Induced map between two product manifolds

Question

Let $f: S^2 \times S^5 \longrightarrow S^3 \times S^4$ be a smooth map. Show that the induced map $f^*: H^7_{dR}(S^3 \times S^4) \longrightarrow H^7_{dR}(S^2 \times S^5)$ is not surjective.

I know that both cohomology groups are isomorphic to $\mathbb{R}$, and a 7 form on $S^3\times S^4$ is a 3 form wedge a 4 form. Is the 7 form 0 after pullback? How can I prove it?

Answer 1

$H^7(S^2\times S^7)$ is generated by the product of a form of degree $2$ and a form of degree 5 (kunneth) the image of such a form by $f$ is zero since $H^2(S^3\times S^4)=0$.