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

Abstract

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

We find some sufficient and necessary conditions for the pro-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.