I’ve just uploaded to arXiv my paper “The homology of ${\mathrm{tmf}}$,” which is an outgrowth of a project I was working on last summer. The main result of the paper is a description, well-known in the field but never written down in detail, of the mod ${2}$ cohomology of the spectrum ${\mathrm{tmf}}$ of (connective) topological modular forms, as a module over the Steenrod algebra: one has

$\displaystyle H^*(\mathrm{tmf}; \mathbb{Z}/2) \simeq \mathcal{A} \otimes_{\mathcal{A}(2)} \mathbb{Z}/2,$

where ${\mathcal{A}}$ is the Steenrod algebra and ${\mathcal{A}(2) \subset \mathcal{A}}$ is the 64-dimensional subalgebra generated by ${\mathrm{Sq}^1, \mathrm{Sq}^2,}$ and ${ \mathrm{Sq}^4}$. This computation means that the Adams spectral sequence can be used to compute the homotopy groups of ${\mathrm{tmf}}$; one has a spectral sequence

$\displaystyle \mathrm{Ext}^{s,t}( \mathcal{A} \otimes_{\mathcal{A}(2)} \mathbb{Z}/2, \mathbb{Z}/2) \simeq \mathrm{Ext}^{s,t}_{\mathcal{A}(2)}(\mathbb{Z}/2, \mathbb{Z}/2) \implies \pi_{t-s} \mathrm{tmf} \otimes \widehat{\mathbb{Z}_2}.$

Since ${\mathcal{A}(2) \subset \mathcal{A}}$ is finite-dimensional, the entire ${E_2}$ page of the ASS can be computed, although the result is quite complicated. Christian Nassau has developed software to do these calculations, and a picture of the ${E_2}$ page for ${\mathrm{tmf}}$ is in the notes from André Henriques‘s 2007 talk at the Talbot workshop. (Of course, the determination of the differentials remains.)

The approach to the calculation of ${H^*(\mathrm{tmf}; \mathbb{Z}/2)}$ in this paper is based on a certain eight-cell (2-local) complex ${DA(1)}$, with the property that

$\displaystyle \mathrm{tmf} \wedge DA(1) \simeq BP\left \langle 2\right\rangle,$

where ${BP\left \langle 2\right\rangle = BP/(v_3, v_4, \dots, )}$ is a quotient of the classical Brown-Peterson spectrum by a regular sequence. The usefulness of this equivalence, a folk theorem that is proved in the paper, is that the spectrum ${BP\left \langle 2\right\rangle}$ is a complex-orientable ring spectrum, so that computations with it (instead of ${\mathrm{tmf}}$) become much simpler. In particular, one can compute the cohomology of ${BP\left \langle 2\right\rangle}$ (e.g., from the cohomology of ${BP}$), and one finds that it is cyclic over the Steenrod algebra. One can then try to “descend” to the cohomology of ${\mathrm{tmf}}$. This “descent” procedure is made much simpler by a battery of techniques from Hopf algebra theory: the cohomologies in question are graded, connected Hopf algebras. (more…)

In this post, I’d like to describe a toy analog of the Sullivan conjecture. Recall that the Sullivan conjecture considers (pointed) maps from ${BG}$ into a finite complex, and states that the space of such is contractible if $G$ is finite. The stable version replaces ${BG}$ with the Eilenberg-MacLane spectrum:

Theorem 13 Let ${H \mathbb{F}_p}$ be the Eilenberg-MacLane spectrum. Then the mapping spectrum

$\displaystyle (S^0)^{H \mathbb{F}_p}$

is contractible. In particular, for any finite spectrum ${F}$, the graded group of maps ${[H \mathbb{F}_p, F] = 0}$.

In the previous post, I sketched a proof (from Ravenel’s “Localization” paper) of this result based on a little chromatic technology. The spectrum ${H \mathbb{F}_p}$ is dissonant: that is, the Morava ${K}$-theories don’t see it. However, any finite spectrum is harmonic: that is, local with respect to the wedge of Morava ${K}$-theories. It follows formally that the spectrum of maps ${H \mathbb{F}_p \rightarrow S^0}$ is contractible (and thus the same with ${S^0}$ replaced by any finite spectrum). The non-formal input was the fact that ${S^0}$ is in fact harmonic, which requires a little work.

In this post, I’d like to sketch an earlier proof of the above theorem. This proof is based on the Adams spectral sequence. In fact, the proof runs parallel to Miller’s proof of the Sullivan conjecture. There is a classical Adams spectral sequence for computing ${[H \mathbb{F}_p, S^0]}$, with ${E_2}$ page given by

$\displaystyle \mathrm{Ext}^{s,t}_{\mathcal{A}}(\mathbb{F}_p, \mathcal{A}) \implies [ H \mathbb{F}_p, S^0]_{t-s} ,$

with ${\mathcal{A}}$ the (mod ${p}$) Steenrod algebra.

It turns out, however, for purely algebraic reasons, that the ${E_2}$ term is trivial. Miller’s proof of the Sullivan conjecture relies on more complicated algebra to show that the unstable version of all this has the same vanishing property at ${E_2}$. Most of this material is from Margolis’s Spectra and the Steenrod algebra. (more…)

One of the really nice pictures in homotopy theory is the “chromatic” one, relating the structure of the stable homotopy category to the geometry of formal groups (or rather, the geometry of the moduli stack of formal groups). A while back, I did a series of posts trying to understand a little about the relationship between formal groups and complex cobordism; the main result I was able to get to was Quillen’s theorem on the formal group of $MU$. I didn’t understand too much of the picture then, but I spent the summer engaging with it and think I have a slightly better feel for it now. In this post, I’ll try to give a description of how a natural attack on the homotopy groups of a spectrum via descent leads very naturally to the moduli stack of formal groups and to the Adams-Novikov spectral sequence. (There are other approaches to Adams-type spectral sequences, for instance in these notes of Haynes Miller.)

1. Descent

Let’s start with some high-powered generalities that I don’t really understand, and then come back to earth. Consider an ${E_\infty}$-ring ${R}$; the most important examples will be ${R = H \mathbb{Z}/2}$ or ${R = MU}$. There is a map of ${E_\infty}$-rings ${S \rightarrow R}$, where ${S}$ is the sphere spectrum.

Let ${X}$ be a plain spectrum. Then, equivalently, ${X}$ is a module over ${S}$. Tensoring with ${R}$ gives an ${R}$-module spectrum ${ R \otimes X}$, where the smash product of spectra is written ${\otimes}$. In fact, we have an adjunction

$\displaystyle \mathrm{Mod}(S) \rightleftarrows \mathrm{Mod}(R)$

between ${R \otimes }$ and forgetting the ${R}$-module structure. As in ordinary algebra, we might try to apply the methods of flat descent to this adjunction. In other words, given a spectrum ${X}$, we might try to recover ${X}$ from the ${R}$-module ${R \otimes X}$ together with the “descent data” on ${X}$. The benefit is that while the homotopy groups ${\pi_* X}$ may be intractable, those of ${R \otimes X}$ are likely to be much easier to compute: they are the ${R}$-homology groups of ${X}$.

Let’s recall how this works in algebra. Given a faithfully flat morphism of rings ${A \rightarrow B}$ and an ${A}$-module ${M}$, then we can recover ${M}$ as the equalizer of

$\displaystyle M \otimes_A B \rightrightarrows M \otimes_A B \otimes_A B.$

How does one imitate this construction in homotopy? One then has a cosimplicial ${E_\infty}$-ring given by the cobar construction

$\displaystyle R \rightrightarrows R \otimes R \dots .$

The ${\mathrm{Tot}}$ (homotopy limit) of a cosimplicial object is the homotopyish version of the 1-categorical notion of an equalizer. In particular, we might expect that we can recover the spectrum ${X}$ as the homotopy limit of the cosimplicial diagram

$\displaystyle R \otimes X \rightrightarrows R \otimes R \otimes X \dots .$ (more…)

Let ${MSO }$ be the Thom spectrum for oriented cobordism, so ${MSO}$ can be obtained as a homotopy colimit

$\displaystyle MSO = \varinjlim MSO(n) = \varinjlim \Sigma^{-n}\mathrm{Th}(BSO(n))$

where the ${n}$th space is the Thom space of the (universal) vector bundle over ${BSO(n)}$. By the Thom-Pontryagin theorem, the homotopy groups ${\pi_* MSO}$ are isomorphic (as a graded ring) to the cobordism ring ${\Omega_{SO}}$ of oriented manifolds. The goal of the next few posts is to discuss some of the classical results on the oriented cobordism ring. This post will handle the easiest case; since it is somewhat analogous to the situation for complex cobordism, it is a bit brief.

In the past, we described Milnor’s computation of ${\pi_* MU}$ (the complex cobordism ring), and showed that ${\pi_* MU \simeq \mathbb{Z}[x_2, x_4, \dots ]}$. The situation for ${MSO}$ is somewhat more complicated, as there will be torsion; as we will see, however, it is only 2-torsion that occurs, and not too wild of a sort. Let’s start by noting what ${\Omega_{SO} \otimes \mathbb{Q}}$ is.

Theorem 1 (Thom) We have

$\displaystyle \pi_* MSO \otimes \mathbb{Q} \simeq \mathbb{Q}[x_4, x_8, x_{12}, \dots ],$

where the ${x_i}$ can be taken to be the even-dimensional ${\mathbb{CP}^{2i}}$.

This result is a corollary of Serre’s work, which shows that rational stable homotopy is equivalent to rational homology. In other words, we have an isomorphism

$\displaystyle \pi_* MSO \otimes \mathbb{Q} \simeq H_*(MSO; \mathbb{Q}) \simeq H_*(BSO; \mathbb{Q})$

(where the last isomorphism is the Thom isomorphism). Now ${H_*(BSO; \mathbb{Q})}$ is a polynomial ring on variables as above; this follows from the following computation:

Proposition 2 Let ${R}$ be a ring containing ${1/2}$. Then ${H_*(BSO; R)}$ is a polynomial ring on generators ${\alpha_{i} \in H_{4i}(BSO; \mathbb{R})}$. These generators are obtained as the image of generating elements in ${H_{4i}(BSO(2); R) = H_{4i}(\mathbb{CP}^\infty; R)}$. (more…)

This is a continuation of yesterday’s post, which used the Adams spectral sequence to compute the first two stable homotopy groups of spheres (only as a toy example for myself: one can use more elementary tools). In this post, I’d like to describe the third stable stem.The claim is that the first four columns of the ${E_2}$-page of the Adams spectral sequence for the sphere look like:

Furthermore, we have the relation ${h_0^2 h_2 = h_1^3}$. This is the complete picture for the first four columns.

Note that there can be no nontrivial differentials in this range for dimensional reasons. Since ${h_0}$ corresponds to multiplication by 2 in the stable stems, this corresponds to the fact that ${\pi_3(S^0) = \mathbb{Z}/8}$: in fact, we find that ${\pi_3(S^0)}$ has a three-term filtration with successive quotients ${\mathbb{Z}/2}$, and that passage down each step of the filtration is given by multiplication by ${2}$. The relation ${h_0^2 h_2 = h_1^3}$ corresponds to the fact that the Hopf map ${\nu \in \pi_3(S^0)}$ (which corresponds to the element of Hopf invariant one in ${\pi_3(S^0)}$) satisfies

$\displaystyle 4 \nu = \eta^3,$

for ${\eta}$ the element of Hopf invariant one in ${\pi_1(S^0)}$ represented by ${h_1}$. (more…)

I’d like to use the next couple of posts to compute the first three stable stems, using the Adams spectral sequence. Recall from the linked post that, for a connective spectrum ${X}$ with appropriate finiteness hypotheses, we have a first quadrant spectral sequence

$\displaystyle \mathrm{Ext}^{s,t}_{\mathcal{A}_2^{\vee}}(\mathbb{Z}/2, H_*( X; \mathbb{Z}/2)) \implies \widehat{\pi_{t-s} X} ,$

where the ${\mathrm{Ext}}$ groups are computed in the category of comodules over ${\mathcal{A}_2^{\vee}}$ (the dual of the Steenrod algebra), and the convergence is to the ${2}$-adic completion of the homotopy groups of ${X}$. In the case of ${X}$ the sphere spectrum, we thus get a spectral sequence

$\displaystyle \mathrm{Ext}^{s,t}_{\mathcal{A}_2^{\vee}}(\mathbb{Z}/2, \mathbb{Z}/2) \implies \widehat{\pi_{t-s} S^0},$

converging to the 2-torsion in the stable stems. In this post and the next, we’ll compute the first couple of ${\mathrm{Ext}}$ groups of ${\mathcal{A}_2^{\vee}}$, or equivalently of ${\mathcal{A}_2}$ (this is usually called the cohomology of the Steenrod algebra), and thus show:

1. ${\pi_1 S^0 = \mathbb{Z}/2}$, generated by the Hopf map ${\eta}$ (coming from the Hopf fibration ${S^3 \rightarrow S^2}$).
2. ${\pi_2 S^0 = \mathbb{Z}/2}$, generated by the square ${\eta^2}$ of the Hopf map.
3. ${\pi_3 S^0 = \mathbb{Z}/8}$, generated by the Hopf map ${\nu}$ (coming from the Hopf fibration ${S^7 \rightarrow S^4}$). We have ${\eta^3 = 4 \nu}$. (This is actually true only mod odd torsion; there is also a ${\mathbb{Z}/3}$, so the full thing is a ${\mathbb{Z}/24}$.)

In fact, we’ll be able to write down the first four columns of the Adams spectral sequence by direct computation. There are numerous fancier tools which let one go further. (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.

Let’s try to do some (baby) examples of the Adams spectral sequence. The notation used will be that of yesterday’s post.

1. ${H\mathbb{Z}}$

So let’s start with a silly example, where the answer is tautological: ${H \mathbb{Z}}$. We could try to compute the homotopy groups of this using the Adams spectral sequence. At a prime ${p}$, this means that we should get the ${p}$-adic completion ${\mathbb{Z}_p}$ in degree zero, and nothing elsewhere.

It turns out that we can write down a very explicit Adams resolution for ${H \mathbb{Z}}$. To start with, we need a map ${f: H \mathbb{Z} \rightarrow X}$ where ${X}$ is a wedge of ${H \mathbb{Z}/p}$ and shifts, and such that ${f}$ is a monomorphism on ${\mathbb{Z}/p}$-homology. We can take the map

$\displaystyle f: H \mathbb{Z} \rightarrow H \mathbb{Z}/p;$

the fact that ${f}$ is a monomorphism on ${\mathbb{Z}/p}$-homology follows because ${f}$ is an epimorphism on ${\mathbb{Z}/p}$-cohomology, by Serre’s computation of the cohomology of Eilenberg-MacLane spaces. Serre’s computation tells us, in fact, that the cohomology of ${H \mathbb{Z}}$ is the Steenrod algebra mod the ideal generated by the Bockstein. (more…)

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. (more…)