Two oriented -manifolds and are called cobordant if there is an oriented manifold with boundary such that where is with orientation reversed. The triple is called a oriented cobordism. Cobordism is an equivalence relation, and a very coarse invariant of manifolds. For example, all surfaces are cobordant to the empty set (and hence to each other).

There is a cobordism category, where the objects are manifolds, and the morphisms are cobordisms between them. This category is important in topological quantum field theory.