---
tag: C02WQY819R
kind: concept
name: Cobordism ring
msc: [57R75, 55N22]
sources:
  - url: https://en.wikipedia.org/wiki/Cobordism_ring
same_as:
  wikidata: Q19596423
  wikipedia: Cobordism ring
---

## Summary

The cobordism ring is the graded ring whose elements in degree $n$ are [[cobordism|T8BVGPPSZP]] classes of closed smooth $n$-manifolds.
Its addition is disjoint union and its multiplication is cartesian product.
There are versions for unoriented, oriented, complex and other structured manifolds.
Thom computed the unoriented ring and showed that cobordism groups are homotopy groups of Thom spectra.
This brought homotopy theory into the classification of manifolds.

## Explanations

### L1 human

Two closed manifolds are cobordant if together they form the boundary of a compact manifold one dimension higher.
For example, two circles are cobordant, since they bound a cylinder.
Cobordism classes can be added by putting manifolds side by side and multiplied by taking products.
The result is a ring that records which manifolds can be turned into one another by such a filling.