Open books and configurations of symplectic surfaces

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

Mathematical Subject Classification 2000

Primary: 57R17

Secondary: 57N10, 57N13

References

Forward citations

Publication

Received: 27 January 2003

Accepted: 23 October 2002

Published: 20 June 2003

Authors

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

Volume 3, issue 1 (2003)

David T Gay