  We investigate the relation between the realization of center symmetry and
the dependence on the topological parameter $\theta$ in $SU(N)$ Yang-Mills
theories, exploiting trace deformations as a tool to regulate center symmetry
breaking in a theory with a small compactified direction. We consider, in
particular, $SU(4)$ gauge theory, which admits two possible independent
deformations, and study, as a first step, its phase diagram in the deformation
plane for two values of the inverse compactified radius going up to $L^{-1}
\sim 500$ MeV, comparing the predictions of the effective 1-loop potential of
the Polyakov loop with lattice results. The $\theta$-dependence of the various
phases is then addressed, up to the fourth order in $\theta$, by numerical
simulations: results are found to coincide, within statistical errors, with
those of the standard confined phase iff center symmetry is completely restored
and independently of the particular way this happens, i.e. either by local
suppression of the Polyakov loop traces or by long range disorder.
