A Characterization of Virtual Poincare Duality Groups

摘要

An important open question in geometric group theory and the topology of manifolds asks whether every Poincare duality group is realized as the fundamental group of a closed aspherical manifold. This note establishes an analogous property for virtual Poincare duality groups following ideas and results of [10; 11; 12] and gives an approximate solution to the realization question for Poincare duality groups. We recall that a group Γ is a Poincare duality group if Γ is of type FP and, for some n > 0, H~i(Γ; ZΓ) = 0 for i ≠ n and H~n(Γ; ZΓ) is an infinite cyclic group [1; 7; 4].