  The ability to pump quantised amounts of charge is one of the hallmarks of
topological materials. An archetypical example is Laughlin's gauge argument for
transporting an integer number of electrons between the edges of a quantum Hall
cylinder upon insertion of a magnetic flux quantum. This is mathematically
equivalent to the equally famous suggestion of Thouless' that an integer number
of electrons are pumped between two ends of a one-dimensional quantum wire upon
sliding a charge-density wave over a single wave length. We use the
correspondence between these descriptions to visualise the detailed dynamics of
the electron flow during a single pumping cycle, which is difficult to do
directly in the quantum Hall setup, because of the gauge freedom inherent to
its description. We find a close correspondence between topological edge states
and the mobility edges in charge-density wave, quantum Hall, and other
topological systems. We illustrate this connection by describing an
alternative, non-adiabatic mode of topological transport that displaces
precisely the opposite amount of charge as compared to the adiabatic pump. We
discuss possible experimental realisations in the context of ultracold atoms
and photonic waveguide experiments.
