Monodromy group

Definition
Suppose $$X$$ is a path-connected locally path-connected topological space and $$\tilde{X}$$ is a covering space of $$X$$ with covering map $$p:\tilde{X} \to X$$. Suppose $$x_0\in X$$. Consider the set $$p^{-1}(x_0)$$. There is a natural group action of the fundamental group $$\pi_1(X,x_0)$$ on the set $$p^{-1}(x_0)$$ defined as follows:

Any loop starting and ending at $$x_0$$ defines, for each point in $$p^{-1}(x_0)$$, a unique path starting at the point. Consider the endpoint of this path. This gives a set map $$p^{-1}(x_0) \to p^{-1}(x_0)$$. The inverse loop gives the inverse set map, so the set map is a permutation. Thus, for each loop, we get an element of $$\operatorname{Sym}(p^{-1}(x_0))$$. In other words, we have a map:

$$\Omega(X,x_0) \to \operatorname{Sym}(p^{-1}(x_0))$$

Two homotopic loops (in the based homotopy sense) induce the same permutation, so this descends to a map:

$$\pi_1(X,x_0) \to \operatorname{Sym}(p^{-1}(x_0))$$

Finally, we can verify that the map is a group homomorphism.

The image of this map, viewed as a subgroup of $$\operatorname{Sym}(p^{-1}(x_0))$$ (i.e., not just as an abstract group, but in the context of the action) is termed the monodromy group for the covering map at $$x_0$$. The group action itself is termed the monodromy action.