Regular covering

Deck transformation group of a regular covering as quotient

Let be a connected and path-connected regular covering, , and . Let be the basepoint-invariant characteristic subgroup and be the deck transformation group of . Then¹ homotopy

Proof

We will show that an isomorphism is given by

where is a path from to and .

If is an alternative path from to then and thus

so is independent of the choice of .

It is also clear that is a homomorphism, since and for any

Let such that . Then and thus (by First lemma Uniqueness). Then , so since the deck transformation group acts properly discontinuously . Therefore is a group monomorphism.

Let and let be the lift of with . Since acts transitively on fibres there exists a with , and thus . Therefore is a group epimorphism and thus an isomorphism.

In particular, if is simply connected then — see Universal covering.

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Footnotes

1. 2010, Algebraische Topologie, ¶2.3.39, p. 97 ↩