  We show that for families of 1d lattice systems in an invertible phase, the
cohomology class of the higher Berry curvature can be refined to an integral
degree-3 class on the parameter space. Similarly, for families of
U(1)-invariant 2d lattice systems in an invertible phase, the higher Thouless
pump can be refined to an integral degree-2 class on the parameter space. We
show that the 2d Thouless pump can be identified with an excess Berry curvature
of a flux insertion.
