  Topological insulators can be characterized alternatively in terms of bulk or
edge properties. We prove the equivalence between the two descriptions for
two-dimensional solids in the single-particle picture. We give a new
formulation of the $\mathbb{Z}_{2}$-invariant, which allows for a bulk index
not relying on a (two-dimensional) Brillouin zone. When available though, that
index is shown to agree with known formulations. The method also applies to
integer quantum Hall systems. We discuss a further variant of the
correspondence, based on scattering theory.
