On the deleted product criterion for embeddability of manifolds in \(\mathbb{R}^m\)
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 \textit{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 \textit{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 \(m\geq 6+d\).
scientific article; zbMATH DE number 1129397
scientific article