  The link between chemical orbitals described by local degrees of freedom and
band theory, which is defined in momentum space, was proposed by Zak several
decades ago for spinless systems with and without time-reversal in his theory
of "elementary" band representations. In Nature 547, 298-305 (2017), we
introduced the generalization of this theory to the experimentally relevant
situation of spin-orbit coupled systems with time-reversal symmetry and proved
that all bands that do not transform as band representations are topological.
Here, we give the full details of this construction. We prove that elementary
band representations are either connected as bands in the Brillouin zone and
are described by localized Wannier orbitals respecting the symmetries of the
lattice (including time-reversal when applicable), or, if disconnected,
describe topological insulators. We then show how to generate a band
representation from a particular Wyckoff position and determine which Wyckoff
positions generate elementary band representations for all space groups. This
theory applies to spinful and spinless systems, in all dimensions, with and
without time reversal. We introduce a homotopic notion of equivalence and show
that it results in a finer classification of topological phases than approaches
based only on the symmetry of wavefunctions at special points in the Brillouin
zone. Utilizing a mapping of the band connectivity into a graph theory problem,
which we introduced in Nature 547, 298-305 (2017), we show in companion papers
which Wyckoff positions can generate disconnected elementary band
representations, furnishing a natural avenue for a systematic materials search.
