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…)

This is part of a series of posts intended to understand some of the basic structure of the category ${\mathcal{U}}$ of unstable modules over the (mod ${2}$) Steenrod algebra, to prepare for the proof of the Sullivan conjecture. Here’s what we’ve seen so far:

• ${\mathcal{U}}$ is a Grothendieck abelian category, with a set of compact, projective generators ${F(n)}$ (the free unstable module on a generator in degree ${n}$). (See this post.)
• ${F(n)}$ has a tautological class ${\iota_n}$ in degree ${n}$, and has a basis given by ${\mathrm{Sq}^I \iota_n}$ for ${I}$ an admissible sequence of excess ${\leq n}$. (This post explained the terminology and the proof.)
• ${F(1)}$ was the subspace ${\mathbb{F}_2\left\{t, t^2, t^4, \dots\right\} \subset \widetilde{H}^*(\mathbb{RP}^\infty; \mathbb{F}_2)}$.

Our goal in this post is to describe some of the additional structure on the category ${\mathcal{U}}$, which will eventually enable us to prove (and make sense of!) results such as ${F(n) \simeq (F(1)^{\otimes n})^{ \Sigma_n}}$. We’ll start with the symmetric monoidal tensor product and the suspension functor, and then connect this to the Frobenius maps (which will be defined below).

1. The symmetric monoidal structure

Our first order of business is to describe the symmetric monoidal structure on ${\mathcal{U}}$, which will be given by the ${\mathbb{F}_2}$-linear tensor product. In fact, recall that the Steenrod algebra is a cocommutative Hopf algebra, under the diagonal map

$\displaystyle \mathrm{Sq}^n \mapsto \sum_{i+j = n} \mathrm{Sq}^i \otimes \mathrm{Sq}^j.$

The Hopf algebra structure is defined according to the following rule: we have that ${\theta}$ maps to ${\sum \theta' \otimes \theta''}$ if and only if for every two cohomology classes ${x,y }$ in the cohomology of a topological space, one has

$\displaystyle \theta(xy) = \sum \theta'(x) \theta''(y).$

The cocommutative Hopf algebra structure on ${\mathcal{A}}$ gives a tensor product on the category of (graded) ${\mathcal{A}}$-modules, which is symmetric monoidal. It’s easy to check that if ${M, N}$ are ${\mathcal{A}}$-modules satisfying the unstability condition, then so does ${M \otimes N}$. This is precisely the symmetric monoidal structure on ${\mathcal{U}}$. (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…)

In the previous post, I described the Sullivan conjecture and gave a vague outline of its proof by Miller. Shortly after Miller’s paper, various applications of the theorem to other problems in homotopy theory were discovered.

The intuition here is that there are two ways a space might appear to be finite: one is that its homotopy groups might be bounded and the other is that its homology groups might be bounded. It’s generally very hard for the two to happen at once, at least in the simply connected case. For instance, Eilenberg-MacLane spaces — the basic examples of spaces with bounded homotopy groups — have very messy cohomology. Similarly, finite complexes — spaces with bounded homology — generally have very complicated homotopy groups.

The purpose of this post is to explain a proof of the following result, conjectured by Serre:

Theorem 5 (McGibbon, Neisendorfer) Let ${X}$ be a simply connected finite complex such that ${\widetilde{H}_*(X; \mathbb{Z}/p) \neq 0}$. Then ${\pi_i X}$ contains ${p}$-torsion for infinitely many ${i}$.

As we’ll see, this result can be proved using the Sullivan conjecture. (more…)

I’ve been reading lately about the Sullivan conjecture and its proof (which is the subject of a course that Kirsten Wickelgren is teaching next semester). The resolution of this conjecture and work related to it led to a great deal of interesting algebra in the 1980s and 1990s, which I’ve been trying to understand a little about. Some useful references here are Haynes Miller’s 1984 paper, Lionel Schwartz’s book, and Jacob Lurie’s course notes.

1. Motivation

Let ${X}$ be a variety over the complex numbers. The complex points ${X(\mathbb{C})}$ are a topological space that has a homotopy type, which is often of interest. Étale homotopy theory (a refinement of étale cohomology) allows one to give a purely algebraic description of the profinite completion ${\widehat{X(\mathbb{C})}}$ of the homotopy type of ${X(\mathbb{C})}$. If ${X}$ is defined over the real numbers, though, then one can also study the topological space ${X(\mathbb{R})}$ of real points of ${X}$; one has

$\displaystyle X(\mathbb{R}) = X(\mathbb{C})^{\mathbb{Z}/2}$

for the conjugation action on ${X(\mathbb{C})}$. (more…)