  We study a topological obstruction of a very stringy nature concerned with
deforming the target space of an $N=2$ non-linear \sm. This target space has a
singularity which may be smoothed away according to the conventional rules of
geometry but when one studies the associated conformal field theory one sees
that such a deformation is not possible without a discontinuous change in some
of the correlation functions. This obstruction appears to come from torsion in
the homology of the target space (which is seen by deforming the theory by an
irrelevant operator). We discuss the link between this phenomenon and orbifolds
with discrete torsion as studied by Vafa and Witten.
