The next goal of this series of posts is to understand the computation of the complex cobordism ring ${\pi_* MU}$. In the computation of the unoriented cobordism ring ${\pi_* MO}$, we did this by working out ${H_*(MO; \mathbb{Z}/2)}$ as a comodule over the dual Steenrod algebra ${\mathcal{A}_2^{\vee}}$, and observing that it was cofree. We were able to conclude that ${MO}$ is actually a direct sum of Eilenberg-MacLane spectra ${H \mathbb{Z}/2[i]}$, after which it was easy to get the homotopy groups.

${MU}$ is a little harder, because it does not split in the same way. We can still compute ${H_*(MU; \mathbb{Z}/p)}$ for each prime ${p}$, but it won’t exactly be cofree. So the strategy will be to approximate ${MU}$ by a sort of complex of cofree things. This is the idea of the Adams spectral sequence.

1. Motivation

Let ${X}$ be a connective spectrum. Suppose ${p}$ is a prime number, and that we know the cohomology ${H^*(X; \mathbb{Z}/p)}$ as a module over the Steenrod algebra ${\mathcal{A}_2}$ (or, equivalently, that we know ${H_*(X; \mathbb{Z}/p)}$ as a comodule over ${\mathcal{A}_2^{\vee}}$). Our goal is to glean from this information about the homotopy groups ${\pi_* X}$.

To understand the Adams spectral sequence (ASS), let’s try to recall how spectral sequences arise. One of the most basic examples comes from a filtered complex ${F^0 K^\bullet \subset \dots \subset F^n K^\bullet= K^\bullet}$ of abelian groups. In this case, one has a spectral sequence starting with the homology of the associated graded ${H_* ( \mathrm{gr} K)}$ and converging to the homology of the actual thing. Many spectral sequences in homological algebra (for instance, the Grothendieck spectral sequence for the derived functors of a composite), arise from filtering a suitable complex in this way. At least the initial page might be computable.

Unfortunately, many of the spectral sequences in topology don’t arise in this way. Nonetheless, they arise similarly.

Idea: a spectral sequence arises, more generally, when one has a filtered object (for instance, a filtered spectrum), starting with the homotopy groups of the associated graded, and converging to the homotopy groups of the whole thing.

Let’s say ${X}$ is a spectrum, and let’s say we have a filtration

$\displaystyle X = X_0 \supset X_1 \supset \dots \supset X_n \supset \dots.$

Under good conditions, we might know something about the homotopy groups of the “associated graded” pieces: that is, of the cofibers or quotients ${X_i/X_{i+1}}$. We’d like to go from there to ${\pi_* X}$. When ${n =2}$ this is just a long exact sequence in homotopy groups: for higher ${n}$, we get a spectral sequence.

How might we set up this spectral sequence? The theory of exact couples provides one method. There is a cofiber sequence

$\displaystyle X_{i+1} \rightarrow X_i \rightarrow X_i/X_{i+1} \rightarrow X_{i+1}[1]$

for each ${i}$, and, taking all these together, we get an exact triangle of homotopy groups

The associated spectral sequence (which is a sequence of bigraded groups starting with the bottom term in the above triangle at ${E_1}$) starts from the homotopy of the “associated graded” ${\bigoplus X_{i+1}/X_i}$ and, in good cases, converges to the homotopy groups of ${X}$.

One can, in fact, write down explicit descriptions of the terms of the spectral sequence, and see that they are better and better approximations to the homotopy groups of $X$.

2. Setting up the Adams spectral sequence

If we want to find a suitable filtration ${X = X_0 \supset X_1 \supset \dots}$ to compute ${\pi_* X}$, we then need to know the homotopy groups of the successive cofibers. A good way to arrange this, of course, is to have the cofibers be Eilenberg-MacLane spaces. This is very different from the Atiyah-Hirzebruch spectral sequence, where one filters a space by its CW filtration, and the successive cofibers are spheres (whose homology is easy but whose homotopy is hard).

Definition 1 An Adams resolution for the spectrum ${X}$ is a sequence

$\displaystyle X = X_0 \leftarrow X_1 \leftarrow \dots$

such that the cofiber of each map ${X_{s+1} \rightarrow X_s}$ is a wedge of Eilenberg-MacLane spaces, and such that the map on ${\mathbb{Z}/p}$-homology ${H_*(X_{s+1}; \mathbb{Z}/p) \rightarrow H_*(X_s; \mathbb{Z}/p)}$ is zero.

The existence of an Adams resolution reduces to the following claim: given a spectrum ${X}$, there is a wedge of ${H \mathbb{Z}/p}$-spectra ${Y}$ and a map ${X \rightarrow Y}$ which is a monomorphism on ${\mathbb{Z}/p}$-homology. We can do this very explicitly: take ${Y = H \mathbb{Z}/p \wedge X}$, for instance.

Using the sequence ${\left\{X_i\right\}}$ (which we can think of as a “filtration” of ${X}$ by turning all the maps into cofibrations), we get a spectral sequence starting from the ${\pi_*}$ of the cofibers. This is called the Adams spectral sequence. In good cases, this will converge to the ${p}$-adic completion of ${\pi_* X}$. The idea is that the condition that ${H_*(X_{s+1}; \mathbb{Z}/p) \rightarrow H_*(X_s; \mathbb{Z}/p)}$ be zero will guarantee that the inverse limit ${\varprojlim X_i }$ will have no ${p}$-adic homology, and consequently we can think of the filtration of ${X}$ as exhaustive at least when completed at ${p}$.

I don’t want to worry too much about these types of convergence issues. They’re a little technical, and they are probably not all that well suited for a blog. However, let’s try to see what the ${E_2}$ page of this spectral sequence looks like.

The spectral sequence associated to the filtration ${\left\{X_i\right\}}$ is obtained from an exact couple. Let ${K_i}$ be the cofiber of ${X_{i+1} \rightarrow X_i}$, so there are maps ${X_i \rightarrow K_i}$ for each ${i}$, which are monomorphisms on ${\mathbb{Z}/p}$-homology. At the beginning (the ${E_1}$ page), the exact couple which gives rise to the ASS looks like

where the map ${f}$ comes from the maps ${X_{i+1} \rightarrow X_i}$ (so it is of bidegree ${(0, -1)}$). The map ${g}$ comes from the maps ${X_i \rightarrow K_i}$, and the map ${h}$ comes from the maps ${K_i \rightarrow \Sigma X_{i+1}}$.

3. The ${E_2}$ page

Let’s figure out the ${E_2}$ page of this spectral sequence. The ${E_2}$ page is the homology of the differential ${gh}$ on ${\bigoplus \pi_* K_i}$. Another way to say this is that the ${K_i}$ form a “complex” in the homotopy category of spectra: that is, there are maps

$\displaystyle X \rightarrow K_0 \rightarrow \Sigma^{-1} K_1 \rightarrow \Sigma^{-2} K_2 \rightarrow \dots,$

any two of which are nullhomotopic. These come from chasing around the cofiber sequences, and are the homotopyish version of the ${gh}$ map. Because we are working with an Adams resolution, we find that the sequence

$\displaystyle 0 \rightarrow H_*(X; \mathbb{Z}/p) \rightarrow H_*(K_0; \mathbb{Z}/p) \rightarrow \dots$

is exact and is, in particular, a resolution of ${H_*(X;\mathbb{Z}/p)}$ by cofree ${\mathcal{A}_p^{\vee}}$-comodules.

Now ${\pi_* K_i = \hom_{\mathcal{A}_2^{\vee}}(\mathbb{Z}/p, H_*(K_i; \mathbb{Z}/p))}$ because maps of ${\mathcal{A}_p^{\vee}}$-comodules ${\mathbb{Z}/p \rightarrow H_*(K_i; \mathbb{Z}/p)}$ are precisely the primitive elements in ${H_*(K_i; \mathbb{Z}/p)}$. As ${K_i}$ is a sum of Eilenberg-MacLane spectra, we can get the homotopy groups as the primitive elements in homology. Consequently, if we want to take the homology of the bigraded group ${\pi_* K_i}$, this is the same as the homology of the bigraded group

$\displaystyle \hom_{\mathcal{A}_p^{\vee}}(\mathbb{Z}/p, H_*(K_i; \mathbb{Z}/p)).$

But we have just seen that ${H_*(K_i; \mathbb{Z}/p)}$ is a resolution of ${H_*(X; \mathbb{Z}/p)}$ by cofree ${\mathcal{A}_p^{\vee}}$-comodules. The conclusion is that the homology of this bigraded group, or the ${E_2}$ term, is

$\displaystyle \mathrm{Ext}^{*, *}_{\mathcal{A}_p^{\vee}}(\mathbb{Z}/p, H_*(X; \mathbb{Z}/p)) .$

The big theorem is:

Theorem 2 (Adams) Let ${X}$ be a connective spectrum whose homotopy groups are finitely generated. Then there is a spectral sequence whose ${E_2}$ page is

$\displaystyle \mathrm{Ext}^{s,t}_{\mathcal{A}_p^{\vee}}(\mathbb{Z}/p, H_*(X; \mathbb{Z}/p))$

(where ${\mathrm{Ext}}$ is computed in the category of comodules over ${\mathcal{A}_p^{\vee}}$). This spectral sequence converges to the ${p}$-adic completion ${\widehat{(\pi_{t-s} X)}}$, in the sense that there is a filtration on ${\widehat{\pi_{t-s} X}}$ (compatible with the ${p}$-adic filtration) whose successive quotients are the ${E_\infty}$ terms.

I don’t feel like I have enough to say about the proof of this theorem to make it worth spending much time on it at this point—it’s annoyingly technical because the convergence is generally not that strong. The thing that really does need explanation is why ${p}$-adic completion should be relevant at all. That comes in because the ${X_i}$ aren’t really an exhaustive filtration of ${X}$—if they were, then the spectral sequence would just converge to the homotopy groups of ${X}$. But they are an exhaustive filtration if ${X }$ is “${p}$-adically complete.”

The goal for the next few posts is to use the Adams spectral sequence, rather than worrying about the theory behind it.