Abstract
In previous works concerning Atiyah-Hirzebruch spectral sequence (AHSS) as a useful tool in the context of band theory, people get the complete list of topological invariants for 230 space groups and even the K groups for nearly 59% of symmetry settings in three spatial dimensions.
The generalization of the Brillouin Klein bottle to three dimensions has led to various attempts. Here, we propose a complete framework, ten Brillouin platycosms，which are the elementary shapes of the 3D momentum space units, to classify the topological insulators over them.Applying the Atiyah-Hirzebruch spectral sequence (AHSS), we show that the reduced K group of each platycosm is isomorphic to the second cohomology group of the corresponding Bieberbach group. Thus, by calculating the updating pages of the AHSS, we get the following reduced K groups.
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