----- > [!proposition]+ Proposition. ([[fibers of path-connected space under covering map are in bijection]]) > Let $p: \widetilde{X} \to X$ be a [[covering space]] and $X$ be [[path-connected]]. Then the sets $p ^{-1}(x)$ for $x \in X$ are all in [[bijection]]. ^proposition > [!proof]+ Proof. ([[fibers of path-connected space under covering map are in bijection]]) > Fix $x_{0},x_{1} \in X$ and $\gamma:[0,1] \to X$ a [[parameterized curve]] between them. For each $y_{0} \in p ^{-1}(x_{0})$ let $\tilde{\gamma}_{y_{0}}$ be [[the homotopy lifting lemma|the lift]] of $\gamma$ starting at $y_{0}$. Define the function $\gamma_{*}: p ^{-1}(x_{0}) \to p ^{-1}(x_{1}), y_{0} \mapsto \tilde{\gamma}_{y_{0}}(1).$ Similarly, the inverse path $\overline{\gamma}$ defines a function $\overline{\gamma_{*}}:p ^{-1}(x_{1}) \to p ^{-1}(x_{0})$ which takes a point $y_{1} \in p ^{-1} (x_{1})$ to the endpoint of the [[lifting|lift]] of $\overline{\gamma}$ starting at $y_{1}$. We claim the two functions are [[inverse map|inverses]]. Indeed, $\begin{align} \overline{\gamma_{*}} \circ {\gamma_{*}}(y_{0}) = & \overline{\gamma_{*}} (\tilde{\gamma}_{y_{0}}(1)) \\ = & \text{endpoint of lift of $\overline{\gamma}$ starting at } \tilde{\gamma}_{y_{0}}(1) \\ = & \text{endpoint of lift of } \overline{\gamma} * \gamma \text{ starting at } y_{0}\\ =& \text{endpoint of lift of } c_{x_{0}} \text{ starting at } y_{0} =& y_{0} \end{align}$ and an entirely analogous computation witnesses the right inverse. ( $*$ denotes the [[fundamental groupoid|path concatenation operation]]. ) Note the relationship between $\gamma_{*}$ here and the function defined in [[lifting correspondence derived from a covering map|lifting correspondence]]. ![[CleanShot 2024-06-06 at [email protected]|300]] ^proof ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` #reformatrevisebatch01