Boundary winding representation of Yang-Mills topological charge ($\frac{1}{8{\pi }^2}\int \mathrm{tr}(F\wedge F)=\mathrm{deg}g$)

For an anti-Hermitian SU(2) connection on ${\mathbb{R}}^4$ tending to $g^{-1}dg$ at infinity, the Chern-Simons 3-form and Maurer-Cartan equation give $(8{\pi }^2{)}^{-1}\int \mathrm{tr}(F\wedge F)=-(24{\pi }^2{)}^{-1}{\int }_{S^3}\mathrm{tr}(g^{-1}dg{)}^3=\mathrm{deg}g$. This is the second-Chern convention with the boundary orientation. Defining instanton charge with the opposite trace sign reverses this integer.