Thesis title: 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 has traditionally relied on K-theory and the periodic table of topological insulators and superconductors, which
associates topological invariants with 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 invariants, and employs stable equivalence notions 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 more 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 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.