Deck Transformation

Deck transformations, also called covering transformations, are defined for any cover p:A->X. They act on A by homeomorphisms which preserve the projection p. Deck transformations can be defined by lifting paths from a space X to its universal cover X^~, which is a simply connected space and is a cover of pi:X^~->X. Every loop in X, say a function f on the unit interval with f(0)=f(1)=p, lifts to a path f^~ in X^~, which only depends on the choice of f^~ in pi^(-1)(p), i.e., the starting point in the preimage of p. Moreover, the endpoint f^~(1) depends only on the homotopy class of f and f^~(0). Given a point q in X^~, and alpha, a member of the fundamental group of X, a point alpha·q is defined to be the endpoint of a lift of a path f which represents alpha.

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

Deck transformation

For example, when X is the square torus then X^~ is the plane and the preimage pi^(-1)(p) is a translation of the integer lattice {(n,m)} subset R^2. 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 Z×Z on R^2 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.

See also

Cover, Fundamental Group, Group Action, Universal Cover

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Deck Transformation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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