Let $\pi: E \to M$ be a fibration of smooth manifolds, where $M$ is simply connected. How can I show that $\pi^*:\Omega^*(M) \to \Omega^*(E)$ is injective?
This fact was used in the proof of proposition 18.13 in Bott and Tu, but I don't know why this is true. I've also tried looking up some sources but haven't seen this result stated elsewhere. On a side note, I realized that if there is a section $s:M \to E$, $\pi^*$ will be injective but I believe fibrations don't generally have sections(?).
Please note that I have asked the same question at Induced homomorphism on cohomology by fibration but sadly had made a mistake in mistyping $\Omega^*$ as $H^*$. Since MikeMiller has provided a nice counterexample for that problem it seems appropriate for me ask the right version of my question in another topic.