On genus theory for 3-manifolds in arithmetic topology

TL;DR

This paper develops a topological analogue of Hilbert's Satz 90 for idele groups, applying it to derive a genus formula for finite abelian branched covers over integral homology 3-spheres, inspired by number theory.

Contribution

It introduces a novel topological analogue of a fundamental number theory result and applies it to derive a genus formula for 3-manifolds, bridging topology and arithmetic.

Findings

Established a topological analogue of Hilbert's Satz 90.

Derived a genus formula for abelian branched covers of 3-spheres.

Connected arithmetic topology concepts with 3-manifold invariants.

Abstract

Based on the analogies of arithmetic topology, we show a topological analogue of Hilbert's Satz 90 for idele groups and utilize our previously established Hasse norm principle to present a proof of an Iyanaga--Tamagawa type genus formula for finite abelian branched covers over integral homology 3-spheres in a very parallel manner to the case of number theory.