The phases before "+" come from H d + 1 [ G , U ( 1 ) ] . The phases after "+" come from ⊕ ⊕ k = 1 d H k [ G , i T O d + 1 − − k ] . Just like group theory can give us 230 crystal structures in 3+1D, group cohomology theory can give us various SPT phases in any dimensions with any on-site symmetry groups.