We provide an extensive look at Bott periodicity in the context of complex gapped topological phases of free fermions. In doing so, we remark on the existence of a product structure in the set of inequivalent phases induced by the external tensor product of vector bundles -- a structure which has not yet been explored in condensed-matter literature. Bott periodicity appears in the form of a generalized Dirac monopole built out of a given phase, which is equivalent to the product of a Dirac monopole phase with that same given phase. The complex emph{K}-theory cohomology ring is presented as a natural way to store the information of these phases, with a grading corresponding to the number of Clifford symmetries modulo $2$. The K"unneth formula allows us to derive the result that, for band insulators, the Su-Schrieffer-Heeger (SSH) chain in one dimension allows one to generate the emph{K}-cohomology of the $d$-dimensional Brillouin zone. In particular, we find that the product of two SSH chains in independent momentum directions yields a two-dimensional Chern insulator. The results obtained relate the associated topological phases of charge-conserving band insulators and their topological invariants in all spatial dimensions in a unified way.