Quasi embeddings and embeddings of polyhedra in \(\mathbb{R}{}^ m\)
Let \(X\) be a metric space and \(\varepsilon>0\). A map \(f: X\to Y\) is said to be an \(\varepsilon\)-map if for each \(y\in f(X)\), \(\text{diam }f^{- 1}(y)<\varepsilon\). If for each \(\varepsilon>0\) there is an \(\varepsilon\)-map of \(X\) into \(Y\) then \(X\) is said to be quasi-embeddable in \(Y\). For each pair \((m,k)\), \(m\geq k\), of positive integers, one can ask the following question: Does every \(k\)-dimensional compact polyhedron which quasi-embeds in \(\mathbb{R}^ m\), also embed in \(\mathbb{R}^ m\)? This problem has a long history. Among the best results so far is a theorem of \textit{C. Weber} [\(\varepsilon\)-applications dans une variété, Comment. Math. Helv. 49, 125-135 (1974; Zbl 0291.54011)] asserting that the question above has an affirmative answer for \(2m\geq 3(k+1)\). Weber's theorem has certain dimension restrictions due to the use of the Freudenthal suspension isomorphism theorem and general position problems. The main purpose of the paper under review is to show that Weber's restrictions are necessary in all but a finite number of cases. This is done by constructing a particular example of a polyedron, using the higher-dimensional analogue of the classical Casson finger moves, which were first introduced by \textit{A. N. Dranishnikov}, \textit{E. V. Shchepin} and the reviewer in [On intersections of compacta in Euclidean space: The metastable case, Tsukuba J. Math. (in print)].
scientific article; zbMATH DE number 90163
scientific article