Singer observed in 1978 (Comm.Math.Phys. 60, 7-12) that the homotopy group of the space of gauge fields modulo gauge equivalence with gauge group $G$ on $S^4$ is given by

$\pi_n({\cal A}/{\cal G}) = \pi_{n-1}{\cal G} = \pi_{n+3} G$

Does anyone know what the corresponding expression is if the base manifold $S^4$ is replaced by $T^4$?