Symplectic fillings and positive scalar curvature

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 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

References

Forward citations

Publication

Received: 27 February 1998

Accepted: 9 July 1998

Published: 12 July 1998

Proposed: Dieter Kotschick

Seconded: Tomasz Mrowka, John Morgan

Authors

Paolo Lisca
Dipartimento di Matematica Università di Pisa I-56127 Pisa Italy

Volume 2, issue 1 (1998)

Paolo Lisca