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regular covering
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(Definition)
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Theorem 1 Let $p\co E\to X$ be a covering map where $E$ and $X$ are connected and locally path connected and let $X$ have a basepoint $*$ . The following are equivalent:
- The action of $\Au(p)$ , the group of covering transformations of $p$ , is transitive on the fiber $p^{-1}(*)$ ,
- for some $e\in p^{-1}(*)$ , $p_*\left(\pi_1(E,e)\right)$ is a normal subgroup of $\pi_1(X,*)$ , where $p_*$ denotes $\pi_1(p)$ ,
- $\forall e,e'\in p^{-1}(*), \quad p_*\left(\pi_1(E,e)\right)=p_* \left(\pi_1(E,e')\right)$ ,
- 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.
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"regular covering" is owned by Dr_Absentius.
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(view preamble | get metadata)
| Other names: |
normal covering, Galois covering |
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Cross-references: term, normal, regular, properties, covering, deck transformations, monodromy action, contained, theorem, proof, group, discrete, normal subgroup, fiber, transitive, group of covering transformations, action, the following are equivalent, basepoint, locally path connected, connected, covering map
There are 2 references to this entry.
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 6794 times total.
Classification:
| AMS MSC: | 55R05 (Algebraic topology :: Fiber spaces and bundles :: Fiber spaces) |
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Pending Errata and Addenda
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