In Definition 1.7.14 we introduced the notion of weak and strong symplectic filling of a contact 3—manifold. In the first section of the present chapter we extend these definitions to contact manifolds of arbitrary dimension, and we discuss other notions of fillings and their interrelations.

If a contact manifold (M, ξ) admits a strong symplectic filling (W, ω), one also says that (M, ξ) is the ω—convex boundary of (W, ω). With the appropriate changes of signs in the definition, one arrives at the notion of ω—concave boundary. A symplectic manifold with a concave and a convex boundary component can then be interpreted as a symplectic cobordism. In particular, a strong filling of (M, ξ) may be interpreted as a symplectic cobordism from the empty set to (M, ξ). In Section 5.2 we discuss the gluing of symplectic cobordisms, which proves the transitivity of the cobordism relation.

With this discussion we prepare the ground for the following chapter. There we show how to perform surgery on a given contact manifold by regarding it as the convex boundary component of a symplectic manifold and then attaching symplectic handles to that boundary.