Imbedding Free Cyclic Group Actions in Circle Group Actions

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Suppose a closed, orientable, irreducible $3$-manifold $M$ admits a free cyclic group action of prime order. We consider the problem of determining when $M$ admits an effective action of the circle group $SO(2)$ in which the cyclic action is imbedded. The main result is that if the ${Z_k}$ action is “$Z$-classified", then it is weakly equivalent to a ${Z_k}$ action imbedded in an effective action of $SO(2)$ if and only if some homeomorphism generating the first ${Z_k}$ action is homotopic to the identity homeomorphism on $M$.