Covering Space - Monodromy Action

Monodromy Action

Again suppose p: C → X is a covering map and C (and therefore also X) is connected and locally path connected. If x is in X and c belongs to the fiber over x (i.e. p(c) = x), and γ: → X is a path with γ(0) = γ(1) = x, then this path lifts to a unique path in C with starting point c. The end point of this lifted path need not be c, but it must lie in the fiber over x. It turns out that this end point only depends on the class of γ in the fundamental group π₁(X, x). In this fashion we obtain a right group action of π₁(X, x) on the fiber over x. This is known as the monodromy action.

There are two actions on the fiber over x: Aut(p) acts on the left and π₁(X, x) acts on the right. These two actions are compatible in the following sense: for all f in Aut(p), c in p−1(x) and γ in π₁(X, x).

If p is a universal cover, then Aut(p) can be naturally identified with the opposite group of π₁(X, x) so that the left action of the opposite group of π₁(X, x) coincides with the action of Aut(p) on the fiber over x. Note that Aut(p) and π₁(X, x) are naturally isomorphic in this case (as a group is always naturally isomorphic to its opposite through g ↦ g−1).

If p is a regular cover, then Aut(p) is naturally isomorphic to a quotient of π₁(X, x).

In general (for good spaces), Aut(p) is naturally isomorphic to the quotient of the normalizer of p_*(π₁(C, c)) in π₁(X, x) over p_*(π₁(C, c)), where p(c) = x.