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

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 $\mathbb{C} P^2$ with knot group $\mathbb{Z}_2$ are ambiently isotopic if they are homologous. This combines with work of Tristram and Lee-Wilczy\'{n}ski, as well as the classification of $\mathbb{Z}$-surfaces, to complete a proof of the statement: a class $d \in H_2(\mathbb{C} P^2) \cong \mathbb{Z}$ is represented by a locally flat $2$-sphere with abelian knot group if and only if $|d| \in \lbrace 0,1,2\rbrace$; and this sphere is unique up to ambient isotopy.