  We show that a slab of a three-dimensional inversion-symmetric higher-order
topological insulator (HOTI) in class A is a 2D Chern insulator, and that in
class AII is a 2D $Z_2$ topological insulator. We prove it by considering a
process of cutting the three-dimensional inversion-symmetric HOTI along a
plane, and study the spectral flow in the cutting process. We show that the
$Z_4$ indicators, which characterize three-dimensional inversion-symmetric
HOTIs in classes A and AII, are directly related to the $Z_2$ indicators for
the corresponding two-dimensional slabs with inversion symmetry, i.e. the Chern
number parity and the $Z_2$ topological invariant, for classes A and AII
respectively. The existence of the gapless hinge states is understood from the
conventional bulk-edge correspondence between the slab system and its edge
states. Moreover, we also show that the spectral-flow analysis leads to another
proof of the bulk-edge correspondence in one- and two-dimensional
inversion-symmetric insulators.
