Space of kink solutions in SU(N)×Z₂

We find (N + 1)/2 distinct classes ("generations" of kink solutions in an SU(N)×Z₂ field theory. The classes are labeled by an integer q. The members of one class of kinks will be globally stable while those of the other classes may be locally stable or unstable. The kink solutions in the q^th class have a continuous degeneracy given by the manifold Σ_q=H/K_q, where H is the unbroken symmetry group and K_qCH is the group under which the kink solution remains invariant. The space Σ_q is found to contain two incontractable spheres for some values of q, indicating the possible existence of certain incontractable spherical structures in three dimensions. We explicitly construct the three classes of kinks in an SU(5) model with a quartic potential and discuss the extension of these ideas to magnetic monopole solutions in the model.