  The interplay between symmetry and topology leads to a rich variety of
electronic topological phases, protecting states such as the topological
insulators and Dirac semimetals. Previous results, like the Fu-Kane parity
criterion for inversion-symmetric topological insulators, demonstrate that
symmetry labels can sometimes unambiguously indicate underlying band topology.
Here we develop a systematic approach to expose all such symmetry-based
indicators of band topology in all the 230 space groups. This is achieved by
first developing an efficient way to represent band structures in terms of
elementary basis states, and then isolating the topological ones by removing
the subset of atomic insulators, defined by the existence of localized
symmetric Wannier functions. Aside from encompassing all earlier results on
such indicators, including in particular the notion of filling-enforced quantum
band insulators, our theory identifies symmetry settings with previously hidden
forms of band topology, and can be applied to the search for topological
materials.
