Consider covering spaces \(p: \tilde{X} \to X\) with \(\tilde{X}\) and \(X\) connected CW complexes, the cells of \(\tilde{X}\) projecting homeomorphically onto cells of \(X\). Restricting \(p\) to the 1-skeleton then gives a covering space \(\tilde{X}^1 \to X^1\) over the 1-skeleton of \(X\). Show:
Two such covering spaces \(\tilde{X}_1 \to X\) and \(\tilde{X}_2 \to X\) are isomorphic if the restrictions \(\tilde{X}_1^1 \to X^1\) and \(\tilde{X}_2^1 \to X^1\) are isomorphic.
\(\tilde{X} \to X\) is a normal covering space if \(\tilde{X}^1 \to X^1\) is normal.
The groups of deck transformations of the coverings \(\tilde{X} \to X\) and \(\tilde{X}^1 \to X^1\) are isomorphic, via the restriction map.
Solution
Choose a basepoint \(x_0\) in the \(0\)-skeleton and write \[ j:X^1\hookrightarrow X. \] Since \(X\) is obtained from \(X^1\) by attaching cells of dimension at least \(2\), \[ j_*:\pi_1(X^1,x_0)\twoheadrightarrow\pi_1(X,x_0) \] is surjective.
1. If a connected cover \(p:\widetilde X\to X\) corresponds to the subgroup\[ H\le\pi_1(X,x_0), \]then its restricted \(1\)-skeleton cover corresponds to\[ j_*^{-1}(H)\le\pi_1(X^1,x_0). \]
A loop in \(X^1\) lifts closed to \(\widetilde X^1\) exactly when the same loop, viewed in \(X\), lifts closed to \(\widetilde X\). By the lifting criterion for connected covers, this is exactly the condition that its class lie in \(j_*^{-1}(H)\).
2. If the restricted covers\[ \widetilde X_1^1\to X^1, \qquad \widetilde X_2^1\to X^1 \]are isomorphic, then the full covers\[ \widetilde X_1\to X, \qquad \widetilde X_2\to X \]are isomorphic.
Let \(H_1,H_2\le\pi_1(X)\) be the subgroups corresponding to the two connected covers. Isomorphism of the unpointed restricted covers means that \[ j_*^{-1}(H_1) \] and \[ j_*^{-1}(H_2) \] are conjugate in \(\pi_1(X^1)\). Applying the surjection \(j_*\) shows that \(H_1\) and \(H_2\) are conjugate in \(\pi_1(X)\). The connected-covering classification therefore gives an isomorphism of the full covers. This proves (a).
3. If the restricted cover \(\widetilde X^1\to X^1\) is normal, then \(\widetilde X\to X\) is normal.
Normality of the restricted cover means \[ j_*^{-1}(H)\triangleleft\pi_1(X^1). \] Let \(g\in\pi_1(X)\) and \(h\in H\). Choose lifts \[ \hat g,\hat h\in\pi_1(X^1) \] with \[ j_*(\hat g)=g, \qquad j_*(\hat h)=h, \] possible because \(j_*\) is surjective. Since \(\hat h\in j_*^{-1}(H)\) and this subgroup is normal, \[ \hat g\hat h\hat g^{-1}\in j_*^{-1}(H). \] Applying \(j_*\) gives \[ ghg^{-1}\in H. \] Thus \(H\triangleleft\pi_1(X)\), so the full cover is normal. This proves (b).
4. Restriction gives an injective homomorphism\[ \operatorname{Deck}(\widetilde X/X) \longrightarrow \operatorname{Deck}(\widetilde X^1/X^1). \]
A deck transformation preserves cells and hence restricts to the \(1\)-skeleton. If its restriction is the identity, then it fixes a point of the connected covering space \(\widetilde X\). A deck transformation of a connected cover is determined by the image of one point, so it is the identity.
5. The restriction homomorphism in 4 is surjective.
For the full cover corresponding to \(H\le G=\pi_1(X)\), \[ \operatorname{Deck}(\widetilde X/X)\cong N_G(H)/H. \] For the restricted cover, put \[ G_1=\pi_1(X^1), \qquad H_1=j_*^{-1}(H). \] Then \[ \operatorname{Deck}(\widetilde X^1/X^1) \cong N_{G_1}(H_1)/H_1. \] Because \(j_*:G_1\twoheadrightarrow G\) is surjective, \[ j_*^{-1}(N_G(H))=N_{G_1}(H_1). \] Indeed, conjugating \(H_1\) into itself is equivalent after applying \(j_*\) to conjugating \(H\) into itself, and the reverse implication follows from taking preimages under \(j_*\). Therefore \(j_*\) induces an isomorphism \[ N_{G_1}(H_1)/H_1 \xrightarrow{\sim} N_G(H)/H, \] which is exactly the deck transformation obtained by extending a deck transformation from the \(1\)-skeleton. Thus restriction is surjective.
6. Hence the deck groups of the two covers are naturally isomorphic.
5.This proves (c).