As I self-study Bott and Tu 's Differential Forms in Algebraic Topology, I am stopped by some quite basic questions:
1) Let $M$ and $F$ are real manifolds, and let $\pi : M\times F \rightarrow M$ and $\mu: M\times F \rightarrow F$ to the projection maps. Let $\psi :\Omega^*(M)\otimes \Omega^*(F) \rightarrow \Omega^*(M\times F)$ be the map given by $\omega\otimes\phi \mapsto \pi^*(\omega)\wedge \mu^*(\phi)$. Is the map $\psi$ injective?
2) What exactly is the distinction bewteen $\omega\otimes\phi$ and $\pi^*(\omega)\wedge \mu^*(\phi)$? Before I really start thinking about the first question, I tend to think them as one same thing. Appreciated for any help.