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Using a dimensional reduction scheme based on scattering theory, we show thatthe classification tables for topological insulators and superconductors withreflection symmetry can be organized in two period-two and four period-eightcycles, similar to the Bott periodicity found for topological insulators andsuperconductors without spatial symmetries. With the help of the dimensionalreduction scheme the classification in arbitrary dimensions $d \ge 1$ can beobtained from the classification in one dimension, for which we present aderivation based on relative homotopy groups and exact sequences to classifyone-dimensional insulators and superconductors with reflection symmetry. Theresulting classification is fully consistent with a comprehensiveclassification obtained recently by Shiozaki and Sato [Phys.\ Rev.\ B {\bf 90},165114 (2014)]. The use of a scattering-matrix inspired method allows us toaddress the second descendant $\bZ_2$ phase, for which the topologicalnontrivial phase was previously reported to be vulnerable to perturbations thatbreak translation symmetry.

Bott periodicity for the topological classification of gapped states of matter with reflection symmetry