## Algebraic & Geometric Topology

### Indecomposable $\mathrm{PD}_3$–complexes

Jonathan A Hillman

#### Abstract

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 $ℤ$ nor $ℤ⊕ℤ∕2ℤ$ 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 $PD3$–complex. Otherwise, one edge group may be $ℤ∕6ℤ$. 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

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

Dates
Revised: 23 October 2011
Accepted: 28 October 2011
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715335

Digital Object Identifier
doi:10.2140/agt.2012.12.131

Mathematical Reviews number (MathSciNet)
MR2916272

Zentralblatt MATH identifier
1252.57009

#### Citation

Hillman, Jonathan A. Indecomposable $\mathrm{PD}_3$–complexes. Algebr. Geom. Topol. 12 (2012), no. 1, 131--153. doi:10.2140/agt.2012.12.131. https://projecteuclid.org/euclid.agt/1513715335

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