oriented cobordism

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