Definition 17.1:

Let $p: {\tilde X}\longrightarrow X$ be a covering projection and $x_0 \in X$ be a given point. For a loop $\gamma$ in $X$ based at $x_0$, define the right-action of $\pi_1(X, x_0)$ on the fiber $p^{-1}(x_0)$ as follows. For ${\tilde x}_1\in p^{-1}(x_0)$,

$${\tilde x}_1\cdot[\gamma] = {\tilde \gamma}(1),\eqno(17.1)$$

where $\tilde \gamma$ is the unique lift of $\gamma$ starting at ${\tilde x}_1$.