Lifts of Paths
We begin by defining a lift of a continuous function $f : Y \to X$ from a topological space $Y$ to a topological space $X$.
Definition: Let $X$ and $Y$ be topological spaces and let $f : Y \to X$ be a continuous map. Let $(\tilde{X}, p)$ be a covering space of $X$. A Lift of $f$ is a continuous map $\tilde{f} : Y \to \tilde{X}$ such that the following diagram is commutative:
That is, $f = p \circ \tilde{f}$. |
We first begin with a very general theorem which tells us exactly when we can obtain a lift of a continuous function $f : Y \to X$. We need quite a few conditions. First we must select a point $y_0 \in Y$. We then denote $x_0 = f(y_0)$, and choose one element $\tilde{x_0} \in p^{-1}(x_0)$. The theorem below tells us that a unique lift of $f$ exists if and only if the image of the fundamental group $\pi_1(Y, y_0)$ under $f_*$ is a subset of the image of the fundamental group $\pi_1(\tilde{X}, \tilde{x_0})$ under $p_*$
Theorem 1 (The Lifting of Continuous Functions Theorem): Let $X$ be a topological space and let $(\tilde{X}, p)$ be a covering space of $X$. Let $Y$ be a topological space that is path connected and locally path connected and let $f : Y \to X$ be a continuous map. Let $y_0 \in Y$ and let $x_0 = f(y_0)$. Let $\tilde{x_0} \in p^{-1}(x_0)$. Then there exists a unique lift $\tilde{f} : Y \to \tilde{X}$ of $f$ with $\tilde{f}(y_0) = \tilde{x_0}$ if and only if $f_*(\pi_1(Y, y_0))$ is a subset of $p_*(\pi_1(\tilde{X}, \tilde{x_0}))$. |
Let $Y = [0, 1]$. Then a continuous function $\alpha : [0, 1] \to X$ is a path in $X$ and we can talk about lifts of paths in $X$. The following theorem is crucially important. It tells us that if $X$ is a topological space and $(\tilde{X}, p)$ is a covering space of $X$ then a path $\alpha$ in $X$ can be uniquely lifted provided we specify a starting point for the lift.
Theorem 2 (The Lifting of Paths Theorem): Let $X$ be a topological space and let $(\tilde{X}, p)$ be a covering space of $X$. Let $\alpha$ be a path in $X$ that starts at $x \in X$. Then for every $y \in p^{-1}(x)$ there exists a unique lift of $\alpha$ that starts at $y$. |
Theorem 3: Let $X$ be a topological space and let $(\tilde{X}, p)$ be a covering space of $X$. Let $\alpha$ and $\beta$ be paths in $X$ that both start at $x_1$ and end at $x_2$. Let $y \in p^{-1}(x)$ and let $\tilde{\alpha}$ and $\tilde{\beta}$ be the unique lifts of $\alpha$ and $\beta$ starting at $y$. If $\alpha \simeq_{\{0, 1 \}} \beta$ then $\tilde{\alpha} \simeq_{\{ 0, 1 \}} \tilde{\beta}$, and $\tilde{\alpha}$ and $\tilde{\beta}$ end at the same point. |