Deck transformation

Orbit space of a properly discontinuous effective group action

Let be connected and locally path-connected topological space, act properly discontinuously on 1, and be the orbit space with projection . Then is a regular covering and is its deck transformation group. #m/thm/homotopy

Proof

That is a covering follows directly from Orbit space of a properly discontinuous group action. It is clear by construction that each satisfies the following commutative diagram and is thereby a deck transformation, s o .

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

It is also clear by construction that acts transitively on every fibre of (since the fibres of are precisely the orbits of ). Now let , and choose an arbitrary . Since acts transitively on fibres, there exists a such that , but both and are lifts of over itself, so it follows by uniqueness that . Hence , and since A covering is regular iff its deck transformation group acts transitively on fibres, is a regular covering.

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

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

Footnotes

  1. This is equivalent to saying acts effectively on . ↩