So, last time we were talking about Brown representability. We were, in particular, trying to show that a contravariant functor 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 sends filtered colimits into filtered limits, just that for coproducts. In fact, generally 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…)