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We study non-interacting electrons in disordered one-dimensional materialswhich exhibit a spectral gap, in each of the ten Altland-Zirnbauer symmetryclasses. We define an appropriate topology on the space of Hamiltonians so thatthe so-called strong topological invariants become complete invariants yieldingthe one-dimensional column of the Kitaev periodic table, but now derivedwithout recourse to K-theory. We thus confirm the conjecture regarding aone-to-one correspondence between topological phases of gapped non-interacting1D systems and the respective Abelian groups$\{0\},\mathbb{Z},2\mathbb{Z},\mathbb{Z}_2$ in the spectral gap regime. Themain tool we develop is an equivariant theory of homotopies of local unitariesand orthogonal projections. Moreover, we discuss an extension of the unitarytheory to partial isometries, to provide a perspective towards theunderstanding of strongly-disordered, mobility-gapped materials.

Topological Classification of Insulators: I. Non-interacting Spectrally-Gapped One-Dimensional Systems