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Homeomorphisms of Quaternion space and projective planes in four space

Published online by Cambridge University Press:  09 April 2009

T. M. Price
T. M. Price
Affiliation:
Department of Mathematics, University of Melbourne, Victoria 3052, Australia and University of Iowa, Iowa City, Iowa 52242, U.S.A.
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Abstract

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It is known that all locally flat projective planes in S4 have homeomorphic normal disk bundles. In this paper we investigate the homeomorphisms of Q3 (= boundary of the normal disk bundle) on to itself. We show that a homeomorphisms of Q3 onto itself is determined, up to isotopy, by the outerautomorphism of π1(Q3) that it induces. Since Q3 is an irreducible, not sufficiently large 3-manifold with finite fundamental group this characterization is interesting in its own right. The characterization of homeomorphisms is then used to study certain questions about embeddings of the projective plane in S4. One result is that there are at most two distinct projective planes in S4 with a given complement.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 23 , Issue 1 , February 1977 , pp. 112 - 128
Copyright
Copyright © Australian Mathematical Society 1977

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