When can we take the Brillouin zone to be a sphere? When reading some literatures on topological insulators, I've seen authors taking Brillouin zone(BZ) to be a sphere sometimes, especially when it comes to strong topological insulators. Also I've seen the usage of spherical BZ in these answers(1,2) by SE user Heidar. I can think of two possibilities:
(1)Some physical system has a spherical BZ. This is hard to imagine, since it seems to me that all lattice systems with translational symmetries will have a torodal BZ, by the periodicity of Bloch wavefunctions. The closest scenario I can imagine is a continuous system having $\mathbf{R}^n$ as BZ, and somehow(in a way I cannot think of) acquires an one-point compactification.
(2)A trick that makes certain questions easier to deal with, while the true BZ is still a torus.
Can someone elaborate the idea behind a spherical BZ for me?
Update: I recently came across these notes(pdf) by J.Moore. In the beginning of 
page 9 he mentioned 

We need to use one somewhat deep fact: under some assumptions, if $π_1(M)
= 0$ for some target space $M$, then maps from the torus $T^
2\to M$ are contractible to maps from the sphere $S^2
 \to M$

I think this is a special case of the general math theorem I want to know, but unfortunately Moore did not give any reference so I'm not sure where to look.
EDIT: The above math theorem is intuitively acceptable to me although I'm not able to prove it. I can take this theorem as a working hypothesis for now, what I'm more interested in is, granted such theorem, what makes a $\pi_1(M)=0$ physical system candidate for strong topological insulators(robust under local perturbations), and why in $\pi_1(M)\neq 0$ case we can only have weak topological insulators.
Crossposted: When can we take the Brillouin zone to be a sphere?
 A: The answer to your question is covered in these two papers:
Homotopy and Quantization in Condensed Matter Physics
J. E. Avron, R. Seiler, and B. Simon
http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.51.51
and
Homotopy Theory of Strong and Weak Topological Insulators
Ricardo Kennedy, Charles Guggenheim
http://arxiv.org/abs/1409.2529
The first one explains how to use maps from the sphere in the case of the quantum Hall effect, whereas the second one discusses a more general context of all symmetry classes and dimensions.

Let $X$ be some topological space and $\mathbb{T}^2$ be the $2$-torus.
Consider the task of classifying the set $[\mathbb{T}^2\to X]$, the set of homotopy classes the maps $\mathbb{T}^2\to X$.
Then as Avron, Seiler and Simon argue the following:
As you should know, $\pi_1(\mathbb{T}^2)\cong\mathbb{Z}\times \mathbb{Z}$, that is, there are two basic loops $S^1\to \mathbb{T}^2$ which are not homotopic to each other. Call them $\gamma_1$ (blue) and $\gamma_2$ (red). 
 
Then any map $f:\mathbb{T}^2\to X$ induces two maps $S^1\to X$ via $$f\circ\gamma_i:S^1\to X$$
Then one can prove that for two maps $f_1:\mathbb{T}^2\to X$ and $f_2:\mathbb{T}^2\to X$, if the either $f_1\circ\gamma_1$ is not homotopic to $f_2\circ\gamma_1$ OR $f_1\circ\gamma_2$ is not homotopic to $f_2\circ\gamma_2$, then $f_1$ is not homotopic to $f_2$.
However, "even if the two induced maps $S^1\to X$ are pair-wise homotopic, there could still be a left over piece, which is a map $S^2\to X$.". In this way, you could classify maps $\mathbb{T}^2\to X$ by two pieces of $\pi_1(X)$, one piece of $\pi_2(X)$, etc...
For our case, $X$ is the set of gapped Hamiltonians, which topologically is the Grassmannian ($G_m(\mathbb{C}^n)$ for an $n$ level system with the Fermi energy above $m$ levels). 
For instance, for a two-level system with 1 occupied level, we have $$G_1(\mathbb{C}^2)\cong S^2$$. But the the $2$-sphere is simply connected (that is, $\pi_1(S^2)=0$), so the two aforementioned loops $\gamma_1$ and $\gamma_2$ will not contribute to the classification, and we would only have the "left over" piece $$S^2\to S^2$$ And it is well known that $$[S^2\to S^2]\equiv\pi_2(S^2)\cong\mathbb{Z}$$ The integer being the degree of the map $\mathbb{T}^2\to S^2$ which would be the same degree as the induced map $S^2\to S^2$ which in a more general setting (of a complex line bundles over $\mathbb{T^2}$) would be the Chern number of that line bundle. Indeed, isomorphism classes of line bundles over $\mathbb{T^2}$ are isomorphic to homotopy classes of maps $\mathbb{T^2}\to Gr_1\left(\mathbb{C}^2\right)\cong S^2$.
