Geometry & Topology 2 (1998) 103–116

DOI: 10.2140/gt.1998.2.103

arXiv: math.GT/9807188

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 b₂⁺(X) > 0 or the boundary of X is disconnected. As an application we show that the Poincaré homology 3–sphere, oriented as the boundary of the positive E₈ 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 Frøyshov, we also prove a result constraining the topology of symplectic fillings of rational homology 3–spheres having positive scalar curvature metrics.

Keywords

contact structures, monopole equations, Seiberg–Witten equations, positive scalar curvature, symplectic fillings

Mathematical Subject Classification

Primary: 53C15

Secondary: 57M50, 57R57

References

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Publication

Received: 27 February 1998
Accepted: 9 July 1998
Published: 12 July 1998
Proposed: Dieter Kotschick
Seconded: Tomasz Mrowka, John Morgan

Authors