exact, the Liouville vector field associated to the global primitive is not inwards pointing along the negative ends.…”

“…Notice however that we do not claim that the global 1-form ν in Item 2 defines a contact structure at the concave boundary; in other words, the cobordism we give is not claimed to be Liouville, but just pseudo-Liouville (as defined in the introduction). Lastly, we point out that Theorem 3.1 can be thought of as a "stabilized" version of [2, Proposition 8.3] and [35,Theorem 1]; in fact, smoothly (but not symplectically), the cobordism C is just the product of the cobordism from [2,35] with T 2 . For the reader's convenience, we start by giving a topological description of the cobordism C as obtained by gluing two "cobordisms with corners", C bot and C top , and then describe the symplectic structures on these pieces in more detail.…”

“…to give a detailed proof of Theorem 3.1. We will follow closely, with some adaptations, the proof given in [35].…”

On symplectic fillings of spinal open book decompositions I: Geometric constructions

“…Spine removal surgery. The following makes precise the non-exact cobordism construction that was sketched in Theorem D of the introduction, generalizing previous constructions from [Eli04, GS12, Wen13b] (see also the higher-dimensional analogues in [MNW13,DGZ14,Klu18]). For this discussion, it is useful to allow a slight loosening of the main definition of this paper: we will say that a generalized spinal open book is an object satisfying all the conditions of Definition 1.2 except that fibers of the paper π P : M P Ñ S 1 are allowed to have components with no boundary.…”

“…We will see in the proof that the disk D 2 in the symplectic handle Σ rem ˆD2 can freely be replaced by any other compact oriented surface with connected boundary; more generally, one could equally well remove several spine components at once and replace them with Σ rem ˆS for a compact oriented surface S with the right number of boundary components. The key intuition is to view S as a symplectic cap for the appropriate disjoint union of fibers in the contact circle fibration π Σ : M Σ Ñ Σ, and this is also the right perspective in higher-dimensional cases such as [DGZ14,Klu18]. We will not comment any further on these generalizations since the applications in this paper do not require them.…”