# Generalized homology and Atiyah-Hirzebruch spectral sequence in crystalline symmetry protected topological phenomena

@article{Shiozaki2018GeneralizedHA, title={Generalized homology and Atiyah-Hirzebruch spectral sequence in crystalline symmetry protected topological phenomena}, author={Ken Shiozaki and Charles Zhaoxi Xiong and Kiyonori Gomi}, journal={arXiv: Strongly Correlated Electrons}, year={2018} }

We propose that symmetry protected topological (SPT) phases with crystalline symmetry are formulated by equivariant generalized homologies $h^G_n(X)$ over a real space manifold $X$ with $G$ a crystalline symmetry group. The Atiyah-Hirzebruch spectral sequence unifies various notions in crystalline SPT phases such as the layer construction, higher-order SPT phases and Lieb-Schultz-Mattis type theorems. Our formulation is applicable to interacting systems with onsite and crystalline symmetries as…

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