So, today I am going to talk about the Brown representability theorem. This is a central fact in algebraic topology, proved in the 1950s by Edgar Brown in a paper in the Annals. It states that under mild conditions a contravariant functor on the homotopy category of pointed CW complexes is representable. As we saw yesterday, this guarantees the existence of Eilenberg-Maclane spaces. More importantly, it guarantees—at least for CW complexes—things like universal bundles. I will have more to say about these applications in the future. First, let us try to understand the result itself.

So let be a contravariant functor to the category of pointed sets. We require that satisfy two axioms below. First, like any representable functor, it should send coproducts to products. Since the coproduct in is the wedge sum, we require that (for ranging over some index set)

under the canonical map that arises by taking the product (over ) of the maps .

Second, we require that satisfy the following **Mayer-Vietoris axiom**. If for subcomplexes , then if and “glue together,” i.e. become the same element of , they both come from a fixed element of . This is a sheaf-theoretic condition.

The first observation is that any representable functor must satisfy the coproduct axiom and the Mayer-Vietoris (i.e. sheafish) axiom. The coproduct axiom is automatic for any category.

The sheaf axiom is less trivial. Let as before. Suppose that is represented by a pointed complex . Given “gluable” elements and , by assumption the restrictions are equivalent (i.e. homotopic). This does not immediately mean we can glue the maps. However, by the homotopy extension property for CW pairs, we can homotope to some such that become *equal* (not merely homotopic) on . Together these define a map that becomes equivalent to .

These axioms are not too difficult to check in many interesting cases. For instance, they are true for singular cohomology. This is why the following is highly important:

Theorem 1 (Brown)If is a contravariant functor satisfying the coproduct and Mayer-Vietoris axioms, then is representable.

This is the result that I would like to begin to prove today.

** -universal elements **

The idea behind the proof of Brown representability is that one is working with CW complexes. CW complexes are built up in a piece-by-piece manner from the comparatively simple objects of -cells. And in a sense, the way to construct the representing object is to construct it for the spheres.

To prove Brown representability, we will need to find a space and an element such that if is any space, then the elements of are in bijection with the for ranging over the homotopy classes of maps .

As I stated above, the strategy will be to do this for the spheres. Namely, we shall find a space such that the above claim is true for the spheres of any dimension. We shall then show that the space will work as a representing object in general.

Definition 2Let be a pair with and . The pair is said to be-universalif the map sending to is bijective for and epimorphic for . A pair is-universalif it is -universal for all .

The point is that we will find a universal pair, and show that this is indeed a representing object (a “universal object” in the categorical sense).

The process of finding universal elements is contained in the following lemma. It is an approach that comes up over and over: to add enough spheres to make the map surjective, and then to attach various cells to reduce the number of homotopy classes of maps.

Now, to make notation clearer, let us note that is a **presheaf** on the category , that is a contravariant functor to . Motivated by standard notation, let us fix the following convention: if is a subcomplex of a CW complex, and , we write for the pull-back of to .

Lemma 3Let be a -universal pair. Then there is a space obtained by attaching cells and an -universal element such that .

*Proof:* The first thing to note that is that is a *group* in a natural way. This actually formally follows from general nonsense. In fact, sends coproducts in to products, so sends cogroup objects to group objects in (i.e. groups). This group structure is deduced from the cogroup law in . This is the same way that has a group structure.

In particular, we find that the map

is a *group homomorphism*. To say that it is injective is thus to say that it has trivial kernel.

We already know that this injectivity is true for . We want to attach various -cells to make this true for .

Namely, we let be a collection of maps that forms a system of representatives for the kernel of . We can assume that the are *cellular*. We attach -cells to by these maps to get a new space . Doing so makes all the maps nullhomotopic in .

The claim is that extends to the bigger space . The way to see this is that is the (reduced) mapping cone of . Now we will use a general fact:

Lemma 4Suppose satisfies the Mayer-Vietoris axiom Let be a cellular map in . Suppose pulls back to the trivial (basepoint) of . Then extends to the mapping cone .

This lemma follows by the cover of by and .

We see that in our case, there is such that pulls back to . The claim is that is -universal, and moreover

is injective (thus an isomorphism). Indeed, if pulls back to the trivial element, we can first assume by homotoping that lands inside , thanks to the cellular approximation theorem. In that case, it follows that pulls to the trivial element, so it is one of the (at least up to homotopy). But these are trivial in .

On the other hand, we still have to get an -universal element. We haven’t done that yet.

Namely, we have to get surjectivity on the -level. For this, we just augment the space by wedging along a whole bunch of spheres.

Let be a listing of elements in . We can form

Let us choose an element extending such that (this is what extension means) and restricted to the th factor of is . This is possible by the coproduct condition.

This last condition means that the inclusion of on the th factor pulls back to . In particular,

is surjective. Since we have adjoined only cells of dimension , the cellular approximation lemma implies that any map comes from a map . So pulls back to the trivial element if only if the associated map does. As before, we conclude that this implies that is homotopically trivial in .

The point is that, using this, we will be able to construct -universal elements by a filtered colimit argument, and we will be able to show that these are actually representing pairs by the Whitehead theorem. This, however, needs to be finished next time.

December 15, 2010 at 11:29 pm

[…] in fact we have proved a pretty general result about representable functors on the pointed homotopy category of CW complexes. Indeed, Brown representability can be used to […]