Indecomposable $\mathrm{PD}_3$–complexes

Abstract

We show that if $X$ is an indecomposable ${PD}_3$–complex and ${\pi }_1\left(X\right)$ is the fundamental group of a reduced finite graph of finite groups but is neither $\mathbb{Z}$ nor $\mathbb{Z}\oplus \mathbb{Z}∕2\mathbb{Z}$ then $X$ is orientable, the underlying graph is a tree, the vertex groups have cohomological period dividing 4 and all but at most one of the edge groups is $\mathbb{Z}∕2\mathbb{Z}$. If there are no exceptions then all but at most one of the vertex groups is dihedral of order $2m$ with $m$ odd. Every such group is realized by some ${PD}_3$–complex. Otherwise, one edge group may be $\mathbb{Z}∕6\mathbb{Z}$. We do not know of any such examples.

We also ask whether every ${PD}_3$–complex has a finite covering space which is homotopy equivalent to a closed orientable 3-manifold, and we propose a strategy for tackling this question.