Instanton number as a winding number at infinity
= Instanton number as a winding number at infinity
For a finite-action $SU(2)$ connection on $\mathbb R^4$ that approaches $A=-dg\,g^{-1}$ at infinity,
$$
k=\frac1{24\pi^2}\int_{S^3_\infty}
\operatorname{Tr}\left[(dg\,g^{-1})^{\wedge3}\right]
$$
under the convention $k=(8\pi^2)^{-1}\int\operatorname{Tr}(F\wedge F)$. This integer is the <degree of a map between oriented manifolds> $g:S^3_\infty\to SU(2)\cong S^3$; orientation and trace conventions may reverse its sign.