Exotically knotted 2-spheres and the fundamental groups of their complements

TL;DR

This paper constructs 4-manifolds with 2-links whose complement groups realize any finitely presented group, revealing complex relationships between knot theory, 4-manifolds, and fundamental groups.

Contribution

It demonstrates the existence of simply connected 4-manifolds containing infinite families of topologically isotopic but smoothly inequivalent 2-links with prescribed complement groups.

Findings

Existence of 4-manifolds with specified fundamental groups of 2-link complements.

Construction of infinite families of inequivalent 2-links with the same topological type.

Conditions under which these 2-links have nullhomotopic components.

Abstract

We show that for any finitely presented group $G$, there is a simply connected closed 4-manifold containing an infinite family of topologically isotopic but smoothly inequivalent 2-links whose 2-link group is $G$. We also show that, if $G$ satisfies the necessary topological conditions, these 2-links have nullhomotopic components.