Maps of manifolds into the plane which lift to standard embeddings in codimension two

Let f :M→R2 be a smooth map of a closed n-dimensional manifold (n⩾2) into the plane and let πn+22 :Rn+2→R2 be an orthogonal projection. We say that f has the standard lifting property, if every embedding f̃ :M→Rn+2 with πn+22∘f̃=f is standard in a certain sense. In this paper we give some sufficient conditions for a generic smooth map f to have the standard lifting property when M is a closed surface or an n-dimensional homotopy sphere.