Date: 2024-03-24 | Article ID: 44113 |

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.

**2024-03-24**(corrected spelling Steifel -> Stiefel, added a reference about Stiefel-Whitney numbers and boundaries, added a reference about another definition of linking number)**2024-02-27**(removed an unused reference)**2024-02-25**(first version)

www.cphysics.org | © 2018-2024 Randy S For noncommercial use only |
updated 2024-04-08 |