  We construct aspherical closed orientable 5-manifolds with perfect
fundamental group. This completes part of our study (with D.H.Kochloukova and
I.Lima) of $PD_n$-groups with pro-$p$ completion a pro-$p$ Poincar\'e duality
group of dimension $\leq{n-2}$. We also consider the question of whether there
are any examples with "dimension drop" 1.
