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:

1. real-oriented ring spectrum ${E}$ is a ring spectrum together with a functorial, multiplicative choice of Thom classes for real vector bundles; equivalently, there is a morphism of ring spectra

$\displaystyle MO \rightarrow E.$

That is, the universal real-oriented ring spectrum is unoriented cobordism, whose homotopy groups can be completely computed.

2. If ${E}$ is real-oriented, then ${\pi_* E}$ is a ${\mathbb{Z}/2}$-vector space, and all the usual computations of ${H_*( BO; \mathbb{Z}/2)}$ and so forth work just fine for ${E}$, and there is a theory of Stiefel-Whitney classes in ${E}$-cohomology.
3. Given a real-oriented spectrum ${E}$, we thus have ${E^*(\mathbb{RP}^\infty) = \pi_* E [[t]]}$ where ${t}$ is the Stiefel-Whitney class of the tautological bundle. Similiarly, ${E^*(\mathbb{RP}^\infty \times \mathbb{RP}^\infty) = \pi_* E [[t_1, t_2]]}$.

Since ${\mathbb{RP}^\infty}$ is the classifying space for real line bundles, there is a monoidal product

$\displaystyle \mathbb{RP}^\infty \times \mathbb{RP}^\infty \rightarrow \mathbb{RP}^\infty,$

classifying the tensor product of line bundles. As before, this means that we can extract a formal group law over ${\pi_* E}$. This formal group law ${f(\cdot, \cdot) \in \pi_* E [[x, y]]}$ has the property that if ${\mathcal{L}_1, \mathcal{L}_2}$ are two line bundles over a finite-dimensional space ${X}$, then in ${E^*(X)}$,

$\displaystyle w_1( \mathcal{L}_1 \otimes \mathcal{L}_2) = f( w_1(\mathcal{L}_1), w_1(\mathcal{L}_2)).$

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 realline bundles ${\mathcal{L}}$, we have that ${\mathcal{L}^{\otimes 2}}$ is always trivial. (The analog is very false for complex line bundles.) This means that the formal group law ${f}$ must satisfy

$\displaystyle f(x, x) = 0.$

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

With this in mind, the analog of Quillen’s theorem for ${MU}$ becomes:

Theorem 1 (Quillen) The formal group law for ${MO}$ is the universal formal group law over a satisfying ${f(x, x) = 0}$. (more…)