Symplectic fillings and positive scalar curvature

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.

14 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol2/paper6.abs.html

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• Ozsvath-Szabo invariants and tight contact three-manifolds, I
• On the existence of tight contact structures on Seifert fibered 3-manifolds
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• The topology of Stein fillable manifolds in high dimensions I
• A Hierarchy of Local Symplectic Filling Obstructions for Contact 3-Manifolds
• Seifert fibered contact three-manifolds via surgery
• Stein fillable Seifert fibered 3-manifolds
• An infinite family of tight, not semi-fillable contact three-manifolds
• Contact structures and reducible surgeries