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

Jonathan A. Hillman

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

Geometric Topology 2026-06-28 v1

Authors: Jonathan A. Hillman

Abstract

Let $X$ be a $PD_4$ -complex and let $\pi=\pi_1(X)$ . If $\pi$ is torsion-free and $\pi_2(X)$ is a finitely generated projective $\mathbb{Z}[\pi]$ -module then either $\pi$ is free or $\pi$ is $FP$ and $c.d.\pi=4$ . If, moreover, $H^3(\pi;\mathbb{Z}[\pi])=0$ then $\pi$ is a free product of $PD_4$ -groups and a free group.

Cite

@article{arxiv.2606.29356,
  title  = {$PD_4$-complexes with $π_2$ a projective $\mathbb{Z}[π_1]$-module},
  author = {Jonathan A. Hillman},
  journal= {arXiv preprint arXiv:2606.29356},
  year   = {2026}
}