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…)

This post is part of a series (started here) of posts on the structure of the category ${\mathcal{U}}$ of unstable modules over the mod ${2}$ Steenrod algebra ${\mathcal{A}}$, which plays an important role in the proof of the Sullivan conjecture (and its variants).

In the previous post, we introduced some additional structure on the category ${\mathcal{U}}$.

• First, using the (cocommutative) Hopf algebra structure on ${\mathcal{A}}$, we got a symmetric monoidal structure on ${\mathcal{U}}$, which was an algebraic version of the Künneth theorem.
• Second, we described a “Frobenius” functor

$\displaystyle \Phi : \mathcal{U} \rightarrow \mathcal{U},$

which was symmetric monoidal, and which came with a Frobenius map ${\Phi M \rightarrow M}$.

• We constructed an exact sequence natural in ${M}$,

$\displaystyle 0 \rightarrow \Sigma L^1 \Omega M \rightarrow \Phi M \rightarrow M \rightarrow \Sigma \Omega M \rightarrow 0, \ \ \ \ \ (4)$

where ${\Sigma}$ was the suspension and ${\Omega}$ the left adjoint. In particular, we showed that all the higher derived functors of ${\Omega}$ (after ${L^1}$) vanish.

The first goal of this post is to use this extra structure to prove the following:

Theorem 39 The category ${\mathcal{U}}$ is locally noetherian: the subobjects of the free unstable module ${F(n)}$ satisfy the ascending chain condition (equivalently, are finitely generated as ${\mathcal{A}}$-modules).

In order to prove this theorem, we’ll use induction on ${n}$ and the technology developed in the previous post as a way to make Nakayama-type arguments. Namely, the exact sequence (4) becomes

$\displaystyle 0 \rightarrow \Phi F(n) \rightarrow F(n) \rightarrow \Sigma F(n-1) \rightarrow 0,$

as we saw in the previous post. Observe that ${F(0) = \mathbb{F}_2}$ is clearly noetherian (it’s also not hard to check this for ${F(1)}$). Inductively, we may assume that ${F(n-1)}$ (and therefore ${\Sigma F(n-1)}$) is noetherian.

Fix a subobject ${M \subset F(n)}$; we’d like to show that ${M}$ is finitely generated. (more…)

The purpose of this post (like the previous one) is to go through some of the basic properties of the category ${\mathcal{U}}$ of unstable modules over the (mod ${2}$) Steenrod algebra. An analysis of ${\mathcal{U}}$ will ultimately lead to the proof of the Sullivan conjecture. Most of this material, again, is from Schwartz’s Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture; another useful source is Lurie’s notes.

1. The modules ${F(n)}$

In the previous post, we showed that the category ${\mathcal{U}}$ had enough projectives. More specifically, we constructed — using the adjoint functor theorem — an object ${F(n)}$, for each ${n}$, which we called the free unstable module on a class of degree ${n}$.The object ${F(n)}$ had the universal property

$\displaystyle \hom_{\mathcal{U}}(F(n), M) \simeq M_n,\quad M \in \mathcal{U}.$

To start with, we’d like to have a more explicit description of the module ${F(n)}$.

To do this, we need a little terminology. A sequence of positive integers

$\displaystyle i_k, i_{k-1}, \dots, i_1$

$\displaystyle i_j \geq 2 i_{j-1}$

for each ${j}$. It is a basic fact, which can be proved by manipulating the Adem relations, that the squares

$\displaystyle \mathrm{Sq}^I \stackrel{\mathrm{def}}{=} \mathrm{Sq}^{i_k} \mathrm{Sq}^{i_{k-1}} \dots \mathrm{Sq}^{i_1}, \quad I = (i_k, \dots, i_1) \ \text{admissible}$

form a spanning set for ${\mathcal{A}}$ as ${I}$ ranges over the admissible sequences. In fact, by looking at the representation on various cohomology rings, one can prove:

Proposition 29 The ${\mathrm{Sq}^I}$ for ${I }$ admissible form a basis for the Steenrod algebra ${\mathcal{A}}$. (more…)

The purpose of this post is to introduce the basic category that enters into the unstable Adams spectral sequence and the proof of the Sullivan conjecture: the category of unstable modules over the Steenrod algebra. Throughout, I’ll focus on the (simpler) case of ${p=2}$.

1. Recap of the Steenrod algebra

Let ${X}$ be a space. Then the cohomology ${H^*(X; \mathbb{F}_2)}$ has a great deal of algebraic structure:

• It is a graded ${\mathbb{F}_2}$-vector space concentrated in nonnegative degrees.
• It has an algebra structure (respecting the grading).
• It comes with an action of Steenrod operations

$\displaystyle \mathrm{Sq}^i: H^*(X; \mathbb{F}_2 ) \rightarrow H^{*+i}(X; \mathbb{F}_2), \quad i \geq 0.$

The Steenrod squares, which are constructed from the failure of strict commutativity in the cochain algebra ${C^*(X; \mathbb{F}_2)}$, are themselves subject to a number of axioms:

• ${\mathrm{Sq}^0}$ acts as the identity.
• ${\mathrm{Sq}^i}$ is compatible with the suspension isomorphism between ${H_*(X; \mathbb{F}_2), \widetilde{H}_*(\Sigma X; \mathbb{F}_2)}$.
• One has the Adem relations: for ${i < 2j}$,

$\displaystyle \mathrm{Sq}^i \mathrm{Sq}^j = \sum_{0 \leq 2k \leq i} \binom{j-k-1}{i-2k} \mathrm{Sq}^{i+j-k}\mathrm{Sq}^k. \ \ \ \ \ (3)$

In other words, there is a (noncommutative) algebra of operations, which is the Steenrod algebra ${\mathcal{A}}$, such that the cohomology of any space ${X}$ is a module over ${\mathcal{A}}$. The Steenrod algebra can be defined as

$\displaystyle \mathcal{A} \stackrel{\mathrm{def}}{=} T( \mathrm{Sq}^0, \mathrm{Sq}^1, \dots )/ \left( \mathrm{Sq}^0 = 1, \ \text{Adem relations}\right) .$ (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…)

This is the third in a series of posts on oriented cobordism. In the first post, we analyzed the spectrum $MSO$ at odd primes; in this post, we will analyze the prime $2$. After this, we’ll be able to deduce various classical geometric facts about manifolds.

The next goal is to  determine the structure of the homology ${H_*(MSO; \mathbb{Z}/2)}$ as a comodule over ${\mathcal{A}_2^{\vee}}$. Alternatively, we can determine the structure of the cohomology ${H^*(MSO; \mathbb{Z}/2)}$ over the Steenrod algebra ${\mathcal{A}_2}$: this is a coalgebra and a module.

Theorem 8 (Wall) As a graded ${\mathcal{A}_2}$-module, ${H^*(MSO; \mathbb{Z}/2)}$ is a direct sum of shifts of copies of ${\mathcal{A}_2}$ and ${\mathcal{A}_2/\mathcal{A}_2\mathrm{Sq}^1}$.

This corresponds, in fact, to a splitting at the prime 2 of $MSO$ into a wedge of Eilenberg-MacLane spectra.

In fact, this will follow from the comodule structure theorem of the previous post once we can show that if ${t \in H^0(MSO; \mathbb{Z}/2)}$ is the Thom class, then the action of ${\mathcal{A}_2}$ on ${t}$ has kernel ${J = \mathcal{A}_2 \mathrm{Sq}^1}$: that is, the only way a cohomology operation can annihilate ${t}$ if it is a product of something with ${\mathrm{Sq}^1}$. Alternatively, we have to show that the complementary Serre-Cartan monomials in ${\mathcal{A}_2}$ applied to ${t}$,

$\displaystyle \mathrm{Sq}^{i_1} \mathrm{Sq}^{i_2} \dots \mathrm{Sq}^{i_n} t, \quad i_k \geq 2i_{k-1}, \quad i_n \neq 1,$

are linearly independent in ${H^*(MSO; \mathbb{Z}/2)}$. (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…)

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…)

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…)