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

Let $𝑝 : ˜ 𝑋 ↠ 𝑋$ be a connected and locally path-connected regular covering and $Γ = A u t 𝖢 𝗈 𝗏 𝑋 (𝑝)$ be its deck transformation group. Let $˜ 𝑋 / Γ$ be the orbit space with projection $𝑞 : ˜ 𝑋 ↠ ˜ 𝑋 / Γ$ . Then there exists an isomorphism $Φ : ˜ 𝑋 / Γ \to 𝑋$ such that the following diagram commutes¹: #m/thm/homotopy

Proof

First note that $Γ$ acts properly discontinuously (The deck transformation group acts properly discontinuously) and $𝑞$ is a regular covering (Orbit space of a properly discontinuous effective group action). Since $𝑝$ is clearly constant for each fibre of $𝑞$ , there exists a function $Φ$ such that $𝑝 = Φ 𝑞$ , and by Universal property this is continuous. Since $𝑝$ is surjective so is $𝑞$ , and since $𝑝$ is regular and thus $Γ$ is transitive $Φ$ is injective, because if $Φ [˜ 𝑥 1] = Φ [˜ 𝑥 2]$ it follows

𝑝 (˜ 𝑥 1) = Φ 𝑞 (˜ 𝑥 1) = Φ 𝑞 (˜ 𝑥 2) = 𝑝 (˜ 𝑥 2)

and thus there exists $𝛾 \in Γ$ with $˜ 𝑥 2 = 𝛾 (˜ 𝑥 1)$ , implying $[˜ 𝑥 2] = [˜ 𝑥 1]$ . Since both $𝑝$ and $𝑞$ are local homeomorphisms, so is $Φ$ , in particular it is open. Therefore $Φ$ is a homeomoprhism.

#state/tidy | #lang/en | #SemBr

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

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

Footnotes