  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 is related to the quantized isotropic magnetoelectric coefficient of
its bulk. The boundary of a $N$-band Hopf insulator can be fully gapped, and we
find that there is no unique way of dividing a finite system into bulk and
boundary. Despite this non-uniqueness, we find that the magnetoelectric
coefficient of the bulk and the anomalous Hall conductivity of the boundary are
quantized to the same integer value. We propose an experiment where the
quantized boundary effect can be measured in a non-equilibrium state.
