# regular covering

###### Theorem 1.

Let $p\colon\thinspace E\to X$ be a covering map where $E$ and $X$ are connected and locally path connected and let $X$ have a basepoint $*$. The following are equivalent:

1. 1.

The action of $\operatorname{Aut}(p)$$p$, is transitive on the fiber $p^{-1}(*)$,

2. 2.

for some $e\in p^{-1}(*)$, $p_{*}\left(\pi_{1}(E,e)\right)$ is a normal subgroup of $\pi_{1}(X,*)$, where $p_{*}$ denotes $\pi_{1}(p)$,

3. 3.

$\forall e,e^{\prime}\in p^{-1}(*),\quad p_{*}\left(\pi_{1}(E,e)\right)=p_{*}% \left(\pi_{1}(E,e^{\prime})\right)$,

4. 4.

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

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

###### Definition 2.

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

Title regular covering RegularCovering 2013-03-22 13:27:38 2013-03-22 13:27:38 Dr_Absentius (537) Dr_Absentius (537) 6 Dr_Absentius (537) Definition msc 55R05 normal covering Galois covering