  We consider a lattice implementation of the eta-invariant, using the complex
phase of the determinant of the simplified domain-wall fermion, which couples
to an interpolating five-dimensional gauge field. We clarify the relation to
the effective action for chiral Ginsparg-Wilson fermions. The integrability,
which holds true for anomaly-free theories in the classical continuum limit, is
not assured on the lattice with a finite spacing. A lattice expression for the
five-dimensional Chern-Simons term is obtained.
