Algebraic and Geometric Topology 3 (2003), paper no. 19, pages 569-586.

Open books and configurations of symplectic surfaces

David T. Gay

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.

Keywords. Symplectic, contact, concave, open book, plumbing, fillable

AMS subject classification. Primary: 57R17. Secondary: 57N10, 57N13.

Note: There is an erratum to this paper which should be read alongside it.

DOI: 10.2140/agt.2003.3.569

E-print: arXiv:math.GT/0209153

Submitted: 27 January 2003. Accepted: 23 April 2003. Published: 20 June 2003.

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David T. Gay
Department of Mathematics, University of Arizona
617 North Santa Rita, PO Box 210089
Tucson, AZ 85721, USA

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