[[Covering]]
# Main theorem of coverings

Let $(X,x_{0})$ be a [[Local (path) connectedness|locally path-connected]], [[Connectedness|connected]], and [[Semilocal simple connectedness|semilocally simply connected]] topological space.
Then for every subgroup $H \sube \pi_{1}(X,x_{0})$ there exists a covering $p : (\tilde{X},\tilde{x}_{0}) \twoheadrightarrow (X,x_{0})$ unique up to equivalence with [[Characteristic subgroup of a covering|characteristic subgroup]] $H$. #m/thm/homotopy

> [!tip]+ Construction
> Take the universal covering $\hat{p} : \hat{X}\twoheadrightarrow X$ and consider $\Phi (H) \sube \Gamma$ where $\Phi : \pi_{1}(\tilde{X},x_{0}) \to \Gamma$ is an isomorphism.
> The covering is given by $\tilde X = \hat{X} / \Phi(H)$ with 
> $$
> \begin{align*}
> p : (\tilde{X}, \tilde{x}_{0}) &\twoheadrightarrow (X,x_{0}) \\ 
> \hat{x} \Phi(H) &\mapsto \hat{p}(\hat{x}_{0})
> \end{align*}
> $$

> [!check]- Proof
> Uniqueness up to equivalence follows from [[equivalence of coverings criterion]].
> Since $(X, x_{0})$ is semilocally simply connected, [[A locally path-connected, connected space is semilocally simply connected iff it has a universal covering|it has a universal covering]] $\hat{p} : (\hat{X}, \hat{x}_{0}) \twoheadrightarrow (X,x_{0})$.
> Let $\Gamma = \Aut_{\Cov_{X}}(\hat{p})$ 
> According to [[Deck transformation group of a regular covering as quotient]]
> $$
> \begin{align*}
> \Phi: \pi_{1}(X,x_{0}) &\to \Gamma \\
> [\alpha] &\mapsto (\hat{x}_{0} \mapsto \hat{\alpha}(1))
> \end{align*}
> $$
> is an isomorphism, where $\hat{\alpha}$ is the unique [[Lift of a map to a covering space|lift]] of $\alpha$ with $\hat{\alpha}(0) = \hat{x}_{0}$, and $(\hat{x}_{0} \mapsto \hat{\alpha}(1))$ denotes a unique deck transformation with this property.
> 
> Now take the [[orbit space]] $\tilde{X} = \hat{X} / \Phi(H)$ with the canonical projection
> $$
> \begin{align*}
> f: (\hat{X},\hat{x}_{0}) &\twoheadrightarrow (\tilde{X}, \tilde{x}_{0}) \\
> \hat{x} &\mapsto \Phi(H)\hat{x}
> \end{align*}
> $$
> Since [[the deck transformation group acts properly discontinuously]], so too does $\Phi(H) \sube \Gamma$,
> and the [[orbit space of a properly discontinuous effective group action]] forms a covering,
> which in this case is universal.
> Thus
> $$
> \begin{align*}
> \Aut_{\Cov_\tilde{X}}(f) \cong \pi_{1}(\tilde{X},\tilde{x}_{0}) \cong \Phi(H) \cong H
> \end{align*}
> $$
> We now define
> $$
> \begin{align*}
> p : (\tilde{X},\tilde{x}_{0}) &\to (X,x_{0}) \\
> f(\hat{x}) &\mapsto \hat{p}(\hat{x})
> \end{align*}
> $$
> which is well-defined since $f(\hat{x}) = f(\hat{x}')$ iff $\hat{x}' = \gamma(\hat{x})$ for some $\gamma \in H \sube \Gamma$, and then $\hat{p} \circ \gamma(\hat{x}) = \hat{p}(\hat{x})$;
> and continuous by [[Quotient topology#Universal property]].
> 
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> 
> Now let $x \in X$ and let $U$ be a neighbourhood of $x$ evenly covered by $\hat{p}$ with sheets $\{ \hat{U}_{i} \}_{i \in I}$.
> Let $J \sube I$ such that for all $i \in I$ there exists exactly one $j \in J$ such that $f(\hat{U}_{i}) = f(\hat{U}_{j})$,
> and let $\tilde{U}_{j} = f(\hat{U}_{j})$.
> Then
> $$
> \begin{align*}
> p^{-1}(U) = \coprod_{j \in J} \tilde{U}_{j}
> \end{align*}
> $$
> and $(p \restriction \tilde{U}_{j})^{-1} = (f \restriction \hat{U}_{j}) \circ (\hat{p} \restriction \hat{U}_{j})^{-1}$,
> therefore $p$ is a covering.
> Then by construction
> $$
> \begin{align*}
> [\alpha] \in H &\iff (\hat{x}_{0} \mapsto\hat{\alpha}(1)) \in \Phi(H) \\
> &\iff \tilde{\alpha}(1) = f \circ \hat{\alpha}(1) = \tilde{x}_{0} \\
> &\iff [\alpha] = \pi_{1}p[\tilde{\alpha}]
> \end{align*}
> $$
> so $H = \pi_{1}p(\pi_{1}(\tilde{X},\tilde{x}_{0}))$ as required.
> <span class="QED"/>

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