  Strong invariants of even-dimensional topological insulators of independent
Fermions are expressed in terms of an invertible operator on the Hilbert space
over the boundary. It is given by the Cayley transform of the boundary
restriction of the half-space resolvent. This dimensional reduction is routed
in new representation for the $K$-theoretic exponential map. It allows to
express the invariants via the reflection matrix at the Fermi energy, for the
scattering set-up of a wire coupled to the half-space insulator.
