Pro-$p$ completions of $PD_n$-groups

Abstract: We sharpen earlier work on the pro-$p$ completions of orientable $PD_3$-groups. There are four cases, and we give examples of aspherical 3-manifolds representing each case. In three of the four cases the new results are best possible. We also consider the pro-$p$ completion of some orientable $PD_n$ groups for $n \leq 5$, including surface-by-surface groups.