Embeddings of quaternion space in S⁴

Consider a (real) projective plane which is topologically locally flatly embedded in S⁴. It is known that it always admits a 2-disk bundle neighborhood, whose boundary is homeomorphic to the quaternion space Q, the total space of the nonorientable S¹-bundle over RP² with Euler number ±2, with fundamental group isomorphic to the quaternion group of order eight. Conversely let f : Q → S⁴ be an arbitrary locally flat topological embedding. Then we show that the closure of each connected component of S⁴ - f(Q) is always homeomorphic to the exterior of a topologically locally flatly embedded projective plane in S⁴. We also show that, for a large class of embedded projective planes in S⁴, a pair of exteriors of such embedded projective planes is always realized as the closures of the connected components of S⁴ - f(Q) for some locally flat topological embedding f : Q → S⁴.