From time to time, I pretend to be an algebraic topologist. But I'm not really hard-core and some of the deeper mysteries of the subject are still … mysterious. One that came up recently is the exact role of CW-complexes. I'm very happy with the mantra "CW-complexes Good, really horrible pathological spaces Bad." but there's a range in the middle there where I'm not sure if the classification is "Good" or just "Pretty Good". These are the spaces with the homotopy type of a CW-complex.
In the algebraic topology that I tend to do then I treat CW-complexes in the same way that I treat Riemannian metrics when doing differential topology. I know that there's always a CW-complex close to hand if I really need it, but what I'm actually interested in doesn't seem to depend on the space actually being a CW-complex. But, as I said, I'm only a part-time algebraic topologist and so there may be whole swathes of this subject that I'm completely unaware of where actually having a CW-complex is of extreme importance.
Thus, my question:
In algebraic topology, if I have a space that actually is a CW-complex, what can I do with it that I couldn't do with a space that merely had the homotopy type of a CW-complex?
Best Answer
Those of us who do algebraic topology too much should remember occasionally that topological spaces are, in general, terrible to work with.
CW-complexes have a lot of properties that make them nice to work with in homotopy theory, such as being amenable to study by homotopy groups and such as being able to define maps out inductively. These properties are still possessed by objects with the homotopy type of a CW-complex.
However, CW-complexes are also nice on the point-set level. They're compactly generated, locally contractible, and every compact subset is contained in a finite CW-subcomplex - and finite CW-complexes have almost every space-level regularity property that one can name. If one has the goal of doing homotopy theory, the fact that we have this large class of nice spaces means that (to a certain extent) we can set many point-set considerations aside. And sometimes these point-set considerations can be irritating (such as the smash product being nonassociative).
Objects just in the homotopy type can be terrible. Take the cone on your favorite pathological space and you find something with the homotopy type of a nice CW-complex but terrible point-set behavior.