In my topology assignment I came across the following problem:
True or false? Let $E$ and $X$ be path-connected. For every covering map $p:E\rightarrow X$ and continuous map $f:E\rightarrow E$ such that $pf=p$ holds $f$ is homeomorphism.
I couldn't think of any counterexample, so my guess is that assertion is true. Since $X$ is not locally path-connected, I cannot apply lifting theorem to obtain $f$ and its inverse, so I guess I should manually prove properties of homeomorphism on $f$, using path lifting theorem several times.
Recently I had to deal with the operation of a fundamental group operation on a set $p^{-1}(x)$ for $x\in X$ such that $p^{-1}(x)\times \pi_1(X,x)\rightarrow \pi_1(X, x): [w]\cdot e=w(1)$, that is each point in $E$ maps to the endpoint of a unique lift of the path $w$ (similar to the one in Hatcher). I played around a little trying to prove injection / surjection property of $f$, but didn't got far. The only thing I proved is that $f([w]\cdot e) = [w]\cdot f(e)$ but nothing more.
So I ask what would be an adequate approach to solve this question.
Thanks in advance!
Hint 1: Here's a hint for showing that $f$ is a local homeomorphism: The condition $pf=p$ implies that points in $e$ are carried to points in the fiber $p^{-1}(p(e))$. We can find neighborhoods $U$ of $e$ and $W$ of $f(e)$ such that $p$ restricts to a homeomorphism on each neighborhood and $f(U) \subset W$. See if you can use this to show that $f$ in fact carries $U$ homeomorphically into a subset of $W$.
Hint 2: [Courtesy of the OP's own suggestion in the comments below.] For both injectivity and surjectivity: The action of $\pi_1(X,x_0)$ on $p^{-1}(x_0)$ can be shown to satisfy $f(e) \cdot [\gamma]=f(e \cdot [\gamma])$. Playing around with this fact and a couple well-chosen paths between points in $p^{-1}(x_0)$ will give both injectivity and surjectivity.