The four-dimensional $SU(N)$ Yang-Mills Lagrangian is given by $$\mathcal{L}=\frac{1}{2e^2}\mathrm{Tr}F_{\mu\nu}F^{\mu\nu}$$

and gives rise to the Euclidean equations of motion $\mathcal{D}_\mu F^{\mu\nu}=0$ with covariant derivative $\mathcal{D}_\mu$. Finite action solutions $A_\mu$ satisfy the condition that,

$$A \to ig^{-1}\partial_\mu g$$

as we approach $\partial \mathbb{R}^4 \cong \mathbb{S}^3_{\infty}$, with $g$ an element of $SU(N)$. These provide a map from $\mathbb{S}^3_\infty$ to $SU(N)$, and are classified by homotopy theory. In Tong's lecture notes on solitons, he states without proof that the second Chern class, or Pontryagin number $k \in \mathbb{Z}$ is given by

$$k = \frac{1}{24\pi^2}\int_{S^3_\infty}\mathrm{d}S^3_\mu \, \, \mathrm{Tr} \,(\partial_\nu g)g^{-1}(\partial_\rho g)g^{-1} (\partial_\sigma g)g^{-1}\epsilon^{\mu \nu \rho \sigma}$$

As I understand them, Chern classes are characteristic classes of bundles on manifolds; in this case what bundle and manifold is $k$ associated with? Tong states "The integer... counts how many times the group wraps itself around spatial $\mathbb{S}^3_\infty$."

In addition, Tong states without rigorous proof the metric of the moduli space (the space of all solutions to the equations of motion which are self-dual):

$$g_{\alpha \beta} = \frac{1}{2e^2} \int \mathrm{d}^4 x \mathrm{Tr} \, \, (\delta_\alpha A_\mu)(\delta_\beta A_\mu)$$

with $\delta_\alpha A_\mu = \partial A_\mu / \partial X^{\alpha} + \mathcal{D}_\mu \Omega_\alpha$ where $\Omega_\alpha$ is an infinitesimal transformation, and $X^{\alpha}$ are the collective coordinates. How does one compute such a metric of a moduli space? Why should it be given by the sum of all zero modes?

I would prefer an answer which utilizes arguments from differential geometry, and topology. Resource recommendations which are more rigorous or explicit would also be appreciated.

(Tong's notes may be found here: http://www.damtp.cam.ac.uk/user/tong/tasi/tasi.pdf).