A deleted product criterion for approximability of maps by embeddings

We prove the following theorem: Suppose that m ⩾ 3(n + 1)2 and that ƒ : K → Rm is a PL map of an n-dimensional finite polyhedron K. Then ƒ is approximable by embeddings if and only if there exists an equivariant homotopical extension Φ : K̃ → Sm−1 of the map \̃tf : K̃ƒ → Sm−1, defined by f̃(x,y) = (ƒ(x) − ƒ(y))(∥ƒ(x) − ƒ(y)∥), where K̃ƒ = {(x, y) ε K × K ¦ ƒ(x) ≠ ƒ(y)}. Our result is a controlled version of the classical deleted product criterion of embeddability of n-dimensional polyhedra in Rm. The proof requires additional (compared with the classical result) general position arguments, for which the restriction m ⩾ 3(n + l)2 is again necessary. We also introduce the van Kampen obstruction for approximability by embeddings.