During my Basic Algebraic Topology class, my professor left an exercise for us to make, but I'm having some trouble getting ahold of it. Here it is:
Exercise: Let $p:\tilde X\to X$ be a covering map of $m$ sheets, with $m$ prime, and $\tilde X$ simply connected. Prove that $\Pi_1(X)\cong\Bbb Z_m$.
There was a previous result that tells us that the fiber $p^{-1}(x)$ has the discrete topology and $\#p^{-1}(x)=\#L$, being $L$ the set of slices of our covering space. Could I use this in order to follow with anything? Or is there another way I can approach this exercise?
I still want to do the exercise by myself, so could anyone lend me some help to carry on without spoiling the whole result? Thanks in advance.
Choose a base point $b$ in $X$ and one from $p^{-1}(b)$, say $a$. Since $\bar X$ is path connected there exists a path from $a$ to every point in $p^{-1}(b)$. The composition of these paths with $p$ gives non homotopic loops in $X$. Also since $\bar X$ is simply connected every loop in $X$ will be homotopic to one of these. Hence there are exactly $m$ elements in $\pi_{1}(X)$ and hence it is $\Bbb Z_{m}$.