Obstructions to approximating maps of n-manifolds into R2n by embeddings

We prove a criterion for approximability by embeddings in R2n of a general position map f :K→R2n−1 from a closed n-manifold (for n⩾3). This approximability turns out to be equivalent to the property that f is a projected embedding, i.e., there is an embedding f̄ :K→R2n such that f=π∘f̄, where π :R2n→R2n−1 is the canonical projection. We prove that for n=2, the obstruction modulo 2 to the existence of such a map f̄ is a product of Arf-invariants of certain quadratic forms.