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](http://aip.scitation.org/doi/pdf/10.1063/1.3149481). 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?



Relation between the homotopy classes of maps on a torus, and maps on a sphere
One case in which you can establish a simple relationship is when $Y$ is a loop space. Suppose that $Y\simeq \Omega Z=\mbox{map}\_\*(S^1, Z)$. Then there is a bijection $[{\mathbb T}^d, Y]\_\*\cong [\Sigma{\mathbb T}^d, Z]\_\*$. On the other hand, there is an equivalence $$\Sigma{\mathbb T}^d \simeq \bigvee\_{i\ge 1} \bigvee\_{d\choose i} S^{i+1}.$$


This follows, using induction, from the equivalence $\Sigma X\times Y\simeq \Sigma X\vee \Sigma Y \vee \Sigma X\wedge Y$ (Proposition 4I.1 in Hatcher's book). It follows that there is a bijection


$$[{\mathbb T}^d, Y]\_\*\cong [\bigvee\_{1\le i\le d} \bigvee\_{d\choose i} S^{i}, Y]\_\*\cong \prod\_{i=1}^d\prod\_{d\choose i} \pi\_i(Y).$$


It is easy to extend this calculation to unpointed homotopy classes of maps, because $[{\mathbb T}^d, Y]\cong [{\mathbb T}^d\_+, Y]\_\*$ and there is an equivalence $\Sigma {\mathbb T}^d\_+\simeq \Sigma {\mathbb T}^d \vee S^1$.


Homogeneous spaces are usually not loop spaces, but for example topological groups are loop spaces, including the orthogonal and the unitary groups.


By the way, this answer on math.SE has a very nice description of the set $[{\mathbb T}^2, X]$ for a general space $X$. <https://math.stackexchange.com/questions/36488/how-to-compute-homotopy-classes-of-maps-on-the-2-torus>


