Hecke's Theorem on the Different for $3$-Manifolds

TL;DR

This paper establishes an analog of Hecke's theorem for branched covers of closed 3-manifolds, demonstrating that the branch divisor is a square in the first homology group, extending number field results to topology.

Contribution

It proves a topological analog of Hecke's theorem, relating branch divisors to squares in the first homology group of 3-manifolds.

Findings

Branch divisor is a square in the first homology group

Analog of Hecke's theorem for number fields applied to 3-manifolds

Extension of classical number theory results to topology

Abstract

Hecke has shown that the different of an extension of number fields is a square in the class group. We prove an analog for branched covers of closed $3$-manifolds saying that the branch divisor is a square in the first homology group.