Algebraic & Geometric Topology

Indecomposable $\mathrm{PD}_3$–complexes

Jonathan A Hillman

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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.

Article information

Algebr. Geom. Topol., Volume 12, Number 1 (2012), 131-153.

Received: 27 January 2009
Revised: 23 October 2011
Accepted: 28 October 2011
First available in Project Euclid: 19 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M05: Fundamental group, presentations, free differential calculus 57M99: None of the above, but in this section
Secondary: 57P10: Poincaré duality spaces

degree–$1$ map Dehn surgery graph of groups indecomposable $3$–manifold $\mathrm{PD}_3$–complex $\mathrm{PD}_3$–group periodic cohomology virtually free


Hillman, Jonathan A. Indecomposable $\mathrm{PD}_3$–complexes. Algebr. Geom. Topol. 12 (2012), no. 1, 131--153. doi:10.2140/agt.2012.12.131.

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