  We apply the homotopy group theory in classifying the topological defects in
atomic spin-1 and spin-2 Bose-Einstein condensates. The nature of the defects
depends crucially on the spin-spin interaction between the atoms. We find the
topologically stable defects both for spin-1 ferromagnetic and
anti-ferromagnetic states, and for spin-2 ferromagnetic and cyclic states. With
this rigorous approach we clarify the previously controversial identification
of symmetry groups and order parameter spaces for the spin-1 anti-ferromagnetic
state, and show that the spin-2 cyclic case provides a rare example of a
physical system with non-Abelian line defects, like those observed in biaxial
nematics. We also show the possibility to produce vortices with fractional
winding numbers of 1/2, 1/3 and their multiples in spinor condensates.
