I know that the pull-back of a surjective smooth submersion is injective on k-forms on a manifold. That is, if $f:M\rightarrow N$ is surjective and a smooth submersion (for every $p\in M$ the push-forward/differential at p, $f_{*,p}:T_{p}M\rightarrow T_{f(p)}N$ is surjective) then the pull-back
$$f^{*}:\Omega^{k}(N)\rightarrow\Omega^{k}(M)$$ is injective. I was wondering on which level the condition that f is surjective is neccesary. Is there a counterexample that illustrate the necessity of this condition? Thank you in advance.
Suppose $M$ is a manifold and $N=M_0\cup M_1$ is the union of two disjoint copies of $M$, The embedding $i_0:M_0\rightarrow M$ is a smooth submersion, but it is not injective on $k$-forms, let $\alpha$ be a $k$-form on $M_1$, it can be extended to $N$ by setting $\alpha_{\mid M_0}=0$ and the pullback of $\alpha$ is zero.
Suppose that $U$ and $V$ are open disjoint subsets of $N$, using cut-off function, we can defined a function $f$ such that there exists an open subset $U'\subset U$ such that $f_{\mid U'}=1$ and the restriction of $f$ on $N-U$ is zero. Let $M=V$ and $i:V\rightarrow N$ the embedding. Let $\alpha$ be a non zero form on $U$, non zero on $U'$ (can be constructed with cut off functions) $f\alpha$ can be extended to $N$ by setting $f\alpha_{\mid N-U}=0$, the pullback of this extension by $i$ is zero.