Deck transformation

A deck transformation maps sheets to sheets

Let be a covering and be a Deck transformation. Let , be an evenly covered open neighbourhood of , and be a sheet over . Then is also a sheet over . #m/thm/homotopy

Proof

Given a deck transformation , the following diagram commutes.

https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXHRpbGRlIFgiXSxbNCwwLCJcXHRpbGRlIFgiXSxbMiwyLCJYIl0sWzYsMF0sWzAsMiwicCIsMix7InN0eWxlIjp7ImhlYWQiOnsibmFtZSI6ImVwaSJ9fX1dLFsxLDIsInAiLDAseyJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJlcGkifX19XSxbMCwxLCJcXGdhbW1hIiwwLHsiY3VydmUiOi0xfV0sWzEsMCwiXFxnYW1tYV57LTF9IiwwLHsiY3VydmUiOi0xfV1d

Let be an evenly covered connected open set and be a sheet over . Then is connected and , so for some sheet over . Let . Then so there exists some such that . But is injective in so Therefore .

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