Homotopy Theory of Topological Defects in Spinor Condensates

We investigate the topological defects in atomic spin-1 and spin-2 Bose-Einstein condensates by applying the homotopy group theory. With this rigorous approach we clarify the previously controversial identification of symmetry groups and order parameter spaces for the spin-1 case, and show that the spin-2 case provides a rare example of a physical system with non-Abelian line defects, and the possibility to have winding numbers of 1/3 and its multiples.