Topologically isotopic and smoothly inequivalent 2-spheres in simply connected 4-manifolds whose complement has a prescribed fundamental group

TL;DR

This paper introduces a method to construct infinitely many smoothly inequivalent 2-spheres in simply connected 4-manifolds that are topologically isotopic, with complements having specific fundamental groups, revealing new knotting phenomena in 4-manifolds.

Contribution

It provides the first known examples of knotting phenomena in 4-manifolds with prescribed fundamental groups, including finite cyclic and binary icosahedral groups.

Findings

Constructed infinite sets of inequivalent 2-spheres with prescribed fundamental groups.

Demonstrated topological isotopy despite smooth inequivalence.

Presented examples in non-smoothable 4-manifolds.

Abstract

We describe a procedure to construct infinite sets of pairwise smoothly inequivalent 2-spheres in simply connected 4-manifolds, which are topologically isotopic and whose complement has a prescribed fundamental group that satisfies some conditions. This class of groups include finite cyclic groups and the binary icosahedral group. These are the first known examples of knotting phenomena in 4-manifolds with such properties. Examples of locally flat embedded 2-spheres in non-smoothable 4-manifolds are also given.