----- > [!proposition]+ Proposition. ([[pushforward morphism of covering map is an embedding]]) > Let $p:\widetilde{X} \to X$ be a [[covering space|covering map]], $x_{0} \in X$, and $\tilde{x}_{0} \in p ^{-1}(x_{0})$. Then the [[fundamental group]] [[homomorphism of fundamental groups induced by a continuous map|pushforward]] map $p_{*}: \pi_{1}(\widetilde{X}, \tilde{x}_{0}) \to \pi_{1}(X,x_{0})$ is [[injection|injective]]. > ^proposition > [!proof]+ Proof. ([[pushforward morphism of covering map is an embedding]]) > We'll use [[group homomorphism is injective iff kernel is trivial]]. Let $[\gamma] \in \pi_{1}(\widetilde{X}, \tilde{x}_{0})$ such that $p_{*}([\gamma])=[p \circ \gamma]=[e_{x_{0}}]$. Let $H$ be a [[path homotopy]] between $p \circ \gamma$ and $e_{x_{0}}$, and [[lifting|lift]] $e_{x_{0}}$ to a [[the homotopy lifting lemma|unique path]] in $\widetilde{X}$ starting at $\tilde{x}_{0}$, this [[parameterized curve]] must be $e_{\tilde{x}_{0}}$. Since [[path homotopies lift uniquely under covering maps]], we conclude $\gamma$ is [[path homotopy|path homotopic]] to $e_{\tilde{x}_{0}}$. > ^proof #[](group%20homomorphism%20is%20injective%20iff%20kernel%20is%20trivial%20iff%20is%20a%20monomorphism.md)ile.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as 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