I wonder if there is any relation between the homotopy classes of map $S^n \to S^2$, and the homotopy classes of map $T^n \to S^2$, where $n=2,3$.
In particular, I want to understand the below quotes from the physics paper https://arxiv.org/abs/1407.3427 :
The classification of the map $T^2 \to S^2$ is equivalent to the homotopy group $\pi_2(S^2)$ classifying the map $S^2\to S^2$, because the nontrivial cycles on the torus do not affect the homotopy classes of the map. It is well known that the winding number of this map is the same as the first Chern number.
However, there is a subtle difference between the map $T^3 \to S^2$ and the map $S^3 \to S^2$. The maps from $T^3\to S^2$ is more complicated than the maps $S^3\to S^2$, because there are 3 independent 2D torus inside $T^3$ and each of them may has nonzero Chern numbers. For $\mu = x,y,z$, let $C_\mu$ be the Chern number when taking $k_\mu$ to be constant. It is pointed out by Pontryagin that if $C_\mu \neq 0$, then the Hopf index, defined as $$\mathcal H =\frac{1}{32\pi^2} \int_{T^3} A \wedge dA,$$ (I will not quote the definition of $A$) is no longer integer-valued and take values in the finite group $\mathbb Z_{2\times \mathrm{gcd}(C_x, C_y, C_z)}$.
Could you explain why the bold part holds? Also, (mathematical) references on this topic is appreciated. I have basic knowledge of algebraic topology and differential geometry.