  We propose that symmetry protected topological (SPT) phases with crystalline
symmetry are formulated by equivariant generalized homologies $h^G_n(X)$ over a
real space manifold $X$ with $G$ a crystalline symmetry group. The
Atiyah-Hirzebruch spectral sequence unifies various notions in crystalline SPT
phases such as the layer construction, higher-order SPT phases and
Lieb-Schultz-Mattis type theorems. Our formulation is applicable to interacting
systems with onsite and crystalline symmetries as well as free fermions.
