  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 operators
up to equivariant homotopy. The equivariance is with respect to a time-reversal
involution on $X$ and an involution on $\mathcal{H}_n^*$ defining a certain
symmetry class. Furthermore we study certain controlled degenerations of such
maps and describe how one can implement jumps from one homotopy class to
another while allowing only "small" degenerations.
