2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/221992We continue our study of ends non-compact manifolds. The over-arching aim is to provide an appropriate generalization of Siebenmann's famous collaring theorem that applies to manifolds having non-stable fundamental group systems at infinity. In this paper a primary goal is finally achieved; namely, a complete characterization of pseudo-collarability for manifolds of dimension at least 6.This is the version published by Geometry & Topology on 27 April 2006Geometric Topology57N15, 57Q12, 57Q10, 57R65text