Give the definition of a covering space \(\widetilde{X}\) (and covering map \(p: \widetilde{X} \to X\)) for a topological space \(X\).
State the homotopy lifting property of covering spaces. Use it to show that a covering map \(p: \widetilde{X} \to X\) induces an injective group homomorphism \[ p_*: \pi_1(\widetilde{X}, \widetilde{x}_0) \to \pi_1(X, p(\widetilde{x}_0)) \] on fundamental groups.
Let \(p: \widetilde{X} \to X\) be a covering map with \(\widetilde{X}\) and \(X\) path-connected. Suppose that the induced homomorphism \(p_*\) on fundamental groups is an isomorphism. Prove that \(p\) is a homeomorphism.
Solution
Goal: Define covering spaces and the homotopy lifting property, prove that covering maps induce injections on \(\pi_1\), and prove that a \(\pi_1\)-isomorphism between path-connected spaces makes the covering map a homeomorphism.
1. Part (a): Definition of covering space and covering map.Proof:
1.1. Let \(X\) be a topological space. A covering space of \(X\) consists of a topological space \(\widetilde{X}\) and a continuous surjective map \(p: \widetilde{X} \to X\) such that every point \(x \in X\) has an open neighborhood \(U \subseteq X\) that is evenly covered by \(p\).
1.2. An open set \(U \subseteq X\) is evenly covered if the preimage \(p^{-1}(U)\) can be expressed as a disjoint union of open subsets in \(\widetilde{X}\):\[p^{-1}(U) = \bigsqcup_{\alpha \in A} V_\alpha,\]such that for each index \(\alpha \in A\), the restriction \(p|_{V_\alpha}: V_\alpha \to U\) is a homeomorphism.
1.3. The map \(p\) is called a covering map, \(X\) is the base space, and the sets \(V_\alpha\) are called the sheets of the covering over \(U\).
2. Part (b): Statement of the Homotopy Lifting Property (HLP).Proof:
2.1. Let \(p: \widetilde{X} \to X\) be a covering map, let \(Y\) be a topological space, let \(f: Y \to \widetilde{X}\) be a continuous map, and let \(H: Y \times [0, 1] \to X\) be a continuous homotopy such that \(H(y, 0) = p(f(y))\) for all \(y \in Y\).
2.2. Then there exists a unique continuous homotopy \(\widetilde{H}: Y \times [0, 1] \to \widetilde{X}\) such that \(\widetilde{H}(y, 0) = f(y)\) for all \(y \in Y\) and \(p \circ \widetilde{H} = H\).
3. Part (b): Proof that \(p_*: \pi_1(\widetilde{X}, \widetilde{x}_0) \to \pi_1(X, x_0)\) is injective.Proof:
3.1. Let \(x_0 = p(\widetilde{x}_0)\). Let \([\gamma] \in \pi_1(\widetilde{X}, \widetilde{x}_0)\) be represented by a continuous loop \(\gamma: [0, 1] \to \widetilde{X}\) with \(\gamma(0) = \gamma(1) = \widetilde{x}_0\).
3.2. Suppose \([\gamma] \in \ker p_*\), which means \(p_*([\gamma]) = [p \circ \gamma] = [c_{x_0}] \in \pi_1(X, x_0)\), where \(c_{x_0}\) is the constant loop at \(x_0\).
3.3. This means there is a basepoint-preserving homotopy of paths \(H: [0, 1] \times [0, 1] \to X\) such that:\[H(s, 0) = (p \circ \gamma)(s), \quad H(s, 1) = x_0, \quad H(0, t) = x_0, \quad H(1, t) = x_0 \quad \text{for all } s, t \in [0, 1].\]
3.4. Applying the Homotopy Lifting Property with \(Y = [0, 1]\) and initial map \(\gamma: [0, 1] \to \widetilde{X}\), there exists a unique continuous lift \(\widetilde{H}: [0, 1] \times [0, 1] \to \widetilde{X}\) with \(\widetilde{H}(s, 0) = \gamma(s)\) and \(p \circ \widetilde{H} = H\).
3.5. Consider the path \(t \mapsto \widetilde{H}(0, t)\). It is a lift of the constant path \(t \mapsto H(0, t) = x_0\) starting at \(\widetilde{H}(0, 0) = \gamma(0) = \widetilde{x}_0\). By uniqueness of path lifting, \(\widetilde{H}(0, t) = \widetilde{x}_0\) for all \(t \in [0, 1]\).
3.6. Similarly, the path \(t \mapsto \widetilde{H}(1, t)\) is a lift of the constant path \(t \mapsto H(1, t) = x_0\) starting at \(\widetilde{H}(1, 0) = \gamma(1) = \widetilde{x}_0\), so \(\widetilde{H}(1, t) = \widetilde{x}_0\) for all \(t \in [0, 1]\).
3.7. At \(t = 1\), \(s \mapsto \widetilde{H}(s, 1)\) is a lift of the constant path \(s \mapsto H(s, 1) = x_0\) starting at \(\widetilde{H}(0, 1) = \widetilde{x}_0\), so \(\widetilde{H}(s, 1) = \widetilde{x}_0\) for all \(s \in [0, 1]\).
3.8. Thus \(\widetilde{H}\) is a basepoint-preserving homotopy of loops in \(\widetilde{X}\) between \(\gamma\) and the constant loop \(c_{\widetilde{x}_0}\), proving \([\gamma] = 0 \in \pi_1(\widetilde{X}, \widetilde{x}_0)\).
3.9. Therefore \(\ker p_* = \{0\}\), so \(p_*\) is injective.
4. Part (c): Bijectivity of the covering map \(p\).Proof:
4.1. Fix a basepoint \(\widetilde{x}_0 \in \widetilde{X}\) and let \(x_0 = p(\widetilde{x}_0)\). We show that the fiber \(p^{-1}(x_0)\) consists only of the single point \(\widetilde{x}_0\).
4.2. Let \(\widetilde{x}_1 \in p^{-1}(x_0)\) be any point in the fiber.
4.3. Since \(\widetilde{X}\) is path-connected, there exists a continuous path \(\widetilde{\alpha}: [0, 1] \to \widetilde{X}\) from \(\widetilde{x}_0\) to \(\widetilde{x}_1\) (so \(\widetilde{\alpha}(0) = \widetilde{x}_0\) and \(\widetilde{\alpha}(1) = \widetilde{x}_1\)).
4.4. The projected path \(\alpha = p \circ \widetilde{\alpha}: [0, 1] \to X\) satisfies \(\alpha(0) = p(\widetilde{x}_0) = x_0\) and \(\alpha(1) = p(\widetilde{x}_1) = x_0\), so \(\alpha\) is a loop based at \(x_0\), representing \([\alpha] \in \pi_1(X, x_0)\).
4.5. By hypothesis, \(p_*: \pi_1(\widetilde{X}, \widetilde{x}_0) \to \pi_1(X, x_0)\) is an isomorphism, so it is surjective. Thus there exists a loop class \([\beta] \in \pi_1(\widetilde{X}, \widetilde{x}_0)\) such that \(p_*([\beta]) = [\alpha]\).
4.6. This means the loop \(p \circ \beta\) is path-homotopic to \(\alpha\) in \(X\) via a path-homotopy \(H: [0, 1] \times [0, 1] \to X\) with \(H(s, 0) = \alpha(s)\) and \(H(s, 1) = (p \circ \beta)(s)\), keeping endpoints fixed at \(x_0\).
4.7. By the Homotopy Lifting Property, \(H\) lifts to a homotopy \(\widetilde{H}: [0, 1] \times [0, 1] \to \widetilde{X}\) with \(\widetilde{H}(s, 0) = \widetilde{\alpha}(s)\).
4.8. The lift of \(p \circ \beta\) starting at \(\widetilde{x}_0\) is the loop \(\beta\), so \(\widetilde{H}(1, 1) = \beta(1) = \widetilde{x}_0\).
4.9. Since \(\widetilde{H}(1, t)\) must be constant in the discrete fiber \(p^{-1}(x_0)\), \(\widetilde{x}_1 = \widetilde{\alpha}(1) = \widetilde{H}(1, 0) = \widetilde{H}(1, 1) = \widetilde{x}_0\).
4.10. Thus \(p^{-1}(x_0) = \{\widetilde{x}_0\}\) is a singleton.
4.11. Since \(X\) is path-connected and the cardinality of the fiber \(p^{-1}(x)\) is locally constant, \(|p^{-1}(x)| = 1\) for all \(x \in X\). Thus \(p\) is a bijection.
5. Part (c): Bijective covering map is a homeomorphism.Proof:
5.1. The map \(p: \widetilde{X} \to X\) is continuous and bijective by
4.
4.2. To show that \(p\) is a homeomorphism, it suffices to prove that \(p\) is an open map.
4.3. Let \(W \subseteq \widetilde{X}\) be an open set, and let \(x \in p(W)\). Choose \(\widetilde{w} \in W\) such that \(p(\widetilde{w}) = x\).
4.4. Let \(U\) be an evenly covered open neighborhood of \(x\) in \(X\). Since \(|p^{-1}(x)| = 1\), \(p^{-1}(U) = V\) consists of a single open sheet \(V \subseteq \widetilde{X}\), and \(p|_V: V \to U\) is a homeomorphism.
4.5. The set \(W \cap V\) is an open neighborhood of \(\widetilde{w}\) in \(V\).
4.6. Since \(p|_V\) is a homeomorphism, \(p(W \cap V)\) is open in \(U\), and hence open in \(X\).
4.7. Since \(x = p(\widetilde{w}) \in p(W \cap V) \subseteq p(W)\), \(p(W)\) is a neighborhood of each of its points, so \(p(W)\) is open in \(X\).
4.8. Therefore \(p\) is an open continuous bijection, so \(p\) is a homeomorphism.
6. Conclusion:Proof:Covering maps induce injective homomorphisms on fundamental groups via the homotopy lifting property, and every covering map between path-connected spaces that induces an isomorphism on \(\pi_1\) is a homeomorphism.