  The Topological Hypothesis states that phase transitions should be related to
changes in the topology of configuration space. The necessity of such changes
has already been demonstrated. We characterize exactly the topology of the
configuration space of the short range Berlin-Kac spherical model, for spins
lying in hypercubic lattices of dimension d. We find a continuum of changes in
the topology and also a finite number of discontinuities in some topological
functions. We show however that these discontinuities do not coincide with the
phase transitions which happen for d >= 3, and conversely, that no topological
discontinuity can be associated to them. This is the first short range,
confining potential for which the existence of special topological changes are
shown not to be sufficient to infer the occurrence of a phase transition.
