The paper by J. E. Moore and L. Balents explores the topological invariants of time-reversal-invariant (TRI) band structures in two and three dimensions. In two dimensions, these invariants are multiple copies of the $\mathbb{Z}_2$ invariant introduced by Kane and Mele, which protects the topological insulator and gives rise to a spin Hall effect. 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 determine transitions between ordinary and topological insulators as the bands are occupied by electrons. The authors derive these invariants using maps from the Brillouin zone to the space of Bloch Hamiltonians and clarify their connections to integer invariants underlying the integer quantum Hall effect and previous invariants of $\mathcal{T}$-invariant Fermi systems. The paper also discusses the implications of these invariants for physical systems, such as the adiabatic connection between different topological insulators.The paper by J. E. Moore and L. Balents explores the topological invariants of time-reversal-invariant (TRI) band structures in two and three dimensions. In two dimensions, these invariants are multiple copies of the $\mathbb{Z}_2$ invariant introduced by Kane and Mele, which protects the topological insulator and gives rise to a spin Hall effect. 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 determine transitions between ordinary and topological insulators as the bands are occupied by electrons. The authors derive these invariants using maps from the Brillouin zone to the space of Bloch Hamiltonians and clarify their connections to integer invariants underlying the integer quantum Hall effect and previous invariants of $\mathcal{T}$-invariant Fermi systems. The paper also discusses the implications of these invariants for physical systems, such as the adiabatic connection between different topological insulators.