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A locally flatly embedded 2-sphere in a compact 4-manifold $X$ is called aspine if the inclusion map is a homotopy equivalence. A spine is called simpleif the complement of the 2-sphere has abelian fundamental group. We prove thatif two simple spines represent the same generator of $H_2(X)$ then they areambiently isotopic. In particular, the theorem applies to simple shake-slicing2-spheres in knot traces.

Simple spines of homotopy 2-spheres are unique