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-Wilczyński, 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 $|d|\in \{0,1,2\}$; and this sphere is unique up to ambient isotopy.

13 pages. v2: minor changes incorporating suggestions of a referee. To appear in Bull. Lond. Math. Soc