Orbit space of a properly discontinuous group action

Let $𝐺$ be a (discrete) group acting continuously and properly discontinuously on a topological space $˜ 𝑋$ , and let $𝑋 = ˜ 𝑋 / 𝐺$ be the orbit space with the quotient topology and projection $𝑝 : ˜ 𝑋 ↠ 𝑋$ . Then $𝑝$ is a covering.¹ topology

Proof

Let $𝑥 = 𝑝 (˜ 𝑥) = 𝐺 ˜ 𝑥 \in 𝑋$ . Since $𝐺$ acts properly discontinuously, $˜ 𝑥$ has an open neighbourhood $˜ 𝑈$ such $𝛾 1 ˜ 𝑈 \cap 𝛾 2 ˜ 𝑈 = \emptyset$ for any $𝛾 1, 𝛾 2 \in 𝐺$ with $𝛾 1 \neq 𝛾 2$ . Then $𝑈 = 𝑝 (˜ 𝑈) = 𝐺 ˜ 𝑈$ is an evenly covered open neighbourhood of $𝑥$ : $𝑝 - 1 (𝑈) = 𝑝 - 1 (𝑝 (˜ 𝑈)) = ⨿ 𝛾 \in 𝐺 𝛾 ˜ 𝑈$ and $𝑝 ↾ 𝛾 ˜ 𝑈 : 𝛾 ˜ 𝑈 \to 𝑈$ is a homeomorphism for each $𝛾 \in 𝐺$ , since it is surjective and continuous (by construction), injective (no to points in $˜ 𝑈$ lie in the same orbit by proper continuity), and open (in fact $𝑝$ is open, since $˜ 𝑉$ open implies $𝛾 ˜ 𝑉$ open for all $𝛾$ , so $⋃ 𝛾 𝛾 ˜ 𝑉 = 𝑝 - 1 𝑉$ is open and thus $𝑉 = 𝑝 (˜ 𝑉)$ is open). Thus every $𝑥 \in 𝑋$ has an evenly covered neighbourhood, so $𝑝$ is a covering of the orbit space $𝑋$ with $˜ 𝑋$ itself.

Properties

A stronger result is obtained for the Orbit space of a properly discontinuous effective group action: $𝐺$ is itself the deck transformation group.
If $˜ 𝑋$ is simply connected $𝜋 1 (𝑋, 𝑥 0) = 𝐺$ and $𝑝$ is a Universal covering.

tidy | en | SemBr

2010, Algebraische Topologie, pp. 81–82 ↩

Table of Contents

Orbit space of a properly discontinuous group action

Properties

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Backlinks

Table of Contents

Orbit space of a properly discontinuous group action

Properties

Footnotes

Graph View

Backlinks