  We study the generalization of the three-dimensional two-band Hopf insulator
to the case of many bands, where all the bands are separated from each other by
band gaps. The obtained $\mathbb{Z}$ classification of such a $N$-band Hopf
insulator and the formulation of its bulk-boundary correspondence are our main
results. We find that the quantized boundary effect can only be measured in a
non-equilibrium state.
