In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a filling of a manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

 X is a cobordism

Cobordism

In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds are cobordant if their disjoint union is the boundary of a manifold one dimension higher. The name comes...

 W between X and the empty set

Empty set

In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...

. More to the point, the n-dimensional topological manifold

Topological manifold

In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...

 X is the boundary

Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...

 of an n+1-dimensional manifold W. Perhaps the most active area of current research is when , where one may consider certain types of fillings.

There are many types of fillings, and a few examples of these types (within a probably limited perspective) follow.

• An oriented
  Orientation (mathematics)
  In mathematics, orientation is a notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary dimensions...
  
   filling of any orientable manifold X is another manifold W such that the orientation of X is given by the boundary orientation of W, which is the one where the first basis vector of the tangent space
  Tangent space
  In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
  
   at each point of the boundary is the one pointing directly out of W, with respect to a chosen Riemannian metric. Mathematicians call this orientation the outward normal first convention.

All the following cobordisms are oriented, with the orientation on W given by a symplectic structure. Let ξ denote the kernel

Kernel (mathematics)

In mathematics, the word kernel has several meanings. Kernel may mean a subset associated with a mapping:* The kernel of a mapping is the set of elements that map to the zero element , as in kernel of a linear operator and kernel of a matrix...

 of the contact form α.

• A weak symplectic
  Symplectic manifold
  In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...
  
   filling of a contact manifold (X,ξ) is a symplectic manifold
  Symplectic manifold
  In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...
  
   (W,ω) with W = X such that .
• A strong symplectic filling of a contact manifold (X,ξ) is a symplectic manifold (W,ω) with W = X such that ω is exact near the boundary (which is X) and α is a primitive for ω. That is, ω = dα in a neighborhood of the boundary W = X.
• A Stein filling of a contact manifold (X,ξ) is a Stein manifold
  Stein manifold
  In mathematics, a Stein manifold in the theory of several complex variables and complex manifolds is a complex submanifold of the vector space of n complex dimensions. The name is for Karl Stein.- Definition :...
  
   W which has X as its strictly pseudoconvex boundary and ξ is the set of complex tangencies to X - that is, those tangent planes to X that are complex with respect to the complex structure on W. The canonical example of this is the 3-sphere
  3-sphere
  In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...
  
  

where the complex structure on is multiplication by in each coordinate and W is the ball {|x|<1} bounded by that sphere.

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. Further, it can be shown that each type of filling is an example of the one preceding it, so that a Stein filling is a strong symplectic filling, for example. 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).