Our goal is now to return to topology, and in particular to study the formal group law of the universal complex-oriented theory (complex cobordism). As we computed using the Adams spectral sequence,

This is the Lazard ring, by the computations of the previous couple of posts. On the other hand, it is not at all clear that the map

classifying the formal group law over (arising from the complex orientation) is actually an isomorphism: in other words, that the formal group law of is the universal one. The fact that it *is* in fact an isomorphism is the content of Quillen’s theorem, which will be proved in this post.

**1. The Hurewicz map**

There is a Hurewicz map

In particular, the map leads to a map classifying a formal group law over . This looks suspiciously like the map earlier, and in fact it is the same thing. Once we know this, we’ll be able to get a good handle on the formal group law for , because and are close.

So we have:

**Claim**: The formal group law on classified by the map is the formal group law where .

We will in fact prove something more general. Let be a complex-oriented ring spectrum. Then the smash product is a complex-oriented ring spectrum in *two different ways*:

- There is a complex orientation coming from the ring-spectra morphism and the complex orientation of .
- There is a complex orientation coming from the ring-spectra morphism and the complex orientation of .

Each of these classify two different formal group laws over the ring . The formal group law is just the formal group law of . The formal group law is a little different.

**Claim**: the formal group law over is obtained from by the “change of coordinates” .

In particular, smashing a complex-oriented spectrum with can be thought of as a universal change of coordinates of the associated formal group law.

This is, of course, a generalization of the previous claim: in that case, and the formal group law of is the additive one.

**2. The proof of the claim**

Let’s now try to prove this claim. Let be a complex-oriented ring spectrum. We know then that

for any choice of complex orientation of (that is, an element of ). Of course, we have two such choices of :

- We have , which comes from the orientation of .
- We have , which comes from the orientation of .

The two formal group laws on are both manifestations of the map

and just depend on the choice of complex orientation to be expressed in terms of a formal power series.

So, in other words, our goal is to express in terms of . Our goal is to prove that

If we prove this, it will follow that the two formal group laws over are connected by the “change of coordinates” , which will prove the claim.

How might we evaluate in terms of ? The strategy is to pair against classes in . This is spanned by classes . We choose the as coming the complex orientation of : that is, they are the image of the basis of under the map

In other words, is the dual basis to .

So, let’s pair against the classes . First, we should recall what “pairing” means. The class is represented by the map

of degree two. Each is represented by a map

In order to do the pairing, we have to form

where the last map is multiplication. Because both and came from maps and , we find that the big composite is equal to

where is identified with the analogous element in and with the analogous element in .

But this is just saying that is the *image* of under the map (of degree two) in -homology. In other words,

because the are precisely the image of the basis of in . Since is the dual basis in to , we get

This is precisely the claim about the generator .

As we said before, this means that the formal group law of in -coordinates is just a twist of that of the FGL of , under the power series .

**3. Quillen’s theorem**

Now we are almost at the end of the proof of Quillen’s theorem. What do we know?

- is a polynomial ring . There is a map classifying the formal group law of .
- There is a Hurewicz map . The map classifies the formal group law where .
- We have studied the map before: in particular, we have seen that it is an injection, and determined that, on indecomposables, the map
(where denotes indecomposables) is an isomorphism if is not a power of a prime, and is given by multiplication by if .

Observe now that the map is injective, and its image *contains* the image of the Lazard ring . If we show, conversely, that the image of is *contained in* the image of the Lazard ring, then we will be done. We can check this on indecomposables.

In particular, we are reduced to showing that if for a prime , then the map induces

whose image is contained in .

*Proof:* It is equivalent to say that the mod Hurewicz homomorphism

is zero. But this actually follows from the Adams spectral sequence for done a few blog posts back. In fact, we saw that in the ASS converging to , the indecomposables in degree had Adams filtration (i.e., in the spectral sequence) at least one. For a class to have Adams filtration is precisely to say that it is annihilated by the Hurewicz map—this is a consequence of how one sets up the spectral sequence with Adams resolutions.

So we can finally state:

Theorem 1 (Quillen)The formal group law of is the universal one.

It’s sort of striking, actually, how computational the proof of Quillen’s theorem was, when the result itself is very conceptual: that the universal complex-oriented theory should correspond to the universal formal group law. However, there appears to be no non-computational proof, even though, for instance, a construction of itself from formal group laws would be extremely awesome.

May 28, 2012 at 10:11 pm

[…] purely algebraic terms: was the Lazard ring, and the Hurewicz map was the map classifying the formal group law for the “change of […]