#### Volume 8, issue 1 (2004)

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A few remarks about symplectic filling

### Yakov Eliashberg

Geometry & Topology 8 (2004) 277–293
 arXiv: math.SG/0308183
##### Abstract

We show that any compact symplectic manifold $\left(W,\omega \right)$ with boundary embeds as a domain into a closed symplectic manifold, provided that there exists a contact plane $\xi$ on $\partial W$ which is weakly compatible with $\omega$, i.e. the restriction $\omega {|}_{\xi }$ does not vanish and the contact orientation of $\partial W$ 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.

##### Keywords
contact manifold, symplectic filling, symplectic Lefschetz fibration, open book decomposition
Primary: 53C15
Secondary: 57M50
##### Publication
Revised: 13 January 2004
Accepted: 2 January 2004
Published: 14 February 2004
Proposed: Leonid Polterovich
Seconded: Peter Ozsváth, Dieter Kotschick
##### Authors
 Yakov Eliashberg Department of Mathematics Stanford University Stanford California 94305-2125 USA