Examples 19.1:

(i) For the covering space ex$: \mathbb{R} \longrightarrow S^1$ given by ex$(t) = \exp(2\pi it)$ the deck transformations are the maps

$$T_n :\mathbb{R} \longrightarrow \mathbb{R}, \quad T_n(x) = x + n,\quad n\in\mathbb{Z}$$

(ii) For the two sheeted covering $p : S^n \longrightarrow \mathbb{R}P^n$ the deck transformations are the identity map and the antipodal map.

The following theorem summarizes the most basic properties of the group of deck transformations.