"Homotopy" group with the torus $\mathbb{T}^2$ as a domain and the sphere $\mathbb{S}^2$ as codomain

Question

I'm watching this lecture in Condensed Matter physics that concerns topological aspects of materials. In particular, the lecturer is considering the homotopy groups of various spaces. At one particular point (around 50 min), he claims that the classification of maps from the torus $\mathbb{T}^2$ to the sphere $\mathbb{S}^2$ is the same as the classification from the sphere $\mathbb{S}^2$ to itself. I understand the latter concept is the homotopy group $\pi_2(\mathbb{S}^2) = \mathbb{Z}$. However, I don't know what exactly is the analogous classification for a torus: I understand the basic idea, but I don't know the terminology nor where to look for a proof of this statement.

My question then boils down to: is there a terminology for this sort of classification? Where can I find a proof of this claim?

Let me add that I have a very poor background in algebraic topology, although I'm somewhat comfortable with groups and topology separately.

Answer 1

I think you can take two loops corresponding to generators of $T^2$ (the 1 cells in the CW decomposition), since $\pi_1$ of $S^2$ is trivial, the images of these loops can be homotoped to a point. Then, homotopy extension property says this homotopy extends to $T^2$ so every map $T^2$ to $S^2$ can be homotoped to a map from $T^2/$circles to $S^2$, but $T^2/$circles is $S^2$.

Answer 2

The Hopf Degree Theorem states that if $X$ is a compact, oriented $n$-dimensional manifold, then two (continuous) maps $X\to S^n$ are homotopic if and only if they have the same degree.