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 π: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^~ element X^~, which only depends on the choice of f^~ element π^(-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 element X^~, and α, a member of the fundamental group of X, a point α·q is defined to be the endpoint of a lift of a path f which represents α.

cover | fundamental group | group action | universal cover