Analogies between group actions on 3-manifolds and number fields

Abstract

Let a cyclic group G act either on a number field double-struck L sign or on a 3-manifold M. Let Sdouble-struck L sign be the number of ramified primes in the extension double-struck L signG ⊂ double-struck L sign and sM be the number of components of the branching set of the branched covering M → M/G. In this paper, we prove several formulas relating sdouble-struck L sign and sM 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.