Wall's manifold-realization question for finitely presented Poincare duality groups

Theorem 1.1: there is a finitely presented group $G$ with a finite oriented four-dimensional aspherical Poincare complex as classifying space, $H^i(G;\mathbb ZG)=0$ for $i\ne4$ and $H^4(G;\mathbb ZG)\cong\mathbb Z$ with trivial action, such that no closed aspherical topological 4-manifold has fundamental group $G$. This answers Wall's question negatively for finitely presented integral $PD$ groups with finite classifying space. Not shown: a counterexample in dimension 3 or in dimensions at least 5, or anything about smooth or homology-manifold realizations beyond the topological category.

Theorem 1.1: there is a finitely presented group $G$ with a finite oriented four-dimensional aspherical Poincare complex as classifying space, $H^i(G;\mathbb ZG)=0$ for $i\ne4$ and $H^4(G;\mathbb ZG)\cong\mathbb Z$ with trivial action, such that no closed aspherical topological 4-manifold has fundamental group $G$. This answers Wall's question negatively for finitely presented integral $PD$ groups with finite classifying space. Not shown: a counterexample in dimension 3 or in dimensions at least 5, or anything about smooth or homology-manifold realizations beyond the topological category.