Geodesic
Indecomposable PD3–complexes
Hillman, Jonathan A1
1School of Mathematics and Statistics F07, University of Sydney, Sydney NSW 2006, Australia
Algebraic and Geometric Topology, Tome 12 (2012) no. 1, pp. 131-153
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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 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.

DOI : 10.2140/agt.2012.12.131
Keywords: 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  1

1 School of Mathematics and Statistics F07, University of Sydney, Sydney NSW 2006, Australia 
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Hillman, Jonathan A. Indecomposable PD3–complexes. Algebraic and Geometric Topology, Tome 12 (2012) no. 1, pp. 131-153. doi: 10.2140/agt.2012.12.131

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