# Monodromy group

## Definition

Suppose is a path-connected locally path-connected topological space and is a covering space of with covering map . Suppose . Consider the set . There is a natural group action of the fundamental group on the set defined as follows:

Any loop starting and ending at defines, for each point in , a unique path starting at the point. Consider the endpoint of this path. This gives a set map . The inverse loop gives the inverse set map, so the set map is a permutation. Thus, for each loop, we get an element of . In other words, we have a map:

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

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

The image of this map, viewed as a subgroup of (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 . The group action itself is termed the **monodromy action**.