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Symplectic fillings and positive scalar curvature

By Paolo Lisca

Abstract

Let X be a 4-manifold with contact boundary. We prove that the monopole invariants of X introduced by Kronheimer and Mrowka vanish under the following assumptions: (i) a connected component of the boundary of X carries a metric with positive scalar curvature and (ii) either b_2^+(X)>0 or the boundary of X is disconnected. As an application we show that the Poincare homology 3-sphere, oriented as the boundary of the positive E_8 plumbing, does not carry symplectically semi-fillable contact structures. This proves, in particular, a conjecture of Gompf, and provides the first example of a 3-manifold which is not symplectically semi-fillable. Using work of Froyshov, we also prove a result constraining the topology of symplectic fillings of rational homology 3-spheres having positive scalar curvature metrics.Comment: 14 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol2/paper6.abs.htm

Topics: Mathematics - Geometric Topology, Mathematics - Differential Geometry, Mathematics - Symplectic Geometry, 53C15, 57M50, 57R57
Publisher: 'Mathematical Sciences Publishers'
Year: 1998
DOI identifier: 10.2140/gt.1998.2.103
OAI identifier: oai:arXiv.org:math/9807188

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