# 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.