Poincaré duality complexes with highly connected universal cover

Bleile, Beatrice; Bokor, Imre; Hillman, Jonathan

doi:10.2140/agt.2018.18.3749

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Poincaré duality complexes with highly connected universal cover

Beatrice Bleile, Imre Bokor and Jonathan A Hillman

Algebraic & Geometric Topology 18 (2018) 3749–3788

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 $n \geq 3$ . 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 $ω \in H^{1} (G; ℤ ∕ 2 ℤ)$ and a homology class $μ \in H_{n} (G; ℤ^{ω})$ , 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.

Keywords

Poincaré duality complex (or PD-complex), $(n{-}2)$–connected, fundamental triple, realization theorem, splitting theorem, indecomposable, graph of groups, periodic cohomology, virtually free

Mathematical Subject Classification 2010

Primary: 57N65, 57P10

References

Publication

Received: 31 October 2016

Revised: 7 July 2018

Accepted: 27 July 2018

Published: 11 December 2018

Authors

Beatrice Bleile
Mathematics School of Science and Technology University of New England Armidale, NSW Australia

Imre Bokor
Mathematics School of Science and Technology University of New England Armidale, NSW Australia

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

Volume 18, issue 7 (2018)