Let a cyclic group act either on a number field or on a -manifold . Let be the number of ramified primes in the extension and be the number of components of the branching set of the branched covering . In this paper, we prove several formulas relating and to the induced -action on and respectively. We observe that the formulas for -manifolds and number fields are almost identical, and therefore, they provide new evidence for the correspondence between -manifolds and number fields postulated in arithmetic topology.
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Adam S. Sikora, Analogies between group actions on 3-manifolds and number fields. Comment. Math. Helv. 78 (2003), no. 4, pp. 832–844DOI 10.1007/S00014-003-0781-X