Prove that any map $f: S^2 \times S^2 \to S^2$ is null-homotopic on one coordinate, which is to say, either $f|: S^2 \times \left\{0\right\} \to S^2$ or $f|: \left\{0\right\} \times S^2 \to S^2$ is null homotopic.
I know this can be proved by considering the induced map on the cohomology ring $f^{\ast}: H^{\ast}(S^2)\to H^{\ast}(S^2\times S^2)$.
But I think there should be a more elementary way to prove this. Does anyone have an elementary proof simpler than the cohomology ring?