Open books and configurations of symplectic surfaces

Abstract

We study neighborhoods of configurations of symplectic surfaces in symplectic 4-manifolds. We show that suitably `positive' configurations have neighborhoods with concave boundaries and we explicitly describe open book decompositions of the boundaries supporting the associated negative contact structures. This is used to prove symplectic nonfillability for certain contact 3-manifolds and thus nonpositivity for certain mapping classes on surfaces with boundary. Similarly, we show that certain 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 configurations 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. Erratum (added December 2003): We correct the main theorem and its proof. As originally stated, the theorem gave conditions on a configuration of symplectic surfaces in a symplectic 4-manifold under which we could construct a model neighborhood with concave boundary and describe explicitly the open book supporting the contact structure on the boundary. The statement should have included constraints on the areas of the surfaces.