Topological Classification of Insulators: III. Non-interacting Spectrally-Gapped Systems in All Dimensions

We study non-interacting electrons in disordered materials which exhibit a spectral gap, in each of the ten Altland-Zirnbauer symmetry classes, in all space dimensions. We define an appropriate space of Hamiltonians and a topology on it so that the so-called strong topological invariants become \emph{complete} invariants yielding the Kitaev periodic table, but now derived as the set of path-connected components of the space of Hamiltonians, rather than as $K$-theory groups. We thus confirm the conjecture (phrased e.g. in \cite{KatsuraKoma2018}) regarding a one-to-one correspondence between topological phases of gapped non-interacting systems and the respective Abelian groups $\{0\},\ZZ,2\ZZ,\ZZ_2$ in the spectral gap regime. A central conceptual achievement of the paper is the identification of the natural notions of locality and bulk non-triviality for this classification problem. Once these are in place, the main technical step is to lift the relevant $K$-theory calculations to $\pi_0$ of unitaries and projections.