Symplectic fillings and positive scalar curvature

Details Symplectic fillings and positive scalar curvature Symplectic fillings and positive scalar curvature

1998-07-12

arxiv.org/abs/math/9807188v1

Geom. Topol. 2 (1998), 103-116

Mathematics

Geometric Topology

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

Scientific paper

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.

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