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, Gay-Stipsicz and the third author. 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.

Figures (11)

• Figure 1: A spinal open book with two spine components, which are $S^1$-fibrations over a genus $1$ surface with one boundary component and an annulus respectively. They are connected to each other by an $S^1$-family of pages with genus $2$, and we can also see a fragment of a second $S^1$-family of pages attached to the annular spine component.
• Figure 2: The path $\gamma_\epsilon$ and region $\Gamma_\epsilon \subset (-1,0] \times (-1,0]$ used for smoothing corners.
• Figure 3: The smoothed corner of $M_\epsilon = \partial W_\epsilon$ with the transverse vector field $V_K$.
• Figure 4: A schematic picture of the closed manifold $M'$ with compact subdomains $M = M_P \cup M_\Sigma \subset M'$ and $M_P' \subset M'$, together with the various collar neighborhoods ${\mathcal{N}}(\partial M)$ and ${\mathcal{N}}(\partial M_P')$ of $\partial M_P$.
• Figure 5: The domain $E'$ with its boundary faces and collar neighborhoods, shown together with a portion of the fibration $\Pi_h : {\mathcal{N}}(\partial_h E) \to \Sigma$. In this example, $M_P$ contains at least two connected components, one (shown at the right) that touches two separate spinal components but not the boundary, and another (at the left) that does touch $\partial M$.

Theorems & Definitions (135)

• Theorem A: LisiVanhornWendl2
• Theorem B: LisiVanhornWendl2
• Theorem C: see Theorem \ref{['thm:SteinHomotopy']}
• Theorem D: see Theorem \ref{['thm:spineRemoval']}
• Remark 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Definition 1.5
• Remark 1.6