Simple spines of homotopy 2‐spheres are unique

Abstract

AbstractA 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.