Volume 12, issue 1 (2012)

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Indecomposable $\mathrm{PD}_3$–complexes

Jonathan A Hillman

Algebraic & Geometric Topology 12 (2012) 131–153

We show that if X is an indecomposable PD3–complex and π1(X) is the fundamental group of a reduced finite graph of finite groups but is neither nor 2 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 2. 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 PD3–complex. Otherwise, one edge group may be 6. We do not know of any such examples.

We also ask whether every PD3–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.

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
Received: 27 January 2009
Revised: 23 October 2011
Accepted: 28 October 2011
Published: 24 February 2012
Jonathan A Hillman
School of Mathematics and Statistics F07
University of Sydney
Sydney NSW 2006