Classifying spaces for knots: New bridges between knot theory and algebraic number theory

Abstract

A powerful way to study groups is via their actions on suitable spaces. Classifying spaces for families of subgroups are a type of these spaces, obtained by imposing some strict conditions on the fixed-point sets. We show how in the framework of knot theory the notion of classifying spaces for families pops out naturally and we provide explicit models for them. The study of their topological properties hints at the existence of arcane connections between knot theory and algebraic number theory. Trying to track these connections, we discovered a behavioural analogy between the fundamental group of our classifying spaces and the ideal class group of an algebraic number field, a 9-term exact sequence for knots mimicking Poitou-Tate exact sequence for algebraic number fields and, in some cases, a second analogy between the first (co)homology group of the classifying spaces and the first Shafarevich group of a number field extension.