Suppose \(g:Y\to S^1\) is a path-connected 3-sheeted covering space. Prove that this covering space is unique up to isomorphism.
Solution
1. Connected covering spaces of \(S^1\) correspond, up to isomorphism, to subgroups of \(\pi_1(S^1)\cong\ZZ\), and the number of sheets equals the index of the subgroup.
\(S^1\) is path-connected, locally path-connected, and semilocally simply connected, so the classification of connected covering spaces applies; conjugacy classes of subgroups of the abelian group \(\ZZ\) are single subgroups. The number of sheets of a connected covering equals the index of \(p_*\pi_1\) of the total space.
2. The only subgroup of index \(3\) in \(\ZZ\) is \(3\ZZ\).
Every subgroup of \(\ZZ\) is \(n\ZZ\) for a unique \(n\ge0\), of index \(n\) when \(n\ge1\).
3. Q.E.D.
By steps 1 and 2, every path-connected \(3\)-sheeted covering of \(S^1\) is isomorphic to the covering with subgroup \(3\ZZ\), namely \(S^1\to S^1\), \(z\mapsto z^3\).