nLab Milnor-Quillen theorem on MU





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The Milnor-Quillen theorem on MUMU determines the structure of the graded ring π (MU)\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 [y 1,y 2,]\mathbb{Z}[y_1, y_2, \cdots] on generators y n+1y_{n+1} in degree 2(n+1)2(n+1) for all nn \in \mathbb{N}.

Notice that by Thom's theorem, this is also isomorphic to the cobordism ring Ω Uπ (MU)\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 LL.

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

Lπ (MU)Ω U[y 1,y 2,]. 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).


Write MUMU 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

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

from the Lazard ring LL.


This canonical homomorphism is an isomorphism

Lπ (MU). 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 π (MU)\pi_\bullet(MU) is finitely generated so that it is now sufficient to prove it over the p-adic integers for all pp.

Now use the H𝔽 pH\mathbb{F}_p-Adams spectral sequence, which, on its second page, expresses these homotopy groups by the H𝔽 pH\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 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


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


The computation of π (MU)\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 Lπ (MU)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

But see

Other review includes


Last revised on March 4, 2024 at 23:06:09. See the history of this page for a list of all contributions to it.