Regular covering

Correspondence between regular coverings and orbit spaces of their deck transformation groups

Let be a connected and locally path-connected regular covering and be its deck transformation group. Let be the orbit space with projection . Then there exists an isomorphism such that the following diagram commutes¹: homotopy

Proof

First note that acts properly discontinuously (The deck transformation group acts properly discontinuously) and is a regular covering (Orbit space of a properly discontinuous effective group action). Since is clearly constant for each fibre of , there exists a function such that , and by Universal property this is continuous. Since is surjective so is , and since is regular and thus is transitive is injective, because if it follows

and thus there exists with , implying . Since both and are local homeomorphisms, so is , in particular it is open. Therefore is a homeomoprhism.

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Footnotes

1. 2010, Algebraische Topologie, ¶2.3.38, pp. 96ff ↩