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We sharpen earlier work on the pro-$p$ completions of orientable$PD_3$-groups. There are four cases, and we give examples of aspherical3-manifolds representing each case. In three of the four cases the new resultsare best possible. We also consider the pro-$p$ completion of some orientable$PD_n$ groups for $n \leq 5$, including surface-by-surface groups.

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