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 ${X(\mathbb{C})}$.

One might try to get at the homotopy type of ${X(\mathbb{R})}$ purely algebraically as well. Since ${X}$ (as a scheme) comes with an involution given by complex conjugation, one gets a ${\mathbb{Z}/2}$-action on the étale homotopy type ${\widehat{X(\mathbb{C})}}$. Unfortunately, taking ${\mathbb{Z}/2}$-fixed points isn’t homotopy invariant (unless you work in “genuine” equivariant homotopy theory). The homotopy invariant thing to do would be to take homotopy fixed points of the ${\mathbb{Z}/2}$-action. However, it’s not at all clear that this process should say anything at all about the (honest, non-homotopical) ${\mathbb{Z}/2}$-fixed points ${X(\mathbb{R})}$. Sullivan conjectured that this process would recover the 2-adic information:

Conjecture: (Sullivan) ${X(\mathbb{C})^{\mathbb{Z}/2} \rightarrow X(\mathbb{C})^{h \mathbb{Z}/2}}$ becomes an equivalence after 2-adic completion.

In particular, the Sullivan conjecture states that the 2-adic completion of the real points ${X(\mathbb{R})}$ can be recovered from the algebraic data of ${X}$ as a scheme over the real numbers, i.e., using étale homotopy theory.

One can’t expect to recover the ${p}$-adic completion of ${X(\mathbb{R})}$ for ${p}$ odd. For example, if ${X = \mathbb{P}^1_{\mathbb{C}}}$ (with the usual real structure), then ${X(\mathbb{R}) = X^{\mathbb{Z}/2} = \mathbb{P}^1(\mathbb{R}) = S^1}$. However, ${X(\mathbb{C}) \simeq S^2}$, and the homotopy fixed points of ${\mathbb{Z}/2}$ acting on ${X(\mathbb{C})}$ completed at ${p}$ for ${p}$ odd is simply connected. In fact, one has

$\displaystyle \pi_* (X(\mathbb{C})_{p})^{h \mathbb{Z}/2} = ( \pi_* X(\mathbb{C})_{p})^{\mathbb{Z}/2},$

when ${p > 2}$.

More generally, let ${G}$ be a finite ${p}$-group, and let ${X}$ be a finite ${G}$-CW complex. One has a map

$\displaystyle X^G \rightarrow X^{hG}$

where ${X^G}$ is the ${G}$-fixed point set and ${X^{hG}}$ is the homotopy fixed point set: that is, the operation which depends only on the “naive” ${G}$-action on ${X}$ (and not the structure of ${X}$ as a ${G}$-CW complex). One generally can’t expect this to be an equivalence at primes different from ${p}$, as above. The Sullivan conjecture states that the map is an equivalence after completing at ${p}$.

Let’s consider the case where ${G}$ acts on ${X}$ trivially. Then

$\displaystyle X^{hG} = \mathrm{Fun}_G(EG, X) = \mathrm{Fun}(BG, X) = X^{BG}$

is the space of maps from ${BG \rightarrow X}$. In this case, one doesn’t even have to ${p}$-adically complete. One has:

Theorem 1 (H.R. Miller) Let ${G}$ be a finite group and ${X}$ a finite complex. Then the space of pointed maps

$\displaystyle \mathrm{map}_*(BG, X)$

is contractible. In other words, the map

$\displaystyle X \rightarrow X^{BG}$

is a homotopy equivalence.

This is the main result of Miller’s 1984 paper “The Sullivan conjecture on maps from classifying spaces.” In the next couple of posts, I’d like to go through some of the ideas in its proof (as well in later proofs). Other than being a remarkable result in its own right — computing homotopy types of spaces of maps is remarkably hard — the Sullivan conjecture shows how different ${K(G, 1)}$ spaces (except in rare cases like ${S^1}$ and compact surfaces of genus ${>0}$) are from finite complexes.

For example, the Sullivan conjecture can be used to give a proof of the following conjecture of Serre:

Theorem 2 Let ${X}$ be a simply connected finite complex. Then if ${X}$ has nontrivial mod ${p}$ homology, then ${\pi_i(X)}$ has ${p}$-torsion for infinitely many ${i}$.

2. The unstable Adams spectral sequence

Miller’s proof of the Sullivan conjecture is based on the unstable Adams spectral sequence. Namely, to start, let’s make some simplifications: let ${G = \mathbb{Z}/p}$ and ${X}$ a finite, simply connected complex. One wants to show

$\displaystyle \mathrm{map}_*(B\mathbb{Z}/p, X) \simeq \ast,$

and one can do this by computing the homotopy groups of ${\mathrm{map}_*(B\mathbb{Z}/p, X)}$. Let’s replace ${X}$ by its ${p}$-adic completion and show that

$\displaystyle \mathrm{map}_*(B \mathbb{Z}/p, X^{\hat{}}_p) \simeq \ast.$

The strategy is now to compute the homotopy groups of ${\mathrm{map}_*(B \mathbb{Z}/p, X^{\hat{}}_p)}$ using the unstable Adams spectral sequence. This is a spectral sequence that computes unstable homotopy classes of maps, but computing the ${E_2}$ term alone requires fearsome ${\mathrm{Ext}}$ calculations. Miller’s argument shows that in this case, everything miraculously collapses at the ${E_2}$ page, for purely algebraic reasons.

The unstable Adams spectral sequence, like many other spectral sequences in topology, is based on resolving a given space ${X}$ by nicer ones. Let ${\mathcal{S}_*}$ be the category of pointed spaces. Consider the adjunction:

$\displaystyle HR \otimes \Sigma^\infty, \Omega^\infty: \mathcal{S}_* \rightrightarrows \mathrm{Mod}(R)$

where ${R = H \mathbb{Z}/p}$ (so ${\mathrm{Mod}(R)}$ can be identified with the derived category of ${\mathbb{Z}/p}$-modules). The left adjoint sends a space to its singular chain complex over ${\mathbb{Z}/p}$. The right adjoint takes the infinite loop space associated to a chain complex of ${R}$-modules.

The adjunction gives a monad ${T}$ on ${\mathcal{S}_*}$ with ${TX = \Omega^\infty( HR \otimes \Sigma^\infty X)}$. In particular, (as with monads in general) one gets a means of resolving a given space ${X}$ by a cosimplicial object

$\displaystyle TX \rightrightarrows T^2 X \begin{smallmatrix} \rightarrow \\ \rightarrow \\ \rightarrow \end{smallmatrix} \dots.$

Under good conditions, the totalization of this cosimplicial object is precisely the ${p}$-adic completion ${\widehat{X}_p}$. Given a space ${Y}$, one therefore has a homotopy equivalence:

$\displaystyle \mathrm{map}_*(Y, \hat{X}_p) \simeq \mathrm{Tot} \mathrm{map}_*(Y, T^{s+1} X).$

The advantage is that the spaces ${T^{s+1} X}$ are Eilenberg-Maclane spaces (in fact, Eilenberg-MacLane spaces determined in terms of the homology of ${X}$), so that the space of maps ${\mathrm{map}_*(Y, T^{s+1} X)}$ is easy to describe in terms of homology. Whenever one has a cosimplicial space ${Z^\bullet}$ (let’s say pointed), there is a homotopy spectral sequence

$\displaystyle E_2^{s,t} = \pi^s \pi_t Z^\bullet \implies \pi_{t-s} \mathrm{Tot}(Z^\bullet)$

The upshot is:

Theorem 3 (Bousfield-Kan) There is a spectral sequence, converging to ${\pi_{t-s} \mathrm{map}_*(Y, \hat{X}_p)}$, with ${E_2}$ page given by

$\displaystyle E_2 = \mathrm{Ext}^s_{CA}( \Sigma^t H_{*} Y, H_* X) \implies \pi_{t-s} \mathrm{map}_*(Y, \hat{X}_p),$

where homology is mod ${p}$ homology. The ${\mathrm{Ext}}$ groups are computed in the category of unstable coalgebras over the Steenrod algebra.

The meaning of the ${\mathrm{Ext}}$ groups is as follows: one defines a functorial cosimplicial resolution of ${H_* X}$ in the category of unstable coalgebras over the Steenrod algebra, from a given monad on this category. The ${\mathrm{Ext}}$ groups are computed as triple cohomology.

Note the resemblance to the stable Adams spectral sequence. The stable Adams spectral sequence takes a spectrum ${X}$ and resolves it via the cosimplicial spectrum

$\displaystyle H R \otimes X \rightrightarrows HR \otimes HR \otimes X \begin{smallmatrix} \rightarrow \\ \rightarrow \\ \rightarrow \end{smallmatrix} \dots ,$

3. Idea of the proof

The unstable Adams spectral sequence offers a method of computing ${\pi_* \mathrm{map}_*(B\mathbb{Z}/p, \widehat{X}_p)}$ for ${X}$ a simply connected finite complex, based on resolving ${X}$ by Eilenberg-MacLane spaces. The ${E_2}$ term was given by

$\displaystyle \mathrm{Ext}^s( \Sigma^t H_*( B\mathbb{Z}/p), H_*(X)),$

and was computed cosimplicially in the category of coalgebras in unstable modules over the Steenrod algebra.

For psychological reasons, let’s dualize and consider

$\displaystyle \mathrm{Ext}^s( H^*(X), \Sigma^t H^*(B \mathbb{Z}/p)),$

where now ${\mathrm{Ext}}$ is computed in the category of unstable algebras over the Steenrod algebra. This is the category in which the cohomology of spaces takes its values.

That’s not easily computable. Miller’s observation is that these groups are completely null. This is a striking feature of the algebraic category of unstable modules over the Steenrod algebra. The first hint that this should happen comes from:

Theorem 4 The object ${H^*( B \mathbb{Z}/p)}$ is injective in the category of unstable modules over the Steenrod algebra.

This is something that can be checked directly, but which also admits an interpretation and proof in the language of “generic representation theory,” which realizes a quotient category of “unstable modules over the Steenrod algebra” as “generic representations of general linear groups over finite fields.” This is definitely something I’d like to understand better.

Anyway, the above theorem, while it suggests that ${\mathrm{Ext}}$ groups should vanish, is very far from being sufficient to prove the Sullivan conjecture! The ${\mathrm{Ext}}$ groups that enter into the unstable Adams spectral sequence come from unstable algebras rather than unstable modules. Sullivan’s strategy here is to develop another spectral sequence going from ${\mathrm{Ext}}$‘s in modules to ${\mathrm{Ext}}$‘s in algebras. The spectral sequence is essentially a Grothendieck spectral sequence, and the result that makes this powerful is a boundedness result that Miller proves.