---
tag: 4E518M24MA
kind: statement
role: theorem
name: Quillen's theorem on complex cobordism
msc: [55N22, 14L05]
uses: [5CZHFNY9F3, TNHP7XXYZT, BAW0T68GWK]
about: [5CZHFNY9F3, TNHP7XXYZT]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_cobordism
---

## Statement

Let $L$ be the Lazard ring, the commutative ring generated by elements $a_{ij}$, $i, j \ge 0$, subject to the relations that make $F_L(x, y) = \sum a_{ij} x^i y^j$ a [[formal group law|TNHP7XXYZT]].
Give $a_{ij}$ the degree $2(i + j - 1)$.
Let $MU$ be the [[complex cobordism|5CZHFNY9F3]] spectrum with its standard complex orientation, and let $F_{MU}$ be its formal group law over $MU_* = \pi_* MU$, in homological grading.
Let $\theta \colon L \to MU_*$ be the unique ring homomorphism that carries $F_L$ to $F_{MU}$.
Then $\theta$ is an isomorphism of graded rings.

## Remarks

Both rings are polynomial rings $\Z[x_1, x_2, \ldots]$ with one generator in each degree $2i$: this is Lazard's theorem for $L$ and the theorem of Milnor and Novikov for $MU_*$.
So, degree by degree, $\theta$ is a map between free abelian groups of the same finite rank, and it suffices to show that it is surjective.
Quillen proved surjectivity in 1969 with his geometric description of cobordism classes and with cohomology operations in cobordism.
Rationally, the logarithm of $F_{MU}$ is Mishchenko's series $\sum_{n \ge 0} \frac{[\C P^n]}{n + 1} x^{n + 1}$.

At work: a ring map $MU_* \to R$ is the same as a formal group law over $R$.
For instance, the additive law over $\Z$ corresponds to the map that kills all positive-degree classes, and the multiplicative law $x + y - xy$ over $\Z$ corresponds to the Todd genus.

## Explanations

### L1 human

Complex cobordism classifies manifolds with a complex structure on their stable normal bundle up to cobordism.
Formal group laws are an algebraic notion, power series that act like addition.
Quillen's theorem says that the ring of complex cobordism classes of manifolds is exactly the ring that classifies formal group laws.
It is a surprising bridge between geometry and algebra.

### L2 human

The theorem turns questions about $MU$ into questions about formal groups.
Since $MU_*MU = MU_*[b_1, b_2, \ldots]$, the pair $(MU_*, MU_*MU)$ is the Hopf algebroid that classifies formal group laws and their strict isomorphisms.
This identifies the $E_2$-page of the [[Adams–Novikov spectral sequence|D34QHZVPES]] with cohomology over the moduli of formal groups.
Localizing at a prime $p$ leads to Brown–Peterson cohomology and $p$-typical laws, and the height filtration of formal groups organizes chromatic homotopy theory.