  We study the $\theta$ dependence of the continuum limit of 2d $U(N)$ gauge
theories defined on compact manifolds, with special emphasis on spherical
($g=0$) and toroidal ($g=1$) topologies. We find that the coupling between
$U(1)$ and $SU(N)$ degrees of freedom survives the continuum limit, leading to
observable deviations of the continuum topological susceptibility from the
$U(1)$ behavior, especially for $g=0$, in which case deviations remain even in
the large $N$ limit.
