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 […]