  We propose a non-perturbative formulation of the Atiyah-Patodi-Singer(APS)
index in lattice gauge theory, in which the index is given by the $\eta$
invariant of the domain-wall Dirac operator. Our definition of the index is
always an integer with a finite lattice spacing. To verify this proposal, using
the eigenmode set of the free domain-wall fermion, we perturbatively show in
the continuum limit that the curvature term in the APS theorem appears as the
contribution from the massive bulk extended modes, while the boundary $\eta$
invariant comes entirely from the massless edge-localized modes.
