Indecomposable nonorientable \(\mathrm{PD}_3\)-complexes (Q518808)

It is known that indecomposable, orientable 3-manifolds are either aspherical, have finite fundamental group or have fundamental group \(\mathbb Z\). This is not the case for \(\mathrm{PD}_3\)-complexes, although \textit{J. Crisp} [Comment. Math. Helv. 75, No. 2, 232--246 (2000; Zbl 0961.57020)] has shown that in the orientable case the indecomposables, that is those that are not the connected sum of simpler non-sphere complexes, are either aspherical or have virtually free fundamental group. This paper considers the case of nonorientable \(\mathrm{PD}_3\)-complexes. The main result is that if \(X\) is an indecomposable, nonorientable \(\mathrm{PD}_3\)-complex with fundamental group \(\pi\) having infinitely many ends then \(\pi \cong \pi^{+} \rtimes {\mathbb Z}/2{\mathbb Z}^{-}\) and \(\pi^{+}\) is torsion free.