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.
Let be a variety over the complex numbers. The complex points 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 of the homotopy type of . If is defined over the real numbers, though, then one can also study the topological space of real points of ; one has
for the conjugation action on .
One might try to get at the homotopy type of purely algebraically as well. Since (as a scheme) comes with an involution given by complex conjugation, one gets a -action on the étale homotopy type . Unfortunately, taking -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 -action. However, it’s not at all clear that this process should say anything at all about the (honest, non-homotopical) -fixed points . Sullivan conjectured that this process would recover the 2-adic information:
Conjecture: (Sullivan) becomes an equivalence after 2-adic completion.
In particular, the Sullivan conjecture states that the 2-adic completion of the real points can be recovered from the algebraic data of as a scheme over the real numbers, i.e., using étale homotopy theory.
One can’t expect to recover the -adic completion of for odd. For example, if (with the usual real structure), then . However, , and the homotopy fixed points of acting on completed at for odd is simply connected. In fact, one has
More generally, let be a finite -group, and let be a finite -CW complex. One has a map
where is the -fixed point set and is the homotopy fixed point set: that is, the operation which depends only on the “naive” -action on (and not the structure of as a -CW complex). One generally can’t expect this to be an equivalence at primes different from , as above. The Sullivan conjecture states that the map is an equivalence after completing at .
Let’s consider the case where acts on trivially. Then
is the space of maps from . In this case, one doesn’t even have to -adically complete. One has:
Theorem 1 (H.R. Miller) Let be a finite group and a finite complex. Then the space of pointed maps
is contractible. In other words, the map
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 spaces (except in rare cases like and compact surfaces of genus ) 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 be a simply connected finite complex. Then if has nontrivial mod homology, then has -torsion for infinitely many .
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 and a finite, simply connected complex. One wants to show
and one can do this by computing the homotopy groups of . Let’s replace by its -adic completion and show that
The strategy is now to compute the homotopy groups of using the unstable Adams spectral sequence. This is a spectral sequence that computes unstable homotopy classes of maps, but computing the term alone requires fearsome calculations. Miller’s argument shows that in this case, everything miraculously collapses at the page, for purely algebraic reasons.
The unstable Adams spectral sequence, like many other spectral sequences in topology, is based on resolving a given space by nicer ones. Let be the category of pointed spaces. Consider the adjunction:
where (so can be identified with the derived category of -modules). The left adjoint sends a space to its singular chain complex over . The right adjoint takes the infinite loop space associated to a chain complex of -modules.
The adjunction gives a monad on with . In particular, (as with monads in general) one gets a means of resolving a given space by a cosimplicial object
Under good conditions, the totalization of this cosimplicial object is precisely the -adic completion . Given a space , one therefore has a homotopy equivalence:
The advantage is that the spaces are Eilenberg-Maclane spaces (in fact, Eilenberg-MacLane spaces determined in terms of the homology of ), so that the space of maps is easy to describe in terms of homology. Whenever one has a cosimplicial space (let’s say pointed), there is a homotopy spectral sequence
The upshot is:
Theorem 3 (Bousfield-Kan) There is a spectral sequence, converging to , with page given by
where homology is mod homology. The groups are computed in the category of unstable coalgebras over the Steenrod algebra.
The meaning of the groups is as follows: one defines a functorial cosimplicial resolution of in the category of unstable coalgebras over the Steenrod algebra, from a given monad on this category. The groups are computed as triple cohomology.
Note the resemblance to the stable Adams spectral sequence. The stable Adams spectral sequence takes a spectrum and resolves it via the cosimplicial spectrum
using an analogous monad.
3. Idea of the proof
The unstable Adams spectral sequence offers a method of computing for a simply connected finite complex, based on resolving by Eilenberg-MacLane spaces. The term was given by
and was computed cosimplicially in the category of coalgebras in unstable modules over the Steenrod algebra.
For psychological reasons, let’s dualize and consider
where now 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 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 groups should vanish, is very far from being sufficient to prove the Sullivan conjecture! The 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 ‘s in modules to ‘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.