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

Abstract

A spinal open book decomposition on a contact manifold is a generalization of a supporting open book which exists naturally e.g. on the boundary of a symplectic filling with a Lefschetz fibration over any compact oriented surface with boundary. In this first paper of a two-part series, we introduce the basic notions relating spinal open books to contact structures and symplectic or Stein structures on Lefschetz fibrations, leading to the definition of a new symplectic cobordism construction called spine removal surgery, which generalizes previous constructions due to Eliashberg [Eli04], Gay-Stipsicz [GS12] and the third author [Wen13b]. As an application, spine removal yields a large class of new examples of contact manifolds that are not strongly (and sometimes not weakly) symplectically fillable. This paper also lays the geometric groundwork for a theorem to be proved in part II, where holomorphic curves are used to classify the symplectic and Stein fillings of contact 3-manifolds admitting a spinal open book with a planar page.

A landscape of contact manifolds via rational SFT

“…Proof. We first note that Y admits a supporting (trivial) SOBD, as considered in [52,58] We view the first one as a contact fibration over a Liouville domain, and the second, as a Liouville fibration over a contact manifold (its fibers are called the pages).…”

“…the ruling curve defines a map that is visible on homology; in particular, such curve can not be eliminated by perturbing the contact form. On the other hand, not every contact manifold with finite planarity is iterated planar: for example P(T 3 , ξ std ) = 2 by Corollary 6.9, while (T 3 , ξ std ) is not supported by a planar open book by [31] (it is, however, supported by a planar spinal open book [72,52]). By functoriality, if there is an exact cobordism from Y to Y with P(Y ) < ∞, then the Weinstein conjecture holds for Y .…”

Lefschetz-Bott fibrations on line bundles over symplectic manifolds

“…In this appendix, we show that the total space of a symplectic Lefschetz-Bott fibration over the unit disk D serves as a strong symplectic filling of a contact manifold supported by an open book induced by the fibration. To prove this, we mainly follow a recent paper of Lisi, Van Horn-Morris and Wendl [18] where they discussed a relation between 4-dimensional Lefschetz fibrations and various symplectic fillings of contact 3-manifolds.…”

On symplectic fillings of spinal open book decompositions II: Holomorphic curves and classification

“…Since we will need to make extensive use of J-holomorphic curves, §2 gives a review of some of the essential technical results-its contents are mostly standard, but it also includes (in §2.4) some useful lemmas about coherent orientations that may not have appeared in writing before. Section 3 continues the development (begun in [LVW,§3]) of a precise model for the half-symplectization of a contact manifold supported by a spinal open book, including an explicit foliation by J-holomorphic curves with specific properties that are needed for the proofs of the main results. The analytical properties of this holomorphic foliation are then studied in §4, including existence and uniqueness results that are needed for the computations of algebraic contact invariants carried out in §5.…”