I'm trying to answer this question without the machinery of the cohomology ring/Kunneth/Poincare:
Let $X$ and $Y$ be path-connected and locally contractible spaces such that $H^1(X;\mathbb{Q})\neq0$ and $H^1(Y;\mathbb{Q})\neq0$. Show that $X\vee Y$ is not a retract of $X\times Y$.
I don't see how to do this. I would start by supposing $X\vee Y$ were a retract of $X\times Y$. Then we have an induced injective homomorphism $i_\ast\colon H_1(X)\oplus H_1(Y)\to H_1(X\times Y)$. Now, $H_1(X)\neq0$ and $H_1(Y)\neq0$ by the Universal Coefficient Theorem, and $X\times Y$ is path-connected and locally contractible.
