  Building on the 10-way symmetry classification of disordered fermions, the
authors have recently given a homotopy-theoretic proof of Kitaev's "Periodic
Table" for topological insulators and superconductors. The present paper offers
an introduction to the physical setting and the mathematical model used. Basic
to the proof is the so-called Diagonal Map, a natural transformation akin to
the Bott map of algebraic topology, which increases by one unit both the
momentum-space dimension and the symmetry index of translation-invariant ground
states of gapped free-fermion systems. This mapping is illustrated here with a
few examples of interest.
