Relation between the homotopy classes of maps on a torus, and maps on a sphere

Question

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?

Answer 1

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

Answer 2

It would be interesting if the paper "Homotopy Groups and Torus Homotopy Groups", Ralph H. Fox, Annals of Mathematics, Second Series, Vol. 49, No. 2 (Apr., 1948), pp. 471-510, could be seen as relevant to aspects of physics!

Actually I was told by Brian Griffiths that Fox was looking for higher order versions of the van Kampen theorem for the fundamental group but did not achieve that. That aim has been achieved by working on higher versions of groupoids.

I have also remembered the relevany paper "Generalizations of Fox homotopy groups, Whitehead products, and Gottlieb groups", M Golasinski, D Gonçalves, P Wong - Ukrainian Mathematical Journal, 2005 - Springer.