Id\`elic class field theory for 3-manifolds and very admissible links

TL;DR

This paper develops a topological analogue of idelic class field theory for 3-manifolds, introducing concepts like very admissible links, idèles, and class groups, and establishing reciprocity laws similar to number theory.

Contribution

It introduces the notion of very admissible links in 3-manifolds and constructs an idelic class field theory framework analogous to arithmetic cases.

Findings

Defined very admissible links in 3-manifolds

Constructed idèle class groups for these pairs

Established global reciprocity law and existence theorem

Abstract

We study a topological analogue of id\`elic class field theory for 3-manifolds, in the spirit of arithmetic topology. We firstly introduce the notion of a very admissible link $\mathcal{K}$ in a 3-manifold $M$, which plays a role analogous to the set of primes of a number field. For such a pair $(M,\mathcal{K})$, we introduce the notion of id\`eles and define the id\`ele class group. Then, getting the local class field theory for each knot in $\mathcal{K}$ together, we establish analogues of the global reciprocity law and the existence theorem of id\`elic class field theory.