Let (X,*) be a connected and locally connected based space and p:EX a covering map. We will denote p-1(*), the fiber over the basepoint, by F, and the fundamental groupMathworldPlanetmathPlanetmath π1(X,*) by π. Given a loop γ:IX with γ(0)=γ(1)=* and a point eF there exists a unique γ~:IE, with γ~(0)=e such that pγ~=γ, that is, a lifting of γ starting at e. Clearly, the endpoint γ~(1) is also a point of the fiber, which we will denote by eγ.

Theorem 1.

With notation as above we have:

  1. 1.

    If γ1 and γ2 are homotopicMathworldPlanetmathPlanetmath relative I then

  2. 2.

    The map


    defines a right action of π on F.

  3. 3.

    The stabilizerMathworldPlanetmath of a point e is the image of the fundamental group π1(E,e) under the map induced by p:

  1. 1.

    Let eF, γ1,γ2:IX two loops homotopic relative I and γ~1,γ~2:IE their liftings starting at e. Then there is a homotopyMathworldPlanetmath H:I×IX with the following properties:

    • H(,0)=γ1,

    • H(,1)=γ2,

    • H(0,t)=H(1,t)=*,tI.

    According to the lifting theorem H lifts to a homotopy H~:I×IE with H(0,0)=e. Notice that H~(,0)=γ~1 (respectively H~(,1)=γ~2) since they both are liftings of γ1 (respectively γ2) starting at e. Also notice that that H~(1,) is a path that lies entirely in the fiber (since it lifts the constant path *). Since the fiber is discrete this means that H~(1,) is a constant path. In particular H~(1,0)=H~(1,1) or equivalently γ~1(1)=γ~2(1).

  2. 2.

    By (1) the map is well defined. To prove that it is an action notice that firstly the constant path * lifts to constant paths and therefore


    Secondly the concatenation of two paths lifts to the concatenation of their liftings (as is easily verified by projecting). In other words, the lifting of γ1γ2 that starts at e is the concatenation of γ~1, the lifting of γ1 that starts at e, and γ~2 the lifting of γ2 that starts in γ1(1). Therefore

  3. 3.

    This is a tautologyMathworldPlanetmath: γ fixes e if and only if its lifting starting at e is a loop.

Definition 2.

The action described in the above theorem is called the monodromy action and the corresponding homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath


is called the monodromy of p.

Title monodromy
Canonical name Monodromy
Date of creation 2013-03-22 13:26:20
Last modified on 2013-03-22 13:26:20
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 8
Author mathcam (2727)
Entry type Definition
Classification msc 55R05
Related topic MonodromyGroup
Defines monodromy
Defines monodromy action
Defines monodromy homomorphism