$PD_3$-pairs with compressible boundary

TL;DR

This paper generalizes the classification of $PD_3$-pairs with aspherical and spherical boundary components by relaxing previous injectivity assumptions on the boundary's fundamental group.

Contribution

It extends prior work to include $PD_3$-pairs with spherical boundaries and lower cohomological dimension of the fundamental group.

Findings

Relaxed the $ ext{pi}_1$-injectivity hypothesis in $PD_3$-pair classification.

Extended classification to pairs with spherical boundary components.

Included cases with $ ext{c.d.} ext{pi}_1(P) extless= 2$.

Abstract

We extend work of Turaev and Bleile to relax the $\pi_1$-injectivity hypothesis in the characterization of the fundamental triples of $PD_3$-pairs with aspherical boundary components. This is further extended to pairs $(P,\partial{P})$ which also have spherical boundary components and with $c.d.\pi_1(P)\leq2$.