Homotopy of periodic 2 × 2 matrices.

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 R 3 / { 0 }. Since closed curves in R 3 / { 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. [ABSTRACT FROM AUTHOR]