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 $b_2^{+}\left(X\right)>0$ 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.