I am currently trying to understand the proof of Brown's Representability Theorem, which says that any generalized cohomology theory is represented by an $\Omega$-spectrum. Can anyone point me to some interesting applications of this theorem, within or outside of algebraic topology?

One example that Brown provides in his original paper "Cohomology Theories" is to show that the functor $$ CW_*\to Sets_*; X\mapsto \text{isomorphism classes of principal $G$-bundles on $X$} $$

satisfies his axioms, and hence is must have a classifying space $BG$. Here $CW_*$ and $Sets_*$ denote the category of pointed CW-complexes and sets respectively.

Are there any others? Thanks!


Here's one application, I don't know if it's the kind of thing that you had in mind.

"Cohomology with coefficients in $A$" $H^*(-; A)$ is represented by the Eilenberg-MacLane spectrum $K(A, n)$, because if $X_n$ represents $H^n(-; A)$, then $$\pi_i(X_n) = [S^i, X_n] = H^n(S^i; A) = \begin{cases} A & i = n \\ 0 & i \neq n\end{cases}$$

So "cohomology operations", that is functors $H^n(-; A) \to H^m(-; B)$ are, by the Yoneda lemma, in bijection with elements of $[K(A, n), K(B, m)] \cong H^m(K(A,n); B)$. In particular you can apply the Hurewicz theorem to find them all if $m \leq n$.

This applies to any cohomology theories, not just ordinary cohomology, of course: if $Y_n$ represents $h^n$ and $Z_n$ represents $k^n$ (where $h^*$, $k^*$ are cohomology theories) then: $$\operatorname{Nat}(h^n, k^m) \cong \operatorname{Nat}([-, Y_n], [-, Z_m]) \cong [Y_n, Z_m] \cong k^m(Y_n)$$

So to compute cohomology operations you "just" need to compute cohomology groups of one space. (The catch is that the spaces in question have very complicated homotopy type, generally)


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