Monodromy action of a covering space

Lifting loops based at $x_0$ permutes the fibre $p^{-1}(x_0)$ and defines the monodromy action of ${\pi }_1(X,x_0)$ on that fibre. A connected covering has transitive monodromy.

For a nondegenerate embedded smooth algebraic curve over the complex numbers, the cover that chooses one point of a transverse hyperplane section has transitive monodromy. The analogous incidence space for two distinct ordered points is irreducible, giving two-transitivity. A loop around an ordinary tangent hyperplane exchanges two intersection points. Two-transitivity conjugates this transposition to every transposition, so the monodromy is the full symmetric group. In particular its action on fixed-size subsets is transitive.

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.