Monodromy acts by fibre bijections, and its orbits are the intersections of path components with the fibre

Given: The objects, hypotheses, and choice principles stated above.

Fix a covering $p:E\to B$, a basepoint $b_0\in B$, and $e\in p^{-1}(b_0)$. For $[\alpha ]\in {\pi }_1(B,b_0)$, define $e\cdot [\alpha ]$ as the endpoint of the unique lift of $\alpha$ beginning at $e$ (thm-path-lifting-for-covering-maps). Endpoint homotopy invariance makes this well defined (cor-lifted-path-endpoints-depend-only-on-path-homotopy). With the library's traversal-order product this is a right action; the corresponding left action is $[\alpha ]\cdot e:=e\cdot [\alpha {]}^{-1}$ (def-group-action). (The monodromy right action on a covering fibre and its equivalent left-action convention).

For every pointed topological space $(X,x_0)$, the product $[\alpha ][\beta ]=[\alpha \ast \beta ]$ is well defined and makes ${\pi }_1(X,x_0)$ a group. Its identity is the class of the constant loop $c_{x_0}$, and $[\alpha {]}^{-1}=[\bar{\alpha }]$. (Loop classes form the group ${\pi }_1(X,x_0)$ under concatenation).

Throughout, $I:=[0,1]=\{t\in \mathbb{R}:0\le t\le 1\}$ (def-interval) carries the subspace topology inherited from $\mathbb{R}$ with its usual topology (def-subspace-topology-top, lem-real-line-is-a-metric-space, def-metric-topology, def-metrizable-space). It is called the unit interval. A path in $X$ from $x$ to $y$ is a continuous map $\gamma :I\to X$ with $\gamma (0)=x$ and $\gamma (1)=y$, where $I:=[0,1]$ carries the subspace topology inherited from $\mathbb{R}$; $X$ is path-connected when any two of its points are joined by such a path. (Paths, path-connected spaces and path components).