nLab deck transformation




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory





Generally, given a covering space EXE \to X, a deck transformation is a bundle-automorphism, hence a continuous map δ:EE\delta \,\colon\, E \xrightarrow{\;\sim\;} E fixing the base XX.

Assuming without real restriction that XX is connected, then EXE \to X is a principal bundle with discrete fibers and the deck transformation determines and is determined by the element δ x:E xE x\delta_x \,\colon\, E_x \xrightarrow{\;\sim\;} E_x of the permutation group Aut(E x)Aut(E_x) of its fiber E xE_x over any fixed basepoint xXx \in X.

When the fundamental theorem of covering spaces applies (i.e. when XX is locally path-connected and semi-locally simply-connected topological space) then the fundamental group π 1(X,x)\pi_1(X,x) has a canonical action on this fiber, by a group homomorphism π 1(X,x)Aut(E x)\pi_1(X,x) \xrightarrow{\;} Aut(E_x), and often it is specifically this π 1\pi_1-action which is referred to as being by deck transformations.

In particular, in the case that EXE \to X is the universal cover, then E xπ 1(X)E_x \,\simeq\, \pi_1(X) and EXE \to X is a π 1(X)\pi_1(X)-principal bundle with the fundamental group π 1(X)\pi_1(X) being identified with the group of deck transformations.



A deck transformation or cover automorphism is an automorphism of a covering space relative to the base space.

i.e. if p:EXp\colon E\to X is a cover then a cover automorphism fdeck(p)={f|fAut(E),pf=p}Aut(E)f\in deck(p)=\{f|f\in Aut(E), p\circ f=p\}\subseteq Aut(E) is an automorphism of EE such that pp is invariant under composition with ff.


Any of the references at covering space.

See also:

Last revised on August 18, 2022 at 04:06:50. See the history of this page for a list of all contributions to it.