$\require{physics}\newcommand{\cbrt}[1]{\sqrt[3]{#1}}\newcommand{\sgn}{\text{sgn}}\newcommand{\ii}[1]{\textit{#1}}\newcommand{\eps}{\varepsilon}\newcommand{\EE}{\mathbb E}\newcommand{\PP}{\mathbb P}\newcommand{\Var}{\mathrm{Var}}\newcommand{\Cov}{\mathrm{Cov}}\newcommand{\pperp}{\perp\kern-6pt\perp}\newcommand{\LL}{\mathcal{L}}\newcommand{\pa}{\partial}\newcommand{\AAA}{\mathscr{A}}\newcommand{\BBB}{\mathscr{B}}\newcommand{\CCC}{\mathscr{C}}\newcommand{\DDD}{\mathscr{D}}\newcommand{\EEE}{\mathscr{E}}\newcommand{\FFF}{\mathscr{F}}\newcommand{\WFF}{\widetilde{\FFF}}\newcommand{\GGG}{\mathscr{G}}\newcommand{\HHH}{\mathscr{H}}\newcommand{\PPP}{\mathscr{P}}\newcommand{\Ff}{\mathcal{F}}\newcommand{\Gg}{\mathcal{G}}\newcommand{\Hh}{\mathbb{H}}\DeclareMathOperator{\ess}{ess}\newcommand{\CC}{\mathbb C}\newcommand{\FF}{\mathbb F}\newcommand{\NN}{\mathbb N}\newcommand{\QQ}{\mathbb Q}\newcommand{\RR}{\mathbb R}\newcommand{\ZZ}{\mathbb Z}\newcommand{\KK}{\mathbb K}\newcommand{\SSS}{\mathbb S}\newcommand{\II}{\mathbb I}\newcommand{\conj}[1]{\overline{#1}}\DeclareMathOperator{\cis}{cis}\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}\DeclareMathOperator*{\range}{range}\DeclareMathOperator*{\nul}{null}\DeclareMathOperator*{\Tr}{Tr}\DeclareMathOperator*{\tr}{Tr}\newcommand{\id}{1\!\!1}\newcommand{\Id}{1\!\!1}\newcommand{\der}{\ \mathrm {d}}\newcommand{\Zc}[1]{\ZZ / #1 \ZZ}\newcommand{\Zm}[1]{\left(\ZZ / #1 \ZZ\right)^\times}\DeclareMathOperator{\Hom}{Hom}\DeclareMathOperator{\End}{End}\newcommand{\GL}{\mathbb{GL}}\newcommand{\SL}{\mathbb{SL}}\newcommand{\SO}{\mathbb{SO}}\newcommand{\OO}{\mathbb{O}}\newcommand{\SU}{\mathbb{SU}}\newcommand{\U}{\mathbb{U}}\newcommand{\Spin}{\mathrm{Spin}}\newcommand{\Cl}{\mathrm{Cl}}\newcommand{\gr}{\mathrm{gr}}\newcommand{\gl}{\mathfrak{gl}}\newcommand{\sl}{\mathfrak{sl}}\newcommand{\so}{\mathfrak{so}}\newcommand{\su}{\mathfrak{su}}\newcommand{\sp}{\mathfrak{sp}}\newcommand{\uu}{\mathfrak{u}}\newcommand{\fg}{\mathfrak{g}}\newcommand{\hh}{\mathfrak{h}}\DeclareMathOperator{\Ad}{Ad}\DeclareMathOperator{\ad}{ad}\DeclareMathOperator{\Rad}{Rad}\DeclareMathOperator{\im}{im}\renewcommand{\BB}{\mathcal{B}}\newcommand{\HH}{\mathcal{H}}\DeclareMathOperator{\Lie}{Lie}\DeclareMathOperator{\Mat}{Mat}\DeclareMathOperator{\span}{span}\DeclareMathOperator{\proj}{proj}$ Let $p: (\tilde{X}, \tilde{x}_0) \to (X, x_0)$ be a covering map, as usual. We are given a map $f: (Y, y_0) \to (X, x_0)$. Then, a ==**lift**== of $f$, $\tilde{f}$, is a map such that $p\circ \tilde{f} = f$ and $\tilde{f}(y_0) = \tilde{x}_0$. The lifting property told us all we need to know about lifting homotopies. Let's see if we can find analogous results for the general case. There are two directions: existence and uniqueness. > [!theorem] Lifting Criterion > Suppose $Y$ is ==*path-connected*== and ==*locally path-connected*==. Then, a lift exists iff $ f_*(\pi_1(Y, y_0))\subset p_*(\pi_1(\tilde{X}, \tilde{x}_0)) $ The [[unique lifting]] theorem completes this characterization. > [!idea] Conditions > > - The $f_*(\pi_1(Y))\subset p_*(\pi_1(\tilde{X}))$ condition is required for all of our loops to lift to loops. > > - ==*Path-connectedness*== is required to construct anything at all. > > - ==*Local path-connectedness*== is required to show our construction is continuous. > > > [!proof]- First off, if such a lift exists, then $f_* = p_*\circ \tilde{f}_*$, so the inclusion is obvious. For the other direction, well, we already know how to lift paths, and we know that $Y$ is path-connected. Thus, there is exactly one way to proceed: > > [!part]- > For any $y$, draw some path $\gamma$ connecting $y_0$ to $y$; this is sent to a path $f\gamma$ connecting $f(y_0) = x_0$ to $f(y)$. This guy lifts to a unique path $\widetilde{f\gamma}$ in $\tilde{X}$ starting at $\tilde{x}_0$. The place where this path ends must be $\tilde{f}(y)$, so let's define it as such. > > > > [!part]- This is well-defined > Pick a different path $\gamma'$ from $y_0$ to $y$. Observe that $(f\gamma)\conj{(f\gamma')}$ is a loop in $f_*(\pi_1(Y, y_0))$. Thus, ==*by the condition,*== it must lift to a loop in $\tilde{X}$ based at $\tilde{x}_0$. Staring at the lifts of $f\gamma$ and $f\gamma'$ individually, we conclude that $\widetilde{f\gamma}$ and $\widetilde{f\gamma'}$ end at the same point. **Observe that in general, there will exist a lift of $(f\gamma)\conj{(f\gamma')**$ as a path, but it will not necessarily be a loop.} > > > > [!part]- This is continuous > Let $y$ be arbitrary. For any open neighborhood $\tilde{U}$ of $\tilde{f}(y)$, I need to exhibit an open neighborhood $V$ of $y$ such that $\tilde{f}(V)\in \tilde{U}$. Fine, $f(y)$ is evenly covered, so WLOG $\tilde{U}$ is mapped homeomorphically onto $U$ by $p$. Then, $f^{-1}(U)$ is open in $Y$. By ==*local path-connectedness,*== I can pick a path-connected open neighborhood $V\subset f^{-1}(U)$ of $y$. This literally concludes. Like, now pick any $v\in V$. We can draw a path from $y_0$ to $v$ called $\gamma\eta$, where $\gamma$ is a fixed path from $y_0$ to $y$, and $\eta$ is a path from $y$ to $v$. This gets sent to some path $f\gamma f\eta$ which ends in $U$, which lifts to some path $\widetilde{f\gamma} \widetilde{f\eta}$ which ends in $\tilde{U}$. By definition, the ending point is $\tilde{f}(v)$.