#### Geometry & Topology, Vol. 2 (1998)
Paper no. 6, pages 103--116.

## 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 b_2^+(X)>0 or the boundary of X is disconnected. As an application we show that the Poincare 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 Froyshov, 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

**AMS subject classification**.
Primary: 53C15.
Secondary: 57M50, 57R57.

**DOI:** 10.2140/gt.1998.2.103

**E-print:** `arXiv:math.GT/9807188`

Submitted to GT on 27 February 1998.
Paper accepted 9 July 1998.
Paper published 12 July 1998.

Notes on file formats
Paolo Lisca

Dipartimento di Matematica

Universita di Pisa

I-56127 Pisa, ITALY

Email: lisca@dm.unipi.it

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