#### Volume 2, issue 1 (1998)

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Symplectic fillings and positive scalar curvature

### Paolo Lisca

Geometry & Topology 2 (1998) 103–116
 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 ${b}_{2}^{+}\left(X\right)>0$ 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 ${E}_{8}$ 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 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