Finally, it’s time to try to understand the computation of the cobordism ring . This will be the first step in understanding Quillen’s theorem, that the formal group law associated to is the universal one. We will compute using the Adams spectral sequence.

In this post, I’ll set up what we need for the Adams spectral sequence, which is a little bit of algebraic computation. In the next post, I’ll describe the actual calculation of the spectral sequence, which will complete the description of .

**1. The homology of **

The starting point for all this is, of course, the *homology* , which is a ring since is a ring spectrum. (In the past, I had written reduced homology for spectra, but I will omit it now; recall that for a space , we have .)

Anyway, let’s actually do something more general: let be a complex-oriented spectrum (which gives rise to a homology theory). We will compute .

Proposition 1where each has degree .

The proof of this will be analogous to the computation of . In fact, the idea is essentially that, by the Thom isomorphism theorem,

where the last equality is because is complex-oriented, and consequently the -homology of looks like the ordinary homology of it. (more…)