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On the deleted product criterion for embeddability of manifolds in \(\mathbb{R}^m\). (English) Zbl 0905.57015

For any space \(N\), let \(\widetilde N\) denote the deleted square of \(N\), i.e. the set of all \((x,y)\) in \(N\times N\) with \(x\neq y\). For every embedding \(f: N\to{\mathbb R}^m\) we can define a map \(\widetilde f :\widetilde N\to S^{m-1}\) by \(\widetilde f(x,y)=\bigl(f(x)-f(y)\bigr)\big/\big\| f(x)-f(y)\big\| \), and this map is equivariant with respect to the coordinate switching involution of \(\widetilde N\) and the antipodal involution of \(S^{m-1}\). Similarly, if \(\{f_t: N\to{\mathbb R}^m\mid 0\leq t\leq 1\}\) is an isotopy of embeddings, then \(\{\widetilde f_t\mid 0\leq t\leq 1\}\) is an equivariant homotopy. Under suitable restrictions one can prove a converse: if there exists an equivariant map \(\widetilde N\to S^{m-1}\), then \(N\) embeds in \({\mathbb R}^m\), and if \(f,g: N\to{\mathbb R}^m\) are embeddings such that \(\widetilde f\) and \(\widetilde g\) are equivariantly homotopic, then \(f\) and \(g\) are isotopic. A theorem of this kind was first proved by A. Haefliger [ibid. 37, 155-176 (1962; Zbl 0186.27302)] for the smooth category, i.e. for \(N\) a compact smooth manifold and for smooth embeddings and isotopies, and later by C. Weber [ibid. 42, 1-27 (1967; Zbl 0152.22402)] for the PL category. Their common dimensional restriction was \(m\geq 3(n+1)/2\) for the embedding part and \(m > 3(n+1)/2\) for the isotopy part, where \(n=\dim N\). The bound \(3(n+1)/2\) is sharp in general, but in the present paper the author shows that it can be relaxed in presence of connectivity conditions on \(N\): he proves that if \(N\) is a closed, \(d\)-connected PL \(n\)-manifold with \(d=0,1,2\), then the embedding part of Weber’s theorem remains true if \(m\geq(3n+2-d)/2\) and the isotopy part remains true if \(m\geq(3n+3-d)/2\) and \(m\geq 6+d\).

MSC:

57Q35 Embeddings and immersions in PL-topology
57Q37 Isotopy in PL-topology
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