Let $p\ \colon \widetilde X \longrightarrow X$ be a covering map such that any loop $\gamma$ in $X$ based at $x_0 \in X$ lifts uniquely to a loop based at $\widetilde {x_0} \in p^{-1} (\{x_0\}).$ Assume that $X$ is path connected. For any $x \in X,$ take a path $\gamma$ joining $x_0$ and $x$ (starting at $x_0$ and ending at $x$) and define a map $s : X \longrightarrow \widetilde X$ by $s (x) = \widetilde {\gamma}_{\widetilde {x_0}} (1),$ where $\widetilde {\gamma}_{\widetilde {x_0}}$ is the unique lift of $\gamma$ starting at $\widetilde {x_0}.$ Show that $s$ is well-defined.
My Attempt $:$ Let $x \in X$ be arbitrarily taken. Let $\gamma$ and $\delta$ be two paths in X starting at $x_0$ and ending at $x.$ In order to show that $s$ is well-defined we need only to show that $\widetilde {\gamma}_{\widetilde {x_0}} (1) = \widetilde {\delta}_{\widetilde {x_0}} (1),$ where $\widetilde {\gamma}_{\widetilde {x_0}}$ and $\widetilde {\delta}_{\widetilde {x_0}}$ are the unique lifts of $\gamma$ and $\delta$ to $\widetilde X$ starting at $\widetilde {x_0}.$ For that, consider the loop $\delta \ast \overline {\gamma}$ based at $x_0,$ where $\overline {\gamma}$ is the opposite path traversed by that of $\gamma$ i.e. $\widetilde {\gamma} (t) = \gamma (1 – t),$ $t \in [0,1].$ Let $\widetilde {{\delta} \ast \overline {\gamma}}_{\widetilde {x_0}}$ be the unique lift of $\delta \ast \overline {\gamma}$ starting at $\widetilde {x_0}.$ Then by the virtue of the unique path lifting lemma it follows that $\widetilde {{\delta} \ast \overline {\gamma}}_{\widetilde {x_0}} = \widetilde {\delta}_{\widetilde {x_0}} \ast \widetilde {\overline {\gamma}}_{\widetilde {\delta}_{\widetilde {x_0}} (1)}.$ By the given hypothesis this lift is loop based at $\widetilde {x_0}$ and hence $\widetilde {\overline {\gamma}}_{\widetilde {\delta}_{\widetilde {x_0}} (1)} (1) = \widetilde {x_0}.$
Does it help anyway? Any help would be highly appreciated.
Thanks for investing your valuable time on my question.
Best Answer
Your proof is almost there, you just need to observe that $\overline{(\widetilde{\bar \gamma}_{\widetilde{\delta}_{\widetilde{x_0}}(1)})}$ is a lift of $\gamma$ starting at $\widetilde{x_0}$ and therefore the unique lift of $\gamma$ starting at $\widetilde{x_0}$, ie $\widetilde{\gamma}_{\widetilde{x_0}}$.