Existence of exotic 2-knots (and orientable surfaces) in S^4

Determine whether there exist exotic 2-knots in the 4-sphere S^4, i.e., smoothly embedded 2-spheres in S^4 that are topologically isotopic but not smoothly equivalent; more generally, ascertain whether there exist orientable surfaces in S^4 that are topologically isotopic but smoothly inequivalent.

Background

A central question in 4-manifold topology is whether S^4 contains smoothly knotted surfaces that are topologically unknotted, i.e., topologically isotopic but not smoothly isotopic. In the present work, the author constructs exotic 2-links in simply connected 4-manifolds with prescribed link group, thereby providing broad evidence in larger ambient manifolds.

Despite extensive progress on exotic knotted surfaces in other 4-manifolds, the case of S^4 remains unresolved. The paper highlights that null-homologous exotic surfaces are known to exist in larger simply connected 4-manifolds, but whether any such phenomena occur in S^4 itself is still unknown.