The second Chern class in Spinning System

Topological property in a spinning system should be directly associated with its wavefunction. A complete decomposition formula of SU(2) gauge potential in terms of spinning wavefunction is established rigorously. Based on the $\phi $-mapping theory and this formula, one proves that the second Chern class is inherent in the spinning system. It is showed that this topological invariant is only determined by the Hopf index and Brouwer degree of the spinning wavefunction.