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 $U\left(1\right)$ and reflection symmetry, $2d$ systems with $U\left(1\right)$ and $C_n$ rotation symmetry, and $3d$ systems with $U\left(1\right)$ and inversion symmetry.

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• Electronic structure
• Symmetry protected topological states
• Topological insulators
• Topological phase transition
• Topological phases of matter
• Topological superconductors
• Strongly correlated systems