A deck transformation maps sheets to sheets

Let $𝑝 : ˜ 𝑋 ↠ 𝑋$ be a covering and $𝛾 : ˜ 𝑋 \to ˜ 𝑋$ be a Deck transformation. Let $𝑥 0 \in 𝑋$ , $𝑈 \subseteq 𝑋$ be an evenly covered open neighbourhood of $𝑥 0$ , and $˜ 𝑈 \subseteq ˜ 𝑋$ be a sheet over $𝑈$ . Then $𝛾 (˜ 𝑈)$ is also a sheet over $𝑈$ . #m/thm/homotopy

Proof

Given a deck transformation $𝛾 : ˜ 𝑋 \to ˜ 𝑋$ , the following diagram commutes.

Let $𝑈$ be an evenly covered connected open set and $˜ 𝑈$ be a sheet over $𝑈$ . Then $𝛾 (˜ 𝑈)$ is connected and $𝑝 𝛾 (˜ 𝑈) = 𝑈$ , so $𝛾 (˜ 𝑈) \subseteq ˜ 𝑈'$ for some sheet $˜ 𝑈'$ over $𝑈$ . Let $˜ 𝑥 \in ˜ 𝑈'$ . Then $𝑝 (˜ 𝑥) \in 𝑈$ so there exists some $˜ 𝑥' \in 𝛾 (˜ 𝑈) \subseteq ˜ 𝑈'$ such that $𝑝 (˜ 𝑥') = 𝑝 (˜ 𝑥)$ . But $𝑝$ is injective in $˜ 𝑈'$ so $˜ 𝑥 = ˜ 𝑥' \in ˜ 𝑈'$ Therefore $˜ 𝑈' = ˜ 𝑈$ .

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