---- $X$ and $Y$ are [[topological space|topological spaces]]. > [!theorem]+ Theorem. ([[the homotopy lifting lemma]]) > Let $p: \widetilde{X} \to X$ be a [[covering space|covering map]], $H:Y \times I \to X$ be a [[homotopy]] from $f_{0}$ to $f_{1}$, and $\tilde{f}_{0}$ be a [[lifting|lift]] of $f_{0}$. Then there exists a unique [[homotopy]] $\widetilde{H}:Y \times I \to \widetilde{X}$ between $\tilde{f}_{0}$ and $\widetilde{H}(-, 1)$ such that >- $\widetilde{H}(-,0)=\tilde{f}_{0}(-)$, and >- $p \circ \widetilde{H}=H$. ^theorem > [!proposition]+ Corollary. (Path lifting) — Take $Y=\{ * \}$ > Let $p: \widetilde{X} \to X$ be a [[covering space]], $\gamma:I \to X$ be a [[parameterized curve]], and $\tilde{x}_{0} \in \widetilde{X}$ be such that $p(\tilde{x}_{0})=\gamma(0)$. Then there exists a unique [[parameterized curve]] $\tilde{\gamma}:I \to \widetilde{X}$ such that > - $\tilde{\gamma}(0)=\tilde{x}_{0}$, and > - $p \circ \tilde{\gamma} = {\gamma}$. ^proposition > [!proof]+ Proof. ([[the homotopy lifting lemma]]) > Let $\{ U_{\alpha} \}_{\alpha \in I}$ be an [[cover|open covering]] of $X$ by sets which are [[evenly covered]] by $p$, and write $p ^{-1}(U_{\alpha})=\coprod_{\beta \in I_{\alpha}}V_{\beta}$ with $p |_{V_{\beta}} \xrightarrow{\sim} U_{\alpha}$. Then $\{ H ^{-1}(U_{\alpha}) \}_{\alpha \in I}$ serves as an open cover of $Y \times I$. For each $y_{0} \in Y$ this restricts to an open cover of the [[compact]] space $\{ y_{0} \} \times I$, and by the [[Lebesgue number lemma]] there is a $N=N(y_{0})>0$ such that each [[parameterized curve]] $H |_{\{ y_{0} \} \times \left[ \frac{i}{N}, \frac{i+1}{N} \right]} : \{ y_{0} \} \times I \to X$ has image living entirely inside some $U_{\alpha} \subset X$ for each $i$. In fact, it is not hard to see (tube lemma? or just draw a picture?) that there is a [[neighborhood]] $W_{y_{0}}$ of $y_{0}$ such that $H\left( W_{y_{0}} \times \left[ \frac{i}{N}, \frac{i+1}{N} \right] \right)$ lies entirely inside $U_{\alpha}$. > >We now obtain a [[lifting|lift]] $\widetilde{H} |_{W_{y_{0}} \times I}$ of $H |_{W_{y_{0}} \times I}$ as follows, defining it 'piece by piece'. > >1. We have $H |_{W_{y_{0}} \times \left[ 0, \frac{1}{N} \right]}:W_{y_{0}} \times \left[ 0, \frac{1}{N} \right] \to U_{\alpha}$, and a lift $\tilde{f}_{0} |_{W_{y_{0}}}$ of $f_{0}$ with image in some $V_{\beta}$ which is mapped [[homeomorphism|homeomorphically]] onto $U_{\alpha}$ via $p |_{V_{\beta}}$, so we can define $\widetilde{H} |_{W_{y_{0}} \times \left[ 0, \frac{1}{N} \right]} := (p |_{V_{\beta}})^{-1} \circ H |_{W_{y_{0}} \times \left[ 0, \frac{1}{N} \right]}:W_{y_{0}} \times [ 0, \frac{1}{N}] \to V_{\beta} \subset \widetilde{X}.$ >2. Proceed in the same way, lifting the (short) [[homotopy]] $H |_{W_{y_{0}} \times \left[ \frac{1}{N}, \frac{2}{N} \right]}$ and so on. > >At the end of this process, we obtain a map $\widetilde{H} |_{W_{y_{0}} \times I}$ lifting $H |_{W_{y_{0}} \times I}$ and extending $\tilde{f}_{0} |W_{y_{0}}$. > >We now do this for every point $y_{0} \in Y$. All we have to do is verify that the lifts we have constructed agree on $(W_{y_{0}} \times I) \cap (W_{y_{1}} \times I)$. See the lecture notes for this (short) check, it is a consequence of [[the homotopy lifting lemma]]. ^proof #### #### 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 as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` #reformatrevisebatch01