Simple groups and complements of smooth surfaces in simply connected $4$-manifolds

For each integer $n$ we construct a simply connected $4$-manifold $X$ admitting a smoothly embedded surface $\mathrm{\Sigma }$ 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.