The Smale invariant of a knot
\textit{S. Smale} [Ann. Math., II. Ser. 69, 327-344 (1959; Zbl 0089.182)] associated to each immersion \(f: S^ n\to {\mathbb{R}}^ k\) an element s(f) in \(\pi_ n(V_ n({\mathbb{R}}^ k))\) and asked for a characterization of those elements s(f) where f is an embedding. This problem was solved by Kervaire, Milnor and Haefliger in all cases but \(k=n+2\). The main result of the paper under review may be stated as follows: For \(n>2\), an immersion \(f: S^ n\to {\mathbb{R}}^{n+2}\) is regularly homotopic to an embedding iff the element \(i(f)\in \pi_ nSO(n+2)\) corresponding to \(s(f)\in \pi_ n(V_ n({\mathbb{R}}^{n+2}))\) vanishes under the J- homomorphism into \(\pi_{2n+2}S^{n+2}.\) The proof consists of identifying i(f) and understanding the J- homomorphism geometrically and combining these when f is an embedding to see that \(J(i(f))=0\). The proof of the converse is by construction using examples provided by \textit{E. Brieskorn} [Invent. Math. 2, 1-14 (1966; Zbl 0145.178)].
scientific article; zbMATH DE number 3940280