---
tag: BH32RQBX6A
kind: concept
name: Deck transformation group
msc: [57M10, 55R05]
sources:
  - url: https://en.wikipedia.org/wiki/Covering_space
---

## Summary

The deck transformation group of a [[covering map|3XZRRE165N]] $p \colon \tilde X \to X$ is the group of homeomorphisms of $\tilde X$ that commute with $p$.
These maps permute the points of each fibre, and they measure the symmetry of the covering.
For the [[universal cover|Y1YPE6PM52]] of a reasonable space, the deck transformation group is isomorphic to the [[fundamental group|S1XK3HXHFX]] of the space.

## Explanations

### L1 human

A deck transformation shuffles the sheets of a covering without moving anything in the base.
For the helix $\R$ over the circle, shifting the whole helix up by one full turn is such a symmetry.
The shifts by an integer number of turns are all of them, so the deck transformation group is $\Z$.
This matches the fact that a loop can wind around the circle any integer number of times.