In modern condensed matter physics, one is often interested in the homotopy classes of mappings from a $d$-dimensional torus $$\mathbb{T}^d=\underbrace{S^1\times\ldots \times S^1}_d$$ (corresponding to the Brillouin zone of a $d$-dimensional system) to various topological spaces $Y$, for example the Grassmannians $$O(m+n)/O(m)\times O(n) \qquad \textrm{or} \qquad U(m+n)/U(m)\times U(n) $$ or some other symmetric spaces $-$ see for example Eq. (4) of this publication. Such considerations lie at the hearth of the discussion of all topological phases of non-interacting particles, like the quantum Hall effect, Weyl semimetals or topological superconductors.

Unfortunately, most texts on the topic are rather sloppy and without much explanation replace the torus $\mathbb{T}^d$ by a sphere $S^d$, thus instead considering the homotopy groups $\pi_d(Y)$.

Hence the following questions: Is there any simple relation between the set of homotopy classes of mappings from $T^d$ to a general topological space $Y$ and the homotopy groups $\pi_d(Y)$? If the relation is in general too complex, under what conditions on $Y$ does it simplify? (For example, if $Y$ is a symmetric space?) Is the order of the two homotopy classes related?