Journal of the Australian Mathematical Society - Series A
Vol. 65 Part 3 (1998)
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 with Euler number , with fundamental group isomorphic to the quaternion group of order eight. Conversely let 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 .
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