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. 🙂
Best Answer
The super-classical example would be the use of the (?Serre's?) theorem that $H^n(X;G) = [X,K(G,n)]$ to deduce that co-dimension two knots have Seifert surfaces. This is written up in Kervaire and Weber's article in LNM 685 "A survey of multi-dimensional knots".
The basic idea goes like this: let $C$ be the complement of a co-dimension two knot in $S^n$. Apply Poincare/Alexander duality to deduce that $C$ is a homology $S^1 \times D^{n-1}$. So $H^1 C \simeq \mathbb Z$, and $H^1 \partial C$ is either $\mathbb Z^2$ or $\mathbb Z$ according to whether or not $n=3$ or $n>3$. In either case the restriction map is an injection. Serre's theorem gives you a map $C \to S^1$ which you make transverse to a point (and a standard projection map on the boundary), this makes the preimage of this point a Seifert surface for the knot. By Seifert-surface I mean an orientable co-dimension one submanifold of $S^n$ whose boundary is the knot.
Serre and Thom used these ideas repeatedly in their early attacks on the Steenrod realization problem. Well, the version where you're trying to realize the homology classes by embedded submanifolds. This is a relative version of their arguments.