  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 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 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.
