00 8 Indecomposable Pd 3 - Complexes

We show that if the fundamental group π of an inde-composable orientable P D 3-complex is the fundamental group of a finite graph of finite groups then the vertex groups have periodic cohomology and the edge groups are metacyclic. If the vertex groups all have cohomological period dividing 4 at most one is not dihedral, the edge groups are Z/2Z, and the underlying graph is a tree. We also ask whether every P D 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. It is a well known consequence of the Sphere Theorem that every closed 3-manifold is a connected sum of indecomposable factors, which are either aspherical or have fundamental group Z or a finite group. There is a partial analogue for P D 3-complexes: Turaev showed that a P D 3-complex whose fundamental group is a free product is a connected sum [23], while Crisp showed that every indecomposable P D 3-complex is either aspherical or its fundamental group is the fundamental group of a finite graph of finite groups [3]. However the group may have infinitely many ends, in contrast to the situation for 3-manifolds. The only examples known thus far are orientable and have group S 3 * Z/2Z S 3 [15]. The results given below suggest that any other examples must similarly have group a semidirect product L ⋊ Z/2Z, where L is a free product of cyclic groups of odd order. Crisp showed also that if X is an orientable P D 3-complex then the centralizer of a nontrivial element of prime order must be finite. This result is crucial for our work. In Theorem 6 we show that if X is inde-composable and π 1 (X) = πG for some finite graph G of finite groups then the vertex groups have periodic cohomology and the edge groups are metacyclic. If moreover the vertex groups are metacyclic the edge groups are cyclic. In Theorem 10 we show that if the vertex groups all have cohomological period dividing 4 at most one is not dihedral, the edge groups have order 2 and the underlying graph is a tree.