Liang Fu and C.L. Kane discuss topological insulators with inversion symmetry. These materials have a bulk excitation gap due to spin-orbit interaction and are distinguished by Z₂ topological invariants. In 2D, a single Z₂ invariant separates the quantum spin Hall phase from a conventional insulator, while in 3D, four Z₂ invariants distinguish between strong and weak topological insulators. The presence of inversion symmetry simplifies the evaluation of these invariants by considering the parity of occupied Bloch wavefunctions at time-reversal invariant points in the Brillouin zone. The authors predict that materials like Bi₁₋ₓSbₓ, α-Sn, and HgTe under uniaxial strain are strong topological insulators. They also discuss the implications of these findings for experiments and the topological properties of surface states. The Z₂ invariants are determined by the product of parity eigenvalues at these points, and the results are robust against weak disorder and interactions. The paper provides a detailed formulation of the Z₂ invariants in both 2D and 3D, highlighting the differences between strong and weak topological insulators. The presence of inversion symmetry allows for a simpler calculation of these invariants, which are crucial for identifying topological insulators. The study also explores the connection between topological invariants and surface states, showing how they are protected by time-reversal symmetry. The results have significant implications for the understanding of topological phases of matter and their experimental realization.Liang Fu and C.L. Kane discuss topological insulators with inversion symmetry. These materials have a bulk excitation gap due to spin-orbit interaction and are distinguished by Z₂ topological invariants. In 2D, a single Z₂ invariant separates the quantum spin Hall phase from a conventional insulator, while in 3D, four Z₂ invariants distinguish between strong and weak topological insulators. The presence of inversion symmetry simplifies the evaluation of these invariants by considering the parity of occupied Bloch wavefunctions at time-reversal invariant points in the Brillouin zone. The authors predict that materials like Bi₁₋ₓSbₓ, α-Sn, and HgTe under uniaxial strain are strong topological insulators. They also discuss the implications of these findings for experiments and the topological properties of surface states. The Z₂ invariants are determined by the product of parity eigenvalues at these points, and the results are robust against weak disorder and interactions. The paper provides a detailed formulation of the Z₂ invariants in both 2D and 3D, highlighting the differences between strong and weak topological insulators. The presence of inversion symmetry allows for a simpler calculation of these invariants, which are crucial for identifying topological insulators. The study also explores the connection between topological invariants and surface states, showing how they are protected by time-reversal symmetry. The results have significant implications for the understanding of topological phases of matter and their experimental realization.