Deck transformation

A deck transformation of a covering $p:\tilde{X}\to X$ is a homeomorphism $g:\tilde{X}\to \tilde{X}$ satisfying $p\circ g=p$. Every double covering has a canonical nonidentity deck transformation that exchanges the two points in each fibre.

The deck transformation group consists of all deck transformations under composition. For a connected regular covering corresponding to a normal subgroup $H◃{\pi }_1(X)$, it is isomorphic to ${\pi }_1(X)/H$.

If a finite group $G$ is generated by $h$ elements, send the $a$-generators of the genus-$h$ surface group to those generators and its $b$-generators to the identity. The surface relation is satisfied, so this is a surjective group homomorphism. Its kernel defines a connected regular covering space, on which the deck transformation group $G$ acts freely. The lifted orientation and multiplicativity of the Euler characteristic give genus $g=1+|G|(h-1)$.

Every finitely generated group acts freely and properly discontinuously by Riemannian isometries on a connected manifold in any dimension $d\ge 3$. Choose a surjection $F_r\to T$ and the regular covering of ${\#}_r(S^1\times S^{d-1})$ associated with its kernel. Pulling back a metric makes the deck transformation group isometric. Uniqueness of lifts makes its action free. The cover can be noncompact and need not be simply connected; no claim about realizing every group as a closed three-manifold fundamental group is involved.

For a connected covering, restriction to one fibre identifies the deck transformation group with the centralizer in the fibre's symmetric group of the monodromy action of a covering space. A deck transformation commutes with every lifted loop, and every commuting fibre permutation extends uniquely over the covering by path lifting.

Every two-sheeted covering of a path-connected base has a nonidentity deck transformation that exchanges the two sheets. In monodromy terms, every subgroup of $S_2$ is centralized by its unique transposition.

For a double covering $p:X\to Y$, the mod-two transfer sends a singular simplex of $Y$ to the sum of its two lifts. Together with the chain map $p_{\#}$, it gives a short exact sequence