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).

So ${F}$ is representable. Representable on ${CW_*}$, that is. In particular, there is ${Y \in CW_*}$ such that ${F}$ is naturally equivalent to ${\hom_{CW_*}(-, Y)}$. The fact that the functors of homming into ${Y}$ and homming into ${X}$ are naturally equivalent implies by standard nonsense that there is a map ${Y \rightarrow X}$ such that if ${Z}$ is any CW complex, then the map

$\displaystyle \left\{\mathrm{homotopy \ classes} \ Z \rightarrow Y \right\} \rightarrow \left\{\mathrm{homotopy \ classes} \ Z \rightarrow X \right\}$

is a bijection. Taking ${Z}$ to be the spheres, this implies that

$\displaystyle Y \rightarrow X$

is a weak homotopy equivalence.

Weak homotopy equivalences are not everything. But they can’t distinguish between homology or cohomology, and a weak equivalence between CW complexes is a homotopy equivalence, so they’re at least pretty close.

In fact, though, the actual proof of Brown representability buys us a bit more than that. If ${X}$ contains a subspace ${A}$ which is itself a CW complex, then we can arrange things so that ${Y}$ contains ${A}$ as well, and the map ${Y \rightarrow X}$ is the identity on ${A}$.

To see this, consider the pair ${(A, A \hookrightarrow X)}$ of the CW complex ${A}$ and its inclusion into ${X}$. This pair can be extended, by the proof of Brown representability, to a representing pair ${(Y, Y \rightarrow X)}$. The fact that this pair extends the original pair means that ${Y}$ contains ${A}$ as a subcomplex and ${Y \rightarrow X}$ extends ${A \rightarrow X}$, up to homotopy; since ${A \hookrightarrow Y}$ is a cofibration, we can always make it a precise extension.

Perhaps we ought to state this formally:

Proposition 11 Let ${X}$ be a pointed space containing a pointed subspace ${A}$ which is a CW complex. Then there is a CW complex ${Y}$ containing ${A}$ as a subcomplex, and a weak homotopy equivalence$\displaystyle Y \rightarrow X$

which is the identity on ${A}$.