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2018 Poincaré duality complexes with highly connected universal cover
Beatrice Bleile, Imre Bokor, Jonathan A Hillman
Algebr. Geom. Topol. 18(7): 3749-3788 (2018). DOI: 10.2140/agt.2018.18.3749

Abstract

Turaev conjectured that the classification, realization and splitting results for Poincaré duality complexes of dimension 3 ( PD 3 –complexes) generalize to PD n –complexes with ( n 2 ) –connected universal cover for n 3 . Baues and Bleile showed that such complexes are classified, up to oriented homotopy equivalence, by the triple consisting of their fundamental group, orientation class and the image of their fundamental class in the homology of the fundamental group, verifying Turaev’s conjecture on classification.

We prove Turaev’s conjectures on realization and splitting. We show that a triple ( G , ω , μ ) , comprising a group G , a cohomology class ω H 1 ( G ; 2 ) and a homology class μ H n ( G ; ω ) , can be realized by a PD n –complex with ( n 2 ) –connected universal cover if and only if the Turaev map applied to μ yields an equivalence. We show that a PD n –complex with ( n 2 ) –connected universal cover is a nontrivial connected sum of two such complexes if and only if its fundamental group is a nontrivial free product of groups.

We then consider the indecomposable PD n –complexes of this type. When n is odd the results are similar to those for the case n = 3 . The indecomposables are either aspherical or have virtually free fundamental group. When n is even the indecomposables include manifolds which are neither aspherical nor have virtually free fundamental group, but if the group is virtually free and has no dihedral subgroup of order > 2 then it has two ends.

Citation

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Beatrice Bleile. Imre Bokor. Jonathan A Hillman. "Poincaré duality complexes with highly connected universal cover." Algebr. Geom. Topol. 18 (7) 3749 - 3788, 2018. https://doi.org/10.2140/agt.2018.18.3749

Information

Received: 31 October 2016; Revised: 7 July 2018; Accepted: 27 July 2018; Published: 2018
First available in Project Euclid: 18 December 2018

zbMATH: 07006376
MathSciNet: MR3892230
Digital Object Identifier: 10.2140/agt.2018.18.3749

Subjects:
Primary: 57N65, 57P10

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.18 • No. 7 • 2018
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