In a previous post, we studied the formal group law of {\pi_* MU} in geometric terms: that is, using the interpretation of {\pi_* MU} as the cobordism ring of stably almost-complex manifolds. We found that the logarithm for this formal group law was given by the power series

\displaystyle \log x = \sum_{i \geq 0 } \frac{[\mathbb{CP}^i]}{i+1} x^{i+1}.

In particular, we saw that the complex projective spaces provided a set of independent generators for the rationalization {\pi_* MU \otimes \mathbb{Q}}: that is,

\displaystyle \pi_* MU \otimes \mathbb{Q} \simeq \mathbb{Q}[\{ [\mathbb{CP}^i] \}_{i >0}].

This is analogous to the theorem of Hirzebruch which calculates the oriented cobordism ring {\pi_* MSO \otimes \mathbb{Q}}, and could also have been established directly by arguing that {\pi_* MU \otimes \mathbb{Q} \simeq H_*(MU; \mathbb{Q})}. The structure of the latter ring can be worked out directly, and in fact was.

We might be interested, though, in a set of honest generators for {\pi_* MU} (not generators mod torsion). Such a set is provided by the Milnor hypersurfaces which I would like to discuss in this post. (more…)

In earlier posts, we analyzed the Hurewicz homomorphism

\displaystyle \pi_* MU \rightarrow H_*(MU; \mathbb{Z}) \simeq \mathbb{Z}[b_1, b_2, \dots ]

in purely algebraic terms: {\pi_* MU} was the Lazard ring, and the Hurewicz map was the map classifying the formal group law {\exp( \exp^{-1}(x) + \exp^{-1}(y))} for {\exp(x)} the “change of coordinates”

\displaystyle \exp(x) = x + b_1 x + b_2 x^2 + \dots.

We can also express the Hurewicz map in more geometric terms. The ring {\pi_* MU} is, by the Thom-Pontryagin construction, the cobordism ring of stably almost-complex manifolds. The ring {H_*(MU; \mathbb{Z})} is isomorphic to the Pontryagin ring {H_*(BU; \mathbb{Z})} by the Thom isomorphism. In these terms, we can describe the Hurewicz map explictly:

Theorem 1 The Hurewicz map {\pi_* MU \rightarrow H_*(MU; \mathbb{Z}) \simeq H_*(BU; \mathbb{Z})} sends the cobordism class of a stably almost complex manifold {M} to the element {f_* [M]} where

\displaystyle f_* : M \rightarrow BU

classifies the stable normal bundle of {M} (together with its complex structure), and where {[M]} is the fundamental class of {M}.

In particular, we will be able to work out explicitly where a given complex manifold representing a cobordism class goes under this map. As an application, we’ll show using Lagrange inversion that the complex projective spaces determine the logarithm of the formal group law on {\pi_* MU}. (more…)

Our goal is now to return to topology, and in particular to study the formal group law of the universal complex-oriented theory {MU} (complex cobordism). As we computed using the Adams spectral sequence,

\displaystyle \pi_* MU \simeq \mathbb{Z}[x_1, x_2, \dots ] , \quad \deg x_i = 2i.

This is the Lazard ring, by the computations of the previous couple of posts. On the other hand, it is not at all clear that the map

\displaystyle L \rightarrow \pi_* MU

classifying the formal group law over {\pi_* MU} (arising from the complex orientation) is actually an isomorphism: in other words, that the formal group law of {MU} is the universal one. The fact that it is in fact an isomorphism is the content of Quillen’s theorem, which will be proved in this post. (more…)

We finally have all the computational tools in place to understand Milnor’s computation of {\pi_* MU}, and the goal of this post is to complete it. Let’s recall what we have done so far.

  1. We described the Adams spectral sequence, which ran

    \displaystyle \mathrm{Ext}^{s,t}_{\mathcal{A}_p^{\vee}}(\mathbb{Z}/p, H_*(MU; \mathbb{Z}/p)) \implies \widehat{\pi_{t-s}(MU)},

    where the hat denotes {p}-adic completion.

  2. We worked out the homology of {MU}. {H_*(MU; \mathbb{Z}/p)}, as a comodule over {\mathcal{A} _p^{\vee}}, is {P \otimes \mathbb{Z}/p[y_i]_{i + 1 \neq p^k}} where {P} is a suitable subHopf-algebra of {\mathcal{A}_p^{\vee}}.When {p = 2}, we had that {P = \mathbb{Z}/p[\zeta_1^2, \zeta_2^2, \dots ] \subset \mathcal{A}_2^{\vee}}. When {p} is odd, {P = \mathbb{Z}/p[\zeta_1, \zeta_2, \dots]}.
  3. We worked out a general “change-of-rings isomorphism” for {\mathrm{Ext}} groups.

Now it’s time to put these all together. The {E_2} page of the Adams spectral sequence for {MU} is, as a bigraded algebra,

\displaystyle \mathrm{Ext}^{s,t}_{\mathcal{A}_p^{\vee}}(\mathbb{Z}/p, P ) \otimes \mathbb{Z}/p[y_i]_{i + 1 \neq p^k}.

The polynomial ring can just be pulled out, since it’s not relevant to the comodule structure. Consequently, the {y_i} are in bidegree {(0, 2i)}: the first bidegree comes from the {\mathrm{Ext}}, and the second is because we are in a graded category. The {y_i} only contribute in the second way.

Now, by the change-of-rings result from last time, we have the {E_2} page:

\displaystyle E_2^{s,t} = \mathrm{Ext}^{s,t}_{\mathcal{A}_p^{\vee} // P}(\mathbb{Z}/p, \mathbb{Z}/p ) \otimes \mathbb{Z}/p[y_i]_{i + 1 \neq p^k}.\ \ \ \ \ (1)

This is actually an isomorphism of bigraded algebras. It’s not totally obvious, but the Adams spectral sequence is a spectral sequence of algebras for a ring spectrum like {MU}.

Now observe that {\mathcal{A}_p^{\vee} // P} is an exterior algebra {E} on the generators {\tau_0, \tau_1, \dots} for {p} odd, and on the {\zeta_i} for {p = 2}. Using the formulas for the codiagonal in {\mathcal{A}_p^{\vee}} in this post, we find that the generators for {E} are primitive.


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…)