Let \(p: \tilde{X} \to X\) be a simply-connected covering space of \(X\) and let \(A \subset X\) be a path-connected, locally path-connected subspace, with \(\tilde{A} \subset \tilde{X}\) a path-component of \(p^{-1}(A)\). Show that \(p: \tilde{A} \to A\) is the covering space corresponding to the kernel of the map \(\pi_1(A) \to \pi_1(X)\).
Solution
1. \(p: \tilde A \to A\) is a covering space.
the restriction of a covering map to a path-component of the preimage of a subspace is a covering map.
2. The covering \(p: \tilde A \to A\) corresponds to the subgroup \(p_*(\pi_1(\tilde A)) \le \pi_1(A)\).
the fundamental theorem of covering spaces: a covering corresponds to the image of the fundamental group of the total space.
3. \(p_*(\pi_1(\tilde A)) = \ker(\pi_1(A) \to \pi_1(X))\).
3.1. \(p_*(\pi_1(\tilde A)) \subseteq \ker(\pi_1(A) \to \pi_1(X))\).
the composition \(\pi_1(\tilde A) \xrightarrow{p_*} \pi_1(A) \to \pi_1(X)\) factors through \(\pi_1(\tilde X) = 0\) (since \(\tilde A \subseteq \tilde X\) and \(\tilde X\) is simply connected), so its image is trivial.
3.2. \(\ker(\pi_1(A) \to \pi_1(X)) \subseteq p_*(\pi_1(\tilde A))\).
a loop \(\gamma\) in \(A\) whose image in \(\pi_1(X)\) is trivial lifts to a loop in \(\tilde X\); since \(\gamma\) is based in \(A\) and its lift starts in \(\tilde A\), the lift lies entirely in \(\tilde A\) (a path-component of \(p^{-1}(A)\)), so \(\gamma\) is the image of a loop in \(\tilde A\), i.e. \(\gamma \in p_*(\pi_1(\tilde A))\).
3.3. Hence equality holds.
1 and 2.
4. Q.E.D.
2 and 3.