  I discuss a topological relation of the conduction property of a
many-particle system on a periodic lattice at zero temperature to the energy
spectrum. When the particle number per unit cell is an irreducible fraction
$p/q$, an insulator must have $q$ low-lying states of energy $o(1/L)$ in one
dimension and of energy $o(1)$ in two dimensions, where $L$ is the linear
system size.
