Cobordism equivalence relation - Imported Knowlpedia Content

Definition

Fix a dimension $n$ and a category of closed smooth manifolds, possibly carrying a specified structure such as orientation. Two $n$ -manifolds $M_0$ and $M_1$ are cobordism equivalent, written $M_0\sim_{\mathrm{cob}}M_1$ , if there exists an $(n+1)$ -dimensional cobordism from $M_0$ to $M_1$ over which the specified structure extends. For unoriented smooth cobordism this means a compact $W$ with $\partial W\cong M_0\sqcup M_1$ . For oriented cobordism, the boundary identification must instead respect $\partial W\cong(-M_0)\sqcup M_1$ . The dimension and structural category are fixed before the relation is formed.

Proof that it is an equivalence relation

Reflexivity is represented by the cylinder $M\times[0,1]$ . Symmetry follows by interchanging the incoming and outgoing boundary designations; in the oriented theory one also tracks the induced signs. For transitivity, glue a cobordism $W_{01}$ to $W_{12}$ along their common copy of $M_1$ . The collar neighborhood theorem supplies product neighborhoods that make this gluing smooth Stong, Chapter I.

Classes and null-cobordism

A manifold is null-cobordant if it is cobordism equivalent to the empty manifold, equivalently if it is the entire boundary of a compact manifold with the relevant structure. Disjoint union descends to cobordism classes. In oriented and many structured theories, these classes form abelian groups because orientation reversal or the corresponding inverse structure supplies additive inverses.

Dependence on structure

The relation changes when the permitted structure changes. Two oriented manifolds may be cobordant after forgetting their orientations but fail to be oriented-cobordant. Framed, spin, and complex cobordism similarly require the chosen tangential structure to extend across $W$ . Thus “cobordant” is incomplete unless the ambient bordism theory is understood.

Examples

Every sphere is unoriented and oriented null-cobordant because it bounds a disk with the induced structure. The boundary of any compact manifold is null-cobordant tautologically. Diffeomorphic closed manifolds are cobordant via a cylinder, but cobordism is much coarser than diffeomorphism: a cobordism need not be a product.

References

Robert E. Stong, Notes on Cobordism Theory, Princeton University Press, 1968. Publisher record. Relevant: Chapter I, the cobordism relation and bordism groups.
John Milnor, Topology from the Differentiable Viewpoint, Princeton University Press, revised ed., 1997. DOI record. Relevant: §7, framed cobordism and the geometric use of cobordisms.