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We put the theory of interacting topological crystalline phases on asystematic footing. These are topological phases protected by space-groupsymmetries. Our central tool is an elucidation of what it means to "gauge" suchsymmetries. We introduce the notion of a "crystalline topological liquid", andargue that most (and perhaps all) phases of interest are likely to satisfy thiscriterion. We prove a Crystalline Equivalence Principle, which states that inEuclidean space, crystalline topological liquids with symmetry group $G$ are inone-to-one correspondence with topological phases protected by the samesymmetry $G$, but acting *internally*, where if an element of $G$ isorientation-reversing, it is realized as an anti-unitary symmetry in theinternal symmetry group. As an example, we explicitly compute, using groupcohomology, a partial classification of bosonic symmetry-protected topological(SPT) phases protected by crystalline symmetries in (3+1)-D for 227 of the 230space groups. For the 65 space groups not containing orientation-reversingelements (Sohncke groups), there are no cobordism invariants which maycontribute phases beyond group cohomology, and so we conjecture ourclassification is complete.

Gauging spatial symmetries and the classification of topological crystalline phases