= Monodromy group of a covering
For a <covering map> $p:X\to Y$ and base point $y$, the <path lifting theorem> lifts each <based loop> from each point of the <fibre> $p^{-1}(y)$. Its endpoint defines a <permutation> of that <fibre>; <path reversal> gives the inverse. The <homotopy lifting theorem for a covering map> makes the permutation depend only on the <based homotopy> class. These permutations form the monodromy group, the image of the <fundamental group> action
$$
\pi_1(Y,y)\longrightarrow\operatorname{Sym}(p^{-1}(y)).
$$
The image is independent of the convention for composing routes; one may reverse the loop classes to obtain a left <group action> when route multiplication means following the first route and then the second.
For $p(w)=w^2$ on the <punctured plane>, the <fibre> over $1$ is $\{1,-1\}$, and a once-around <based loop> exchanges its two points. The monodromy group has two elements: only the parity of the <winding number> matters. For the <covering map> $p(w)=e^w$ from the <complex plane> to the <punctured plane>, the <fibre> over $1$ is $2\pi i\mathbb Z$, and a positive turn acts by $2\pi ik\mapsto2\pi i(k+1)$. Its monodromy group is an infinite <cyclic group>. These are the geometric versions of <logarithm and square-root continuation along different paths>: the base point returns, while its lifted endpoint can move to another sheet.
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