"Cohomology classes correspond to homotopy classes of maps to Eilenberg Maclane spaces" and cup product?

Question

I read this in Hatcher. I am especially interested in knowing if the cup product can be understood from this perspective? I would appreciate a reference.

Answer 1

Yes. If $A_1$ and $A_2$ are abelian groups, there is an "external cup product" map

$$H^n(-, A_1) \times H^m(-, A_2) \to H^{n+m}(-, A_1 \otimes A_2)$$

and by the Yoneda lemma this corresponds to a homotopy class of maps

$$B^n A_1 \times B^m A_2 \to B^{n+m} A_1 \otimes A_2$$

(where $B^n A$ is the Eilenberg-MacLane space $K(A, n)$.) If $A$ has a ring structure, then the multiplication map $A \otimes A \to A$ induces a map $B^n A \otimes A \to B^n A$, and hence we get a composite map

$$B^n A \times B^m A \to B^{n+m} A \otimes A \to B^{n+m} A$$

which represents the usual cup product $H^n(-, A) \times H^m(-, A) \to H^{n+m}(-, A)$.