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Open books and configurations of symplectic surfaces

David T Gay

Algebraic & Geometric Topology 3 (2003) 569–586

Erratum: Algebraic & Geometric Topology 3 (2003) 1275–1276

arXiv: math.GT/0209153


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.

symplectic, contact, concave, open book, plumbing, fillable
Mathematical Subject Classification 2000
Primary: 57R17
Secondary: 57N10, 57N13
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Received: 27 January 2003
Accepted: 23 October 2002
Published: 20 June 2003
David T Gay
Department of Mathematics
University of Arizona
617 North Santa Rita
PO Box 210089
Tucson, AZ 85721