Indecomposable Pd 3 - Complexes

We show that if X is an indecomposable P D 3-complex and π 1 (X) is the fundamental group of a reduced finite graph of finite groups but is neither Z nor Z ⊕ Z/2Z then X is orientable, the underlying graph is a tree, the vertex groups have cohomolog-ical period dividing 4 and all but at most one of the edge groups is Z/2Z. 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 P D 3-complex. Otherwise, one edge group may be Z/6Z. We do not know of any such examples. 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.