Orbit space of a properly discontinuous effective group action

Let $˜ 𝑋$ be connected and locally path-connected topological space, $Γ \subseteq A u t 𝖳 𝗈 𝗉 (˜ 𝑋)$ act properly discontinuously on $˜ 𝑋$ ¹, and $𝑋 = ˜ 𝑋 / Γ$ be the orbit space with projection $𝑝 : ˜ 𝑋 ↠ 𝑋$ . Then $𝑝$ is a regular covering and $Γ = A u t 𝖢 𝗈 𝗏 𝑋 (𝑝)$ is its deck transformation group. #m/thm/homotopy

Proof

That $𝑝$ is a covering follows directly from Orbit space of a properly discontinuous group action. It is clear by construction that each $𝛾 \in Γ$ satisfies the following commutative diagram and is thereby a deck transformation, s o $Γ \subseteq A u t 𝖢 𝗈 𝗏 𝑋 (𝑝)$ .

It is also clear by construction that $Γ$ acts transitively on every fibre of $𝑝$ (since the fibres of $𝑝$ are precisely the orbits of $Γ$ ). Now let $𝜙 \in A u t 𝖢 𝗈 𝗏 𝑋 (𝑝)$ , and choose an arbitrary $˜ 𝑥 0 \in ˜ 𝑋$ . Since $Γ$ acts transitively on fibres, there exists a $𝛾 \in Γ$ such that $𝛾 (˜ 𝑥 0) = 𝜙 (˜ 𝑥 0)$ , but both $𝜙$ and $𝛾$ are lifts of $𝑝$ over itself, so it follows by uniqueness that $𝛾 = 𝜙$ . Hence $Γ = A u t 𝖢 𝗈 𝗏 𝑋 (𝑝)$ , and since A covering is regular iff its deck transformation group acts transitively on fibres, $𝑝$ is a regular covering.

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

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

This is equivalent to saying $Γ$ acts effectively on $˜ 𝑋$ . ↩

Orbit space of a properly discontinuous effective group action

Footnotes