Indecomposable PD3–complexes

Hillman, Jonathan A

doi:10.2140/agt.2012.12.131

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Indecomposable $\mathrm{PD}_3$–complexes

Jonathan A Hillman

Algebraic & Geometric Topology 12 (2012) 131–153

DOI: 10.2140/agt.2012.12.131

Abstract

We show that if $X$ is an indecomposable ${PD}_{3}$ –complex and $π_{1} (X)$ is the fundamental group of a reduced finite graph of finite groups but is neither $ℤ$ nor $ℤ \oplus ℤ ∕ 2 ℤ$ 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 $ℤ ∕ 2 ℤ$ . If there are no exceptions then all but at most one of the vertex groups is dihedral of order $2 m$ with $m$ odd. Every such group is realized by some ${PD}_{3}$ –complex. Otherwise, one edge group may be $ℤ ∕ 6 ℤ$ . We do not know of any such examples.

We also ask whether every ${PD}_{3}$ –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.

Keywords

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

References

Publication

Received: 27 January 2009

Revised: 23 October 2011

Accepted: 28 October 2011

Published: 24 February 2012

Authors

Jonathan A Hillman
School of Mathematics and Statistics F07 University of Sydney Sydney NSW 2006 Australia

Volume 12, issue 1 (2012)