It is shown here that a
topological n-sphere which is embedded in Euclidean m-space Rm with a
transverse field of (m − n)-planes (in the sense of Whitehead) bounds a
topological (n+1)-disc in Rm, provided m > n + 2 > 4 and n≠4. On the
other hand, Haefliger has constructed C∞ differentiable embeddings of the
standard (4k−1)-sphere S4k−1 in 6k-space R6k which are differentiably knotted
(i.e. they do not bound differentiably embedded 4k-discs in R6k). However, by
using a sharpened form of the h-cobordism theorem of Smale it is possible
to topologically unknot these spheres. This is achieved by showing that a
differentiably knotted n-sphere in m-space Rm is so knotted because of a single bad
point (provided m > n + 2 > 4). The topological case is then proved by
first approximating the topologically embedded n-sphere by a differentiably
embedded homotopy n-sphere, and thus reducing it to the dfflerentiable
case.
