A discrete formulation for three-dimensional winding number

For a smooth map g X → U(N), where X is a three-dimensional, oriented, and closed manifold, the winding number or the map's degree is defined by W₃ = 1/24π² ∫X Tr[(g⁻¹dg)³]. We introduce a method to compute W₃ using a discrete approximation of X. in which the result is manifestly This approach provides a formulation in which quantization is manifest.