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It is known that all locally flat projective planes in S 4 have homeomorphic normal disk bundles. In this paper we investigate the homeomorphisms of Q' (= boundary of the normal disk bundle) onto itself. We show that a homeomorphism of Q' onto itself is determined, up to isotopy, by the outerautomorphism of ir,(O') that it induces. Since Q* 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 S 4 . One result is that there are at most two distinct projective planes in S" with a given complement.If P and P' are smooth embeddings of the projective plane in S 4 then the normal bundle to P is equivalent to the normal bundle to P' (see Theorem I of Massey (1974)). In fact, using smoothing theory, the above result can be extended to the case where P and P' are PL locally flat embeddings. Furthermore the normal disk bundle is constructed in Massey (1969) and in Price and Roseman (1975). In the latter paper it is shown that the boundary of this normal disk bundle, denoted Q 3 , is a compact 3-manifold whose fundamental group is the quaternion group {a, b: a 2 = b 2 = (ab) 2 } and its universal cover is S 3 . This 3-manifold, which we refer to as quaternion space, arises in a number of problems in topology (see page 198 of Seifert and Threlfall (1934) for an early reference). As mentioned above, it is the quotient space of S 3 modulo the action of the quaternion group on S 3 . The referee has also pointed out that in Borel (1953), Q 3 is identified as 0(3)/Q(3). Using this fibration, as well as some related principal fibrations, Borel shows that S 3 (the symmetric group on 3 letters) acts on Q 3 . In Corollary 1 to Theorem 1 of this paper we show that the group of isotopy classes of homeomorphisms of Q 3 onto itself is, in fact, isomorphic to S 3 .Having classified the homeomorphisms of Q 3 onto itself, we show (analogous to Gluck's results on embeddings of S 2 in S 4 , see Gluck (1962)) 112

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