The ambient classifying space of a classical knot group

TL;DR

This paper constructs an explicit model of the ambient classifying space for prime knot groups as a branched covering of the 3-sphere, exploring its homological properties and duality features with connections to algebraic number theory.

Contribution

It provides an explicit branched covering model of the ambient classifying space for prime knot groups and analyzes its homological and duality properties, linking topology and algebraic number theory.

Findings

Prime knot groups are Bieri-Eckmann duality groups.

The constructed space is a branched covering of the 3-sphere.

Homological properties parallel those in algebraic number theory.

Abstract

For a prime knot group, the classifying space for the family of the subgroups generated by the meridians can be seen as an abstract analogue of the ambient manifold in which the knot lives. An explicit model of this ambient classifying space is constructed as a branched covering space of the 3-sphere branched over the knot; more general branched covering spaces, obtained quotienting the ambient classifying space by finite-index normal subgroups, are studied. Various homological properties of said spaces are established, some of which have parallels in algebraic number theory. In particular, prime knot groups are shown to be Bieri-Eckmann, but not Poincar\'e, duality groups for Bredon cohomology with respect to the family of the meridians.