  The past decade has witnessed significant progress in topological materials
investigation. Symmetry-indicator theory and topological quantum chemistry
provide an efficient scheme to diagnose topological phases from only partial
information of wave functions without full knowledge of topological invariants,
which has resulted in a recent comprehensive materials search. However, not all
topological phases can be captured by this framework, and topological
invariants are needed for a more refined diagnosis of topological phases. In
this study, we present a systematic framework to construct topological
invariants for a large part of symmetry classes, which should be contrasted
with the existing invariants discovered through one-by-one approaches. Our
method is based on the recently developed Atiyah-Hirzebruch spectral sequence
in momentum space. As a demonstration, we construct topological invariants for
time-reversal symmetric spinful superconductors with conventional pairing
symmetries of all space groups, for which symmetry indicators are silent. We
also validate that the obtained quantities work as topological invariants by
computing them for randomly generated symmetric Hamiltonians. Remarkably, the
constructed topological invariants completely characterize $K$-groups in 159
space groups. Our topological invariants for normal conducting phases are
defined under some gauge conditions. To facilitate efficient numerical
simulations, we discuss how to derive gauge-independent topological invariants
from the gauge-fixed topological invariants through some examples. Combined
with first-principles calculations, our results will help us discover
topological materials that could be used in next-generation devices and pave
the way for a more comprehensive topological materials database.
