Atiyah-Hirzebruch spectral sequence in band topology: General formalism and topological invariants for 230 space groups

We study the Atiyah-Hirzebruch spectral sequence (AHSS) for equivariant K-theory in context of band theory. Various notions theory such as irreducible representations at high-symmetric points, compatibility relation, topological gapless and singular points naturally fits into AHSS. As an application AHSS, we get complete list invariants 230 space groups without time-reversal or particle-hole invariance. find that a lot torsion appear even symmorphic groups.