Say you have a covering space $C \rightarrow X$ corresponding to some homomorphism $\pi_1(X)\rightarrow S_n$. Suppose you have an arbitrary (continuous) map $f : Y\rightarrow X$. Then we may pull back the cover $C$ to get a cover $C\times_X Y$ of $Y$. Does this cover of $Y$ correspond to the representation
$$\pi_1(Y)\stackrel{f_*}{\rightarrow}\pi_1(X)\rightarrow S_n$$
If so, why?
- will