Embeddings of polyhedra in \(\mathbb R^m\) and the deleted product obstruction
Weber has proved that if \(2m>3(n+1)\) then an \(n\)-dimensional polyhedron \(K\) embeds in \(\mathbb R^m\) if and only if there exists an equivariant map from the deleted product \(K^*\) into the sphere \(S^{m-1}\). As a consequence he has obtained that in the same range of dimension an \(n\)-dimensional polyhedron embeds in \(\mathbb R^m\) if and only if it quasi-embeds in \(\mathbb R^m\). The authors prove that for any pair of integers \((m,n)\) such that \(m\geq \max(4,n)\) and \(2m<3(n+1)\) there exists an \(n\)-dimensional polyhedron \(R\) which is quasi embeddable in \(\mathbb R^m\) but which is not embeddable in \(\mathbb R^m\). The only two cases remaining open are for \(m=3\) and \(n=2,3\). We say that \(X\) quasi embeds in \(\mathbb R^m\) if for each \(\varepsilon >0\) there exists an \(\varepsilon\)-map from \(X\) onto some subspace in \(\mathbb R^m\). If \(X\) is a polyhedron this definition is equivalent to saying that for each triangulation of \(X\) there exists an almost-embedding of \(X\) in \(\mathbb R^m\).
scientific article; zbMATH DE number 1308006
scientific article