[[Category of coverings]]
# Deck transformation

A **deck transformation**[^de1] of a covering $p : \tilde{X} \twoheadrightarrow X$ is an [[automorphism]] of $p$ in [[Category of coverings]].[^loose] #m/def/homotopy 
The set of all deck transformations $\Aut_{\Cov_{X}}(p)$ is clearly a subgroup of the group of homeomorphisms of $\tilde{X}$, called the **deck transformation group**[^de2].
The deck transformation group has a natural [[group action]] on the covering space $\tilde{X}$ by application.

[^loose]: 2010, [[@looseAlgebraischeTopologie2010|Algebraische Topologie]], p. 95
[^de1]: German _Decktransformation_
[^de2]: German _Decktransformationsgruppe_

## Properties

- [[A covering is regular iff its deck transformation group acts transitively on fibres]]
- [[A deck transformation maps sheets to sheets]]
- [[The deck transformation group acts properly discontinuously]]

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