  The topological invariants of a time-reversal-invariant band structure in two
dimensions are multiple copies of the $\mathbb{Z}_2$ invariant found by Kane
and Mele. Such invariants protect the topological insulator and give rise to a
spin Hall effect carried by edge states. Each pair of bands related by time
reversal is described by a single $\mathbb{Z}_2$ invariant, up to one less than
half the dimension of the Bloch Hamiltonians. In three dimensions, there are
four such invariants per band. The $\mathbb{Z}_2$ invariants of a crystal
determine the transitions between ordinary and topological insulators as its
bands are occupied by electrons. We derive these invariants using maps from the
Brillouin zone to the space of Bloch Hamiltonians and clarify the connections
between $\mathbb{Z}_2$ invariants, the integer invariants that underlie the
integer quantum Hall effect, and previous invariants of ${\cal T}$-invariant
Fermi systems.
