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?

  • $\begingroup$ The torus is an extension of a sphere by a wedge of circles, $\bigvee S^1 \to T^d \to S^d$ so you get an exact sequence $\prod \pi_2Y \to \pi_dY \to [T^d, Y] \to \prod \pi_1Y$. Is this what you meant by "sloppily" replacing the torus with a sphere? $\endgroup$ – Dylan Wilson Jan 6 '17 at 21:25
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    $\begingroup$ @DylanWilson This is true for $d=2$. For larger $d$ you need to use the ``fat'' wedge. $\endgroup$ – Gregory Arone Jan 6 '17 at 23:43
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    $\begingroup$ @GregoryArone whoops, thought that looked suspicious! this is what I get for checking one example and stating it for arbitrary d... $\endgroup$ – Dylan Wilson Jan 7 '17 at 1:45
  • $\begingroup$ Dwelling more into the papers, I should perhaps retract my original question. By "sloppily" I mean that some studies replace $[T^d,Y]_*$ by $\pi_d(Y)$ as if it were obvious (which, as you explain, is false!). But other (Sec. 1.2) are more careful: Ideal crystals are described by $[T^d,Y]_*$, but real ones have various imperfections and the Brillouin zone (the T^d) loses meaning. However, based on other arguments, the $\pi_d(Y)$ component survives even in presence of strong crystal "disorder" => a robust physical observable. $\endgroup$ – Tomas Bzdusek Jan 7 '17 at 12:09
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    $\begingroup$ You may find the Ph.D. thesis of Ricardo Kennedy interesting, as he covers the homotopy theory of topological insulators and superconductors in some detail (I'm assuming that's where your interest lies). kups.ub.uni-koeln.de/5873 $\endgroup$ – j.c. Feb 23 '17 at 14:24

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

  • $\begingroup$ Thank you Gregory for the answer. Without being able to follow your argument, I see that the result for $[\mathbb{T}^d,Y]_*$ ($Y$ a loop space) has a nice geometric interpretation: As $\mathbb{T}^d$ is a hypercube with identified opposite faces, the result tells you to study all $ 1 \leq i \leq d$-dimensional "sub-tori" (or "sub-cubes") $\mathbb{T}^i$ of all $\binom{d}{i}$ orientations. This seems to be in line with the concept of "weak topological indices" used in cond-mat-phys. I can see, that the relation is too complex in general. I probably need to check the physics-wise assumptions. $\endgroup$ – Tomas Bzdusek Jan 6 '17 at 9:32
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    $\begingroup$ Did you mean $\Sigma T_+^d\simeq \Sigma T^d\vee S^1$? $\endgroup$ – skd Jan 6 '17 at 16:23
  • $\begingroup$ @GregoryArone: To specify the relevant $Y$ in question: The 10 classes of target spaces are listed in column "$G/H$ (ferm. NL$\sigma$M)" of Table 1 here. I assume that your answer applies well at least to rows $U(n)$, $O(2n)$, $Sp(2n)$, not sure about the rest. Nevertheless, you helped me a lot. Thanks! $\endgroup$ – Tomas Bzdusek Jan 7 '17 at 12:17
  • $\begingroup$ It looks like it should be $S^i$ not $S^{i+1}$ in the first formula, to be consistent with the later ones. Either that or it should be $\pi_{i+1}$. Is this correct? $\endgroup$ – qbt937 Feb 12 '18 at 2:45
  • $\begingroup$ @qbt937 I think no, because of the suspension on the left hand side in the first formula. $\endgroup$ – Gregory Arone Feb 12 '18 at 6:20

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.

  • $\begingroup$ Fox's torus homotopy groups were applied to the classification of defects in ordered materials, see this paper of Nakanishi et al projecteuclid.org/euclid.cmp/1104161659 $\endgroup$ – j.c. Feb 23 '17 at 14:21

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