---- > [!definition] Definition. ([[monodromy action]]) > Suppose $\widetilde{X}$ and $X$ are [[topological space|topological spaces]], with $X$ [[connected]] and [[locally connected, locally path-connected|locally path-connected]] ([[connected components versus path-connected components|hence]] [[path-connected]]). (or maybe just have $X$ [[path-connected]]?) Let $p: \widetilde{X} \to X$ be a [[covering space|covering map]]. > Given a [[parameterized curve|loop]] $\gamma$ [[pointed set|based]] at $x_{0} \in X$, let $\tilde{\gamma}_{y_{0}}$ denote the [[the homotopy lifting lemma|unique lift]] of $\gamma$ starting at $y_{0}$. Define the fiber map $\begin{align} \gamma_{*}: p ^{-1}(x_{0}) \to& p ^{-1}(x_{0}) \\ y_{0} \mapsto \tilde{\gamma}_{y_{0}} (1). \end{align}$ Since [[path homotopies lift uniquely under covering maps]], $\gamma_{*}$ is constant on [[equivalence class|homotopy classes]] [^1], and there determines a (right) [[group action]] of the [[fundamental group]] $\pi_{1}(X,x_{0})$ on the fiber $p ^{-1}(x_{0})$: $\begin{align} \bullet :& p ^{-1}(x_{0}) \times \pi_{1}(X,x_{0}) \\ &(y_{0}, [\gamma]) \mapsto y_{0} \bullet [\gamma] := \gamma_{*}(y_{0}) = \tilde{\gamma}_{y_{0}}(1), \end{align}$ called the **monodromy action of $\pi_{1}(X,x_{0})$ on $p ^{-1}(x_{0})$**. \ The [[currying|curried]] maps $\begin{align} \phi_{y_{0}}: \pi_{1}(X,x_{0}) \to & p ^{-1}(x_{0}) \\ \phi_{y_{0}}([\gamma]):=&y_{0} \bullet [\gamma] = \tilde{\gamma}_{y_{0}}(1) \end{align}$ >are precisely the [[lifting correspondence derived from a covering map|lifting correspondence]] for different choices of $e_{0}=y_{0}$. ^definition ---- #### [^1]: In the sense that if $\gamma$ and $\beta$ are [[path homotopy|path homotopic]] then so are $\tilde{\gamma}_{y_{0}}$ and $\tilde{\beta}_{y_{0}}$, and thus they have the same endpoint. ---- #### 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 > ``` #reformatrevisebatch01