---
tag: 3JAY4V0GAF
kind: definition
name: Complex cobordism
msc: [55N22, 57R77]
defines: 5CZHFNY9F3
uses: [WSZ40H5E4T, 1HDH716Z24, 5MNYT3AFPH, 1GHF4EKBXS, 6H8AGEN6NY, H28PSZR7XG, T8BVGPPSZP]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_cobordism
---

## Definition

For $n \ge 0$, let $BU(n)$ be the [[classifying space|WSZ40H5E4T]] of the unitary group $U(n)$.
Let $\gamma_n \to BU(n)$ be the [[universal|1HDH716Z24]] complex vector bundle of rank $n$, and let $MU(n)$ be its [[Thom space|5MNYT3AFPH]].
The bundle $\gamma_n \oplus \C$ is classified by a map $BU(n) \to BU(n+1)$ covered by a bundle map $\gamma_n \oplus \C \to \gamma_{n+1}$, which induces a map $\Sigma^2 MU(n) \to MU(n+1)$ of Thom spaces.
The *complex cobordism spectrum* $MU$ is the [[spectrum|1GHF4EKBXS]] with
$$MU_{2n} = MU(n), \qquad MU_{2n+1} = \Sigma MU(n),$$
whose structure maps are the identity $\Sigma MU_{2n} = MU_{2n+1}$ and the map $\Sigma MU_{2n+1} = \Sigma^2 MU(n) \to MU(n+1) = MU_{2n+2}$.
*Complex cobordism* is the [[generalized cohomology theory|H28PSZR7XG]] represented by $MU$: for a pointed CW complex $X$ and $k \in \Z$,
$$\widetilde{MU}^k(X) = \operatorname{colim}_n \, [\Sigma^n X, MU_{n+k}]_*.$$

## Remarks

$MU$ is a [[Thom spectrum|6H8AGEN6NY]], and it is a commutative ring spectrum, with multiplication induced by the Whitney sum maps $BU(n) \times BU(m) \to BU(n+m)$.
By the Pontryagin–Thom construction, $\pi_n(MU)$ is the [[cobordism|T8BVGPPSZP]] group of closed smooth $n$-manifolds with a complex structure on the stable normal bundle.
Milnor and Novikov showed that $\pi_*(MU) \cong \Z[x_1, x_2, \ldots]$ with $\deg x_i = 2i$; in particular $\pi_n(MU) = 0$ for odd $n$.

Example: $\pi_2(MU) \cong \Z$, generated by the class of $\C P^1$ with its complex structure.

Non-example: $\R P^2$ is a closed smooth surface, but it is not orientable, so it has no stably complex structure and gives no class in $\pi_2(MU)$.