A few remarks about symplectic filling

Abstract

We show that any compact symplectic manifold (W, ω) with boundary embeds as a domain into a closed symplectic manifold, provided that there exists a contact plane ξ on ∂W which is weakly compatible with ω, i.e. the restriction ω|ξ does not vanish and the contact orientation of ∂W and its orientation as the boundary of the symplectic manifold W coincide. This result provides a useful tool for new applications by Ozsváth-Szabó of Seiberg-Witten Floer homology theories in three-dimensional topology and has helped complete the Kronheimer-Mrowka proof of Property P for knots.