---- > [!theorem] Theorem. ([[homotopic paths are preserved by liftings under covering maps]]) > Let $p: E \to B$ be a [[covering space|covering map]] between [[topological space|topological spaces]] $E$ and $B$; let $p(e_{0})=b_{0}$. Let $f$ and $g$ be two [[parameterized curve|paths]] in $B$ from $b_{0}$ to $b_{1}$, let $\tilde{f}$ and $\tilde{g}$ be their respective [[lifting|liftings]] to [[parameterized curve|paths]] in $E$ beginning at $e_{0}$. If $f$ and $g$ are [[path homotopy|path homotopic]], then $\tilde{f}$ and $\tilde{g}$ end at the same point of $E$ are are [[path homotopy|path homotopic]]. > [!proof]- Proof. ([[homotopic paths are preserved by liftings under covering maps]]) > Let $F:I \times I \to B$ be the [[path homotopy]] between $f$ and $g$. Then $F(0,0)=b_{0}$. Let $\tilde{F}$ be the [[lifting]] of $F$ to $E$ such that $\tilde{F}(0,0)=e_{0}$. By [[path homotopies lift uniquely under covering maps]] $\tilde{F}$ is a [[path homotopy]], so that $\tilde{F}(\{ 0 \} \times I)=\{ e_{0} \}$ and $\tilde{F}(\{ 1 \} \times I)$ is a singleton $\{ e_{1} \}$. We still need to show that in particular it is a [[path homotopy]] between $\tilde{f}$ and $\tilde{g}$. > > The restriction $\tilde{F}|_{I \times \{ 0 \}}$ of $\tilde{F}$ to the bottom edge of $I \times I$ is a [[parameterized curve]] on $E$ beginning at $e_{0}$ that is a [[lifting]] of $F |_{I \times \{ 0 \}}$. By [[paths lift uniquely under covering maps|uniqueness of path liftings]], we must have $\tilde{F}(s,0)=\tilde{f(s)}$. Same reasoning on top edge gives $\tilde{F}(s,1)=\tilde{g}(s)$. Therefore, both $\tilde{f}$ and $\tilde{g}$ end at $e_{1}$ and $\tilde{F}$ is a [[path homotopy]] between them. ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.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 GROUP BY file.tags as Tag