----- > [!proposition] Proposition. ([[the strengthened lifting correspondence]]) > Let $p:E \to B$ be a [[covering space|covering map]]; let $p(e_{0})=b_{0}$. > 1. [[homomorphism of fundamental groups induced by a continuous map|The]] [[group homomorphism|homomorphism]] $p_{*}: \pi_{1}(E, e_{0}) \to \pi_{1}(B,b_{0})$ is an [[injection]]; > 2. Let $H=p_{*}\big( \pi_{1}(E,e_{0}) \big)$. The [[lifting correspondence derived from a covering map|lifting correspondence]] $\phi$ induces an [[injection]] $\Phi: \pi_{1}(B,b_{0}) / H \to p ^{-1}(b_{0})$ > of the right [[coset|cosets]] of $H$ into $p ^{-1}(b_{0})$, which is [[bijection|bijective]] if $E$ is [[path-connected]]; > 3. If $f$ is a [[parameterized curve|loop]] in $B$ based at $b_{0}$, then $[f] \in H$ iff $f$ lifts to a [[parameterized curve|loop]] in $E$ based at $e_{0}$. An overlapping result (with full proof) may be found in [[monodromy action]] and [[the monodromy correspondence]]. So see there. > [!proof]- Proof. ([[the strengthened lifting correspondence]]) > **1.** [[group homomorphism is injective iff kernel is trivial iff is a monomorphism|Suppose]] $\tilde{h}$ is a [[parameterized curve|loop]] in $E$ at $e_{0}$, and $p_{*}([\tilde{h}])=[e_{b_{0}}]$, i.e., $p \circ \tilde{h}=e_{b_{0}}$ is [[path homotopy|path homotopic]] to the constant loop; let $F$ be a [[path homotopy|path homotopy]] witnessing this. If $\tilde{F}$ is the [[paths lift uniquely under covering maps|(unique) lifting]] of $F$ to $E$ such that $\tilde{F}(0,0)=e_{0}$, [[homotopic paths are preserved by liftings under covering maps|then]] $\tilde{F}$ is path homotopic to the constant loop $e_{e_{0}}$, i.e., $[\tilde{F}]$ is trivial. ----- #### ---- #### 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 > ```