  We propose a relation between the $\eta$ invariant on a manifold with
boundary, the $\eta$ invariants of edge states, and the $\eta$ invariant in an
infinite volume limit. With the example of planar fermions with bag and chiral
bag boundary conditions we show that this relation holds whenever edge states
are sufficiently well-localized near the boundary. As a by-product we show that
the spectrum of edge modes for chiral bag boundary conditions is linear but
bounded.
