Vol. 18, No. 1, 1966

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Unknotting spheres via Smale

Rodolfo DeSapio

Vol. 18 (1966), No. 1, 179–183

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 n4. On the other hand, Haefliger has constructed C differentiable embeddings of the standard (4k1)-sphere S4k1 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.

Mathematical Subject Classification
Primary: 57.20
Received: 3 March 1965
Published: 1 July 1966
Rodolfo DeSapio