zbMATH — the first resource for mathematics

On eversion of spheres. (English. Russian original) Zbl 1108.57022
Geometric topology and set theory. Collected papers. Dedicated to the 100th birthday of Professor Lyudmila Vsevolodovna Keldysh. Transl. from the Russian. Moscow: Maik Nauka/Interperiodica. Proceedings of the Steklov Institute of Mathematics 247, 135-142 (2004); translation from Tr. Mat. Inst. Steklova 247, 151-158 (2004).
The main purpose of this article is to give some direct proofs of the “only if” part of the following theorem due to U. Kaiser: The \(n\)-sphere \(S^n\) can be turned inside out in Euclidean \((n+1)\)-space if and only if \(n=0,2,6\).
This theorem is an extension of the case \(n=2\) and \(m=3\) of the Smale classification theorem for immersions. In this case, the eversion is possible. In addition to the proof of the “if” part, the authors give three kinds of the proof of the “only if” part of Kaiser’s theorem.
Furthermore, the authors sketch the proof, by Novikov, of the Smale classification theorem for immersions and give the proof of an immersion theorem by Haefliger-Hirsch of highly connected manifolds (with or without boundary) into Euclidean spaces.
For the entire collection see [Zbl 1087.55002].
57R42 Immersions in differential topology