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regular covering (Definition)
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. The action of $ \operatorname{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 a normal subgroup of $ \pi_1(X,*)$, where $ p_*$ denotes $ \pi_1(p)$,
  3. $ \forall e,e'\in p^{-1}(*), \quad p_*\left(\pi_1(E,e)\right)=p_* \left(\pi_1(E,e')\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 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.



"regular covering" is owned by Dr_Absentius.
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Other names:  normal covering, Galois covering
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Cross-references: term, normal, regular, properties, covering, deck transformations, monodromy action, contained, group, discrete, normal subgroup, fiber, transitive, group of covering transformations, action, the following are equivalent, basepoint, locally path connected, connected, covering map
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This is version 3 of regular covering, born on 2003-02-12, modified 2004-01-24.
Object id is 4027, canonical name is RegularCovering.
Accessed 4826 times total.

Classification:
AMS MSC: 55R05 (Algebraic topology :: Fiber spaces and bundles :: Fiber spaces)
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