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Journal of the Australian Mathematical Society - Series A
Vol. 65 Part 3 (1998)

Embeddings of quaternion space in S4

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

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 ${\bf R}P^2$with Euler number $\pm 2$, with fundamental group isomorphic to the quaternion group of order eight. Conversely let $f : Q \to S^4$ 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 $f : Q \to S^4$.

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© Copyright 1998, Australian Mathematical Society
TeXAdel Scientific Publishing
1998-09-18