Aspherical $PD_4$-pairs

This is the third of three related preprints. We consider here which groups $π$ and $PD_3$-complexes $Y$ are realised by $PD_4$-pairs $(X,Y)$ with $X$ aspherical and $π_1X\congπ$, and show that such a pair may be assembled from $(D^4,S^3)$ and $PD_4$-pairs of groups, by adding 1-handles and mapping cylinders of $\mathbb{Z}π_1N$-homology equivalences over boundary components $N$, if and only if $H^2(π;\mathbb{Z}π)=0$. (This includes all such pairs with $π_1$-injective boundary components and all with $π$ a free group, but none with $c.d.π=2$. Adding a 1-handle includes connected sum.) If there is a finite 2-dimensional $K(π,1)$-complex then $π$ is realisable by some such pair $(X,Y)$, but there are no obvious building blocks analogous to $PD_4$-pairs of groups, except for when $π$ is a $PD_2$-group.