$PD_3$-groups and HNN Extensions

TL;DR

The paper investigates the structure of $PD_3$-groups that split as HNN extensions, revealing how their Poincaré duality relates to the splitting and providing insights into their algebraic properties.

Contribution

It establishes a connection between $PD_3$-groups splitting as HNN extensions and their Poincaré duality, offering new understanding of their algebraic and topological structure.

Findings

Poincaré dual of the homology class $[C]$ corresponds to an epimorphism $f:G obZ$

Kernel of $f$ is the normal closure of $A$

Additional observations on $PD_3$-groups splitting over $PD_2$-groups

Abstract

We show that if a $PD_3$-group $G$ splits as an HNN extension $A*_C\varphi$ where $C$ is a $PD_3$-group then the Poincar\'e dual in $H^1(G;\mathbb{Z})=Hom(G,\mathbb{Z})$ of the homology class $[C]$ is the epimorphism $f:G\to\mathbb{Z}$ with kernel the normal closure of $A$. We also make several other observations about $PD_3$-groups which split over $PD_2$-groups.