Action of Deck $(\widetilde X, X)$ on the fibers $p^{-1}(x_0)$:

We fix a base point $x_0 \in X$. Since each deck transformation is a bijection, it is a permutation of the fiber $p^{-1}(x_0)$ and so acts on $p^{-1}(x_0)$ as a group of permutations:

$$(\phi, \tilde x_0) \mapsto \phi(\tilde x_0)$$

We study this action closely and relate it to the action of $\pi_1(X, x_0)$ on the fiber $p^{-1}(x_0)$. We first look at the case of regular coverings