Open books and configurations of symplectic surfaces

Abstract

We study neighborhoods of configurations of symplectic sur-faces in symplectic 4–manifolds. We show that suitably " positive " con-figurations have neighborhoods with concave boundaries and we explicitly describe open book decompositions of the boundaries supporting the asso-ciated negative contact structures. This is used to prove symplectic nonfil-lability for certain contact 3–manifolds and thus nonpositivity for certain mapping classes on surfaces with boundary. Similarly, we show that cer-tain pairs of contact 3–manifolds cannot appear as the disconnected convex boundary of any connected symplectic 4–manifold. Our result also has the potential to produce obstructions to embedding specific symplectic config-urations in closed symplectic 4–manifolds and to generate new symplectic surgeries. From a purely topological perspective, the techniques in this paper show how to construct a natural open book decomposition on the boundary of any plumbed 4–manifold.