  We present a rigorous and fully consistent $K$-theoretic framework for
studying gapped topological phases of free fermions such as topological
insulators. It utilises and profits from powerful techniques in operator
$K$-theory. From the point of view of symmetries, especially those of time
reversal, charge conjugation, and magnetic translations, operator $K$-theory is
more general and natural than the commutative topological theory. Our approach
is model-independent, and only the symmetry data of the dynamics, which may
include information about disorder, is required. This data is completely
encoded in a suitable $C^*$-superalgebra. From a representation-theoretic point
of view, symmetry-compatible gapped phases are classified by the
super-representation group of this symmetry algebra. Contrary to existing
literature, we do not use $K$-theory to classify phases in an absolute sense,
but only relative to some arbitrary reference. $K$-theory groups are better
thought of as groups of obstructions between homotopy classes of gapped phases.
Besides rectifying various inconsistencies in the existing literature on
$K$-theory classification schemes, our treatment has conceptual simplicity in
its treatment of all symmetries equally. The Periodic Table of Kitaev is
exhibited as a special case within our framework, and we prove that the
phenomena of periodicity and dimension shifts are robust against disorder and
magnetic fields.
