I am interested at the moment in what groups can occur as the fundamental group of a 4-manifold (or more generally, a 4-dimensional CW complex) with prescribed conditions on the intersection form. I have what I am hoping is a basic homotopy theory question:


A (orientable) PD-$n$ group is a group $G$ such that the Eilenberg-Maclane space $K(G,1)$ admits "Poincare duality", i.e. there is an $n$-dimensional integer homology class in $K(G,1)$ (thought of as the "fundamental class") such that cap product with it yields an isomorphism between the corresponding cohomology and homology groups (like for closed oriented manifolds). This is more general than saying that $K(G,1)$ admits the structure of an orientable closed manifold of dimension $n$.


Let $G$ be a PD-3 group. Is there any reason why $G$ cannot be the fundamental group of an orientable PD4 complex $X$ with vanishing second homotopy group, $\pi\_2(X)=0$? 



PD3 groups and PD4 complexes
$G=Z^3$ is such an example. It is $\pi\_1(T^3)$ hence a PD-3 group. If $X$ is a Poincare 4-complex with fund group $Z^3$, then the injective (by Hopf) map on cohomology $H^2(G)\to H^2(X)$ cannot be onto, because its image is lagrangian for the intersection form by naturality of cup products. Dually $H\_2(X)\to H\_2(G)$ is not injective, and so $\pi\_2(X)$ is not zero. Following this kind of idea is what math.GT/0307101 and math.GT/0608103 is based on. 


