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Published online by Cambridge University Press: 24 October 2008
Given any embedding f:Sr × Bn−r → Sn, f ∣ Sr × 0 can be extended to a homeomorphism of Sn, except when r = n − 2 and f is actually knotted (10). Is there any sense in which this extension can be chosen to vary continuously with respect to f? If r = n − 1, there certainly is: this is simply the canonical Schoenflies theorem, proved in varying (but equivalent) forms in (6), (5) and (1). One of the principal results of this paper provides an affirmative answer to this question when r < n − 1. In the usual fashion of such canonical extension theorems, we look for a local version; that is attempt to find a neighbourhood of the inclusion in E(Sr × Bn−r, Sn) on which a map F can be defined with each F(f) a homeomorphism of Sn which extends f and F (inclusion) = identity. Now if the neighbourhood is sufficiently small, any element of it can be viewed as an embedding Sr × Bn−r → Sr × Rn−r, still close to the inclusion, by removing a standard Sn−r−1 from the image space. There are known methods for extending such embeddings. It would not be hard to adapt Kister's methods (8) for extending embeddings of Bk in Rk canonically to homeomorphisms (approximately the method that is sketched for the r = n − 1 case in (5)). Alternatively the existence of a canonical extension in this situation follows explicitly from the more general Theorem 5 of (1). Unfortunately both methods extend the embedding in a ‘radial’ fashion, with the result that there is little control over the extension towards infinity, and the final homeomorphism cannot necessarily be further extended over the missing Sn−r−1 to yield a homeomorphism of Sn. In fact this failure is due to the extension map F, derived in either of these ways, being continuous with respect to the compact-open topology on the image space, H(Sr × Rn−r). The extension map to be constructed here is continuous with respect to the uniform topology on the image space, and so the homeomorphisms F(f) of Sr × Rn−r are bounded and can thus be extended over all of Sn by setting F(f) ∣ Sn−r−1 equal to the identity.