monodromy


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

    eFeγ1=eγ2.
  2. 2.

    The map

    F×πF,(e,γ)eγ

    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:

    Stab(x)=p*(π1(E,e)).
Proof.
  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

    eF,e1=e.

    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

    e(γ1γ2)=(eγ1)γ2.
  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

ρ:πSym(F)

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