Monodromy action of a covering space

Let $p:E\to X$ be a covering space with path-connected base, and fix a base point $x_0\in X$. Lift a based loop $\alpha$ from each $e\in p^{-1}(x_0)$ and let $e\cdot [\alpha ]$ be its endpoint. The path lifting theorem and homotopy lifting property make this a well-defined permutation of the fibre. With loop multiplication $\alpha \ast \beta$ meaning traverse $\alpha$ first and then $\beta$, it is a right group action:Its orbits correspond exactly to the path components of $E$. Indeed, two fibre points are joined by a path in $E$ precisely when projecting such a path gives a based loop whose lift joins them. Every path component meets the fibre, since a path from any base point to $x_0$ can be lifted.

Consequently this monodromy action of a covering space is transitive if and only if $E$ is path-connected. If $X$ is also locally path-connected, then $E$ is locally path-connected too, and connectedness of $E$ suffices. The local condition explains the familiar formulation using a connected cover.

Let $C\subset {\mathbb{P}}_{\mathbb{C}}^r$, $r\ge 2$, be a smooth irreducible nondegenerate projective variety of dimension one and degree $d\ge 2$. Let $U$ be the open subset of the dual projective space consisting of hyperplanes meeting $C$ in $d$ distinct points. The incidence projectionis a $d$-sheeted covering space in the complex analytic topology. Its monodromy is the full symmetric group $S_d$.^[ref]

The incidence space choosing two distinct ordered intersection points is irreducible, so the monodromy action of a covering space is two-transitive. A small loop around an ordinary tangent hyperplane exchanges its two nearby intersection points and fixes the others. Conjugating this transposition by double transitivity gives every transposition, and these generate $S_d$. In particular the action on $k$-element subsets is transitive for each $0\le k\le d$. For a degree-one projective line, a transverse section has one point and the corresponding group is the trivial $S_1$.

Lifting a based loop in a covering space from every point of its finite fibre gives a permutation of that fibre. Composition of lifted loops yields the monodromy action of a covering space. If a closed geodesic of length $\ell$ has a cycle of length $m$ in its monodromy permutation, the corresponding lift closes after $m$ traversals and has length $m\ell$. For a primitive base geodesic, these cycles describe the primitive lift components.