Open books and configurations of symplectic surfaces and erratum
Abstract
We study neighborhoods of configurations of symplectic surfaces in symplectic 4manifolds. 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 3manifolds and thus nonpositivity for certain mapping classes on surfaces with boundary. Similarly, we show that certain pairs of contact 3manifolds cannot appear as the disconnected convex boundary of any connected symplectic 4manifold. Our result also has the potential to produce obstructions to embedding specific symplectic configurations in closed symplectic 4manifolds 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 4manifold. 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 4manifold 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 September 2002
 arXiv:
 arXiv:math/0209153
 Bibcode:
 2002math......9153G
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Symplectic Geometry;
 57R17;
 57N10;
 57N13
 EPrint:
 Version 5 contains an erratum to version 4. The paper and erratum are published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt319.abs.html with erratum at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt345.abs.html