Embeddings of polyhedra in Rm and the deleted product obstruction

Weber has proved that if 2m ⩾ 3(n + 1) then an n-dimensional polyhedron K embeds in Rm if and only if there exists an equivariant map from the deleted product K∗ into the sphere Sm − 1. As a consequence he has obtained that in the same range of dimensions an n-dimensional polyhedron embeds in Rm if and only if it quasi embeds in Rm. We show that for m ⩾ max(4, n) the dimension restrictions in Weberʹs results are necessary in all cases. This leaves only two open cases remaining (namely m = 3 and n = 2 or 3) in related questions about embeddings.