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.

Author: mathtuition88

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