2024-03-24   44113

Submanifolds and Boundaries

Article 93875 reviews the definitions of topological manifold and smooth manifold for manifolds that don't have boundaries. This article explains how those definitions may be extended to allow boundaries. This article also introduces the concept of a submanifold, a manifold S that is a subset of another manifold M with a special relationship between the (topological or smooth) structures of S and M. If M is an n-dimensional manifold and S is an (n-2)-dimensional submanifold without boundary, then S may or may not be the boundary of an (n-1)-dimensional submanifold Σ of M. When such a Σ exists, it is called a Seifert hypersurface for S. This article uses the concept of a Seifert hypersurface to define the linking number of S with a given a closed loop in M. This generalizes the more familiar concept of linking number between two closed loops (knots) when n=3.

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