Resolving acyclic images of three-manifolds

Abstract

Our main result is that a locally contractible Z2-acyclic image of a 3-manifold without boundary is the cell-like image of a 3-manifold without boundary (all mappings being proper). Consequently, such images are generalized 3-manifolds (that is, finite-dimensional and Z-homology 3-manifolds). A refinement of the proof allows the omission of the Z2-acyclicity hypothesis over a zero-dimensional set; an application is the result of D. R. McMillan, Jr., to the effect that a generalized 3-manifold with compact zero-dimensional singular set admits a resolution if and only if a deleted neighbourhood of the singular set embeds in a compact, orientable 3-manifold.