Hatcher's AT Theorem 4.57 is used in both the algebraic topology construction of Seifert surfaces, and the (similarly flavored) proof that given a compact 3-manifold (with or without boundary), we can associate a properly embedded surface $S$ to any $[a] \in H_2(M, \partial M; \mathbb{Z})$ (this is used to define the Thurston Norm).
Theorem 4.57 states (direct quote): there are natural bijections $T: \langle X, K(G,n) \rangle \to H^n(X; G)$, for all CW-complexes $X$ and all $n > 0$, with $G$ any abelian group. Such a $T$ has the form $T([f]) = f^*(\alpha)$ for a certain distinguished class $\alpha \in H^n(K(G,n); G)$ (Hatcher, Algebraic Topology, Page 393). (Notation: $\langle X, K(G,n) \rangle$ is the set of basepoint-preserving homotopy classes of maps from $X$ to a $K(G,n)$).
The proof (to me, at least) is somewhat complicated, and is constructed by creating a cohomology theory, and uses some basic formalisms (that is, once you get through the 8 pages of algebraic topology, the proof of 4.57 falls out in 1 paragraph).
My question: the $T$ above is a bijection; suppose we assume $X$ is something nice, like a smooth manifold, and $K(G,n)$ is also relatively simple, perhaps $S^1$, which is a $K(\mathbb{Z}, 1)$ -- is there some topological or geometric way to see how a function $f \in \langle X, K(G,n)\rangle$ is built in association with $[a] \in H^n(X,\mathbb{Z})$?
A more general (and fluffy) question: again assuming $X, K(G,n)$ are "nice", as above, is there some intuition for why this theorem holds? (I understand that this is perhaps a somewhat unreasonable question).
Thank you!
