Analogies between group actions on 3-manifolds and number fields

Let a cyclic group G act either on a number field double-struck L sign or on a 3-manifold M. Let S_{double-struck L sign} be the number of ramified primes in the extension double-struck L sign^G ⊂ double-struck L sign and s_M be the number of components of the branching set of the branched covering M → M/G. In this paper, we prove several formulas relating s_{double-struck L sign} and s_M to the induced G-action on Cl(double-struck L sign) and H₁(M), respectively. We observe that the formulas for 3-manifolds and number fields are almost identical, and therefore, they provide new evidence for the correspondence between 3-manifolds and number fields postulated in arithmetic topology.