Two oriented $n$ -manifolds $M$ and $M'$ are called cobordant if there is an oriented $n+1$ manifold with boundary $N$ such that $\partial N=M\coprod M'^{opp}$ where $M'^{opp}$ is $M'$ with orientation reversed. The triple $(N,M,M')$ 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.