Two expressions for instanton winding number integral Suppose the winding number $n$ of the Yang-Mills instanton configuration. It ia given by the expression
$$
\tag 1 n = \frac{1}{16\pi^2}\int \limits_{S^4} d^{4}x\text{tr}\big[F_{ij}\tilde{F}^{ij}\big],
$$
where the integration is over 4-sphere $S^4$. I've met two different approaches of calculation of the right hand-side of $(1)$.
Approach 1 - the transition function
The sphere $S^4$ is divided on two semispheres $H^+, H^-$, whose manifolds are 
$$
\partial H^+ = S^3, \ \partial H^- = -S^3
$$
Next, let's use the identity $F\tilde{F} = dK$ on each semisphere. We have definite local bundle patches 
$$
H^{+}\times SU(n), \ H^{-}\times SU(n) \ \ \text{with coordinates} \ \ \{x,f_{+}\},\{x,f_-\}
$$
Along the equatorial intersection $H^{+}\cap H^{-}= S^3$ the coordinates $f_{\pm}$are related by the transition function $g_{-+}$
$$
f_+ = g_{-+} f_-
$$
By using the formalism above, the integral $(1)$ can be rewritten in the form
$$
\tag 2 n = \int \limits_{H^+}dK_{+} + \int \limits_{H^-}dK_{-} = \int \limits_{S^3}\text{tr}\big[(g_{-+}\partial g_{-+}^{-1})^3\big]
$$
Approach 2 - the gauge $A_0 = 0$
One can fix the gauge $A_0 = 0$. Then at unfinities $t\to \pm \infty$ the 4-potential $A_i$ tends to pure gauge 
$$
A_{i}(\mathbf x , t \to \pm \infty) = g_{\pm}\partial_i {g}_{\pm}^{-1},
$$
and on the spatial boundaries $A_{i}(\mathbf x \to \infty, t) = 0$. Then (see the comment section)
$$
\tag 3 n = \int d\sigma_{\mu}K^{\mu} = \int\limits_{S^3}\text{tr}\big[(g_{+}\partial g^{-1}_{+})^{3}\big] - \int\limits_{S^3}\text{tr}\big[(g_{-}\partial g^{-1}_{-})^{3}\big]
$$
The question. How these approaches are related to each other? Precisely, it seems that $(2)$ is obtained in gauge fixing independent way, while $(3)$ is obtained in the fixed gauge. Therefore, is $(2)$ reduced to $(3)$ in the temporary gauge $A_{0} = 0$? Or these expressions aren't equivalent?
 A: The expressions eq. (2) and eq. (3) are both correct, but not equivalent, because they are given in different settings:


*

*Eq. (2) is the winding number on a sphere $S^4$, i.e. the one-point compactification of Minkowski space $\mathbb{R}^{1,3}$. The $g_{+-}$ is a transition function corresponding to the single choice necessary to glue the local gauge potentials on the hemispheres to a global gauge potential on $S^4$. One should note that, technically, the cocycle construction of the principal bundle would require the intersection of the hemispheres to be open in $S^4$, i.e. they would have to overlap on an $S^3\times(-\epsilon,\epsilon)$ for $\epsilon > 0$. Eq. (2) is what results at $\epsilon\to 0$, at which point we do not technically fulfill the conditions for the cocycle construction anymore, yet the value of $n$ remains unchanged because its dependence on $\epsilon$ is smooth and $n$ takes only discrete values, so it is constant in $\epsilon$.

*Eq, (3) is the winding number of an instantonic field configuration on a cylinder $S^3\times[-T,T]$ for $T > 0$. Here, $g_+$ and $g_-$ are gauge functions such that $A(\pm T) = g_\pm^{-1}\mathrm{d} g_\pm$ (again, the physicist likes to take the limit $T\to\infty$). The crucial difference is that the $g_\pm$ are now no transition functions, but related to the gauge potential value. Also, we do not need to take the temporal gauge here, we can obtain eq. (3) without any gauge fixing at all:
$$ n \propto \int_{S^3\times [-T,T]} F\wedge F = \int_{S^3\times [-T,T]} \mathrm{d}K = \int_{S^3\times\{-T\}}K(g_-) - \int_{S^3\times\{T\}}K(g_+)$$
where the sign appears because the $S^n$ at the end of a cylinder are oppositely oriented and $K(g_\pm)$ denotes the Chern-Simons 3-form for the respective gauge field configuration $A_\pm$.
