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I need to prove this but I don't really know where to start:

Let $p:M\to N$ be a smooth covering space between smooth manifolds. Show that $p^*:\Omega(N)\to\Omega(M)$ is injective. Where $\Omega(M)$ is the space of the differential forms.

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    $\begingroup$ $p^*$ is conventionally $\Omega(N) \rightarrow \Omega(M)$. $\endgroup$ – Mindlack Jan 10 at 17:00
  • $\begingroup$ Oh sure, that was a typo. Edited, thanks $\endgroup$ – Gianni Jan 10 at 17:09
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I would proceed as follows:

1) Note that the differential $d_{q}p:T_{q}M\rightarrow T_{p(q)}N$ is a surjective linear map, at any point $q\in M$.

2) Recall that the dual of a surjective linear map is injective.

3) Apply these facts pointwise to get the result.

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  • $\begingroup$ I think your differential is actually an isomorphism. $\endgroup$ – Mindlack Jan 10 at 17:27
  • $\begingroup$ @Mindlack Sure, since $p$ is a local diffeomorphism. But all we need is surjectivity. $\endgroup$ – studiosus Jan 10 at 17:44
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Write your definitions properly and notice that $p$ induces an isomorphism on each fiber of the tangent bundle.

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