Equivalent non-isotopic spheres in 4-manifolds

TL;DR

This paper constructs examples of 4-manifolds with pairs of embedded 2-spheres that are topologically equivalent but not smoothly isotopic, highlighting limitations of Gabai's 4D Lightbulb Theorem.

Contribution

It provides new examples demonstrating that Gabai's theorem does not extend to all 4-manifolds, and distinguishes between topological and smooth isotopy classes of spheres.

Findings

Existence of homotopic, topologically non-isotopic spheres in 4-manifolds.

Counterexamples to the generalization of Gabai's 4D Lightbulb Theorem.

Identification of spheres that are equivalent and topologically isotopic but not smoothly isotopic.

Abstract

We construct infinitely many smooth oriented 4-manifolds containing pairs of homotopic, smoothly embedded 2-spheres that are not topologically isotopic, but that are equivalent by an ambient diffeomorphism inducing the identity on homology. These examples show that Gabai's recent "Generalized" 4D Lightbulb Theorem does not generalize to arbitrary 4-manifolds. In contrast, we also show that there are smoothly embedded 2-spheres that are both equivalent and topologically isotopic, but not smoothly isotopic.