Topological Symmetry Breakdown in Cholesterics, Nematics, andHe3

Cholesterics, uniaxial and biaxial nematics, and the dipole-free $A$ phase of superfluid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ are characterized by order parameters which are left invariant by suitable "symmetry" groups $H$. We show that in the presence of defects, the full group $H$ may not be implementable on the states because of topological obstructions. Thus $H$ is topologically broken in the presence of suitable defects.