---- > [!definition] Definition. ([[lifting correspondence derived from a covering map]]) > Let $p: E \to B$ be a [[covering space|covering map]]; let $b_{0} \in B$. Choose $e_{0} \in p ^{-1}(b_{0})$. Given $[\gamma] \in \pi_{1}(B,b_{0})$, let $\tilde{\gamma}$ be the [[lifting]] of $\gamma$ to a [[parameterized curve]] in $E$ that begins at $e_{0}$. Let $\phi([\gamma])$ denote the endpoint $\tilde{\gamma}(1)$ of $\tilde{f}$. Then $\phi$ is a [[well-defined]] set map $\phi: \pi_{1}(B,b_{0}) \to p ^{-1}(b_{0}).$ > We call $\phi$ the **lifting correspondence of the covering map $p$**. It depends of course on the choice of point $e_{0}$. > \ > There is a relationship between this construction and the [[monodromy action]]. > [!justification] > Well-definition (that $\phi([f])=\phi([g])$ whenever $f$ is [[path homotopy|path homotopic]] to $g$) follows immediately from [[the homotopy lifting lemma]]'s corollary on 'path lifting'. ---- #### ---- #### 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 > ```