Connection between Free-Fermion and Interacting Crystalline Symmetry-Protected Topological Phases

We present a framework for investigating the effects of interactions on crystalline symmetry-protected topological (SPT) phases. Within this framework, one can establish a direct connection between the equivalence classes of free-fermion systems and their corresponding interacting classes. A central component of this framework is the Atiyah-Hirzebruch spectral sequence, which provides a systematic way to represent crystalline SPT phases as SPT phases with internal symmetries on subspaces. We demonstrate the application of this approach through examples in various dimensions: 1d systems with U(1) and reflection symmetry, 2d systems with U(1) and C_{n} rotation symmetry, and 3d systems with U(1) and inversion symmetry.