  We review Kitaev's celebrated "periodic table" for topological phases of
condensed matter, which identifies ground states (Fermi projections) of gapped
periodic quantum systems up to continuous deformations. We study families of
projections which depend on a periodic crystal momentum and respect the
symmetries that characterize the various classes of topological insulators. Our
aim is to classify such families in a systematic, explicit, and constructive
way: we identify numerical indices for all symmetry classes and provide
algorithms to deform families of projections whose indices agree. Aiming at
simplicity, we illustrate the method for 0- and 1-dimensional systems, and
recover the (weak and strong) topological invariants proposed by Kitaev and
others.
