  In this work we investigate the decorated domain wall construction in bosonic
group-cohomology symmetry-protected topological (SPT) phases and related
quantum anomalies in bosonic topological phases. We first show that a general
decorated domain wall construction can be described mathematically as an
Atiyah-Hirzebruch spectral sequence, where the terms on the $E_2$ page
correspond to decorations by lower-dimensional SPT states at domain wall
junctions. For bosonic group-cohomology SPT phases, the spectral sequence
becomes the Lyndon-Hochschild-Serre (LHS) spectral sequence for ordinary group
cohomology. We then discuss the physical interpretations of the differentials
in the spectral sequence, particularly in the context of anomalous SPT phases
and symmetry-enriched gauge theories. As the main technical result, we obtain a
full description of the LHS spectral sequence concretely at the cochain level.
The explicit formulae are then applied to explain Lieb-Schultz-Mattis theorems
for SPT phases, and also derive a new LSM theorem for easy-plane spin model in
a $\pi$ flux lattice. We also revisit the classifications of symmetry-enriched
2D and 3D Abelian gauge theories using our results.
