2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/59324One proves that there exists an obstruction to an open simply connected $n$-manifold of dimension $n\geq 5$ being geometrically simply connected. In particular there exist uncountably many simply connected $n$-manifolds which are not w.g.s.c. One proves that for $n\neq 4$ an $n$-manifold proper homotopy equivalent to a w.g.s.c. polyhedron is w.g.s.c. (for $n=4$ it is only end compressible). We analyze further the case $n=4$ and Poénaru's conjecture.48 pages, one eps figure, to appear IMRNGeometric Topology57R65, 57Q35, 57M35text