regular covering

Theorem 1.

Let p:EX be a covering map where E and X are connected and locally path connected and let X have a basepoint *. The following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    The action of Aut(p), the group of covering transformations of p, is transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath on the fiber p-1(*),

  2. 2.

    for some ep-1(*), p*(π1(E,e)) is a normal subgroupMathworldPlanetmath of π1(X,*), where p* denotes π1(p),

  3. 3.


  4. 4.

    there is a discrete group G such that p is a principal G-bundle.

All the elements for the proof of this theoremMathworldPlanetmath are contained in the articles about the monodromy action and the deck transformationsMathworldPlanetmath.

Definition 2.

A covering with the properties described in the previous theorem is called a regularPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath or normal covering. The term Galois covering is also used sometimes.

Title regular covering
Canonical name RegularCovering
Date of creation 2013-03-22 13:27:38
Last modified on 2013-03-22 13:27:38
Owner Dr_Absentius (537)
Last modified by Dr_Absentius (537)
Numerical id 6
Author Dr_Absentius (537)
Entry type Definition
Classification msc 55R05
Synonym normal covering
Synonym Galois covering