  We construct clusters of classical Heisenberg spins with two-spin
$\vec{S}_i.\vec{S}_j$-type interactions for which the ground state manifold
consists of disconnected pieces. We extend the construction to lattices and
couplings for which the ground state manifold splits into an exponentially
large number of disconnected pieces at a sharp point as the interaction
strengths are varied with respect to each other. In one such lattice we
construct, the number of disconnected pieces in the ground state manifold can
be counted exactly.
