Topologically isotopic and smoothly inequivalent 2–spheres in simply connected 4–manifolds whose complement has a prescribed fundamental group

We describe a procedure to construct infinite sets of pairwise smoothly inequivalent 2-spheres in simply connected 4-manifolds, which are topologically isotopic and whose complement has a prescribed fundamental group that satisfies some conditions.This class of groups include cyclic groups and the binary icosahedral group.These are the first known examples of such exotic embeddings of 2-spheres in 4-manifolds.Examples of locally flat embedded 2-spheres in a non-smoothable 4-manifold whose complements are homotopy equivalent to smoothly embedded ones are also given. Main resultsThe first main result of this note is the following theorem.