Unitary and homotopy equivalences: classification of low-dimensional topological phases of quantum matter

Topological insulators have attracted significant attention across physics and mathematics due to their technological potential and their rich geometric and algebraic structure. Their hallmark property is the bulk–boundary correspondence, whereby non-trivial bulk topology enforces the existence of metallic edge states. The mathematical classification of such phases of matter, culminating in the periodic table of topological insulators and superconductors, associates topological invariants to different symmetry classes defined by time-reversal, particle–hole, and chiral symmetries. However, this framework involves several approximations: it replaces projection-valued maps (PVMs) with vector bundles, collapses the torus to a sphere -- thus overlooking weak, lower-dimensional invariants --, and employs stable equivalence notions from K-theory that do not always capture the full topological content. This thesis develops a direct homotopy-theoretic approach to the classification of symmetric PVMs, aimed at overcoming these limitations. In particular, it investigates the interplay between unitary equivalence and homotopy, identifies weak invariants absent from the Kitaev table, and establishes a general classification scheme applicable to arbitrary periodic models. The analysis focuses on low-dimensional settings (0, 1, and 2-dimensional systems), which provide a tractable yet non-trivial testing ground. Within this framework, we examine models with a single symmetry present, corresponding to the Altland–Zirnbauer classes A, AI, AII, AIII, C, and D. The results clarify in particular the obstructions to constructing symmetric Wannier bases, refine the understanding of strong versus weak topological invariants, and detail the relation between topological phases of matter and the dimerization choice in discrete models.