Analogies between group actions on 3-manifolds and number fields

Abstract

Let a cyclic group $G$ act either on a number field $\mathbb{L}$ or on a $3$-manifold $M$. Let $s_{\mathbb{L}}$ be the number of ramified primes in the extension ${\mathbb{L}}^G\subset \mathbb{L}$ and $s_M$ be the number of components of the branching set of the branched covering $M\to M/G$. In this paper, we prove several formulas relating $s_{\mathbb{L}}$ and $s_M$ to the induced $G$-action on $Cl(\mathbb{L})$ and $H_1(M),$ 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.