Cobordant

Cobordant is a term used in differential topology to describe a relationship between manifolds. Two manifolds, M and N, are considered cobordant if there exists a larger manifold W, called a cobordism, whose boundary consists of two disjoint copies of M and N. More precisely, if M and N are both compact, oriented, (n)-dimensional manifolds, they are cobordant if there is a compact, oriented, (n+1)-dimensional manifold W such that its boundary $\partial W$ is a disjoint union of M and N, denoted as $\partial W = M \sqcup N$, and the orientation of M is induced from the orientation of W, while the orientation of N is the reverse of that induced from W.

The cobordism relation is an equivalence relation. Reflexivity is trivial, as any manifold M is cobordant to itself via the product manifold $M \times [0,1]$. Symmetry holds because if W is a cobordism between M and N, then reversing the orientation of W yields a cobordism between N and M. Transitivity is established by "gluing" cobordisms together.

The set of cobordism classes of n-dimensional manifolds forms a group under the operation of disjoint union, known as the cobordism group. Studying cobordism provides a powerful tool for classifying manifolds and understanding their topological properties. For instance, two manifolds are considered cobordant if they can be "connected" by a higher-dimensional manifold whose "ends" are precisely those two manifolds. This concept is fundamental in areas like geometric topology and algebraic topology.