#### Volume 12, issue 1 (2012)

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Indecomposable $\mathrm{PD}_3$–complexes

### Jonathan A Hillman

Algebraic & Geometric Topology 12 (2012) 131–153
##### 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 $ℤ$ nor $ℤ\oplus ℤ∕2ℤ$ 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 $ℤ∕2ℤ$. 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 ${PD}_{3}$–complex. Otherwise, one edge group may be $ℤ∕6ℤ$. 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.

##### Keywords
degree–$1$ map, Dehn surgery, graph of groups, indecomposable, $3$–manifold, $\mathrm{PD}_3$–complex, $\mathrm{PD}_3$–group, periodic cohomology, virtually free
##### Mathematical Subject Classification 2000
Primary: 57M05, 57M99
Secondary: 57P10
##### Publication
Received: 27 January 2009
Revised: 23 October 2011
Accepted: 28 October 2011
Published: 24 February 2012
##### Authors
 Jonathan A Hillman School of Mathematics and Statistics F07 University of Sydney Sydney NSW 2006 Australia