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On Hasse norm principle for 3-manifolds in arithmetic topology

Hirotaka Tashiro

TL;DR

The paper addresses a topological analogue of the Hasse norm principle within arithmetic topology, connecting finite cyclic coverings of integral homology 3-spheres to idele-like structures. It builds a topological idèle framework $(I_{M,\mathcal{L}}, P_{M,\mathcal{L}})$ via the diagonal map $\Delta_{M,\mathcal{L}}$ and provides an explicit description using Seifert surfaces, enabling a precise formulation of a topological Hasse norm principle. The main result proves that for a cyclic cover $f:N\to M$ branched along a finite sublink, $P_{M,\mathcal{L}}\cap f_{*}(I_{N, f^{-1}(\mathcal{L})})=f_{*}(P_{N, f^{-1}(\mathcal{L})})$, supported by a diagrammatic and homological argument that mirrors class field theory. This work deepens the analogy between number theory and 3-manifold topology, providing a formal mechanism to study abelian coverings in the topological setting and suggesting extensions to rational homology 3-spheres.

Abstract

Following the analogies between knots and primes, 3-manifolds and number rings in arithmetic topology, we show a topological analogue of the Hasse norm principle for finite cyclic coverings of 3-manifolds, which was originally stated for finite cyclic extensions of number fields.

On Hasse norm principle for 3-manifolds in arithmetic topology

TL;DR

The paper addresses a topological analogue of the Hasse norm principle within arithmetic topology, connecting finite cyclic coverings of integral homology 3-spheres to idele-like structures. It builds a topological idèle framework via the diagonal map and provides an explicit description using Seifert surfaces, enabling a precise formulation of a topological Hasse norm principle. The main result proves that for a cyclic cover branched along a finite sublink, , supported by a diagrammatic and homological argument that mirrors class field theory. This work deepens the analogy between number theory and 3-manifold topology, providing a formal mechanism to study abelian coverings in the topological setting and suggesting extensions to rational homology 3-spheres.

Abstract

Following the analogies between knots and primes, 3-manifolds and number rings in arithmetic topology, we show a topological analogue of the Hasse norm principle for finite cyclic coverings of 3-manifolds, which was originally stated for finite cyclic extensions of number fields.
Paper Structure (6 sections, 13 theorems, 30 equations)

This paper contains 6 sections, 13 theorems, 30 equations.

Table of Contents

  1. Topological local class field theory and ideèles
  2. Local class field theory
  3. Idèle group and principal idèle group
  4. A lemma on meridians
  5. Explicit description of the diagonal map $\Delta_{M,\mathcal{L}}$
  6. Hasse norm principle for 3-manifolds

Key Result

Theorem 1

Let $E/F$ be a finite cyclic extension of number fields. Then,

Theorems & Definitions (24)

  • Theorem 1: Hasse Hasse1931Beweis
  • Theorem 3.1
  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3: cf. NiiboUeki
  • Definition 1.4
  • Definition 1.5
  • Definition 1.8
  • Proposition 1.9: NiiboUeki
  • Lemma 1.10
  • ...and 14 more