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We provide an extensive look at Bott periodicity in the context of complexgapped topological phases of free fermions. In doing so, we remark on theexistence of a product structure in the set of inequivalent phases induced bythe external tensor product of vector bundles -- a structure which has not yetbeen explored in condensed-matter literature. Bott periodicity appears in theform of a generalized Dirac monopole built out of a given phase, which isequivalent to the product of a Dirac monopole phase with that same given phase.The complex K-theory cohomology ring is presented as a natural way to store theinformation of these phases, with a grading corresponding to the number ofClifford symmetries modulo $2$. The K\"unneth formula allows us to derive theresult that, for band insulators, the Su-Schrieffer-Heeger (SSH) chain in onedimension allows one to generate the K-cohomology of the $d$-dimensionalBrillouin zone. In particular, we find that the product of two SSH chains inindependent momentum directions yields a two-dimensional Chern insulator. Theresults obtained relate the associated topological phases of charge-conservingband insulators and their topological invariants in all spatial dimensions in aunified way.

The product of two independent Su-Schrieffer-Heeger chains yields a two-dimensional Chern insulator