Cohomological Approach to Class Field Theory in Arithmetic Topology

Abstract

We establish class field theory for three-dimensional manifolds and knots. For this purpose, we formulate analogues of the multiplicative group, the idèle class group, and ray class groups in a cocycle-theoretic way. Following the arguments in abstract class field theory, we construct reciprocity maps and verify the existence theorems.

Olympic links in a Chebotarev link

“…On the other hand, Mihara [16] formulated an analogue of idelic class field theory for 3-manifolds by introducing certain infinite links called stably generic links, refining the notion of very admissible links given by Niibo and the author [20,21], and gave a cohomological interpretation of our previous formulation. Here we describe the definition of a stably generic link, only using ordinary terminology of low dimensional topology: Definition 2.2 (stably generic link).…”

Chebotarev links are stably generic

“…On the other hand, Mihara [16] formulated an analogue of idelic class field theory for 3‐manifolds by introducing certain infinite links called stably generic links, refining the notion of very admissible links given by Niibo and the author [20, 21], and gave a cohomological interpretation of our previous formulation. Here we describe the definition of a stably generic link, only using ordinary terminology of low‐dimensional topology: Definition Let be a 3‐manifold and a link.…”