---
tag: 5CZHFNY9F3
kind: concept
name: Complex cobordism
msc: [55N22, 57R77]
specializes: [H28PSZR7XG]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_cobordism
same_as:
  wikidata: Q5156571
  wikipedia: Complex cobordism
---

## Summary

Complex cobordism $MU^*$ is the [[generalized cohomology theory|H28PSZR7XG]] represented by the Thom spectrum $MU$ of the universal complex vector bundles.
Its coefficient ring is the cobordism ring of stably complex manifolds, a polynomial ring over $\Z$ with one generator in each positive even degree.
By a theorem of Quillen this ring carries the universal formal group law.
This link makes complex cobordism the central theory of modern stable homotopy theory.

## Explanations

### L1 human

On a point, complex cobordism is built from closed manifolds with a complex structure on their stable normal bundle.
Two such manifolds give the same class when together they bound such a manifold.
The theory extends this count of manifolds to a cohomology theory defined for all spaces.
It is the most refined of the complex-oriented cohomology theories, in the sense that it maps to each of them.
For example, the coefficient group $\pi_2(MU)$ is $\Z$, generated by the Riemann sphere $\C P^1$.