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Let $p : E \to B$ be a covering map with $E$ path connected and $B$ simply connected. Then $p$ is a homeomorphism.

My approach : Let $p(e_0)=b_0.$ Since $E$ is path connected, the lifting correspondence $\phi : \pi_1(B,b_0) \to p^{-1}(b_0)$ is surjective. But since $B$ is simply connected, $\pi_1(B,b_0)=\{e\}.$ Thus $|p^{-1}(b_0)|=1.$ Next, connectedness of $B$ implies that $|p^{-1}(b)|=1 \; \forall \; b \in B.$

Thus every $b \in B$ has an evenly covered neighborhood $U$ such that $p^{-1}(U)=V$ for some open neighborhood of $e_0$ and $p|_U : U \to V$ is a homeomorphism.

I think I am close to finishing the proof but I am stuck here. What is the idea to complete this proof?

Error 404
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  • I don't understand what a surjection from $\pi_1(B)$ to a fiber of $p$ is, and I don't understand how you got it. – Andres Mejia Sep 09 '18 at 21:10
  • You proved that $p$ is a bijective local homeomorphism: what more could you want ? – Maxime Ramzi Sep 09 '18 at 21:19
  • @AndresMejia I actually used Theorem 54.4 of Munkres there. It is as follows "Let $p : E \to B $ is a covering map and $p(e_0)=b_0$. If $E$ is path connected, then the lifting correspondence $\phi : \pi (B,b_0) \to p^{-1}(b_0)$ is surjective. If $E$ is simply connected then it is bijective." – Error 404 Sep 09 '18 at 21:21
  • ahh I understand. I didn't realize that was the "correspondence" map. Maybe you should specify what $\phi$ is. – Andres Mejia Sep 09 '18 at 21:22
  • @AndresMejia My bad. I'll add this in OP now. – Error 404 Sep 09 '18 at 21:23

1 Answers1

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the induced map $p_*:\pi_1(E) \to (B)$ will be injective, so $\pi_1(E)$ also needs to be trivial (making $E$ simply connected as well.)

Since $E$ is path connected, suppose that $|p^{-1}(b)|>1$, and take a path between two points: $\alpha:a \mapsto b$. $p \circ \alpha$ is a loop in $B$, $\dots$

Using the theorem in your question, you have injectivity. You should show that a local homeomorphism is open (maps open sets to open sets) and then you will have the result. See here for a hint.

Andres Mejia
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