This came up in the [question about Eilenberg-MacLane spaces](https://mathoverflow.net/questions/2890/cohomology-and-eilenberg-maclane-spaces). Given the definition of `K(G, n)`, it's easy to prove that there is a map `K(G,n) x K(G,n) --> K(G,n)` that endows cohomology with an additive structure.



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> **Question:** what's the most geometric way to show the existence of maps `K(G,n) x K(G,m) --> K(G,n+m)` that endow cohomology with multiplicative structure?
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How to get product on cohomology using the K(G, n)?
If you form the smash product $X = K(A,p) \wedge K(B,q)$ of two Eilenberg-MacLane spaces, then the resulting space is $(p+q-1)$-connected, and the first non-trivial homotopy group in dimension $p+q$ is $A \otimes B$.


To see this "geometrically", I would model the EM spaces as CW-complexes, whose first non-basepoint cells are in dimensions p and q respectively. Then in the smash product $X$, the bottom dimensional cells are products of the bottom dimensional cells of the EM spaces; this gives the connectivity result, and by looking at the attaching maps of the $(p+q+1)$-dimensional cells in $X$, you can compute $\pi\_{p+q}$. Now you can use obstruction theory to produce a map $X \to K(A\otimes B, p+q)$.


