A complete catalogue of high-quality topological materials

"Topological Quantum Chemistry (TQC) links the chemical and symmetry structure of a given material with its topological properties. This field tabulates the data of the 10398 real-space atomic limits of materials, and solves the compatibility relations of electronic bands in momentum space. A material that is not an atomic limit or whose bands do not satisfy the compatibility relations, is a topological insulator/semimetal. We use TQC to find the topological stoichiometric non-magnetic, "high-quality'' materials in the world. We develop several code additions to VASP which can compute all characters of all symmetries at all high-symmetry points in the Brillouin Zone (BZ). Using TQC we then develop codes to check which materials in ICSD are topological. Out of 26938 stoichiometric materials in our filtered ICSD database, we find 2861 topological insulators (TI) and 2936 topological semimetals (2505 and 2560 non-f electron, respectively). Our method is uniquely capable to show that none of the TI's found exhibit fragile topology. We partition the topological materials in different physical classes. For the majority of the 5797 "high-quality'' topological material, we compute: the topological class (equivalence classes of TQC elementary band representations -- equivalent to the topological index), the symmetry(ies) that protects the topological class, the representations at high symmetry points and the direct gap (for insulators), and the topological index. For topological semimetals we then compute whether the system becomes a topological insulator (whose index/class we compute) upon breaking symmetries -- useful for experiments. 2152 more TI's are obtained in this way. For almost all 5065 non-f-electron topological materials, we provide the electronic band structures, allowing the identification of quantitative properties (gaps, velocities). Remarkably, our exhaustive results show that a large proportion ( ~ 24% !) of all materials in nature are topological (confirmed by calculations of "low-quality'' materials). We confirm the topology of several new materials by Wilson loop calculations. We added an open-source code and end-user button on the Bilbao Crystallographic Server (BCS) which checks the topology of any material. We comment on the chemistry of each compound and sample part of the "low-quality'' ICSD data to find more materials."