## Geometry & Topology

### Symplectic fillings and positive scalar curvature

Paolo Lisca

#### 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.

#### Article information

Source
Geom. Topol., Volume 2, Number 1 (1998), 103-116.

Dates
Accepted: 9 July 1998
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513882904

Digital Object Identifier
doi:10.2140/gt.1998.2.103

Mathematical Reviews number (MathSciNet)
MR1633282

Zentralblatt MATH identifier
0942.53050

#### Citation

Lisca, Paolo. Symplectic fillings and positive scalar curvature. Geom. Topol. 2 (1998), no. 1, 103--116. doi:10.2140/gt.1998.2.103. https://projecteuclid.org/euclid.gt/1513882904

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