Cobordism

Cobordism is a relation among smooth manifolds that groups manifolds according to whether they bound a higher-dimensional manifold. More precisely, two closed smooth n-manifolds M and N are cobordant if there exists a compact (n+1)-manifold W with boundary ∂W that is the disjoint union of M and N, after assigning orientations so that the boundary orientation matches the given orientations on M and N (often written ∂W ≅ M ⊔ N with M and N having opposite induced orientations).

The notion is studied in both oriented and unoriented forms. In oriented cobordism, the manifolds M and N come with orientations, and the cobordism W provides an oriented bounding manifold; in unoriented cobordism, orientations are not required. The cobordism classes of dimension n form graded abelian groups under disjoint union, with the zero element represented by the empty manifold and inverses given by orientation reversal in the oriented case.

Key examples illustrate the idea: the n-sphere S^n is null-cobordant because it bounds the (n+1)-disk D^{n+1}; similarly, many familiar closed manifolds bound simple products or handlebodies. Beyond basic definitions, cobordism theory has a deep homotopy-theoretic realization. The Pontryagin–Thom theorem identifies cobordism classes with homotopy groups of Thom spaces, making cobordism into a generalized homology theory. Variants include complex cobordism, spin cobordism, and others, yielding groups often denoted Ω_n^U, Ω_n^SO, Ω_n^O, etc.

Cobordism invariants help distinguish classes. Stiefel–Whitney numbers, Pontryagin numbers, and the L- and Chern numbers give obstructions to cobounding manifolds in oriented or complex settings. These ideas underpin many structural results and computations in topology, homotopy theory, and surgery theory.