Deck transformation - Knowlpedia

For a covering $p:E\to B$ , a deck transformation is a homeomorphism $h:E\to E$ such that $p\circ h=p$ . It moves points within fibers while preserving the covering projection. Deck transformations form a group under composition.

Torus deck translations

For an integer-matrix covering $p_A$ , each translation $[x]\mapsto[x+a]$ with $Aa\in\mathbb Z^n$ is a deck transformation. These are all the deck transformations: the continuous difference $h([x])-[x]$ lies in the finite kernel of $p_A$ , and the torus is connected, so that difference is constant. The group is $A^{-1}\mathbb Z^n/\mathbb Z^n$ . A general covering need not have enough deck transformations to act transitively on every fiber.

References

Hatcher, Algebraic Topology, §1.3 (covering spaces and deck transformations).