Indecomposable PD₃-complexes

J.A.Hillman

Abstract

We show that if the fundamental group π of an indecomposable PD₃-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 then they are dihedral, the edge groups are Z/2Z and the underlting graph is a tree. We also ask whether every PD₃-complex has a finite covering space which is homotopy equivalent to a closed orientable 3-manifold, and suggest a strategy for proving this.

Keywords: degree-1 map, Dehn surgery, graph of groups, normalizer, PD₃-complex, PD₃-group, 3-manifold, virtually free.

AMS Subject Classification: Primary 57M25.