pull back of smooth covering space is injective

Question

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.

Answer 1

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.

Answer 2

Write your definitions properly and notice that $p$ induces an isomorphism on each fiber of the tangent bundle.