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.

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