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.
