For a covering space \(p: \tilde{X} \to X\) with \(X\) connected, locally path-connected, and semilocally simply-connected, show:
The components of \(\tilde{X}\) are in one-to-one correspondence with the orbits of the action of \(\pi_1(X, x_0)\) on the fiber \(p^{-1}(x_0)\).
Under the Galois correspondence between connected covering spaces of \(X\) and subgroups of \(\pi_1(X, x_0)\), the subgroup corresponding to the component of \(\tilde{X}\) containing a given lift \(\tilde{x}_0\) of \(x_0\) is the stabilizer of \(\tilde{x}_0\), the subgroup consisting of elements whose action on the fiber leaves \(\tilde{x}_0\) fixed.
Solution
Let \[ G=\pi_1(X,x_0) \] act on the fiber \(F=p^{-1}(x_0)\) by lifting based loops.
1. Two points \(\tilde x_0,\tilde x_1\in F\) lie in the same \(G\)-orbit if and only if they lie in the same path component of \(\widetilde X\).
Suppose first that \(\tilde x_1\) is obtained from \(\tilde x_0\) by the action of \([\gamma]\in G\). By definition, a lift of \(\gamma\) joins these two fiber points, so they lie in the same path component.
Conversely, suppose a path \[ \tilde\alpha:I\to\widetilde X \] joins \(\tilde x_0\) to \(\tilde x_1\). Then \[ \alpha=p\circ\tilde\alpha \] is a loop at \(x_0\). The path \(\tilde\alpha\) is a lift of this loop, so the monodromy action of \([\alpha]\) sends one endpoint to the other, according to the convention used for the action. Thus the two fiber points lie in the same orbit.
2. Since \(X\) is locally path connected, every component of \(\widetilde X\) is path connected.Hence the components of \(\widetilde X\) are in bijection with the \(G\)-orbits in \(F\).
1.This proves part (a).
3. Fix \(\tilde x_0\in F\) and let \(\widetilde X_0\) be its component.The stabilizer of \(\tilde x_0\) is\[ \operatorname{Stab}_G(\tilde x_0) =\{[\gamma]\in G:\text{the lift of $\gamma$ at $\tilde x_0$ is closed}\}. \]
By definition, \([\gamma]\) fixes \(\tilde x_0\) exactly when the lift of \(\gamma\) beginning at \(\tilde x_0\) ends again at \(\tilde x_0\).
4. One has\[ \operatorname{Stab}_G(\tilde x_0) =p_*\pi_1(\widetilde X_0,\tilde x_0). \]
If \([\tilde\gamma]\in\pi_1(\widetilde X_0,\tilde x_0)\), then \(p\circ\tilde\gamma\) is a loop whose lift beginning at \(\tilde x_0\) is precisely \(\tilde\gamma\), hence is closed. Therefore its class lies in the stabilizer.
Conversely, if \([\gamma]\) stabilizes \(\tilde x_0\), its lift at \(\tilde x_0\) is a loop in the component \(\widetilde X_0\), and \[ p_*[\tilde\gamma]=[\gamma]. \] Thus the two subgroups coincide.
5. Under the connected-covering/subgroup correspondence, the component \(\widetilde X_0\to X\) corresponds exactly to the stabilizer of \(\tilde x_0\).
The subgroup assigned by the Galois correspondence to the pointed connected cover \[ (\widetilde X_0,\tilde x_0)\to(X,x_0) \] is by definition \[ p_*\pi_1(\widetilde X_0,\tilde x_0). \] By 4 this is the stabilizer. This proves part (b).