### Abstract

Consider a (real) projective plane which is topologically locally flatly embedded in S^{4}. 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 S^{1}-bundle over RP^{2} with Euler number ±2, with fundamental group isomorphic to the quaternion group of order eight. Conversely let f : Q → S^{4} be an arbitrary locally flat topological embedding. Then we show that the closure of each connected component of S^{4} - f(Q) is always homeomorphic to the exterior of a topologically locally flatly embedded projective plane in S^{4}. We also show that, for a large class of embedded projective planes in S^{4}, a pair of exteriors of such embedded projective planes is always realized as the closures of the connected components of S^{4} - f(Q) for some locally flat topological embedding f : Q → S^{4}.

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 |

Publication status | Published - Dec 1 1998 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

^{4}.

*Journal of the Australian Mathematical Society*,

*65*(3), 313-325.