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Some m-dimensional compacta admitting a dense set of imbeddings into \(R^{2m}\). (English) Zbl 0715.54026

Consider the following question: Which compacta X have the property that every map \(f:X\to {\mathbb{R}}^ n\) of X into the Euclidean n-space can be approximated by an embedding? If we denote by \({\mathcal C}(X,{\mathbb{R}}^ n)\) the space of all continuous maps of X into \({\mathbb{R}}^ n\), equipped by the standard “sup-norm” metric \(\rho (f,g)=\sup \{d(f(x),g(x))| \quad x\in X\},\) and denote by \({\mathcal E}(X,{\mathbb{R}}^ n)\) the subspace of \({\mathcal C}(X,{\mathbb{R}}^ n)\), consisting of all embeddings of X into \({\mathbb{R}}^ n\), then the question above can be restated as follows: which compacta X have the property that \({\mathcal E}(X,{\mathbb{R}}^ n)\) is dense in \({\mathcal C}(X,{\mathbb{R}}^ n)\). It follows by the classical Nöbeling- Pontryagin embedding theorem that a sufficient condition for X is that \(\dim X<n/2\). Clearly, this condition is also necessary in the case when n is odd. The authors [ibid. 116, 131-142 (1983; Zbl 0552.55003)] published a theorem, asserting that the condition \(\dim X<n/2\) is also necessary in the case when n is even. However, some time later, J. Krasinkiewicz and K. Lorentz [Disjoint membranes in cubes, Bull. Pol. Acad. Sci., Math. 36, 397-402 (1988)] found a gap in the proof of one of their crucial lemmas (but did not determine whether the main result was incorrect).
In the paper under review the authors themselves constructed a counterexample, i.e. for each \(n\geq 2\) they give an example of an n- dimensional compactum X such that \({\mathcal E}(X,{\mathbb{R}}^{2n})\) is dense in \({\mathcal C}(n,{\mathbb{R}}^{2n}).\)
Reviewer’s remark: The authors’ example turns out to possess the property that \(\dim(X\times X)<2n\). This inspired the following theorem, which is independently due to A. N. Dranishnikov, D. Repovš and E. V. Shchepin [On intersections of compacta of complementary dimensions in Euclidean space, Topology Appl. 38, 237-253 (1991)] and to J. Krasinkiewicz and S. Spież [Fund. Math. 133, No.3, 247- 253 (1989); 134, No.2, 105-115 (1990); see the following reviews)]: Let X be a compactum and suppose that \(n=2\dim X\). Then \({\mathcal E}(X,{\mathbb{R}}^ n)\) is dense in \({\mathcal C}(X,{\mathbb{R}}^ n)\) if and only if \(\dim(X\times X)<n\). More recently it was shown by A. N. Dranishnikov and J. E. West that the condition \(n=2\dim X\) is, in fact, unnecessary [On compacta that intersect unstably in Euclidean space, Cornell University, Ithaca, preprint]. On the other hand, J. Luukkainen has generalized the \(n=2\dim X\) case from compacta X to the class of locally compact separable metric spaces, and from \({\mathbb{R}}^ n\) to the class of topological n-manifolds [Embeddings of n-dimensional locally compact metric spaces to 2n-manifolds, University of Helsinki, preprint].)
Reviewer: D.Repovš

MSC:

54F45 Dimension theory in general topology
54C25 Embedding
54E45 Compact (locally compact) metric spaces
54G20 Counterexamples in general topology
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