In mathematics, a filling of a manifold X is a cobordism W between X and the empty set.
More to the point, the n-dimensional topological manifold X is the boundary of an (n + 1)-dimensional manifold W. Perhaps the most active area of current research is when n = 3, where one may consider certain types of fillings.
Let ξ denote the kernel of the contact form α.
It is known that this list is strictly increasing in difficulty in the sense that there are examples of contact 3-manifolds with weak but no strong filling, and others that have strong but no Stein filling.
It used to be that one spoke of semi-fillings in this context, which means that X is one of possibly many boundary components of W, but it has been shown that any semi-filling can be modified to be a filling of the same type, of the same 3-manifold, in the symplectic world (Stein manifolds always have one boundary component).