Volume 2, issue 1 (1998)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Symplectic fillings and positive scalar curvature

Paolo Lisca

Geometry & Topology 2 (1998) 103–116

arXiv: math.GT/9807188


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.

contact structures, monopole equations, Seiberg–Witten equations, positive scalar curvature, symplectic fillings
Mathematical Subject Classification
Primary: 53C15
Secondary: 57M50, 57R57
Forward citations
Received: 27 February 1998
Accepted: 9 July 1998
Published: 12 July 1998
Proposed: Dieter Kotschick
Seconded: Tomasz Mrowka, John Morgan
Paolo Lisca
Dipartimento di Matematica
Università di Pisa
I-56127 Pisa