$PD_4$-complexes with $π_2$ a projective $\mathbb{Z}[π_1]$-module

Abstract

Let $X$ be a $PD_4$-complex and let $π=π_1(X)$. If $π$ is torsion-free and $π_2(X)$ is a finitely generated projective $\mathbb{Z}[π]$-module then either $π$ is free or $π$ is $FP$ and $c.d.π=4$. If, moreover, $H^3(π;\mathbb{Z}[π])=0$ then $π$ is a free product of $PD_4$-groups and a free group.