Simple spines of homotopy 2-spheres are unique

Abstract

A locally flatly embedded 2-sphere in a compact 4-manifold 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 then they are ambiently isotopic. In particular, the theorem applies to simple shake-slicing 2-spheres in knot traces.