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Embeddings of quaternion space in S4

Part of: Topological manifolds PL-topology

Published online by Cambridge University Press:  09 April 2009

Atsuko Katanaga  and
Osamu Saeki
Atsuko Katanaga
Affiliation:
Institute of Mathematics, University of Tsukuba, Tsukuba-city, Ibaraki 305-8571, Japan e-mail: fuji@math.tsukuba.ac.jp
Osamu Saeki
Affiliation:
Department of Mathematics, Faculty of Science, Hiroshima University, Higashi-Hiroshima 739-8526, Japan e-mail: saeki@math.sci.hiroshima-u.ac.jp
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Abstract

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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.

MSC classification

Secondary: 57N35: Embeddings and immersions 57N13: Topology of $E^4$, $4$-manifolds 57N50: $S^{n-1}subset E^n$, Schoenflies problem 57Q45: Knots and links (in high dimensions)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 65 , Issue 3 , December 1998 , pp. 313 - 325
Copyright
Copyright © Australian Mathematical Society 1998

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