  Certain band insulators allow for the adiabatic pumping of quantized charge
or spin for special time-dependences of the Hamiltonian. These "topological
pumps" are closely related to two dimensional topological insulating phases of
matter upon rolling the insulator up to a cylinder and threading it with a time
dependent flux. In this article we extend the classification of topological
pumps to the Wigner Dyson and chiral classes, coupled to multi-channel leads.
The topological index distinguishing different topological classes is
formulated in terms of the scattering matrix of the system. We argue that
similar to topologically non-trivial insulators, topological pumps are
characterized by the appearance of protected gapless end states during the
course of a pumping cycle. We show that this property allows for the pumping of
quantized charge or spin in the weak coupling limit. Our results may also be
applied to two dimensional topological insulators, where they give a physically
transparent interpretation of the topologically non-trivial phases in terms of
scattering matrices.
