---- > [!theorem] Theorem. ([[the monodromy correspondence]]) > Let $\widetilde{X},X$ be [[topological space|topological spaces]] and $p: \widetilde{X} \to X$ a [[covering space|covering map]]. Further assume $X$ is [[path-connected]]. > >1. The [[fundamental group]] $\pi_{1}(X,x_{0})$ [[monodromy action|acts]] [[transitive group action|transitively]] on $p ^{-1}(x_{0})$ if and only if $\widetilde{X}$ is [[path-connected]]; >2. The [[stabilizer]] of $y_{0} \in p ^{-1}(x_{0})$ is the [[homomorphism of fundamental groups induced by a continuous map|pushforward]] image $\text{im}(p_{*}: \pi_{1}(\widetilde{X}, y_{0}) \to \pi_{1}(X,x_{0})) \leq \pi_{1}(X,x_{0});$ >3. Consequently, if $\widetilde{X}$ is [[path-connected]] then the (infinite) [[orbit-stabilizer theorem]] entails a [[bijection]] $\frac{\pi_{1}(X,x_{0})}{p_{*}(\pi_{1}(\widetilde{X}, y_{0}))} \to p ^{-1}(x_{0})$ induced by [[group action|acting]] on the point $y_{0}$. ^theorem > [!proposition] Corollary. ([[path-connected covering space yields surjective lifting correspondence|universal cover yields bijective lifting correspondence]]) > If $p$ is [[simply connected]] (a [[universal cover]]), then $p_{*}$ is the [[group homomorphism|trivial homomorphism]], hence $\text{Stab}(x)$ is trivial and for each $\tilde{x}_{0} \in p ^{-1}(x)$, (3) becomes a [[bijection]] $ \begin{align} \ell:\pi_{1}(X,x_{0}) \xrightarrow{\sim} p ^{-1}(x_{0}) \\ [\gamma] \mapsto \tilde{x}_{0} \bullet [\gamma]. \end{align}$ See [[path-connected covering space yields surjective lifting correspondence|universal cover yields bijective lifting correspondence]] for an alternative derivation of this result. > >> [!note] Remark. (Multiplication on the fiber) > We wish to use this [[bijection]] to understand the [[fundamental group]], but $p ^{-1}(x_{0})$ is not a [[group]] so it must be used carefully. What *is* true is that the [[bijection]] $\ell$ induces a multiplication law $\cdot$ on $p ^{-1}(x_{0})$, in the sense that if $[\gamma_{1}] \mapsto d_{1}$ and $[\gamma_{2}] \mapsto d_{2}$ then $[\gamma_{1}] * [\gamma_{2}] \mapsto d_{1} \cdot d_{2}$ — i.e., in the sense that if endowing $p ^{-1}(x_{0})$ with this [[binary operation]] *makes it into a [[group]]* $(p ^{-1}(x_{0}), \cdot)$, then that [[group]] is [[group isomorphism|isomorphic]] to $\pi_{1}(X,x_{0})$. > > > >To do this, we start with $y_{0},z_{0} \in p ^{-1}(x_{0})$, each in correspondence with $\ell ^{-1}(y_{0})$, $\ell ^{-1}(z_{0}) \in \pi_{1}(X,x_{0})$. Then perform the multiplication $\ell ^{-1}(y_{0}) * \ell ^{-1}(z_{0})$ in $\pi_{1}(X,x_{0})$, and finally map back with $\ell\big( \ell ^{-1}(y_{0}) * \ell ^{-1}(z_{0}) \big)$ Let's unravel this. >> The product $\ell\big( \ell ^{-1}(y_{0}) * \ell ^{-1}(z_{0}) \big)$ is obtained as follows [^2]. >>1. choose a [[parameterized curve]] $\tilde{\gamma}:I \to \widetilde{X}$ from $\tilde{x}_{0}$ to $z_{0}$ ([[path homotopies lift uniquely under covering maps|this is unique up to homotopy]]); >>2. let $\gamma$ be the [[lifting|lift]] of the [[parameterized curve|loop]] $p \circ \tilde{\gamma}$ starting at $y_{0}$; >>3. then $\ell(\ell ^{-1}(y_{0}) * \ell ^{-1}(z_{0}))=\gamma(1)$. ^proposition > [!proof]- Proof. ([[the monodromy correspondence]]) > **1.** Suppose $\widetilde{X}$ is [[path-connected]]; let $y_{0},y_{0}' \in \widetilde{X} \in p ^{-1}(x_{0})$ with $\gamma:I \to \widetilde{X}$ a [[parameterized curve]] between them. Considering $[p \circ \gamma] \in \pi_{1}(X,x_{0})$, we see $y_{0} \bullet [p \circ \gamma]=\gamma_{*}(y_{0})=y_{0}'$ by definition of the [[monodromy action]]. Conversely, contrapositively suppose $\tilde{X}$ is not [[path-connected]]; let $y_{0},z_{0} \in \widetilde{X}$ live in distinct [[path-connected component|path components]]. Obtain [[parameterized curve|paths]] $\gamma_{y},\gamma_{z}$ in $X$ from $p(y_{0})$ to $x_{0}$ and $p(z_{0})$ to $X$ respectively. [[lifting|Lift]] to [[the homotopy lifting lemma|unique]] [[parameterized curve|paths]] $\tilde{\gamma}_{y_{0}}$, $\tilde{\gamma}_{z_{0}}$ in $\widetilde{X}$ starting at $y_{0}$ and $z_{0}$ respectively, note that these paths must still live in different path components. Since of course $p \circ \tilde{\gamma}_{y_{0}}(1)=\gamma_{y}(1)=x_{0}$ and $p \circ \tilde{\gamma}_{z_{0}}(1)=\gamma_{z}(1)=x_{0}$, the endpoints $\tilde{\gamma}_{y_{0}}(1)$ and $\tilde{\gamma}_{z_{0}}(1)$ both live in $p ^{-1}(x_{0})$. No path exists between them, though, which necessitates that the [[monodromy action]] is not [[transitive group action|transitive]] [^1]. > **2.** For any $y_{0} \in p ^{-1}(x_{0})$, $[\gamma ]\in \text{Stab}(y_{0})$ iff $y_{0} \bullet [\gamma]= \tilde{\gamma}_{y_{0}}(1) =y_{0}$, i.e., iff the lift of $\gamma$ starting at $y_{0}$ is actually a loop based at $y_{0}$, i.e., iff $[\tilde{\gamma}_{y_{0}}] \in \pi_{1}(\widetilde{X}, y_{0})$. > This implies $[\gamma]=p_{*}([\tilde{\gamma}_{y_{0}}])$ so for one inclusion we have $[\gamma] \in \text{im}(p_{*}: \pi_{1}(\widetilde{X}, y_{0}) \to \pi_{1}(X,x_{0}))$. > For the other inclusion, let $[\gamma]$ be in the image, so that there exists a loop class $[\tilde{\gamma}_{y_{0}}]$ in $\widetilde{X}$ based at $y_{0}$. By lift uniqueness, $\tilde{\gamma}_{y_{0}}$ is exactly the lift of $\gamma$ to a [[parameterized curve]] based at $y_{0}$; since it is a loop, we have $[\gamma] \in \text{Stab}(y_{0})$. > **3.** The [[orbit-stabilizer theorem]] guarantees the sets have equal [[cardinality]], hence are in [[bijection]]. ---- #### [^1]: Recall that (by definition) the [[monodromy action]] is transitive iff for all $y_{0}, y_{0}' \in p ^{-1}(x_{0})$ there exists a loop class $[\gamma] \in \pi_{1}(X,x_{0})$ such that the lift of $\gamma$ starting at $y_{0}$ ends at $y_{0}'$. Said lift is a path, but we already found two points in $p ^{-1}(x_{0})$ between which no path exists. Thus the action is not transitive. [^2]: For a little more detail, see below. ![[CleanShot 2024-06-07 at [email protected]]] ![[CleanShot 2024-06-07 at 14.28.46@2x 1.jpg]] ----- #### 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