---
tag: T8BVGPPSZP
kind: concept
name: Cobordism
msc: [57R75, 55N22]
sources:
  - url: https://en.wikipedia.org/wiki/Cobordism
same_as:
  wikidata: Q2546666
  wikipedia: 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|C02WQY819R]] they form under products, were computed by René Thom through the [[Thom spaces|5MNYT3AFPH]] of universal bundles.
Cobordism links differential topology with stable homotopy theory, and its variants, oriented, complex or framed, are generalized homology theories.

## Explanations

### L1 human

Imagine two closed surfaces as the two ends of a solid piece; the piece is a cobordism between them.
A circle and a pair of circles are cobordant: a pair of pants has one waist and two leg openings, all circles.
A single circle is the boundary of a disc, so it is cobordant to the empty manifold.
Cobordism ignores many details of shape and keeps only what survives such sweeping.