  We reformulate the mod-two Atiyah-Patodi-Singer (APS) index in a
physicist-friendly way using the domain-wall fermion. Our new formulation is
given on a closed manifold, which is extended from the original manifold with
boundary, where we instead give a fermion mass term changing its sign at the
location of the original boundary. This new setup does not need the APS
boundary condition, which is non-local. A mathematical proof of equivalence
between the two different formulations is given by two different evaluations of
the same index of a Dirac operator on a higher dimensional manifold. The
domain-wall fermion allows us to separate the edge and bulk mode contributions
in a natural but not in a gauge invariant way, which offers a straightforward
description of the global anomaly inflow.
