Embeddings of polyhedra in Rm and the deleted product obstruction

Weber has proved that if 2m greater than or equal to 3(n + 1) then an n-dimensional polyhedron K embeds in R-m 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 R-m if and only if it quasi embeds in R-m. We show that for m greater than or equal to max(4, n) the dimension restrictions in Weber's results are necessary in all cases. This leaves only two open cases remaining (namely m = 3 and n = 2 or 3) in related questions al,out embeddings. (C) 1998 Elsevier Science B.V.