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.