Before moving on, I’d like to work out the analog for *real-oriented* cohomology theories (which is tangential to the rest of the story, though). This is considerably less interesting, but perhaps it’s a toy example of the ideas explained in the last few posts without the full-blown machinery of the Adams spectral sequence and so forth.

So, let’s state what the analogous ideas are in the real-oriented context:

- A
**real-oriented ring spectrum**is a ring spectrum together with a functorial, multiplicative choice of Thom classes for*real*vector bundles; equivalently, there is a morphism of ring spectraThat is, the universal real-oriented ring spectrum is unoriented cobordism, whose homotopy groups can be completely computed.

- If is real-oriented, then is a -vector space, and all the usual computations of and so forth work just fine for , and there is a theory of Stiefel-Whitney classes in -cohomology.
- Given a real-oriented spectrum , we thus have where is the Stiefel-Whitney class of the tautological bundle. Similiarly, .

Since is the classifying space for real line bundles, there is a monoidal product

classifying the tensor product of line bundles. As before, this means that we can extract a formal group law over . This formal group law has the property that if are two line bundles over a finite-dimensional space , then in ,

So far everything has been analogous to the complex-oriented case, but there is an extra feature here which changes the picture drastically. Namely, when one works with *real*line bundles , we have that is always trivial. (The analog is very false for complex line bundles.) This means that the formal group law must satisfy

This is not satisfied by, say, the multiplicative formal group law. (And in fact, -theory—the natural candidate for this—is not real-oriented, only spin-oriented.)

With this in mind, the analog of Quillen’s theorem for becomes:

Theorem 1 (Quillen)The formal group law for is the universal formal group law over a satisfying .

The condition that implies, for instance, that the base ring of such a formal group law is a -algebra. It is equivalent to the condition that have *infinite height*, and implies in particular that is isomorphic to the additive formal group law. This is why the theory of is so much richer.

How can we prove such a result? Let be the ring classifying the universal formal group law satisfying ; then there is a map

for every real-oriented spectrum . We would like to prove that it is an isomorphism when . Fortunately, we already know that

and so we will determine what looks like, in analogy with the analysis of the Lazard ring.

**1. The universal ring**

There is a map

classifying the formal group law over ; since we have introduced -coefficients this formal group law satisfies . In order to understand , we will do something familiar: we will study the map on indecomposables induced.

As before, this is equivalent to studying appropriately graded formal group laws over rings of the form , for a -module. In other words, we are interested in, for each , polynomials of the form

which define formal group laws over of infinite height.

Anyway, the condition that this be a formal group law implies that the polynomial is a symmetric 2-cocycle as in the sense of the previous posts. In particular, we find:

- If is a power of , then for some . Here the polynomial has -coefficients and is regarded as reduced mod 2.
- If is not a power of , then for some .

This is a consequence of the “symmetric 2-cocycle lemma” and the fact that all primes other than are invertible.

We have also seen that the FGL which arise from a change-of-coordinates in this way are those where is a multiple of (this follows from a direct computation).

But we have one more condition: we want that if our formal group law is of infinite height. In case 2 above, this is automatically satisfied. In case 1, it is not: we get . It follows that in case 1, must be zero, and there are *no*nontrivial formal group laws of that form.

Putting it together, we find:

Proposition 2

- if is not a power of and the map is an isomorphism.
- If is a power of , then .

Consequently, we find that

and the “algebraic Hurewicz map” is just given by adding new polynomial generators.

**2. Putting it together**

Now there’s not much left to do. We’ve given a complete description of the ring classifying formal group laws of infinite height at , and we know how looks—exactly the same. The thing to check is that the map

is actually an isomorphism.

As before, we can study this map by studying the composite

and using the same arguments as the previous time, we can argue that the FGL on classified by it is in fact . But since the image of in is the same as that of , and since is isomorphic as a graded vector space to , we find that the map is an isomorphism.

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