Let us consider a (3+1)-dimensional system $\mathcal{H}$ constructed by stacking (2+1)-dimensional integer quantum Hall systems $\mathcal{H}_\text{Hall}$, e.g., $E_8$ bosonic systems (or $\sigma_H=1$ electronic systems): \begin{eqnarray} \mathcal{H}\equiv\cdots\text{- }\mathcal{H}_\text{Hall}\text{ - }\mathcal{H}_\text{Hall}\text{ - }\mathcal{H}_\text{Hall}\text{ -}\cdots. \end{eqnarray} Whether we should consider this system as a nontrivially topological phase?
If we see its boundary, which is formed by infinitely stacking of chiral edge modes with the same chirality, the boundary theory cannot be trivially gapped. Thus it is reasonable to call the bulk system topologically nontrivial. Also from the bulk viewpoint, its ground state wave function should have long-range entanglement, i.e., we cannot transformed it by local unitary to a product state or atomic state.
However, we also have reason to "define" this system $\mathcal{H}$ to be topologically trivial phase because the above nontriviality can be attributed to the fact that the $\mathcal{H}_\text{Hall}$ is a generator of $\mathbb{Z}$-classification of two-dimensional topological phases. Thus if we still take $\mathcal{H}$ to be nontrivial (3+1)-d topological phase, there seems to be an over-counting.
My question is whether there is a better motivation to let $\mathcal{H}$ be trivial or nontrivial?
(Please note that we do not impose any symmetry although $\mathcal{H}$ accidentally has a translation symmetry along stacking.)
