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.

Advertisements

About mathtuition88

http://mathtuition88.com
This entry was posted in math. Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s