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In this paper we classify maps from a torus phase space $X$ to$\mathcal{H}_n^*$, the space of $n \times n$, non-singular hermitian operatorsup to equivariant homotopy. The equivariance is with respect to a time-reversalinvolution on $X$ and an involution on $\mathcal{H}_n^*$ defining a certainsymmetry class. Furthermore we study certain controlled degenerations of suchmaps and describe how one can implement jumps from one homotopy class toanother while allowing only "small" degenerations.

On Time-Reversal Equivariant Hermitian Operator Valued Maps from a 2D-Torus Phase Space