On open 3-manifolds proper homotopy equivalent to geometrically simply-connected polyhedra (Q5925880)

Generalizing a result of Poénaru, it is shown that an open 3-manifold which is proper homotopy equivalent to a geometrically simply connected polyhedron (i.e. the polyhedron can be exhausted by compact simply connected subpolyhedra) is simply connected at infinity. The result is not true for open contractible manifolds of dimension greater than three due to the existence of compact contractible \(n\)-manifolds with non-simply connected boundary, for \(n>3\). Poénaru obtained the above conclusion for an open simply connected 3-manifold \(U\) whose product \(U \times D^n\) with some \(n\)-ball admits a handlebody decomposition without 1-handles (which implies that the projection from \(U \times D^n\) to \(U\) is a proper simple homotopy equivalence).