Topological meaning of the integral of the trace of Cartan-Maurer forms in Anthony Zee's book on QFT in a nutshell

I learnt in S. Sternberg’s book "Curvature in Mathematics & Physics" over the Maurer-Cartan form that if there is a tangent vector $v in TG_g$ at a point $g in G$ ($G$ is supposed to be a Lie-group) there is a unique left-invariant vector field $X$ such that $X(g) =v$. Then the Maurer-Cartan form is the linear map

$$ omega_a: quad TG_a rightarrow mathfrak{g} $$

that is sending the tangent vector $v$ to the Lie-algebra element $xi = omega_a(v) in mathfrak{small{g}}$. Or in other words:

$$omega(v) = (L_g)^{-1}_ast v$$

where $L_g$ means left multiplication by $gin G$.

This is all pure math, but I always wondered about the possible role of Maurer-Cartan form in physics. On the Physics SE there indeed exists a post which poses exactly this question
Maurer-Cartan form in Physics which is very general.

In the meantime I found a concrete example of its use in physics in Anthony Zee’s book on "QFT in a Nutshell" (p.225, p.288, p.497) where he proposes to calculate the integral over $S^3$ or even $S^N$:

$$ Q:= int_{S^3} mathrm{tr}left[ prod_1^3 g dg^daggerright]$$

where $omega = g dg^dagger$ is another representation of the Maurer-Cartan form. He explains further on that the Maurer-Cartan form only needs to be evaluated in the neighbourhood of the identity element of $G$. So if the a group element is represented by $g = exp(itheta^a t^a)$ with $t^a in mathfrak{g}$ we can write taking into account that $g approx e$ in the neighbourhood of the identity element:

$$ g dg^{dagger} =-i dtheta^a t^{a} $$

If we then limit ourselves to the well-known group $mathrm{SU}(2)$ and its Lie-algebra we get ($t^a = sigma^a$, the Pauli-matrices):

$$Q:= int_{S^3} mathrm{tr}left[ prod_1^3 g dg^dagger right] = iint_{S^3} mathrm{tr}left[sigma^isigma^jsigma^kright] dtheta^i dtheta^j dtheta^k = -12 int_{S^3} dtheta^1 dtheta^2 dtheta^3 cong -12 cdot 2pi^2 $$

What can I learn from this – apart from the nice mathematical exercise – in particular what is the topological meaning of $Q$? Zee poses this question in the chapter on "Magnetic Monopoles", so I guess, it should be somewhat related with that. I hope that this question is not only of mathematical interest, but also of significance in physics as Lee proposes.

I would really appreciate if I got to know it.