nLab deck transformation

Idea

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

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

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

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

Definition

Definition

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

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

Related concepts

• branched cover

• twisted de Rham cohomology

• Borel-equivariant rational homotopy theory

References

Any of the references at covering space.

See also:

• Wikipedia, Deck transformation

• MathWorld, Deck transformation