For translationally invariant systems, we can define some topological invariant based on the translational symmetry, which is referred to "weak" topological invariant. For example, according to Kitaev's K-theory classification (https://arxiv.org/abs/0901.2686), the 3D T-invariant insulators are classified by $\tilde{K}^{-1}_\mathbb{R}(T^3) \cong \mathbb{Z} \oplus 3\mathbb{Z}_2$, where $T^3$ is the momentum space which forms a 3-torus. The $\mathbb{Z}_2$ part is the "weak" invariant.
For general systems, the result is given by $\tilde{K}^{-q}_\mathbb{R}(\bar{S}^d)$. My question is: what's the meaning of this manifold $\bar{S}^d$? Kitaev said that it's compactified momentum space (The asymptotic of Hamiltonian is fixed for $|p|\rightarrow \infty$). However, if the translational symmetry is broken, we can't even define "momentum space" by Fourier transform. So, why can we claim that the classification by K-theory over the compactified momentum space $\bar{S}^d$ gives the "strong" topological invariant, which is robust in the presence of disorder breaking the translational symmetry?