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2012 Indecomposable $\mathrm{PD}_3$–complexes
Jonathan A Hillman
Algebr. Geom. Topol. 12(1): 131-153 (2012). DOI: 10.2140/agt.2012.12.131

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 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.

Citation

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Jonathan A Hillman. "Indecomposable $\mathrm{PD}_3$–complexes." Algebr. Geom. Topol. 12 (1) 131 - 153, 2012. https://doi.org/10.2140/agt.2012.12.131

Information

Received: 27 January 2009; Revised: 23 October 2011; Accepted: 28 October 2011; Published: 2012
First available in Project Euclid: 19 December 2017

zbMATH: 1252.57009
MathSciNet: MR2916272
Digital Object Identifier: 10.2140/agt.2012.12.131

Subjects:
Primary: 57M05, 57M99
Secondary: 57P10

Rights: Copyright © 2012 Mathematical Sciences Publishers

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