Approximating maps of 2-manifolds with zero-dimensional nondegeneracy sets (Q1193418)

Let \(f\) be a map of a 2-manifold \(M^ 2\) to a 3-manifold \(M^ 3\) such that \(N_ f=\{x\in M^ 2: f^{-1}f(x)\neq x\}\) is 0-dimensional. This paper attacks the conjecture that \(f\) can be approximated by an embedding, confirming it when \(N_ f\) is closed and otherwise establishing its equivalence with two related conjectures. The key contribution is a complicated but useful and original technical device which shows, among other things, that such a map \(f\) can be approximated by a map having point preimages of arbitrarily small diameter.