regular covering

Theorem 1.

Let $p\mathrm{:}E\mathrm{\to }X$ be a covering map where $E$ and $X$ are connected andlocally path connected and let $X$ have a basepoint $\mathrm{*}$. The following areequivalent:

1. The action of $\mathrm{Aut}(p)$, the group of covering transformations of$p$, is transitive on the fiber $p^{-1}(*)$,

2. for some $e\in p^{-1}(*)$, $p_{*}\left({\pi }_1(E,e)\right)$ is anormal subgroup of ${\pi }_1(X,*)$, where $p_{*}$denotes ${\pi }_1(p)$,

3. $\forall e,e^{\prime }\in p^{-1}(*),p_{*}\left({\pi }_1(E,e)\right)=p_{*}\left({\pi }_1(E,e^{\prime })\right)$,

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 articlesabout the monodromy action and the deck transformations.

Definition 2.

A covering with the properties described in the previous theorem is calleda regular or normal covering. The term Galoiscovering is also used sometimes.