Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 1 / 21H / Solution

The path lifting theorem says that a path $\gamma :[0,1]\to X$ has a unique lift through a covering map once its initial point in the fiber is chosen. The homotopy lifting lemma says that a homotopy $H:Z\times [0,1]\to X$, with a continuous lift specified at time zero, has a unique continuous lifted homotopy. In particular a homotopy of paths with fixed endpoints preserves the endpoint of a chosen lift, since that endpoint varies continuously in a discrete fiber.

Choose $x_0\in X$ and $y_i\in p_i^{-1}(x_0)$. For $y\in Y_1$, choose a path $\gamma$ in $Y_1$ from $y_1$ to $y$, and let $F(y)$ be the endpoint of the lift of $p_1\gamma$ to $Y_2$ starting at $y_2$. Two choices of $\gamma$ are homotopic relative endpoints because $Y_1$ is simply connected. Project that homotopy and apply homotopy lifting: the lifted endpoint remains fixed. Thus $F$ is well-defined and $p_{2}F=p_1$.

To prove continuity without invoking covering-space classification, choose a path-connected open neighborhood $U$ of $p_1(y)$ evenly covered by both maps. Local path connectedness of $X$ permits this choice. On the sheet of $p_1^{-1}(U)$ containing $y$, append paths within that sheet to the chosen path to $y$. Path-lift uniqueness givesthere, for the sheet $V_2$ containing $F(y)$. Hence $F$ is a local homeomorphism and is continuous. Construct $G:Y_2\to Y_1$ in the same way with the basepoints reversed. Lifting the very same projected paths proves $GF$ and $FG$ fix every point. Therefore