Approximating codimension \(\geq 3\) \(\sigma\)-compacta with locally homotopically unknotted embeddings
\textit{M. A. Shtan'ko} [Mat. Sb., Nov. Ser. 83(125), 234-255 (1970; Zbl 0204.228); translated in Math. USSR, Sb. 12(1970), 234-254 (1971)] proved an ''embedding approximation theorem'' which roughly asserts that arbitrarily close to a fixed embedding of a compactum X into Euclidean space \(E^ n\), there is another embedding of X whose image is locally homotopically l-co-connected (1-LCC) in \(E^ n\). A subset Y of \(E^ n\) satisfies 1-LCC in \(E^ n\) if for each point x in Y and for each neighborhood U of x, there is a neighborhood V of x (contained in U) such that every loop in V-Y is nullhomotopic in U-Y. Here is some history. This marvelous paper of Shtan'ko contained many technical details which were very difficult to understand, and thus, many readers started to wonder if there were some manifest errors in the proof. A rare exception was the author of the paper under review, who lucidly expounded and extended Shtan'ko's work in a paper which was circulated 1973/74 but was never published. The 1986 published version is essentially this circulated version with minor changes. During the intervening 13 years, the time elapsed between its inception and publication, this paper made its presence known because of the following. First, its clear and lucid exposition was instrumental in the formalization of the concept of a ''grope'' which was implicitly buried in Shtan'ko's paper. The ''grope technology'' played a decisive role in proving that the double suspension of a homology sphere is a sphere. For additional information on this and related matters see \textit{J. W. Cannon} [Bull. Am. Math. Soc. 84, 832-866 (1978; Zbl 0418.57005)]. Second, Shtan'ko's embedding approximation theorem is extended to a relative version for non-locally compact \(\sigma\)-compact subsets of topological manifolds. This is technically a very difficult and deep extension. It is a pleasure to welcome this long awaited paper.
embedding of a compactum into Euclidean space
non-locally compact \(\sigma \)-compact subsets of topological manifolds
scientific article; zbMATH DE number 3975891
scientific article