Given a group \(G\) and a normal subgroup \(N\), show that there exists a normal covering space \(\tilde{X} \to X\) with \(\pi_1(X) \approx G\), \(\pi_1(\tilde{X}) \approx N\), and deck transformation group \(G(\tilde{X}) \approx G/N\).
Solution
1. Choose a CW complex \(X\) with \(\pi_1(X) \cong G\) (e.g., the presentation complex of \(G\); a \(K(G,1)\)).
every group is the fundamental group of a CW complex (attach \(2\)-cells for relations and kill higher homotopy).
2. Let \(p : \tilde X \to X\) be the connected covering corresponding to the subgroup \(N \le \pi_1(X) \cong G\).
covering space theory (for nice \(X\), connected coverings are classified by conjugacy classes of subgroups of \(\pi_1\)).
3. Then \(\pi_1(\tilde X) \cong N\).
2 (\(p_*(\pi_1(\tilde X)) = N\)).
4. The covering is normal iff \(N\) is normal in \(\pi_1(X)\).
a covering is normal (regular) iff the corresponding subgroup is normal.
5. Hence, since \(N \triangleleft G\), the covering \(\tilde X \to X\) is normal.
4.
6. The group of deck transformations is \(G(\tilde X) \cong \pi_1(X)/p_*(\pi_1(\tilde X)) \cong G/N\).
for a normal covering, the deck group is the quotient of the base fundamental group by the image of the covering.
7. Q.E.D.
3, 5, 6.