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 . (more…)