Profinite completions of orientable Poincaré duality groups of dimension four and Euler characteristic zero

Abstract

Let $p$ be a prime number, $\mathcal{T}$ be a class of finite groups closed under extensions, subgroups and quotients and the cyclic group of order $p$ is in $\mathcal{T}$.

We find some sufficient and necessary conditions for the pro-$\mathcal{T}$ completion of an abstract orientable Poincaré duality group $G$ of dimension 4 and Euler characteristic 0 to be a profinite orientable Poincaré duality group of dimension 4 at the prime $p$ with Euler $p$-characteristic 0. In particular we show that the pro-$p$ completion ${\hat{G}}_p$ of $G$ is an orientable Poincaré duality pro-$p$ group of dimension 4 and Euler characteristic 0 if and only if $G$ is $p$-good.