Deck Transformation

Deck transformations, also called covering transformations, are defined for any cover . They act on  by homeomorphisms which preserve the projection . Deck transformations can be defined by lifting paths from a space  to its universal cover , which is a simply connected space and is a cover of . Every loop in , say a function  on the unit interval with , lifts to a path , which only depends on the choice of , i.e., the starting point in the preimage of . Moreover, the endpoint  depends only on the homotopy class of  and . Given a point , and , a member of the fundamental group of , a point  is defined to be the endpoint of a lift of a path  which represents .

The deck transformations of a universal cover  form a group , which is the fundamental group of the quotient space

For example, when  is the square torus then  is the plane and the preimage  is a translation of the integer lattice . Any loop in the torus lifts to a path in the plane, with the endpoints lying in the integer lattice. These translated integer lattices are the group orbits of the action of  on  by addition. The above animation shows the action of some deck transformations on some disks in the plane. The spaces are the torus and its universal cover, the plane. An element of the fundamental group, shown as the path in blue, defines a deck transformation of the universal cover. It moves around the points in the universal cover. The points moved to have the same projection in the torus. The blue path is a loop in the torus, and all of its preimages are shown.