Cobordism

Summary

Two closed manifolds of the same dimension are cobordant if together they form the boundary of a compact manifold of one dimension more. Cobordism is an equivalence relation, much coarser than diffeomorphism, and its classes form groups under disjoint union. The unoriented groups, and the cobordism ring they form under products, were computed by René Thom through the Thom spaces of universal bundles. Cobordism links differential topology with stable homotopy theory, and its variants, oriented, complex or framed, are generalized homology theories.