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We describe the homotopy classes of loops in the space of 2 × 2 simple (=non-degenerate) matrices with various symmetries. This turns out to be an elementary exercise in the homotopy of closed curves in R3/0. Since closed curves in R3/0 can be readily visualized, no advanced tools of algebraic topology are needed. The matrices represent gapped Bloch Hamiltonians in 1D with a two dimensional Hilbert space per unit cell.
Avron et al. (Wed,) studied this question.