Finally, it’s time to try to understand the computation of the cobordism ring ${\pi_* MU}$. This will be the first step in understanding Quillen’s theorem, that the formal group law associated to ${MU}$ is the universal one. We will compute ${\pi_* MU}$ 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 $\pi_* MU$.

1. The homology of ${MU}$

The starting point for all this is, of course, the homology ${H_*(MU)}$, which is a ring since ${MU}$ is a ring spectrum. (In the past, I had written reduced homology ${\widetilde{H}_*(MU)}$ for spectra, but I will omit it now; recall that for a space ${X}$, we have ${\widetilde{H}_*(X) = H_*(\Sigma^\infty X)}$.)

Anyway, let’s actually do something more general: let ${E}$ be a complex-oriented spectrum (which gives rise to a homology theory). We will compute ${E_*(MU) = \pi_* E \wedge MU}$.

Proposition 1 ${E_*(MU) = \pi_* E [b_1, b_2,\dots]}$ where each ${b_i}$ has degree ${2i}$.

The proof of this will be analogous to the computation of ${H_* (MO; \mathbb{Z}/2)}$. In fact, the idea is essentially that, by the Thom isomorphism theorem,

$\displaystyle E_*(MU) = E_*(BU) \simeq \pi_* E [b_1, \dots, ]$

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

(This is the second in a series of posts intended for me to try to understand the connection between stable homotopy theory and formal group laws.)

Last time, we introduced the notion of a complex-oriented cohomology theory and saw that we could imitate the classical theory of Chern classes in one such. In this post, I’d like to describe the universal example of a complex-oriented cohomology theory: complex cobordism. This is going to play a very special role in the next few posts.

To unravel this, let’s try to recall what a complex orientation was. It was a choice of Thom classes of complex vector bundles, functorial in the bundle and multiplicative. For starters, let’s focus now on the functoriality. Let ${E}$ be a cohomology theory represented by a spectrum ${E}$. Then since there is a universal ${n}$-dimensional vector bundle ${\zeta_n \rightarrow BU(n)}$, it follows that a functorial choice of Thom classes for ${n}$-dimensional vector bundles is the same as a Thom class for ${\zeta_n}$. So, all we need to give is an element of ${\widetilde{E}^*(T(\zeta_n))}$. If we (and we henceforth do this) normalize things such that the Thom class of the ${n}$-th degree element is in degree ${n}$, then we have to give an element of ${\widetilde{E}^n(T(\zeta_n))}$.

Definition 1 The spectrum ${MU(n)}$ is ${\Sigma^{-2n}T(\zeta_n)}$.

So another way of saying this is that we should have a map of spectra

$\displaystyle MU(n) \rightarrow E.$

There is a map ${S^{2n} \rightarrow T(\zeta_n)}$ which comes from fixing a basepoint in ${BU(n)}$. So, in other words, to give a functorial complex orientation for ${n}$-dimensional complex vector bundles is to give an element of ${\widetilde{E}^n(T(\zeta_n))}$ which restricts to the generator of ${\widetilde{E}^n(S^{2n})}$. (To check that an element is a Thom class, we only need restrict it to one fiber in each connected component of the base.) (more…)