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.

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Paolo Lisca
Dipartimento di Matematica
Universita di Pisa
I-56127 Pisa, ITALY

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