Aspherical $P{D}_4$-pairs

Abstract

This is the third of three related preprints. We consider here which groups $\pi$ and $P{D}_3$-complexes $Y$ are realised by $P{D}_4$-pairs $(X,Y)$ with $X$ aspherical and ${\pi }_{1}X\cong \pi$, and show that such a pair may be assembled from $(D^4,S^3)$ and $P{D}_4$-pairs of groups, by adding 1-handles and mapping cylinders of $\mathbb{Z}{\pi }_{1}N$-homology equivalences over boundary components $N$, if and only if $H^2(\pi ;\mathbb{Z}\pi )=0$. (This includes all such pairs with ${\pi }_1$-injective boundary components and all with $\pi$ a free group, but none with $c.d.\pi =2$. Adding a 1-handle includes connected sum.) If there is a finite 2-dimensional $K(\pi ,1)$-complex then $\pi$ is realisable by some such pair $(X,Y)$, but there are no obvious building blocks analogous to $P{D}_4$-pairs of groups, except for when $\pi$ is a $P{D}_2$-group.

See also arXiv: 2501.125092 and arXiv:2501.12512