$PD_4$-complexes and 2-dimensional duality groups

TL;DR

This paper investigates the classification of $PD_4$-complexes with fundamental group of cohomological dimension two and one end, showing their homotopy types are determined by algebraic invariants under certain conditions.

Contribution

It extends previous work by identifying conditions under which the homotopy type of these complexes is uniquely determined by algebraic invariants.

Findings

Homotopy types are determined by $ ext{pi}$, Stiefel-Whitney classes, and intersection pairing.

Provides conditions on $ ext{pi}$ for classification.

Synthesizes and extends earlier results on $PD_4$-complexes.

Abstract

This paper is a synthesis and extension of three earlier papers on $PD_4$-complexes $X$ with fundamental group $\pi$ such that $c.d.\pi=2$ and $\pi$ has one end. Our goal is to show that the homotopy types of such complexes are determined by $\pi$, the Stiefel-Whitney classes and the equivariant intersection pairing on $\pi_2(X)$. We achieve this under further conditions on $\pi$.