Geometry & Topology

Symplectic fillings and positive scalar curvature

Paolo Lisca

Full-text: Open access


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

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

Received: 27 February 1998
Accepted: 9 July 1998
First available in Project Euclid: 21 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.)
Secondary: 57M50: Geometric structures on low-dimensional manifolds 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]

contact structures monopole equations Seiberg–Witten equations positive scalar curvature symplectic fillings


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

Export citation


  • M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry and Riemannian geometry: I, Math. Proc. Cambridge Philos. Soc. 77 (1975) 43–69
  • D Bennequin, Entrelacements et equations de Pfaff, Astérisque 107–108 (1983), 83–161
  • S K Donaldson, Connections, cohomology and the intersection forms of four–manifolds, Jour. Diff. Geom. 24 (1986) 275–341
  • S K Donaldson, The Seiberg–Witten equations and $4$–manifold topology, Bull. AMS 33 (1996) 45–70
  • Y Eliashberg, Topological characterization of Stein manifolds of dimension $>2$, Intern. Journal of Math. 1, No. 1 (1990) 29–46
  • Y Eliashberg, Filling by holomorphic discs and its applications, London Math. Soc. Lecture Notes Series 151 (1991) 45–67
  • Y Eliashberg, Contact $3$–manifolds twenty years since J. Martinet's work, Ann. Inst. Fourier 42 (1992) 165–192
  • Y Eliashberg, Unique holomorphically fillable contact structure on the $3$–torus, Intern. Math. Res. Not. 2 (1996) 77–82
  • N Elkies, A characterization of the $\Z^n$ lattice, Math. Res. Lett. 2 (1995) 321–326
  • N Elkies, Lattices and codes with long shadows, Math. Res. Lett. 2 (1995) 643–652
  • N Elkies, personal communication
  • J B Etnyre, Symplectic convexity in low dimensional topology, Top. Appl. (to appear)
  • K A Frøyshov, The Seiberg–Witten equations and four–manifolds with boundary, Math. Res. Lett. 3 (1996) no. 3, 373–390
  • E Giroux, Topologie de contact en dimension $3$, Séminaire Bourbaki 760 (1992-93), 7–33
  • R E Gompf, Handlebody construction of Stein surfaces, Ann. of Math. (to appear)
  • R Kirby, Problems in Low-Dimensional Topology. In W H Kazez (Ed.), Geometric Topology, Proc of the 1993 Georgia International Topology Conference, AMS/IP Studies in Advanced Mathematics, pp. 35–473, AMS & International Press (1997)
  • P B Kronheimer, T S Mrowka, Monopoles and contact structures, Invent. Math. 130 (1997) 209–256
  • D Kotschick, J W Morgan, C H Taubes, Four–manifolds without symplectic structures but with non-trivial Seiberg–Witten invariants, Math. Res. Lett. 2 (1995) 119–124
  • F Laudenbach, Orbites périodiques et courbes pseudo-holomorphes, application à la conjecture de Weinstein en dimension $3$ [d'après H. Hofer et al.], Astérisque 227 (1995) 309–333
  • P Lisca, G Matić, Tight contact structures and Seiberg–Witten invariants, Invent. math. 129 (1997) 509–525
  • J Martinet, Formes de contact sur les variètès de dimension $3$, Lect. Notes in Math. 209, Springer–Verlag (1971) 142–163
  • J Milnor, D Husemoller, Symmetric bilinear forms, Ergebnisse der Mathematik und Ihrer Grenzgebiete, Band 73, Springer–Verlag (1973)
  • J W Morgan, T S Mrowka, D Ruberman, The $L^2$–moduli space and a vanishing theorem for Donaldson polynomial invariants, Monographs in Geometry and Topology, no. II, International Press, Cambridge, MA, 1994
  • J W Morgan, T S Mrowka, Z Szabó, Product formulas along $T^3$ for Seiberg–Witten invariants, preprint (1997)
  • C H Taubes, The Seiberg–Witten invariants and symplectic forms, Math. Res. Lett. 1 (1995) 809–822
  • C H Taubes, More constraints on symplectic manifolds from Seiberg–Witten equations, Math. Res. Lett. 2 (1995) 9–14