In mathematics, cobordism is a fundamental equivalence relation on compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds are cobordant if their 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 W is an n-dimensional manifold partial W which 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 closed manifolds and those which are boundaries. The theory was originally developed for smooth manifolds, but there are now also versions for piecewise-linear and topological manifolds. By definition, two closed n-dimensional manifolds M,N are cobordant if the disjoint union M sqcup N is the boundary partial W of a compact n + 1-dimensional manifold W. A cobordism is a manifold W with boundary whose boundary is partitioned in two, partial W=M sqcup N.

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