Embeddings of quaternion space in S4

Abstract

AbstractConsider 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.