Approximating resolutions by cell-like maps with codimension-three point inverses

Let X be a resolvable generalized n-manifold and M be a (PL) n-manifold, where n≥6. We claim every resolution from M to X can be approximated, arbitrarily closely, by cell-like maps with (n−3)-dimensional point preimages. The key instrument of the proof is the disjoint Pontryagin disks property.