Stable isotopy in four dimensions

TL;DR

This paper constructs infinite families of knotted spheres in 4-manifolds that are topologically but not smoothly equivalent, becoming smoothly isotopic after stabilization, and extends these ideas to links and metrics.

Contribution

It introduces new methods to produce infinite families of knotted spheres and links in 4-manifolds that stabilize to smooth isotopy, advancing understanding of smooth structures in four dimensions.

Findings

Existence of infinite families of knotted spheres with stabilization

Construction of links with isotopic proper sublinks after stabilization

Implications for metrics of positive scalar curvature

Abstract

We construct infinite families of topologically isotopic but smoothly distinct knotted spheres in many simply connected 4-manifolds that become smoothly isotopic after stabilizing by connected summing with $S^2 \times S^2$, and as a consequence, analogous families of diffeomorphisms and metrics of positive scalar curvature for such 4-manifolds. We also construct families of smoothly distinct links, all of whose corresponding proper sublinks are smoothly isotopic, that become smoothly isotopic after stabilizing.