Deck transformation group

Remarks

Example: for $p \colon \R \to S^1$, $p(t) = (\cos 2\pi t, \sin 2\pi t)$, each translation $t \mapsto t + n$ with $n \in \Z$ is a homeomorphism with $p(t + n) = p(t)$. These are all the deck transformations, so the group is isomorphic to $\Z$.

Non-example: the translation $t \mapsto t + \tfrac12$ is a homeomorphism of $\R$, but $p(t + \tfrac12) = -p(t) \ne p(t)$, so it is not a deck transformation.

The group is also called the covering transformation group and is written $\operatorname{Deck}(p)$ or $G(\tilde X)$. If $\tilde X$ is connected, a deck transformation that fixes one point is the identity. The covering is called normal, or regular, when the group acts transitively on each fibre.