Probably you can "google" this question, but I can't find anything relevant. The classical Brown Representability Theorem states: Denote $hCW_*$ the homotopy category of pointed CW-complexes. Let $F : hCW_* \to Set_*$ be a contravariant functor. Then $F$ is representable if and only if

- $F$ respects coproducts, i.e. $F(\vee_{i \in I} X_i) = \prod_{i \in I} F(X_i)$ for all families $X_i$ of pointed CW-complexes.
- $F$ satisfies a sort of mayer-vietoris-axiom: If $X$ is a pointed CW-complex which is the union of two pointed subcomplexes $A,B$, then the canonical map $F(X) \to F(A) \times_{F(A \cap B)} F(B)$ is surjective.

I just know two applications: classifying spaces for $G$-principal bundles (in particular, vector bundles) for a locally compact topological group $G$ and for generalized cohomology theories on $CW_*$; the yoneda-lemma also yields functorial relations (cf. Switzer, Algebraic Topology). I'm interested in other explicit applications. I've read that there are categorical generalizations, but in this question I'm just asking whether there are explicit functors defined on CW-complexes, whose representabilty is of interest and can be shown with the theorem above. Also, these examples should really differ from the two ones mentioned above. :-)

Set-valued rather thanabelian group-valued.". Is that the correct interpretation? – Loop Space Jan 12 '10 at 12:47