Deck Transformations

Consider a covering space $p:\widetilde{X}\to X$ . The isomorphisms $\widetilde{X}\to\widetilde{X}$ are called deck transformations, and they form a group $G(\widetilde{X})$ under composition.

For the covering space $p:\mathbb{R}\to S^1$ projecting a vertical helix onto a circle, the deck transformations are the vertical translations mapping the helix onto itself, so $G(\widetilde{X})\cong\mathbb{Z}$ , where a vertical translate of $n$ “steps” upwards/downwards corresponds to the integer $\pm n$ respectively.