Vortices, disclinations, and homotopy classification
Topological defects are classified by the fundamental homotopy group π₁(M) of the order-parameter manifold M. XY spins have M=S¹, so π₁(S¹)=ℤ — vortices with integer winding numbers. Nematic liquid crystals have M=RP²=S²/ℤ₂, giving π₁(RP²)=ℤ₂ — allowing ±1/2 disclinations. The defect charge is a topological invariant: it cannot be removed by local fluctuations.