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 from the French word bord for boundary. The boundary of an (n + 1)-dimensional manifold is an -dimensional manifold that is closed, i.e., with empty boundary. In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed for smooth (i.e., differentiable) manifolds, but there are now also versions for piecewise-linear and topological manifolds.