Aspherical $P{D}_3$-pairs

Abstract

We extend two results known for aspherical 3-manifolds to $P{D}_3$-pairs $(P,\partial P)$ with aspherical ambient space $P$. Every such $P{D}_3$-pair may be assembled by attaching 1-handles to $P{D}_3$-pairs with aspherical; ambient space and $\ddot{I}{€}_1$-injective boundary. (Thus the study of such pairs reduces to the study of $P{D}_3$-pairs of groups.) If $\ddot{I}€$ is a group of type $FP$ whose indecomposable factors $G$ each have $\ddot{I}‡(G_i)=0$ then there are only finitely many such $P{D}_3$-pairs with $\ddot{I}{€}_1(P)\cong \ddot{I}€$.