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

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 $C{P}^2$ with knot group $Z_2$ are ambiently isotopic if they are homologous. This combines with work of Tristram and Lee–Wilczyński, as well as the classification of $Z$-surfaces, to complete a proof of the statement: a class $d\in H_2\left(C{P}^2\right)\cong Z$ is represented by a locally flat sphere with abelian knot group if and only if $\left|d\right|\in \{0,1,2\}$; and this sphere is unique up to ambient isotopy.