# pull back of smooth covering space is injective

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.

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

## 2 Answers

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.

• I think your differential is actually an isomorphism. – Mindlack Jan 10 at 17:27
• @Mindlack Sure, since $p$ is a local diffeomorphism. But all we need is surjectivity. – studiosus Jan 10 at 17:44

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