Edmund H. North was born in New York on March 12, 1911.
In mathematics , a filling of a manifold X is a cobordism W between X and the empty set .. mathematics. mathematics. manifold. manifold. cobordism. cobordism. empty set. empty set. More to the point , the n-dimensional topological manifold X is the boundary of an -LRB- n + 1 -RRB- - dimensional manifold W. Perhaps the most active area of current research is when n = 3 , where one may consider certain types of fillings .. manifold. manifold. topological manifold. topological manifold. boundary. boundary ( topology ). There are many types of fillings , and a few examples of these types -LRB- within a probably limited perspective -RRB- follow .. An oriented 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 at each point of the boundary is the one pointing directly out of W , with respect to a chosen Riemannian metric .. manifold. manifold. boundary. boundary ( topology ). oriented. orientation ( mathematics ). tangent space. tangent space. Riemannian metric. 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 .. oriented. orientation ( mathematics ). symplectic. symplectic manifold. Let ξ denote the kernel of the contact form α .. kernel. kernel ( linear algebra ). contact form. contact form. A weak symplectic filling of a contact manifold -LRB- X , ξ -RRB- is a symplectic manifold -LRB- W , ω -RRB- with W = X such that .. manifold. manifold. symplectic. symplectic manifold. contact manifold. contact manifold. symplectic manifold. symplectic manifold. A strong symplectic filling of a contact manifold -LRB- X , ξ -RRB- is a symplectic manifold -LRB- W , ω -RRB- with W = X such that ω is exact near the boundary -LRB- which is X -RRB- and α is a primitive for ω .. manifold. manifold. boundary. boundary ( topology ). symplectic. symplectic manifold. contact manifold. contact manifold. symplectic manifold. symplectic manifold. exact. exact differential form. That is , ω = dα in a neighborhood of the boundary W = X.. boundary. boundary ( topology ). neighborhood. neighborhood ( mathematics ). A Stein filling of a contact manifold -LRB- X , ξ -RRB- is a Stein manifold 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 .. manifold. manifold. boundary. boundary ( topology ). contact manifold. contact manifold. Stein manifold. Stein manifold. strictly pseudoconvex boundary. strictly pseudoconvex boundary. The canonical example of this is the 3-sphere. 3-sphere. 3-sphere. where the complex structure on is multiplication by in each coordinate and W is the ball x  |  < 1 -RCB- 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 .. symplectic. symplectic manifold. 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 -LRB- Stein manifolds always have one boundary component -RRB- .. manifold. manifold. boundary. boundary ( topology ). symplectic. symplectic manifold. boundary component. boundary component