Let ${X}$ be a connective spectrum with finitely generated homotopy groups. Then the lowest homotopy group in ${\pi_*(X \wedge X)}$ is the tensor square of the lowest homotopy group in ${\pi_*(X)}$: in particular, $X \wedge X$ is never zero (i.e., contractible). The purpose of this post is to describe an example of a nontrivial spectrum ${I}$ with ${I \wedge I \simeq 0}$. I learned this example from Hovey and Strickland’s “Morava ${K}$-theories and localization.”

1. A non-example

To start with, here’s a spectrum which does not work: ${H \mathbb{Q}/\mathbb{Z}}$. This is a natural choice because $\displaystyle \mathbb{Q}/\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Q}/\mathbb{Z} = 0.$

On the other hand, from the cofiber sequence $\displaystyle H \mathbb{Z} \rightarrow H \mathbb{Q} \rightarrow H \mathbb{Q}/\mathbb{Z} \rightarrow \Sigma H \mathbb{Z},$

we obtain a cofiber sequence $\displaystyle H \mathbb{Z} \wedge H \mathbb{Q}/\mathbb{Z} \rightarrow 0 \rightarrow (H \mathbb{Q}/\mathbb{Z})^{\wedge 2} \rightarrow \Sigma H \mathbb{Z} \wedge H \mathbb{Q}/\mathbb{Z}$

which shows in particular that $\displaystyle (H \mathbb{Q}/\mathbb{Z})^{\wedge 2} \simeq \Sigma H \mathbb{Z} \wedge H \mathbb{Q}/\mathbb{Z};$

in particular, its ${\pi_1}$ is isomorphic to ${\mathbb{Q}/\mathbb{Z}}$, not zero. (more…)

There are a lot of forms of the Brown representability theorem, which all basically assert that a functor on a suitable homotopy category which plays well with arbitrary coproducts and satisfies a weak condition on push-outs, is representable.

The form proved by Brown was the following. Let ${\mathbf{hCW}}$ be the homotopy category of pointed CW complexes

Theorem 1 (Brown, c. 1950) Let ${F: \mathbf{hCW} \rightarrow \mathbf{Sets}}$ be a contravariant functor such that ${F}$ sends coproducts to products. Suppose that if ${(X_1, X_2, A) \subset X}$ is a proper triad—i.e., that ${(X_1, A)}$ and ${(X_2, A)}$ were CW pairs with ${X_1 \cap X_2 = A, X_1 \cup X_2 = X}$—then the map $\displaystyle F(X) \rightarrow F(X_1) \times_{F(A)} F(X_2)$

is surjective. Then ${F}$ is representable. (more…)

Now that we have the powerful tool of Brown representability, let us use it to prove several basic results in homotopy theory. The first one is that any space admits a “CW approximation,” i.e. a CW complex which is weakly homotopy equivalent to it.

In the theory of model categories, which I hope to say more about later, any object has a “cofibrant replacement,” which is such a CW approximation when one uses the standard Quillen model structure for topological spaces. One of the consequences of this is that the homotopy category of ${CW_*}$ is equivalent to the homotopy category of all pointed topological spaces (where homotopy category means something slightly different than it usually does, namely what you get by localizing at weak equivalences).

Proposition 10 Let ${X }$ be any pointed space. Then there is a pointed CW complex ${Y \in CW_*}$ and a weak homotopy equivalence ${Y \rightarrow X}$.

For simplicity, let us just assume ${X}$ is path-connected. Else one can do this for each path component.

Proof: Indeed, we have a functor ${F}$ on ${CW_*}$ sending ${Z \rightarrow \left\{\mathrm{pt \ homotopy \ classes} \ Z \rightarrow X \right\}}$. This is a contravariant functor to ${\mathbf{Sets}_*}$ on the homotopy category. Now the claim is that it satisfies the two axioms of coproducts and Mayer-Vietoris. But we basically checked this right before beginning the proof of Brown representability, and is essentially the gluability of homotopy classes of maps (instead of just functions).
(more…)

So, last time we were talking about Brown representability. We were, in particular, trying to show that a contravariant functor $F$ on the homotopy category of pointed CW complexes satisfying two natural axioms (a coproduct axiom and a Mayer-Vietoris axiom) was actually representable. The approach thus far was to construct pairs which were “partially universal,” that is universal for a finite set of spheres, by a messy attaching procedure.

There is much work left to do. The first is to show that we can get a pair which is universal for all the spheres. This will use a filtered colimit argument. However, we don’t know that $F$ sends filtered colimits into filtered limits, just that for coproducts. In fact, generally $F$ will not do this, but it will do something close. So we will have to appeal to a mapping telescope argument which will, incidentally, use the Mayer-Vietoris property.

Next, we will have to show that a pair which is universal for the spheres is universal for all spaces. This will use a bit of diagram-chasing and the fact that, to some extent, CW complexes are determined by the ways you can map spheres into them. This is the Whitehead theorem.

Let’s get to work. (more…)

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 ${CW_*}$ 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 ${F: CW_* \rightarrow \mathbf{Sets}_*}$ be a contravariant functor to the category of pointed sets. We require that ${F}$ satisfy two axioms below. First, like any representable functor, it should send coproducts to products. Since the coproduct in ${CW_*}$ is the wedge sum, we require that (for ${\alpha}$ ranging over some index set) $\displaystyle F(\bigvee X_\alpha) = \prod F(X_\alpha)$

under the canonical map ${F(\bigvee X_\alpha) \rightarrow \prod F(X_\alpha)}$ that arises by taking the product (over ${\beta}$) of the maps ${F(\bigvee X_\alpha) \rightarrow F(X_\beta)}$.

Second, we require that ${F}$ satisfy the following Mayer-Vietoris axiom. If ${X = X_1 \cup X_2}$ for subcomplexes ${X_1, X_2}$, then if ${q_1 \in F(X_1)}$ and ${q_2 \in F(X_2)}$ “glue together,” i.e. become the same element of ${F(X_1 \cap X_2)}$, they both come from a fixed element of ${F(X)}$. This is a sheaf-theoretic condition.

The first observation is that any representable functor ${F}$ 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 ${X = X_1 \cup X_2}$ as before. Suppose that ${F}$ is represented by a pointed complex ${Y}$. Given “gluable” elements ${f_1: X_1 \rightarrow Y}$ and ${f_2: X_2 \rightarrow Y}$, by assumption the restrictions ${f_1|_{X_1 \cap X_2}, f_2|_{X_1 \cap X_2}}$ 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 ${f_2}$ to some ${f_2'}$ such that ${f_1, f_2'}$ become equal (not merely homotopic) on ${X_1 \cap X_2}$. Together these define a map ${X \rightarrow Y}$ that becomes equivalent to ${f_1, f_2}$.

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 ${F: CW_* \rightarrow \mathbf{Sets}_*}$ is a contravariant functor satisfying the coproduct and Mayer-Vietoris axioms, then ${F}$ is representable.

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

So, today I’m going to talk about Eilenberg-MacLane spaces. These are extremely important in algebraic topology for a number of reasons. The first is that, homotopically, they are simple. By definition, they have only one non-zero homotopy group. So they are in some sense simple.

Moreover, the construction of Postnikov towers shows that any space can be thought of as—in some sense—a “twisted product” of these Eilenberg-MacLane spaces, and in that sense they are “building blocks.” By using the Serre spectral sequence for a fibration, one can often prove things for general spaces by proving them for Eilenberg-MacLane spaces. An important example of this phenomenon is the theorem of Serre (a part of his “mod ${C}$ theory”) that a simply connected space has finitely generated homotopy groups if and only if it has finitely generated homology groups.

Another reason is that many functors in algebraic topology are representable on the homotopy category, at least for CW complexes. By definition, the homotopy groups are representable as maps (in the homotopy category) out of spheres into a space. But more generally, this is true for singular cohomology, and in fact for generalized cohomology theories. The precise statement of this phenomenon is called the Brown representability theorem. I plan to blog more about this more, but a consequence of this is that singular cohomology ${H^n(-, G)}$ with coefficients in some abelian group ${G}$ is representable. The representing objects are in fact the Eilenberg-Maclane spaces ${K(G, n)}$. (more…)