The paper by Liang Fu, C.L. Kane, and E.J. Mele explores the generalization of the quantum spin Hall (QSH) effect to three dimensions (3D). Unlike in two dimensions, where the QSH effect is characterized by a single \(Z_2\) topological invariant, in 3D there are four invariants that distinguish 16 "topological insulator" phases. These phases are categorized into two classes: weak (WTI) and strong (STI) topological insulators. WTI states are equivalent to layered 2D QSH states but are fragile due to disorder, which can connect them to band insulators. STI states, on the other hand, are robust and exhibit surface states that realize the 2+1 dimensional parity anomaly without fermion doubling, leading to a novel "topological metal" surface phase. The authors introduce a tight binding model on a distorted diamond lattice that realizes both WTI and STI phases, allowing for the explicit study of surface states. They discuss the relevance of this model to real 3D materials, including bismuth. The paper also reviews the connection between the \(Z_2\) invariant for the bulk QSH phase and the spin-filtered edge states, using a Laughlin-type construction on a cylinder threaded by magnetic flux. The change in the time-reversal polarization (TRP) between different points in the Brillouin zone characterizes the topological properties of the system. In 3D, the TRP changes are related to the bulk band structure and are described by four \(Z_2\) indices. These indices distinguish the phases and determine the connectivity of the edge state spectrum. The paper also discusses the robustness of STI phases and the potential for surface states in these phases, which can be topologically protected against disorder. The authors conclude by suggesting that the exotic surface properties of STI phases could stimulate further experimental and theoretical efforts.The paper by Liang Fu, C.L. Kane, and E.J. Mele explores the generalization of the quantum spin Hall (QSH) effect to three dimensions (3D). Unlike in two dimensions, where the QSH effect is characterized by a single \(Z_2\) topological invariant, in 3D there are four invariants that distinguish 16 "topological insulator" phases. These phases are categorized into two classes: weak (WTI) and strong (STI) topological insulators. WTI states are equivalent to layered 2D QSH states but are fragile due to disorder, which can connect them to band insulators. STI states, on the other hand, are robust and exhibit surface states that realize the 2+1 dimensional parity anomaly without fermion doubling, leading to a novel "topological metal" surface phase. The authors introduce a tight binding model on a distorted diamond lattice that realizes both WTI and STI phases, allowing for the explicit study of surface states. They discuss the relevance of this model to real 3D materials, including bismuth. The paper also reviews the connection between the \(Z_2\) invariant for the bulk QSH phase and the spin-filtered edge states, using a Laughlin-type construction on a cylinder threaded by magnetic flux. The change in the time-reversal polarization (TRP) between different points in the Brillouin zone characterizes the topological properties of the system. In 3D, the TRP changes are related to the bulk band structure and are described by four \(Z_2\) indices. These indices distinguish the phases and determine the connectivity of the edge state spectrum. The paper also discusses the robustness of STI phases and the potential for surface states in these phases, which can be topologically protected against disorder. The authors conclude by suggesting that the exotic surface properties of STI phases could stimulate further experimental and theoretical efforts.