  We classify insulators by generalized symmetries that combine space-time
transformations with quasimomentum translations. Our group-cohomological
classification generalizes the nonsymmorphic space groups, which extend point
groups by real-space translations, i.e., nonsymmorphic symmetries unavoidably
translate the spatial origin by a fraction of the lattice period. Here, we
further extend nonsymmorphic groups by reciprocal translations, thus placing
real and quasimomentum space on equal footing. We propose that group cohomology
provides a symmetry-based classification of quasimomentum manifolds, which in
turn determines the band topology. In this sense, cohomology underlies band
topology. Our claim is exemplified by the first theory of
time-reversal-invariant insulators with nonsymmorphic spatial symmetries. These
insulators may be described as 'piecewise topological', in the sense that
subtopologies describe the different high-symmetry submanifolds of the
Brillouin zone, and the various subtopologies must be pieced together to form a
globally consistent topology. The subtopologies that we discovered include: a
glide-symmetric analog of the quantum spin Hall effect, an hourglass-flow
topology (exemplified by our recently-proposed KHgSb material class), and
quantized non-Abelian polarizations. Our cohomological classification results
in an atypical bulk-boundary correspondence for our topological insulators.
