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We study glide protected topological (GSPT) phases of interacting bosons andfermions in three spatial dimensions certain on-site symmetries. They arecrystalline SPT phases, which are distinguished from a trivial product stateonly in the presence of non-symmorphic glide symmetry. We classify these GSPTphases with various on-site symmetries such as $U(1)$ and time reversal, andshow that they can all be understood by stacking and coupling two-dimensionalshort-range-entangled phases in a glide-invariant way. Using such a coupledlayer construction we study the anomalous surface topological orders of theseGSPT phases, which gap out the two-dimensional surface states without breakingany symmetries. This framework is demonstrated in many examples, including thenon-symmorphic topological insulator with "hourglass fermion" surface states.

Classification and surface anomaly of glide symmetry protected topological phases in three dimensions