Locally flat simple spheres in CP2$\mathbb {C}P^2$

Abstract The fundamental group of the complement of a locally flat surface in a 4‐manifold is called the knot group of the surface. In this article, we prove that two locally flat 2‐spheres in with knot group are ambiently isotopic if they are homologous. This combines with work of Tristram and Lee–Wilczyński, as well as the classification of ‐surfaces, to complete a proof of the statement: a class is represented by a locally flat sphere with abelian knot group if and only if ; and this sphere is unique up to ambient isotopy.