Abstract
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.
Original language | English |
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Pages (from-to) | 313-325 |
Number of pages | 13 |
Journal | Journal of the Australian Mathematical Society |
Volume | 65 |
Issue number | 3 |
DOIs | |
Publication status | Published - Dec 1998 |
Externally published | Yes |
All Science Journal Classification (ASJC) codes
- Mathematics(all)