  The fragmentation of the ground state of a repulsive condensate immersed into
a double-trap potential is found to be a general and critical phenomenon. It
takes place for a given number of bosons if their scattering length is larger
than some critical value or for a given value of the scattering length if the
number of bosons is above some critical number. We demonstrate that the
geometry of the inner trap determines these critical parameters while the
number of the fragments and the fraction of bosons in the various fragments can
be manipulated by the outer trap. There is also a maximal number of bosons for
which the ground state is fragmented. If this number is exceeded, the
fragmented state becomes a very low-lying excited state of the condensate. This
maximal number of bosons can be substantially manipulated by varying the inner
and outer traps. To study three-fold fragmentation we have chosen a potential
well with two barriers as the inner trap and embedded by two types of outer
ones. A many-fold fragmentation is also addressed.
