Simple spines of homotopy 2-spheres are unique

TL;DR

The paper proves that simple spines of homotopy 2-spheres in 4-manifolds are unique up to ambient isotopy when they represent the same homology class, with applications to knot traces.

Contribution

It establishes the uniqueness of simple spines of homotopy 2-spheres in 4-manifolds, extending understanding of their isotopy classes and applications to knot theory.

Findings

Simple spines are unique up to ambient isotopy when representing the same generator of H_2(X).

The result applies to simple shake-slicing 2-spheres in knot traces.

Provides a classification result for certain 2-spheres in 4-manifolds.

Abstract

A locally flatly embedded $2$-sphere in a compact $4$-manifold $X$ is called a spine if the inclusion map is a homotopy equivalence. A spine is called simple if the complement of the $2$-sphere has abelian fundamental group. We prove that if two simple spines represent the same generator of $H_2(X)$ then they are ambiently isotopic. In particular, the theorem applies to simple shake-slicing $2$-spheres in knot traces.