  We derive a map relating the gauge symmetry groups of heterotic strings on
$T^4$ to other components of the moduli space with rank reduction. This
generalizes the results for $T^2$ and $T^3$ which mirror the singularity
freezing mechanism of K3 surfaces in F and M-theory, respectively. The novel
feature in six dimensions is that the map explicitly involves the topology of
the gauge groups, in particular acting only on non-simply-connected ones. This
relation is equivalent to that of connected components of the moduli space of
flat $G$-bundles over $T^2$ with $G$ non-simply-connected. These results are
verified with a reasonably exhaustive list of gauge groups obtained with a
moduli space exploration algorithm.
