Let $\pi:E \to M$ be a fibration, where $M$ is simply connected. How can I show that $\pi^{*}:H^{*}(M) \to H^{*}(E)$ 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 $s:M \to E$ then $\pi^{*}$ will be injective but I believe fibrations don't generally have sections(?). Other than this remark I don't have any ideas how to show this.
I would be grateful if someone can show me a proof of why this is correct