Induced homomorphism is injective

Question

Let $\pi:M\longrightarrow N$ be a surjective submersion or fiber bundle(local trivialization) or fibration. Here $M$ and $N$ are manifolds. Why the induced homomorphism of first cohomology group $\pi^{*}:H^{1}(N)\longrightarrow H^{1}(M)$ is injective?

Is proposition correct for all case?

Answer 1

Let $L$ be a $p$-lens space i.e the quotient of $S^{2n-1}, n>1$ by the action of $\mathbb{Z}/p$, $H_1(L,\mathbb{Z})$ is not trivial implies that $H^1(L,\mathbb{Z}/p)$ is trivial, but $H^1(S^3,\mathbb{Z}/p)$ is trivial.

Integral homology of lens space.