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. 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 .

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.

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Footnotes

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