Resolutions of generalized 3-manifolds whose singular sets have general position dimension one

We show that, modulo the classical Poincaré Conjecture, a closed generalized 3-manifold X is the cell-like image of a closed 3-manifold (i.e., is “resolvable”) if it satisfies the following condition: any map of a disk into X can be approximated by one whose image meets the singular set of X in a zero-dimensional set.