Topological pumps and adiabatic cycles

Topological insulators have gapless states at their boundaries while trivial insulators generically do not. We consider loops in the spaces of Hamiltonians of topologically trivial Bloch insulators, and show that there exist loops for which the boundary gap must necessarily close at some point or points along the loop. We show that some of these loops may be regarded, depending on the symmetry class of the insulator and its physical dimension, as defining pumps of charge, fermion parity, and in more exotic cases of other quantities such as a $Z_2$ parity.