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 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 10Let be any pointed space. Then there is a pointed CW complex and a weak homotopy equivalence .

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

*Proof:* Indeed, we have a functor on sending . This is a contravariant functor to 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 is representable. Representable on , that is. In particular, there is such that is naturally equivalent to . The fact that the functors of homming into and homming into are naturally equivalent implies by standard nonsense that there is a map such that if is any CW complex, then the map

is a bijection. Taking to be the spheres, this implies that

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 contains a subspace which is itself a CW complex, then we can arrange things so that contains as well, and the map is the identity on .

To see this, consider the pair of the CW complex and its inclusion into . This pair can be extended, by the proof of Brown representability, to a representing pair . The fact that this pair *extends* the original pair means that contains as a subcomplex and extends , up to homotopy; since is a cofibration, we can always make it a precise extension.

Perhaps we ought to state this formally:

Proposition 11Let be a pointed space containing a pointed subspace which is a CW complex. Then there is a CW complex containing as a subcomplex, and a weak homotopy equivalence

which is the identity on .

## Leave a Reply