Free cyclic actions on 𝑆¹×𝑆¹×𝑆¹

Hempel, John

doi:10.1090/S0002-9939-1975-0362312-5

Free cyclic actions on $S^{1}\times S^{1}\times S^{1}$
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Proc. Amer. Math. Soc. 48 (1975), 221-227 Request permission

Abstract:

If the cyclic group ${Z_p}$ acts freely on the $3$-torus ${T^3} = {S^1} \times {S^1} \times {S^1}$, then the quotient manifold ${T^3}/{Z_p}$ is shown to be one of seven specifically described $3$-manifolds, each of which is a ${T^2}$ bundle over ${S^1}$. Furthermore the covering projection ${T^3} \to {T^3}/{Z_p}$ can be factored as a standard covering ${T^3} \to {T^3}$ followed by a 1, 2, 3, 4, or 6 sheeted covering ${T^3} \to {T^3}/{Z_p}$. In the process the action of ${Z_p}$ on ${T^3}$ is classified up to equivalence.

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