  We determine conditions on the filling of electrons in a crystalline lattice
to obtain the equivalent of a band insulator -- a gapped insulator with neither
symmetry breaking nor fractionalized excitations. We allow for strong
interactions, which precludes a free particle description. Previous approaches
that extend the Lieb-Schultz-Mattis argument invoked spin conservation in an
essential way, and cannot be applied to the physically interesting case of
spin-orbit coupled systems. Here we introduce two approaches, the first an
entanglement based scheme, while the second studies the system on an
appropriate flat `Bieberbach' manifold to obtain the filling conditions for all
230 space groups. These approaches only assume time reversal rather than spin
rotation invariance. The results depend crucially on whether the crystal
symmetry is symmorphic. Our results clarify when one may infer the existence of
an exotic ground state based on the absence of order, and we point out
applications to experimentally realized materials. Extensions to new situations
involving purely spin models are also mentioned.

Strongly Correlated Electrons;Mesoscale and Nanoscale Physics