  Topological insulators are noninteracting, gapped fermionic systems which
have gapless boundary excitations. They are characterized by topological
invariants, which can be written in many different ways, including in terms of
Green's functions. Here we show that the existence of the edge states directly
follows from the existence of the topological invariant written in terms of the
Green's functions, for all ten classes of topological insulators in all spatial
dimensions. We also show that the resulting edge states are characterized by
their own topological invariant, whose value is equal to the topological
invariant of the bulk insulator. This can be used to test whether a given model
Hamiltonian can describe an edge of a topological insulator. Finally, we
observe that the results discussed here apply equally well to interacting
topological insulators, with certain modifications.
