01 · The idea
In everyday language
A circle bounds a disk, while a cylinder has two circular boundary components. Cobordism turns such pictures into a systematic relation between manifolds: two closed manifolds are cobordant when their disjoint union, with the appropriate orientation convention, is the boundary of a higher-dimensional manifold. Thom connected this geometric relation to homotopy theory and computable characteristic numbers. Rather than classify every possible shape individually, one studies algebraic classes that capture what can be changed through a cobordism. This created a powerful bridge between geometric constructions and algebraic topology.
Why it matters
Cobordism became a central organising tool for manifolds and a model for generalised cohomology theories, influencing subsequent work throughout topology and geometry.
