Abstract
For each integer $n$ we construct a simply connected $4$-manifold $X$
admitting a smoothly embedded surface $\Sigma$ of self intersection number $n$
such that the complement of the surface has non-trivial fundamental group. This
answers a question of Kronheimer in Kirby's 1997 problem list. The proof
combines a topological construction with homological properties of simple
groups such as Thompson's group $V$ and certain sporadic finite simple groups.