  In this lecture for the Nobel symposium, we review previous research on a
class of translational-invariant insulators without spin-orbit coupling. These
may be realized in intrinsically spinless systems such as photonic crystals and
ultra-cold atoms. Some of these insulators have no time-reversal symmetry as
well, i.e., the relevant symmetries are purely crystalline. Nevertheless,
topological phases exist which are distinguished by their robust surface modes.
To describe these phases, we introduce the notions of (a) a halved-mirror
chirality: an integer invariant which characterizes half-mirror planes in the
3D Brillouin zone, and (b) a bent Chern number: the traditional
Thouless-Kohmoto-Nightingale-Nijs invariant generalized to bent 2D manifolds.
Like other well-known topological phases, their band topology is unveiled by
the crystalline analog of Berry phases, i.e., parallel transport across certain
non-contractible loops in the Brillouin zone. We also identify certain
topological phases without any robust surface modes - they are uniquely
distinguished by parallel transport along bent loops, whose shapes are
determined by the symmetry group. Finally, we describe the Weyl semimetallic
phase that intermediates two distinct, gapped phases.
