Aspherical $PD_3$-pairs

TL;DR

This paper generalizes key results from aspherical 3-manifolds to $PD_3$-pairs, showing they can be assembled from simpler pairs and characterizing the finiteness of such pairs under certain group conditions.

Contribution

It extends known results to $PD_3$-pairs with aspherical spaces, demonstrating assembly from simpler pairs and establishing finiteness conditions based on group properties.

Findings

$PD_3$-pairs can be assembled by attaching 1-handles to simpler pairs.

Finiteness of $PD_3$-pairs with a given fundamental group under certain conditions.

Reduction of the study of such pairs to the study of $PD_3$-pair groups.

Abstract

We extend two results known for aspherical 3-manifolds to $PD_3$-pairs $(P,\partial{P})$ with aspherical ambient space $P$. Every such $PD_3$-pair may be assembled by attaching 1-handles to $PD_3$-pairs with aspherical; ambient space and $\pi_1$-injective boundary. (Thus the study of such pairs reduces to the study of $PD_3$-pairs of groups.) If $\pi$ is a group of type $FP$ whose indecomposable factors $G$ each have $\chi(G_i)=0$ then there are only finitely many such $PD_3$-pairs with $\pi_1(P)\cong\pi$.