This paper investigates the
statement (GM): “If X is a compact generalized 3-manifold without boundary, and
whose nonmanifold set is 0-dimensional, then X is the cell-like image of some
closed 3-manifold.” Some necessary and some sufficient conditions on X are
given for (GM) to be true. The question of whether (GM) is true in general
is shown to be inextricably tangled with the Poincaré Conjecture: (1) If
the Poincaré Conjecture fails, then there is an acyclic, monotone union
M of handlebodies whose one-point compactification M is a generalized
3-manifold, yet M is not the cell-like image of any compact 3-manifold. (2)
If X is a compact generalized 3-manifold with zero-dimensional singular
set S and no π1-torsion in any sufficiently tight neighborhood of S, then
(modulo the Poincaré Conjecture) X is the cell-like image of a compact
3-manifold.
