nLab Milnor-Quillen theorem on MU

Idea

The Milnor-Quillen theorem on $MU$ determines the structure of the graded ring $\pi_{\bullet}(MU)$ of stable homotopy groups of the universal complex Thom spectrum MU.

In (Milnor 60) it was shown this is the polynomial ring $\mathbb{Z}[y_1, y_2, \cdots]$ on generators $y_{n+1}$ in degree $2(n+1)$ for all $n \in \mathbb{N}$.

Notice that by Thom's theorem, this is also isomorphic to the cobordism ring $\Omega_\bullet^U \simeq \pi_\bullet(M U)$ of smooth manifolds equipped with stable almost complex structure.

Moreover, by Lazard's theorem, this graded ring is also abstractly isomorphic to the Lazard ring $L$.

But the universal complex orientation on MU induces a preferred ring homomorphism

$$L \longrightarrow \pi_\bullet(M U) \simeq \Omega_\bullet^U \simeq \mathbb{Z}[y_1, y_2, \cdots] \,.$$

A priori it is not clear whether this particular canonical homomorphism exhibits the isomorphism. But it does, this is the result of (Quillen 69).

Statement

Write $MU$ for the E-∞ ring spectrum of complex cobordism cohomology theory. Since this is a complex oriented cohomology theory, by Lazard's theorem there is associated a commutative 1-dimensional formal group law classified by a ring homomorphism of the form

$$L \longrightarrow \pi_\bullet(MU)$$

from the Lazard ring $L$.

Theorem

This canonical homomorphism is an isomorphism

$$L \stackrel{\simeq}{\longrightarrow} \pi_\bullet(MU) \,.$$

This is due to (Quillen 69), based on (Milnor 60), reproduced e.g. as (Kochman 96, theorem 3.7.7, theorem 4.4.13).

Proof strategy: (for Milnor’s part)

Apply the Boardman homomorphism to get the statement over the rational numbers. Deduce that $\pi_\bullet(MU)$ is finitely generated so that it is now sufficient to prove it over the p-adic integers for all $p$.

Now use the $H\mathbb{F}_p$-Adams spectral sequence, which, on its second page, expresses these homotopy groups by the $H\mathbb{F}_p$-homology of MU.

The homology of MU may be computed by reducing, via the Thom isomorphism, to computation of the homology of the classifying space $B U$, which in turn is given by Kronecker pairing from the Conner-Floyd Chern classes.

Using this and applying the change of rings theorem, the Adams spectral sequence is seen to collapse right away, and so the result may now be obtained by explicitly computing the relevant comodule Ext-groups of the homology of MU

Remark

The dual generalized Steenrod algebra $MU_\bullet(MU)$ has a structure of commutative Hopf algebroid over the Lazard ring. This is the content of the Landweber-Novikov theorem.

Related theorems

• Lazard's theorem

• Landweber exact functor theorem

• Landweber-Novikov theorem

References

The computation of $\pi_\bullet(M U)$ was first due to

• John Milnor, On the Cobordism Ring ft* and a Complex Analogue, Amer. J. Math., 82 (1960), 505-521.

the proof that the canonical morphism $L \to \pi_\bullet(M U)$ is an isomorphism is due to

• Daniel Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75 (1969), 1293–1298. (Euclid)

According to

• Mike Hopkins, section 4 of Complex oriented cohomology theories and the language of stacks, course notes 1999 (pdf)

the best reference as of turn of the millennium was still

• Frank Adams, part II, section 6-8 of Stable homotopy and generalised homology, Chicago lectures in mathematics, 1974

But see

• Stanley Kochman, section 4.4 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010,

  Lecture 7 The homology of MU (pdf)

  Lecture 9 The Adams spectral sequence for MU (pdf)

  Lecture 10 The proof of Quillen’s theorem (pdf)

Other review includes

• Stanley Kochman, theorem 3.7.7 (Milnor’s theorem) of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

• Doug Ravenel, Complex cobordism and stable homotopy groups of spheres, chapter 4 $B P$-Theory and the Adams-Novikov spectral sequence, pdf

also

• Charmaine Sia, section 2 of Calculating the $E_2$-term of the Adams spectral sequence (pdf)

• Sam Nolen, sections 1 and 2 of The Adams-Novikov spectral sequence (pdf)

• Akhil Mathew, Quillen’s theorem on the formal group law of MU